### Cats, Dogs and Coin Flips

The solution to yesterday’s rationality test:

This one is much much simpler (and much less infuriating) than some of our earlier rationality puzzles (e.g. here and especially here), but it has a good pedigree, having come to me from my student Tallis Moore, who found it in a paper of Armen Alchian, who attibutes it to the Nobel prizewinner Harry Markowitz.

Several commenters got it exactly right, but whenever possible, I prefer an explanation that invokes cats and dogs. So: Suppose I give you a choice between A) a cat, B) a dog, and C) a coin flip to determine which pet you’ll get:

It’s perfectly rational to prefer the cat to the dog, and perfectly rational to prefer the dog to the cat, but (according to the traditional definition of rationality) quite indefensible to prefer the coin flip to either.

After all, if you like dogs and cats equally, then all three choices are equally good. If you like dogs more than cats, then you’ll prefer the dog to the coin flip. If you like cats more than dogs, you’ll prefer the cat to the coin flip. That seems to cover all the possibilities, and in no case does the coin flip come out on top.

Now replace the cat with yesterday’s Urn A, which contains two red balls (worth \$1000 each) and 1998 black balls (worth \$0). Replace the dog with Urn B, which contains twenty blue balls (worth \$100 each) and 1980 black balls. Which do you prefer: Urn A, Urn B, or a coin flip to choose between the urns? As with the cats and dogs, it’s hard to justify the coin flip.

But Urn C is a coin flip between Urns A and B. Here’s why:

1. Mixing Urns A and B together and then picking a ball is equivalent to flipping a coin to choose between Urns A and B — either way you’ve got a 50/50 chance of a random ball from A and a 50/50 chance of a random ball from B.
2. After you’ve mixed up the balls, you might as well remove exactly half the reds, half the blues and half the blacks — this has zero effect, after all, on any of the relevant probabilities.
3. But mixing Urns A and B, and then taking out half the balls of each color, produces exactly: Urn C.

So Urn C, like the coin flip, is the irrational choice. You can prefer A, you can prefer B, and you can like all three urns equally, but you can’t prefer C.

Of course some people do prefer C, which, as always with these things, leaves room for multiple interpretations. Maybe economists have too crabbed a definition of irrationality — or maybe there’s a lot of irrationality out there, which economists can strive to cure. What’s your take on this?

#### 163 Responses to “Cats, Dogs and Coin Flips”

1. 1 1 Chicago Methods

I don’t understand Steve,

If the constraint is that we just care about the dolla dolla billz at the end of the day, wouldn’t Von Neumann / Morgenstern say that a person would be strictly indifferent between all three choices?

Urn A: 1000(2/2000) = 1
Urn B: 100(20/2000) = 1
Urn C: 1000(1/2000) + 100(10/2000) = 1

I didn’t know we were going to mix urns :O

2. 2 2 TomM

I like cats and dogs equally

Because of this, it would take a lot of brainpower and energy for me to decide between the two (because there’s nothing really to split the two so no decision is ‘right’ or ‘rational’).

I then have an option between:

x Wasting some brainpower on trying to decide between two equal things

or

x Allowing a coin flip to take that choice.

If I like cats and dogs equally, it’s more rational to choose the coin flip because it saves me the mental attention and anguish of trying to choose between two equally attractive options.

3. 3 3 James Knight

Worth mentioning perhaps, that the Wason selection tasks show that the symbols contained within the choices create variance in how rational people’s responses are.

4. 4 4 Dave B

Perhaps you like cats and dogs equally but you also like ‘surprises’ and letting a little randomness into your life. C therefore offers an advantages over either A or B. I don’t see how this can be considered clearly irrational

5. 5 5 Ron

The cats and dogs is a really bad analogy. For one thing, it
completely leaves out the probability of getting nothing. That can
be cured, but there’s an obvious qualitative difference between a
choice of cats and dogs and the choice of money or more money.

Urn A means there’s a large chance of getting nothing and a very
small chance of a large reward.

Urn B means there’s a large chance of getting nothing and a
small chance of getting a moderate reward.

Urn C means there’s a large chance of getting nothing, a very
small chance of getting a large reward, and a fairly small
chance of getting a moderate consolation prize.

Let us suppose that I like money. Let’s further suppose that it’s
linear and 45 degrees – i.e.: I like \$10 exactly 10 times as much as
\$1, and I like \$100 exactly 10 times as much as \$10. I aver that in
this case, any choice among the three urns is rational. Thus, I’ll
vote for the “too crabbed a definition of irrationality”.

6. 6 6 Twofer

I think the 1 dog, 1 cat, coin flip example simplifies the question too much. Instead assume two dogs, two cats, and one dog one cat. Assume that in genearal I slightly prefer dogs to cats and would always pick a dog over a cat (assuming its a cool breed ;) ). But, with diminising utility, a second dog may not provide as much pleasure as a first cat, so the one dog one cat option becomes the best.

I also think Dave B is right – Some people value the possibility/surprise aspect of things and so I guess I come down on the side of economists are too crabbed…

7. 7 7 Ron

Let’s restate the urn problem in a way that makes the choices even
clearer but keeps the restriction of the same expected value.
(yes, these are *big* urns)

Urn A: 1,000,000 balls, of which 2 win \$1,000,000 and 999,998 win nothing.

Urn B: 1,000,000 balls, of which each wins \$2.

Urn C: 1,000,000 balls, of which 1 wins \$1,000,000 and 999,999 win \$1.

Which urn do you choose, now, and are you really irrational?

It’s not really a question of being irrational or being surprised;
it’s a question of would you spend \$1 on the chance of winning
\$1,000,000.

8. 8 8 suckmydictum

Can someone explain to me why Steve’s (and others’) answer does not imply the statement “it is irrational to be somewhere on an indifference curve” ?

9. 9 9 Ken B

If you prefer C I suspect it is because you are valuing somehting the formal mathematics leaves out, like the thrill of gambling. That’s a recurrent problem with these kinds of questions and one reason I am skeptical of “survey science” in general. What a responder reads is often not quite what the surveyor intended.

10. 10 10 Thomas

I think that people who choose Urn 3 are those who don’t like to make decisions.

11. 11 11 Dave

To expand on Dave B’s excellent point.

Even if I prefer dogs to cats, I might like suprises so much that I would take the coinflip regardless.

eg I like dogs with a value of 10 and cats with a value of 5 but I like surprises to a value of 10,000,000 and it’s not worth giving up that kind of utility for the certainty of a preferable pet

12. 12 12 Mark Draughn

suckmydictum, I think you can dispute the irrationality only if it’s truly an indifference curve. In the cats and dogs example, if you truly like both equally, then at first glance flipping a coin (C) would be just as good as A or B.

But here’s where the cats and dogs analogy breaks down: C differs from both A and B because C includes a surprise. As Dave B points out, how you feel about C depends on how you feel about surprises. Even if you have a preference for dogs or cats, you might still prefer C if the amount by which you like being surprised is greater than the amount by which you prefer one pet to another.

The surprise factor is different for the urns full of balls example, because all three offer the possibility of surprise, so C doesn’t particularly stand out.

On the other hand, it may be that part of the fun of drawing balls from an urn for cash prizes comes from fantasizing about winning the prize. If your fantasies about winning \$1000 (or Ron’s \$1 million) are much better than your fantasies about winning \$100, then Urn B is unattractive because choosing it means you can’t fantasize about your great riches.

Then there’s the matter of how much you hate losing. If these Urns are part of some new state lottery game, it probably costs you \$2 to play. Your chance of losing is worst with A and best with B, with C being preferrable to A.

So if you want the least chance of losing that still allows you to fantasize about winning the big prize, you will prefer C.

From what I’ve heard, the theory that some people have strong preferences for fantasy and against losing seems to be consistent with how lotteries are designed. They usually feature a single very large prize and many small prizes, with not too many intermediate prizes. Last year, in an attempt to bring in more players, the Powerball lottery increased both the size of the grand prize (from \$20 to \$40 million minimum) and the odds of winning (from 1/35 to 1/32).

13. 13 13 thomasblair

Ron,

Urn C has a lower EV.

;)

14. 14 14 RPLong

I think Thomas @ #8 has it exactly right.

15. 15 15 Brian

It sounds like you are saying that it is always irrational to use a coin flip to make a decision. Is that right?

In that case, I suppose I’m a pretty irrational person. I often feel incapable of distinguishing whether my preference for thing 1 or thing 2 is stronger, and so I use a coin flip to decide. I suppose this can be called “irrational,” but to me it feels more like a lack of discerning ability: I probably do really prefer one thing to the other, I’m just not insightful enough to tell which one that is.

16. 16 16 ThomasBayes

Here’s an equivalent way of preferring the coin flip over a dog or cat:

Room 1 contains a dog that you can have.
Room 2 contains a dog and a cat from which you can choose one.
A coin flip decides which room you enter. If you prefer for the coin flip to make the decision, they you will choose a cat if you enter room 2.

We could also do this:
Room 1 contains a cat that you can have.
Room 2 contains a dog and a cat from which you can choose one.
A coin flip decides which room you enter. If you prefer for the coin flip to make the decision, they you will choose a dog if you enter room 2.

On one hand we might say you are irrational because you prefer a dog over a cat AND you prefer a cat over a dog.

On the other hand, you might say “No, I am rational. I prefer a dog over a cat if there is a cat in the other room. I prefer a cat over a dog if there is a dog in the other room.”

17. 17 17 Alex

@Al V. #33 from the last post has it right: a hedonic utility function doesn’t get you to Urn C.

I think the cat/dog analogy holds well enough to establish that it would be irrational for one to flip the coin, and then REGRET THE RESULT.

Yesterday, I chose Urn C, because I realized that if I pulled a black from Urn A, my first thought would have been “crap, I should have at least given myself a chance at blue.”

I suppose it’s an exercise for the reader whether a choice that hedges against potential future irrationality can itself be considered rational.

18. 18 18 William

Very interesting. What paper of Alchian’s is this example in?

19. 19 19 Jens Fiederer

I’d pick C myself: a reasonable chance at winning SOMETHING, and at least SOME chance of winning something big. It’s not a STRONG preference, I’d be perfectly happy to pick from any of the urns…but it seems the most satisfying choice. Improve ANY of the urns by just one extra “good” ball and I’d pick that one, so I don’t feel very irrational about this….but choosing C is a matter of FEELING rather than calculation.

I could see reasons for preferring the coin flip to the chosen pet as well: not only the previously mentioned bit about saving decision-making brainpower, but also not having to look the adorable not-chosen pet in the eye and having to say “You simply aren’t worthy”. But I’d have to genuinely not care about cat vs. dog to make the coin flip viable.

20. 20 20 Steve Landsburg

Mark Draughn:

But here’s where the cats and dogs analogy breaks down: C differs from both A and B because C includes a surprise.

I don’t see why you call that a breakdown of the analogy. In the cat/dog case, the coin flip also includes a surprise.

21. 21 21 Steve Landsburg

William: The Alchian paper has the words “measurement” and “utility” in the title, and should be easy to Google. I don’t have it right at hand.

22. 22 22 Mark Draughn

Steve Landsburg:

I don’t see why you call that a breakdown of the analogy. In the cat/dog case, the coin flip also includes a surprise.

I could have written that better. The breakdown is because only the coin flip is a surprise with the cats and dogs example, whereas all three cases include an element of surprise when choosing balls from an Urn.

23. 23 23 Alan Wexelblat

I also vote for “too crabbed a notion” but of people, rather than specifically of rationality. In particular, this example differs significantly from the others in terms of how it applies to people because of the notion of choice.

If I am indifferent to the outcome I may still wish to make a choice, and the degree to which I wish to make that choice is often related to the anticipated consequence. For example, I am indifferent to the choice between paper or plastic bags at the supermarket, but I appreciate being asked and making the choice. I may be indifferent to dogs vs cats but each implies a multi-year commitment for which I wish to exercise agency. I might be OK with flipping a coin myself, as Brian points out, but would object to just being told the results of someone else’s coin flip.

Likewise, if I purchase a lottery scratch ticket it should make no difference whether I scratch off the covers or the clerk does it, but I think you’d find most people want to do it themselves. Likewise, people who buy “pick 5″ or similar tickets have the option of having the machine generate numbers for them or picking their own numbers. I think it’s clear that each choice gives an equally likely (however slim) chance of winning but people overwhelmingly prefer to pick their own numbers. (One might argue that people who play the lottery are a poor sample of rational human beings but then one has to explain why ‘rational’ is a useful definition when it so clearly applies to such a small number of people.)

In a sense this is a problem with any attempt to translate a problem in mathematical probabilities into a set of human actions. The probabilities are what they are, but the farther you remove the problem description from the realm of probability the less well it matches and the less useful it seems. Is that a problem with the notion of “rationality” or with a particular investigative method? I don’t know.

24. 24 24 ThomasBayes

I’ve been thinking about a consequence of rationality this way:

If pA and pB are two probabilities defined over some set of outcomes, and if you prefer pA over pB, then you would be irrational to prefer pC over PA if

pC = s*pA + (1-s)*pB

for any s between 0 and 1 (other than s=1).

In other words, if pA is better than pB, then it should be better than a mixture of pA and pB.

Is this correct? If so, is rationality ever described this way?

25. 25 25 Dan

I think it’s incorrect to label these things as ‘irrational’. The expected utility framework is one of many (e.g. ambiguity aversion) that rests on number of key assumptions/axioms. These are preferences and I can always come with a utility framework using a set of axioms that will justify those preferences.

Also the way the puzzle is worded there does not appear to be a puzzle at all even within the EU framework. Being a finance guy, I am thinking of these as investments where I am trying get a certain risk/return ratio. If you tell me I have the option to invest half my money in A or half in B, or if you tell me I can invest half in A,B,C today and get to face the same gamble tomorrow, the story changes. The way I interpreted the question is that you have to invest all your money in A, B or C now. In which case, if I am targeting a certain variance regardless of the expected payoff such that my utility function is U = – (variance of gamble C – actual variance)^2, then I prefer gamble C.

26. 26 26 Twofer

@ Thomas #10: “I think that people who choose Urn 3 are those who don’t like to make decisions.”

I can imagine scenarios where there is value in not having to make a choice. The movie Sophie’s Choice hinted at some of this. She had 2 children, and was forced to pick which one would live and which one would be killed. She might have preferred to have the option to flip a coin instead – So she wouldn’t have to live with quite so much guilt over her choice.

27. 27 27 Ken Arromdee

If it’s repeated, and the value of picking a cat or a dog diminishes after I’ve picked lots of them, then I should pick dogs up until my preference has gone down enough, then pick cats, etc. Deciding which one to pick has a transaction cost, so I prefer making a single repeatable decision rather than a decision that must be reasoned out from day to day. Therefore picking the one that randomly produces the cat or dog is preferable to picking either the cat or the dog separately.

28. 28 28 iceman

For me what’s not working with the analogy is with the urns we’re not picking an outcome but a distribution profile. (#5 touched on this.) E.g. I may have a clear preference between cats and dogs but prefer either to being petless. So it still seems (without doing the math) that for some sets of preferences, one could rationally favor a ‘medium’ chance of getting a dog while also preserving some chance of getting a cat. The ‘choose your pet’ version removes risk-aversion, which according to Ken A can even depend on what you had for lunch.

29. 29 29 RichardR

If someone likes the excitement of gambling they might choose the coin flip. Or if someone believes in fate they might choose the coin flip. Or if someone is truly indifferent between a cat and a dog they will be like Buridan’s ass and never make a decision so it makes sense to choose a coin toss to make the decision.

30. 30 30 Joel

I like cats and dogs equally, but I’m afraid that after betting on the dog, the pet regulation agency will push for tough anti-dog legislation. After all, I got into a bit of a tizzy with their president over yesterday’s poker game, and she’s sleeping with the chief pet regulator, so there’s a good chance she’ll pull some strings just to spite me.

On the other hand, there’s also a chance that after betting on the cat, they’ll pass anti-cat legislation. (She is a spiteful one.)

If I chose the ticket that gives me a 50-50 chance of winning either the cat or the dog, the best she can do is gamble and pass either an anti-dog law, or an anti-cat law. This choice gives me the best payoff, because at least there’ll be a 50% chance I can enjoy my pet.

See Mike H’s game theoretic solution from yesterday’s post (comment #54). My story is similar to that.

31. 31 31 Bearce

As others have said, the cat, dog, and coin flip analogy doesn’t take into account loss.

Urn A: 0.1% chance of winning.
Urn B: 1% chance of winning.
Urn C: 0.55% chance of winning.

Cat and dogs I get something. Why a relatively risk averse person say they prefer urn C because they prefer to pay the cost (0.45%) in order to increase my chance to win a bigger sum?

32. 32 32 Bearce

Correction: Why *can’t*

33. 33 33 ThomasBayes

For those of you who believe surprise and uncertainty are the key ingredients for this puzzle, I’d like to see your response to this . . .

Suppose you prefer dogs to cats.

Which of the following statements are false?

1. If someone gave you a cat and said, “would you like to trade this cat for a dog?” you would say “yes.”

2. If someone gave you a dog and said, “would you like to trade this dog for a cat?” you would say “no.”

3. If someone gave you a cat when you thought they were going to give you a dog then said “would you like to trade this cat for a dog?” you would say “yes”.

4. If someone gave you a cat based on a coin flip then said “would you like to trade this cat for a dog?” you would say “yes”.

If you believe any of these statements are false, please explain and provide a better definition of what it means to prefer dogs to cats. Thanks in advance.

34. 34 34 John Roccia

This rationality test seems to put all the weight on the outcome, and none on the choice itself.

Imagine I prefer cats to dogs by about 5%. Not very much, but enough to make me pick the cat. However! What if I’m also keenly aware of my opportunity costs; in other words, I see choice A as “Cat; Opportunity Cost: 1 Dog” and choice B as “Dog; Opportunity Cost: 1 Cat.” If I’m also more keenly attuned to losses than I am to gains (like Daniel Kahneman says, we feel losses about twice as much as gains) then either choice is bad for me in terms of the psychological costs of the decision itself.

But if I can shield myself psychologically from the choice by picking C, thus absolving me from the emotional pain of rejecting either a dog or a cat, isn’t that better for me? Couldn’t a rational person choose that?

35. 35 35 Steve Landsburg

ThomasBayes: Bravo.

36. 36 36 Ben

This is a very interesting problem and I don’t think I can come up with any examples where the coin flip is desirable in a purely rational world. But I think there are lots of examples where a purely rational actor would still choose the coin flip in an irrational world. Suppose for instance I love dogs and hate cats and my wife is the opposite. She will make my life very unpleasant if I choose a dog directly (without any possibility of a cat). But, choosing a cat directly is my least favorite option. So the coin flip is my best option in that case. (Of course, this wouldn’t be an issue if she were perfectly rational also)

Or consider something that might be a bit closer to reality: I am risk-averse so I would definitely choose Urn 2. But the red ball industry has some regulatory capture and I risk prosecution if I violate the Red Ball Mandate which specifies that I may not choose any urn that doesn’t contain at least 1 red ball.

37. 37 37 William

For those interested, the Alchian article is called “The Meaning of Utility Measurement.”

It may cause less offense in some people to note that Alchian claims you are irrational only if you simultaneously (1) Do not care about the process for deciding which urn you draw from and (2) Prefer Urn C.

Alchian notes that although (1) seems uncontroversial to those who accept utility maximization, you should remember that slot machines have three wheels instead of one big one with the same attached probability, and in fact players get to watch the wheels go round as they are playing. “Does seeing the wheels go around or seeing how close one came to nearly winning affect the desirability?”

38. 38 38 Dave B

@ #32

I don’t think any of them are false. Given that you prefer dogs to cats you would always want to trade a cat you have for a dog. I am assuming that there is no emotional effects here – it can be perfectly rational to prefer dogs to cats but still prefer THIS cat that I have to an unknown dog, particularly if my preference is not very strong.

However I don’t see where uncertainty and surprise relevently come into these statements as any uncertainty occurs prior to you making your choice. This is quite different from the original question where you are choosing uncertainty. In all these statements you are simply choosing a dog over a cat. Or was that the point?

39. 39 39 Harold

Thomas Bayes 32: I don’t quite get your argument. We are supposing that someone (rather unrealistically) adores surprises to such an extent that this is greater than their preference for a dog over a cat. So for your Q3 and Q4, you would get a huge surprise when they gave you the cat, which you would enjoy immensely, then if they offered to swap for a dog you would say “yes”. However, if there were no swap offered, you would prefer the cat and the surprise in 3 and 4 to the dog in 1 and 2. If we introduce an extra factor that you get with jar C that you don’t get with A and B, surely it could tip the value in favor of jar C?

Another example – instead of a surprise, jar C came with a car. If we put this into your questions, 3 and 4 become “someone gave you a cat and a car…” It would then be rational (for most people) to choose jar C.

Similarly -as in the “Sophies choice” example, jar A and B come with a huge penalty, then jar C becomes rational.

I think it unlikely that most people value surprises to this extent, or that they suffer from picking A or B, so for me it is a poor solution to why people choose jar C, but surely it is rational in extreme circumstances?

I suspect that those that chose “C” generally think along the lines of some posters – that they want to entertain the possibility of winning a large amount, whilst increasing their chances of winning something. Possibly reducing remorse comes into it, but I suspect it is a form of irrationality based on uncertainty. You simply can’t decide if you prefer a small chance at a big prize or a bigger chance at a small prize, so you hedge. As a compromise this sort of reasoning will often get you to a resonable position, so we have evolved heuristics to make this sort of compromise. It may not be rational in particualr circumstances, but overall it may be a winning strategy over indecision.

40. 40 40 DAVE

Thomas bayes.

I prefer tea to coffee. In fact I can’t stand coffee.

I also would love to spend a weekend at the playboy mansion. I would love to spend a weekend at the playboy mansion so much that u would be willing to drink coffee while there to keep me awake the whole time of that’s all that was available.

If you gave me the choice between tea or coffee at my office or a weekend at the playboy mansion that may have only coffee there I wouldn’t even think twice about it. Of course I would marginally prefer if they have tea there but if it turns out I’m drinking coffee, who cares? I’m at the playboy mansion.

Urn A is tea at my office.
Urn B is coffee at my office.
Urn C is a surprise. But I love surprises so much I wouldn’t even consider the greater likelihood of tea or coffee.

41. 41 41 ThomasBayes

Instead of changing the problem in ways that avoid the rationality dilemma, I think it would be instructive for many people to try to understand why there is a dilemma in the first place.

If, for instance, Urn C in the original question contained 1 red and 9 blue balls, then personal preference would rule. And if it was a continuous problem, then you could use 10.00000001 blue balls. But by putting exactly 1 red ball and exactly 10 blue balls in Urn C, we create a very special problem.

If someone said that you can’t solve for x in this equation:

a*x = y

when a=0, we’d probably all agree. I don’t think many people would be arguing that you could solve for x, and then prove it by making ‘a’ some small number different from 0. But I think that is what is happening for this puzzle.

42. 42 42 Fonzy Shazam

1. I am indifferent between cats and dogs.

2. I want an additional cat or dog.

3. I REALLY like games of chance and outcomes where “the Universe” “decides” outcomes for me.

4. I choose option C, the coin flip.

Have I done something in my set up to violate a condition of the puzzle, or am I irrational, or is the puzzle flawed/conclusion about C being an irrational choice flawed?

43. 43 43 Harold

#40. I have been away for a couple of days, so I have not spent so much time mulling this one, although I have read the comments. So I might well have got the wrong end of the stick.

There is no rationality problem unless you pick jar C. It is rational to prefer either a small chance at a big win or a bigger chance at a small win, or a cat or a dog. The argument is that picking Jar C does none of these things, so is not rational. It is the same as choosing a 50:50 chance of Jar A or Jar B. However, quite a lot of people do pick Jar C, so in some sense the puzzle is to work out if there is a rational reason or not. If not, is there an understandable reason.

In puzzles of this type we are usually not “allowed” to introduce extraneous elements not contained in the original puzzle – we can’t say “well, if someone gave me \$100 to pick C then it would be rational to pick C”.

However, we may assume “normal” goals and motivations, which may either eliminate the irrationality, or at least explain it. We assume more or less that people want to obtain the largest amount of money.

One example to remove the irrationality might be that you have absolutely no preference over a small chance of a large win or a larger chance of a smaller win (or a cat or a dog). Jar C then becomes rational as there is no reason to pick any jar in particular. However, it is not likely that people generally have no such preference, so it is a poor solution.

An example to explain the irrationality might be if jar A were re-labelled Jar 13. We might understand that some people who prefer a small chance of high reward would pick jar C as an alternative to jar 13 becasue of superstition. This is not rational, but many find it understandable nonetheless. We would have “explained” the irrationality

I interpret suggesting that surprise is the answer is attempting to smuggle in “extraneous data” disguised as “normal motivation”. It would remove the irrationality (as does the total lack of preference), but is a poor solution. We can come up with various reasons why choosing C would be rational (or understandable) if we introduce extraneous information, but unless this is generally applicable, it fails to explain the number of C answers.

There may exist a solution that IS generally applicable, and would remove the irrationality – that would then be a good solution. I do not think anyone has suggested one so far.

44. 44 44 Peter B

‘Puzzles’ presenting a choice where the average numerical outcome is the same, but the distribution of possible outcomes different, are as old as the hills. You may as well debate which of three ice creams tastes best to you. There are subjective arguments to render any of the choices credible.

45. 45 45 Steve Landsburg

Peter B:

You may as well debate which of three ice creams tastes best to you. There are subjective arguments to render any of the choices credible.

I take it you didn’t bother to read the post before commenting.

46. 46 46 Twofer

William #36 above lists the article. Here it is:

http://www2.uah.es/econ/MicroDoct/Alchian-Utility%20Measurement_1953.pdf

As William mentions, per the article, if you enjoy the process of the coin flip (or find benefit), C is not irrational.

The article also mentions: “A hidden postulate is that the preferences, if transitive and consistent, are stable for the interval involved. Utility for this purpose and by this method is measurable up to a monotonic transformation, i.e. it is ordinal only.” So if you enjoy the possiblity of the first chance at winning \$1000 more than you enjoy the second chance at winning \$1000, then option C becomes a rational choice.

47. 47 47 ThomasBayes

Here’s a question for the people who believe that flipping a coin to make a decision is rational (even when you prefer one outcome over another):

What is an example of a decision that is not rational?

48. 48 48 ThomasBayes

Twofer (45): Does all of this mean that if you enjoy being irrational, then you are being rational?

More seriously, for the original problem to have its intended meaning, you should assume that \$1000 (or \$100) has the same value regardless of how you win it. Allowing the value (or utility or meaning) of \$1000 to be a function of the probability of winning it really does change the problem.

49. 49 49 twofer

ThomasBayes #46 – It’s hard to imagine anybody deriving all that much joy out of a coin flip itself such that it would override a preference of a dog over a cat. But in the first formulation of the phenomena above, Dave B #4 wrote he had no preferece between dogs and cats, but he kind of enjoyed the suspense of a coin flip. That little extra enjoyment of suspense would tip the scale ever so slightly toward C if he is really neutral between dogs and cats. The paper itself indicates there is something to this, mentioning that there must be some enjoyment in watching the 3 wheels spin on a slot machine — even though it does nothing to enhance the odds…

50. 50 50 Dave

#46 – What is an example of a decision that is not rational?

1) I love dogs and living
2) I’m lethally allergic to cats
3) Surprises killed everyone in my immediate family due to delicate heart condition in the genes

It would be irrational to choose the cat urn or the coin flip urn

51. 51 51 ThomasBayes

The world’s most rational man:

52. 52 52 Twofer

ThomasBayes #47 – If that has to be the assumption than your analysis is correct. But I think it’s important to understand the conditions under which the analysis applies, hence the exploration of the edge cases.

Imagine a more real world scenario, one which captures the spirit of the urns, but which actually happened (the decision was real, the dollar amounts fabricated for illustration):

It’s December 1999. You are fortunate enough to be in on the ground floor of a very hot dot com. You own 20,000 stock options. The company has just been given the last round of venture funding, prior to going public. Using the venture funding valuation as a guide, the options are worth \$110 each. You are then given a choice of 3 options:

1) You can keep all your options as is, cash them in after the firm goes public (and the usual 2 year waiting period afterward).

2) You can cash in all your options for \$10 each.

3) You can elect to keep 50% as stock options, and cash in the other 50% at \$10 each.

Are any of these choices really irrational? Is it reasonable to assume that the value of the risky second \$1.1 million is equivalent to the value of the first risky \$1.1M? Is it reasonable to assume the value of the for sure second \$100K is equivalent to the value of the first for sure \$100K? Is it reasonable to assume you might want the first for sure \$100K and leave open the possibility of \$1.1M?

I’ll even give you two more pieces of information with the benefit of hindsight: The dot com phenomena turned into the dot bomb phenomena, AND the technology in question is widely used in the marketplace today…

53. 53 53 ThomasBayes

Twofer (51): I don’t see how to use your scenario to judge rationality in the context of S.L.’s original puzzle. Option 3 is not a coin flip between Options 1 and 2. Urn C is a coin flip between Urn A and Urn B. That is the spirit of the urn problem.

For me, the moral of Steve’s puzzles has been that it is irrational to make decisions based on the toss of a coin. The objection has been that a person might enjoy making decisions by the toss of a coin, so getting a cat is different than getting a cat and knowing that a coin flip came up heads.

——

Q: “Which do you prefer, a cat or a dog?”

A: “Yes.”

I used to think that that was a humorous way to avoid a decision. Now I see it as an exercise in rational thought. ;-)

54. 54 54 Ken B

If U(watching a coin flip) > 0 then one can rationally prefer the coin flip option: all that you need is that the difference between U(cat) and U(dog) be less in absolute value than U(watching a coin flip). This seems both a necessary and sufficient condition. I think the problem setter meant to impose U(watching a coin flip) = 0.

55. 55 55 Frosty

I’m firmly in the camp supporting Urn C as a rational choice.

My perspective is Urn C is merely a hybrid of Urn A and Urn B. Therefore, in deference to the professor’s affinity for dogs, the three analogous choices would be A) Labrador, B) Poodle, C) Labradoodle. I claim it’s rational to prefer the Labradoodle.

My preferences are to have a greater than 0 percent chance to win big (\$1000) and a greater than 0.5 percent chance of winning something. Only Urn C satisfies both preferences and is therefore my most rational choice. Replacing Urn C with a coin flip doesn’t mean my choice becomes irrational. I claim it means it is wrong to assume a coin flip is always irrational.

56. 56 56 Henri Hein

It is time for a Grook:

Whenever you’re called on to make up your mind
and you’re hampered by not having any
the best way to solve the dilemma, you’ll find
is simply by spinning a penny.
No — not so that chance shall decide the affair
while you’re passively standing there moping
but the moment the penny is up in the air
you suddenly know what you’re hoping.

57. 57 57 Steve Landsburg

Frosty: A labradoodle is nothing at all like a coin flip that sometimes delivers a labrador and sometimes a poodle.

58. 58 58 Martin-2

It seems to me that people can do only rough calculations of expected utility in their heads, and the margin of error leaves some people preferring urn C. I won’t say that’s irrational, but I will say that wouldn’t happen to a perfectly rational agent.

59. 59 59 Dave B

@Ken B #53

I think that you are absolutely right.

However, the problem setter is really saying something like: please analyse this scenario taking into account ONLY your preferences for cats and dogs (and no other preferences such as those for uncertainty, surprise, difficulty of decision making or mental calculation, or anything else that might affect a decision) to determine whet the possible ‘rational’ coices should be. In this case I would say that the puzzle is really a mathematical exercise which is fine as a little puzzle but in that case doesn’t tell us much (anything?) about what decisions people could rationally make given that they do have preferences about all those other things.

60. 60 60 James Knight

What about if it is agreed that you’ll pick from urn A, and there is ten of you, and you don’t yet know what any of the 10 will pick – is it an advantage going first, or last, or doesn’t it matter?

61. 61 61 James Knight

Just to clarify, I mean if it is agreed that you’ll pick from urn A, and there is ten of you and all ten are going to pick from the same urn, and you don’t yet know what any of the 10 will pick – is it an advantage going first, or last, or doesn’t it matter?

62. 62 62 Frosty

My position is a person who prefers some chance of winning \$1000 and a greater than 0.5 percent chance of winning something should rationally prefer Urn C because it is the only choice which satisfies both preferences. This seems irrefutable to me. Does anyone disagree so far?

Replacing Urn C with a coin flip doesn’t change the rationality of the choice, it remains the only choice capable of satisfying both preferences. If the traditional economist definition is it always irrational to choose a coin flip, then the definition is “too crabbed”.

63. 63 63 Steve Landsburg

Dave B:

However, the problem setter is really saying something like: please analyse this scenario taking into account ONLY your preferences for cats and dogs (and no other preferences such as those for uncertainty, surprise, difficulty of decision making or mental calculation, or anything else that might affect a decision) to determine whet the possible ‘rational’ coices should be.

No, there’s more to it than that. The Von Neumann/Morgenstern axioms, which assume that choices across lotteries won’t be affected by surprise, difficulty of mental calculation, etc., are intended not as rules for puzzlesolvers, but as the basis of a theory for how actual people actually behave.

64. 64 64 Steve Landsburg

Ken B:

I think the problem setter meant to impose U(watching a coin flip) = 0.

See my response to Dave B above. The vN/M axioms implicitly assume U(watching a coin flip) = 0, and are meant not just as rules for solving puzzles but as a model of how real people make real decisions. If the model systematically makes wrong predictions, that’s a fault of the model.

65. 65 65 James Knight

Steve:

Ok let me rephrase the question…

An urn has 2000 pieces of paper in it: 2 pieces of paper with \$1000 written on them, and 1998 blank pieces. You and 1999 others can have one free pick, and you win the value written on the paper you pick. at the start, each of you has a 1/1000 chance of winning a \$1000 – but if you’re all going to pick in turn, then it is an advantage letting nine people go before you, because each picker before you is more likely to reduce the number of blanks for you. But there comes a point when things swing against you – so how many people of the other 1999 would it be to your advantage to let go before you, before you had your pick, to maximise your highest probability of getting \$1000?

66. 66 66 Steve Landsburg

James Knight: Ex ante, it doesn’t matter where you are in the queue.

67. 67 67 James Knight

Can you elaborate please? I’m sure you’re right, but I’m not quite getting it. It would seem that if you were in that scenario, with such high odds of picking a blank in the early stages, you’d be better off letting a few others go first – as in the early stages the lower the probability of finding one of the 2 winners the better it is for you if others go first. What am I missing?

68. 68 68 Dave B

@62

Steve, if as you say the axioms are meant to be a basis of a theory as to how people actually behave, but as you stated at the beginning a lot of people do choose C which is in conflict with those axioms then it would seem that the axioms clearly do not fully model the way that people actually behave. Especially if people continue with their choice of C even after the ‘irrationality’ of it is explained to them and if they produce plausible arguments for their choice.

This is not to dismiss those axioms which presumably do capture a lot about human rationality. It is just to say (as with all models of reality) that care needs to be taken over whether the assumptions of the model are true before drawing conclusions from it. If the assumptions did hold than a choice of C could be argued as irrational. If the assumptions don’t hold (which appears to be the case under certain circumstances) then we can’t tell.

Of course by examining particular cases and implications then we could present scenarios which clearly show why the non-holding of the assumption is irrational. I don’t think that this example does that though – instead I think it shows that the assumptions do not always hold and can do so quite rationally. In precisely what circumstances the assumptions do then hold is then I think an empirical question.

I would have thought also that it would be good to try and extend the theory so that it could also be applied when the assumptions do not hold even if less strong conclusions could be drawn.

69. 69 69 Frosty

Steven, I prefer you respond to #61, but following are more thoughts…

Given an indifferent person, are each of the choices irrational? If persuaded to choose, wouldn’t choosing the coin flip most accurately reflect indifference? For example, an indifferent person choosing a dog risks being mistakenly interpreted as a revealed preference for dogs.

I completely agree with Dave B (#4, #37, 67) and frankly with so many other comments that is difficult to imagine anyone remains convinced it must be irrational to choose Urn C or the coin flip.

70. 70 70 ThomasBayes

Okay, let’s say we think a theory of rational decision-making allows for the preference of Urn C in the original problem.

Now, consider another problem in which a person is asked to decide between only Urns A and B. They prefer Urn B over Urn A, but, given the choice, they would have preferred Urn C over both. Because they’ve followed this blog, they realize that they don’t have to select Urn A or Urn B. Instead, they can flip a coin and, by doing so, pick from Urn C.

So, unbeknown to us, they flip a coin and select from Urn A. Upon seeing their decision, we add another person to the group of “people who prefer Urn A to Urn B.”

If we allow rational choices to be based on coin flips, then we have to allow this.

Because of this problem, I understand why a theory of rational decision-making would assume people don’t flip coins to make decisions when they prefer one thing over another.

71. 71 71 Dan

@52 Thomas Bayes and @56 Steve

I think the original problem is like the labradoodle or the options examples. With the cats/dogs the problem is turned from one of facing a gamble or a distribution of payoffs to one of certain payoffs.

Suppose I have \$1K to burn in Vegas, and that all the games have the exact same negative expected payoff but different variances. I want to achieve certain level of thrill that is optimal for me (not too much or too little). I can play poker which has low variance or roulette which has high variance. Or I can play blackjack which has the prefect level of thrill for me. I can spend half my time playing poker and half roulette and achieve the same level of variance. Or I can flip a coin and choose to play either roulette or poker and again achieve the same variance. The way I interpreted the problem that last two options may not be possible. But even if they were, they would provide a different level of experience then playing blackjack.

With the cat/dog example, we are equating a gamble with a known/certain payoff. Essentially asking someone who likes money whether they prefer to get \$1000 or \$500 or a coin flip to determine how much to receive. But, even with certain payoffs, I would say the decision to flip would not be ‘irrational’ in some contexts. The poker player Phil Ivey said once that the high from gambling does not come from the large wins but from wins after large losses or potential losses. The thrill comes from that sick feeling you get in your stomach after losing half your bankroll almost like S&M. AS I said before you could always come up with some axioms to justify this type of behavior. Ans its’ better to come up with new model to explain behavior than to say that the behavior is wrong and the model is right.

72. 72 72 Frosty

@69 ThomasBayes – “allow rational choices to be based on coin flips”

The rational choice is to flip a coin, not based on the flip of a coin.

73. 73 73 ThomasBayes

71: “The rational choice is to flip a coin, not based on the flip of a coin.”

I don’t understand. People have been saying that even if they prefer dogs to cats, they would rather have their choice of a cat or dog determined by the flip of a coin. Heads they get a cat. Tails they get a dog. The decision of a dog or cat is based on the flip of a coin, isn’t it?

74. 74 74 nivedita

I think that given the problem as posed, urn C should not rationally be preferred (aside from the edge case where you’re indifferent between A, B and C).

However, consider the following problem instead:
First, let’s get rid of the 1980 black balls that are the same across all three choices.
Next, suppose that you’re at a casino, with a roulette wheel with the numbers 1-20, and you’re offered the following choices of bets:
A. If 1 or 2 come up, you get \$1000, otherwise nothing.
B. You get \$100 no matter what number comes up.
C. If 1 comes up, you get \$1000, if 2-11 come up, you get \$100, otherwise nothing.

Although this choice appears very similar to the original problem, C is no longer equivalent to a coin flip to decide whether to take A or B, and it can now be rational to prefer C to either A or B. For example, I have a very strong aversion to ending up with nothing after the spin. Ordinarily, this would mean I choose B. However, suppose I already have a bet that pays me \$100 if 12-20 come up, and nothing otherwise. In that case, I can prefer C to B, since either way there is no chance I would end up with nothing, and C gives me a chance to win \$1000.

I have a feeling that choices in real life are more similar to the roulette wheel than the urn drawings. That is, the outcomes of events, like the spin of the wheel, are distinguishable in a sense. In the urn draw, if you get a black ball, there is no way to tell if it was black ball #15 or some other black ball, and correspondingly no way to enter into a side bet that pays off on that outcome.

75. 75 75 nivedita

Another way to state the contrast between the original problem and the roulette wheel:

The original problem corresponds to a world in which there are three states – red, blue and black, and a single bet, which pays \$1000 for red, \$100 for blue and \$0 for black. We are asked which of three probability distributions for the world would we prefer.

The roulette wheel corresponds to a world in which there are twenty states and a single probability distribution. We are asked which of three different bets would we prefer.

Stated this way, I think it is clear that real-world choices are more like the roulette wheel problem.

76. 76 76 Frosty

@72 ThomasBayes

In #69, you said “if we allow rational choices to be based on coin flips, then we have to allow this“. I’m pointing out that the choice occurs before the coin flip and therefore cannot be based on the coin flip. The drawing from urn A is based on the coin flip. You seem to be confusing the drawing with the choosing.

What is your response to my position stated in #61? If you disagree, why?

77. 77 77 Dave B

@ThomasBayes #69

I can see why an initial theory of rational decision making would make the assumptions it does. In the same way as I can see why an initial theory of fluid flow might ignore turbulence or a theory of projectile motion ignore air resistance.

If such a theory is successful at explaining phenomenon then it is a useful theory particularly if it gives rise to surprising but verified results. However, one should not be surprised if in certain cases where the assumptions do not hold the theory does not agree with decisions that people make. This all just seems like theories in science generally.

You could even (if you were so inclined) call decisions that arise in circumstances where the assumptions apply ‘Rational’, using Rational as a technical term in much the same way that physicists or mathematicians use ordinary words to express technical terms with very precise meanings e.g. smooth, compact, imaginary.

But now you are on dangerous ground and need to take care. You can claim that such decisions are not Rational (technical term) by definition – they violate the assumptions. But you risk confusing your claim with the claim that they are not rational (ordinary word) which many people would dispute.

78. 78 78 ThomasBayes

Dave B (76): Many people confuse the ordinary word “correlated” with something other than its formal mathematical definition. If I say that X^2 is uncorrelated with X (which it easily could be), and someone argues that they can’t be uncorrelated because we can easily determine X^2 from X, should I change the formal definition of what it means for two random variables to be correlated?

It’s just a word, right? But a lack of correlation in a mathematical sense implies that you cannot benefit by using a linear estimator to predict one variable from the other. This distinction is very important.

So, if a mathematician asks if two variables are correlated, we should be comfortable using the formal definition of correlation when we evaluate the answer. Likewise, when an economist asks if a decision is rational, we should comfortable using the formal definition of rationality when we evaluate the answer.

By the way, many people like to say “correlation does not imply causation” but I rarely hear someone say “causation does not imply correlation.” Both are valid statements.

79. 79 79 ThomasBayes

Frosty (75): I believe several of my earlier posts explain why I disagree with your position in #61.

We probably won’t make much more progress on this unless you understand why my concerns would be eliminated if we added or removed one black ball from Urn C. In that case, I would have no reason to question the rationality of any preference among the three urns. It is the VERY special case that is described in this puzzle that causes the problem.

80. 80 80 Ken B

@78
This goes to my criticism of survey science. You argue C is not rational if the issue is PURELY a master of preferring cats over dogs. I agree given that purely. Frosty notes people need not make decisions purely on that basis. Frosty’s desire to win something, or to have a minimum chance at one, is not part of the preference between a cats and dogs, it’s another factor altogether. Survey science waves these issues away. So I infer does vN/M.

81. 81 81 Henri Hein

@77 (OT):

By the way, many people like to say “correlation does not imply causation” but I rarely hear someone say “causation does not imply correlation.” Both are valid statements.

Agreed, I noticed this too. In general, people underestimate how much rigor it takes to make definitive statements.

82. 82 82 Spike

My apologies if this has already been suggested:

You are offered a choice between a dollar and a lottery ticket with a 1% chance at ~\$100. You choose one of the options.

If you are instead offered \$2 or 2 lottery tickets, is it truly irrational to mix and match?

83. 83 83 Frosty

@78 – ThomasBayes

Alas, it’s disappointing, but agreed, we’re unlikely to make any progress. FWIW, following is what I anticipated as a successful refutation of my position in #61 which boils down to I have two preferences and only C satisfies both.

1) Having the two preferences together is irrational because….
2) C doesn’t satisfy both preferences because…
3) C isn’t the only option which satisfies both preferences because…

The final question I wish someone would answer is whether any of the urns can be rationally chosen by a person indifferent with respect to A and B. By definition, one cannot be indifferent and also prefer A. By definition, one cannot be indifferent and also prefer B. Given that preferring C has been deemed irrational, is it simply irrational to be indifferent?

84. 84 84 Daniel

@66

You’re right that when you pick you would have a higher probability of picking the paper if no one else has picked it yet, but before anyone draws your increased probability of picking the winner if you get to draw is balanced by the accumulated probability that someone else has already picked the winner, so as Steve said it doesn’t matter when you pick.

85. 85 85 Mike H

I’m reminded of an experim,ent described in Freakonomics, where people at a pin exhibition were asked to fill out a survey, then randomly given a Mickey or Donald pin. Then they were asked “would you like to swap your pin?”

Experienced pin collectors swapped about 50% of the time, indicating they rationally identified which pin would be best for them. That is, they never chose Urn C, they never asked to flip a coin to choose their pet.

However, inexperienced pin collectors swapped their randomly selected pin significantly less often than 50%. I don’t recall the exact percentage, let’s say 20% of them swapped their pin. These inexperienced collectors were saying, in effect, “I prefer the coin flip I just had over making a definite choice. I was given Urn C, I’m sticking with it”.

This is irrational in the sense that they are not maximising their expected utility. However, these were inexperienced pin collectors – they don’t know their expected utility from the two types of pins.

I posted a scenario on Steve’s previous post where an individual might not yet know the utility of the Urns A and B, and where Urn C was the strategic choice.

I haven’t thought of a similar scenario involving pins or cats and dogs, but perhaps there is one in the case where people don’t actually know whether they prefer dogs to cats, and want to minimise their risk?

Or perhaps there is a real difference between “Would you like to flip a coin, or just choose a pin/pet?” and “Here before you is a pin/pet you just won in a coin flip. Would you like to swap it for a different, unknown pin/pet?” that affects the choices made by rational people?

86. 86 86 Mike H

For further reading : http://en.wikipedia.org/wiki/Flipism

87. 87 87 Harold

Are Ken B and Dave B related?

Mike H #84. I think this is a good point. In “normal” interactions, someone would usually ask you to swap something if it were to their benefit rather than yours. We therefore have developed a strategy that says “do not swap when asked unless you really know it is a good deal”. This is likely to give us generally good outcomes and explains the pin example.

Flippng a coin is a reasonable strategy where cost of information, or consequences of indecision, are high. In the jar example, choosing jar C is a reasonable strategy if you have not the time or inclination to work out the odds, or you cannot decide whether you have a preference for the higher risk strategy or not. In a sense, while all the information is available, we do not have the capacity to process it, so we are effectively working without full information. Thus not swapping the pin and choosing jar C are the result of applying broadly based strategies to specific situations where they may not produce the best outcome.

What is interesting, and probably more revealing, is that after choosing jar C, we justify it, even after we have taken the time to examine our preferences and study the outcomes.

88. 88 88 Chicago Methods

Ken B @#9,

I think there is a book called, “101 Jokes on how Psychologists are Dumber than Pigeons.”

89. 89 89 Demosthenes

What is rational depends on one’s goals. If the goal were to maximize payout, then even allowing for variable interpretations of that, Urn C makes no sense at all. If you want to try for the most money possible, you have a 100% better chance of getting \$1000 if you select from Urn A than if you select from Urn C. If your goal is to have the best chance to win money, then you have a nearly 50% better chance if you select from Urn B than if you select from Urn C. In neither case is Urn C the preferred choice.

I guess my question is: what goal would make selecting from Urn C the most rational move?

(And I second everyone who disputed the analogy between the examples. There are too many dissimilarities.

90. 90 90 Dram

And what I’ve been wondering from the start, Demosthenes, is why this has to be a binary choice. Why should we assume that the only rational choice is to literally be an extremist and maximize one of the two options as you describe them, while describing anyone who falls on the continuum between those two extremes as irrational?

91. 91 91 Frosty

@88 Demosthenes

Balance. As you said, Urn A offers the best chance to win big and Urn B offers the best chance to win something. However, Urn C offers something unique, a balance; higher probability to win something than Urn A and higher probability to win big than Urn B. A pursuasive argument supporting the claim that choosing C is inherently irrational needs to address at least two questions:

1) Using the urn example, why is it irrational to prefer a balance between winning big and winning something? Do rational people seriously need to prefer one to the other? No middle ground?

2) Using the dog and cat example, if one is truly indifferent between cats and dogs, why is it irrational to prefer to accurately express the indifference via the coin flip? After all, if I’m indifferent, I don’t prefer A to B and I don’t prefer B to A, therefore C is the most accurate choice. I read the footnote in the referenced paper and I infer it says “First, suppose he is indifferent between A and B…” and then paraphrasing, C can be expressed as an indifference toward A or B, therefore “there is no reason for choosing C”. WTF? No reason? One reason is C most accurately reflects his indifference. What am I missing?

92. 92 92 Steve Landsburg

Frosty: If you are truly indifferent between cats and dogs, you’ll be indifferent between cats, dogs, and the coin flip. There’s no circumstance in which the coin flip is preferred to both the cat and the dog.

93. 93 93 Steve Landsburg

Dram:

Why should we assume that the only rational choice is to literally be an extremist and maximize one of the two options as you describe them,

We didn’t assume this; we deduced it.

94. 94 94 Harold

The comparison between dogs/(high risk, high value win) vs cat/(lower risk, lower value) is accurate. It may be easier to see if we introduce an element of possibly getting nothing. Jar A contains 1998 balls and 2 with dog on them. Jar B contains 1998 balls and 2 with cat. Jar C contains 1998 balls, one cat and one dog.

Jar C offers a “balance” – a higher chance to win big than B and a higher chance of winning at all than A. Alternatively, it offers a higher chance of getting a dog than B and a higher chance of getting a cat than A.

We easily see that getting a higher chance of a cat than A is no benefit if we prefer a dog to a cat. Why can we not see that getting a higher chance of winning the small prize is no benefit if what we want is the small chance of the big win? By getting a higher chance of winning the small prize we get a lower chance at the big one, and this is a prefernce we have already expressed.

I see two possibilities. 1) we are indifferent between the choices. 2) The numbers make it difficult for us to see what we are choosing between.

What happens if we substitute two apparently indistinguishable dogs or cats into jar A and B?

95. 95 95 Dram

Steve@92:

Let us describe the urns’ contents as red balls / blue balls / black balls.

In the example you provided, urn A is 2/0/1998, urn B is 0/20/1980, and urn C is 1/10/1989. It is argued that choosing urn C is irrational.

Let us now posit urns X, Y, and Z, respectively. This is trivially identical to the existing problem.

Now let us further posit urns P and Q, respectively 9/110/19881 and 11/90/19899.

If one is presented with urns X, Y, Z, P and Q, is only Z an irrational choice while P and Q are rational, or not?

96. 96 96 Dram

Oops. In my previous comment, X, Y, and Z are 20/0/19980, 0/200/19800, and 10/100/19890.

97. 97 97 Frosty

@91 Steven – Thank you. Your response addresses the second of my two questions and at least we know precisely where we disagree. I attribute some value to C because it accurately expresses indifference, something A and B cannot do, whereas you don’t.

98. 98 98 Frosty

@92 Steven – Thank you again. Your response addresses the first of my two questions. You are claiming a rational person must prefer either winning big or winning something but not both and without any room for balance or middle ground. This is where we disagree.

99. 99 99 Steve Landsburg

Frosty:

You are claiming a rational person must prefer either winning big or winning something but not both and without any room for balance or middle ground.

In other words, you absolutely do not understand the argument, you’re disputing something that nobody has claimed, and your comments are therefore 100% off topic.

100. 100 100 Jack

There are really 4 choices: cat v. dog, selection by choice v. selection by chance.

I still don’t understand why choice C couldn’t be rational provided you have the following preferences:

1. You are entirely indifferent between receiving a cat or a dog,
2. You prefer either a cat or a dog over nothing, and
3. You prefer selection by chance rather than selection by choice (say, because of the excitement of the unknown).

If these 3 preferences apply to you, I don’t see how picking the coin flip could be irrational. You gain from the final object (i.e. receive the value of the object to you) and gain extra through the preferred process by which you receive that object (i.e. receive the value of the selection mechanism to you).

101. 101 101 Dram's brother

“It’s perfectly rational to prefer the cat to the dog, and perfectly rational to prefer the dog to the cat, but (according to the traditional definition of rationality) quite indefensible to prefer the coin flip to either.
After all, if you like dogs and cats equally, then all three choices are equally good.”

The results of the three choices (in terms of non-human companionship) are equally good, and it would be silly to think that this payoff would be improved by a coin flip. But just how am I supposed to pick between two truly equal options? I could acknowledge that they are unlikely to be perfectly equal, and then expend a likely highly incommensurate and dubious effort trying to identify the marginally better alternative. Or I could simply pick one at “random,” i.e. coin flip, i.e. choice C.

So if after an initial deliberation I can tell that I have no preference among A or B, my expected payoffs thus become:
A. Cat minus great mental effort
B. Dog minus great mental effort
C. Cat or dog.

In these terms, C seems quite rational.

102. 102 102 Steve Landsburg

Jack:

If these 3 preferences apply to you, I don’t see how picking the coin flip could be irrational.

The von Neumann Morgenstern axioms do not allow you to have preference 3. You’re entitled to that preference, but by the standards of classical decision theory, if you do have that preference, you’re not rational. Perhaps your point is that this is a lousy definition of rationality, and perhaps you’re right — but the definition is what it is.

103. 103 103 Frosty

@98 100% off topic?…Wow.

I’m under the impression the debate is whether option C can be rationally chosen. You argue no and support your opinion by accurately (I’m not disputing this) demonstrating that one aspect of Urn C is that it is the same in terms of probabilities as a coin flip and subsequent draw from Urn A or Urn B. Because C can accurately be expressed this way, you conclude preferring C must be irrational. Do I understand your argument correctly?

My argument is Urn C offers an attribute of A (chance to win big) AND an attribute of B (better chance to win something). This argument doesn’t apply to the cat and dog scenario because choice C, the coin flip, isn’t offering an aspect of a dog and an aspect of a cat.

After dismissing all attempts by me and others to attribute any value to coin flipping, your argument holds in the dog and cat scenario. I still don’t think it holds in the urn scenario. I am not alone in thinking the two scenarios are not interchangeable.

Good day sir!

104. 104 104 Demosthenes

@99

You offer us three premises (or preference-descriptions, if you would rather). But premise 3 is doing all the work for you. To show you why, consider a person who holds 1 and 2, but not 3. His choices are A (cat), B (dog), and C (coin flip). When 1 is considered, all options are equally good. When 2 is considered, all options are again equally good — since no matter which he chooses, he’ll wind up with something rather than nothing.

Therefore, 3 is the difference-maker…since for a person who holds 1 and 2 (but not 3), there would be no reason to prefer any of the options. Any preference would therefore be based on no reason at all — i.e., it would be irrational. So, in order to make C preferable, you must hold 3. In fact, if you hold 3 strongly, it seems that C would be preferable to A or B regardless of whether you held or didn’t hold either 1 or 2.

But what does 3 gain you? I’m asking sincerely, because the reason you gave (excitement) seems itself to be of an irrational nature. “I prefer C because of how C would make me feel.”

105. 105 105 Jack

Demosthenes,

Whether excitement is of an irrational nature or not is irrelevant, as long as it’s your desired goal (nevertheless I would say that excitement is perfectly rational provided it leads to more overall happiness). Option C would then be optimal if preference 3 (in my example) were true. But as Professor Landsburg states, N&M’s axioms simply do not allow for preference 3 to be true. But I would say that’s incomplete because the axiom is built on an unproven assumption that probabilities themselves don’t have utility, which isn’t always the case. See the example below.

Professor Landsburg,
I read the Alchian paper (or at least the relevant portion) and noted that the von Neumann Morgenstern axioms assume that there is no utility in particular probabilities.

I guess the response is that it’s an overly narrow definition of rationality because humans sometimes do find utility in certain probabilities. As Alchian notes, “Axiom (3)(see p36) is inconsistent with a situation in which the utility of the act of winning itself depends upon the probability of winning, or more generally if probability itself has utility.” E.g. “at Christmas, one does not want to know what gift his wife is going to give him; he prefers ignorance to any hints of certainty as to what his gift will be. This is a type of love for gambling.” (p.36-37)

I think this is where the disagreement is coming from. As for the urn example, a particular type of gambler may find a certain kind of utility in the specific probability of urn C (maybe the mixed probability of drawing a colored ball more accurately aligns with their risk preference?), which N&M’s Axiom 3 does not quantify.

106. 106 106 Demosthenes

“Whether excitement is of an irrational nature or not is irrelevant, as long as it’s your desired goal…”

On this idea, just as long as my goal is to get my left foot (and ONLY my left foot) mangled beyond recognition by a rabid ocelot, any actions I take with the aim of fulfilling that goal are rational.

107. 107 107 Jack

If your goal truly is to get your left foot mangled by an ocelot, if you actually desire this outcome, then I cannot see how any action fulfilling that goal would be irrational.

The only way this analysis differs is if the person is wrong about what they want, in which case it’s not actually their preference.

108. 108 108 Harold

Frosty 102: “My argument is Urn C offers an attribute of A (chance to win big) AND an attribute of B (better chance to win something).” I disputed this in my comment 93. If we look at the cat and dog example, jar C offers an attribute of A (a chance to win a dog) AND an attribute of B (a chance to win a cat). Does this offer an advantage?

Either you have a prefernce for a dog or a cat, in which case the possibility of winning the other is not a benefit, since it comes with an equally reduced chance of getting your preference. Or you are indifferent between cat and dog, in which presumably all jars are equal. It is not possible to rationally have a preference for C.

This works just as well if we substitute “low chance of high reward” for dog and “higher chance of lower reward” for cat.

109. 109 109 Steve Landsburg

Frosty:

Let’s try this to focus the discussion:

Given the assumption (consistent with the vN/M axioms) that there’s no value to coin flipping, consider the following propositions:

I) Given a choice between Urns A, B, and a coin flip between them, it’s irrational to prefer the coin flip.

II) Given a choice between Urn C and a coin flip between Urns A and B, each is equally desirable.

III) Given a choice between Urns A, B, and C, it’s irrational to prefer Urn C.

Do you agree with I)? With II)? With III? With the assertion that III) follows from I) and II)?

110. 110 110 Frosty

I) no, II) yes, III) no

1.) Urn A is the best chance to win big, but worst chance to win something.
2.) Urn B is the best chance to win something, worst chance to win big.
3.) Urn C is middle ground; better (than B) chance to win big and better (than A) chance to win something.

Do you agree with 1? with 2? with 3?

111. 111 111 Steve Landsburg

Frosty: So we disagree on the following question:

I) Given a choice between Urns A, B, and a coin flip between them, it’s irrational to prefer the coin flip.

This, of course, is completely analogous to the dog/cat version:

I’) Given a choice between a dog, a cat, and a coin flip between them, it’s irrational to prefer the coin flip.

So I presume you disagree with this as well. I certainly agree with your statements 1), 2) and 3), so I’) appears to be the entire locus of our disagreement, which means that’s what we ought to focus on.

112. 112 112 Harold

#109. Your answer to 1 indicates to me that you are indifferent between A or B. What if we make it more extreme – instead of 20 balls of \$100 we have 2000 balls of \$1. Expected outcome the same, but much exagerated risk difference. If you had no option of a coin flip, would you choose jar A or Jar B? A small chance of \$1000 or a guaranteed \$1?

This is a bit like asking if you would buy a lottery ticket.

Jar C now contains 1000 x \$1 balls, 1 x \$1000 and 999 no win balls.

This is in contrast to a genuine “mid risk” jar that would contain say 100 x \$50 balls.

113. 113 113 Harold

That last line should read 100 x \$20 balls and 1900 no win balls.

114. 114 114 Ken B

@110:
I think Frosty is saying this is NOT a good analogy. He is arguing that with the urns choice C offers a blend of rewards not directly analogous to any blend of rewards in your cat/dog choice. The rewards he means in C are NOT just expected value but a function of both the expected value and the expected chance of winning. That is not present in your cat/dog coin flip.

I’m not saying who I think is right, I’m just trying to clarify.

115. 115 115 Harold

@113-but is it not just an illusion of a blend? Frosty prefers a coin toss to decide which jar is selected. This means he is indifferent to the jars, is unable to decide, and prefers another method to make the choice. I think my more extreme odds illustrates this. Either you get \$1, or a chance to win \$1000. Tossing a coin does not really offer a blend any more than it offers a blend of cat and dog.

116. 116 116 Ken B

@114: You may be right you may be wrong. I’m just trying to clarify the bone of contention these cogs and dats are fighting over.

117. 117 117 Ken B

Here’s how I see the Frosty-Landsburg dispute, paraphrasing outrageously:

Steve: You can prefer cats or you can prefer dogs, but unless you are perfectly indifferent then C cannot be your pick. On the cat/dog rewards scale C will never ber better than A or B. The best case for C makes you indifferent between A, B, and C; you cannot prefer C.

Frosty: The thing is your problem isn’t about cats or dogs. It’s about chances. My preference is for a particular set of guarantees offered by C that is not offered by either A or B in the original urns problem Your cats/dogs analogy changes the available rewards so cahnges the terms of comparison. Your argument could be perfectly correct re cats and dogs and it would still not apply in the original problem.

118. 118 118 ThomasBayes

If a person prefers to draw from Urn C over drawing from Urn A or Urn B, does that mean that they would pay some amount (however small, but non-zero) to draw from Urn C instead of Urns A or B? (Suppose Urns A and B are free, but Urn C has a small cost that represents the amount by which you prefer Urn C, minus a little bit.)

If not, then what does it mean to prefer Urn C over Urns A and B?

If so, then wouldn’t it be better for them to flip a coin themselves and then draw from Urn A or Urn B according to the coin toss? They will have exactly the same odds of winning as Urn C, but they’ll avoid the cost by drawing from Urn A or Urn B.

119. 119 119 Dan

.

120. 120 120 Dan

As I mentioned before, the preferences are over gambles not known payoffs. The cats/dogs example is therefore not a good analogy. In once case you have U(A) where A is random, in the other U(C) where C is constant. You don’t need all the the von Neumann Morgenstern assumptions to derive U(C)>U(D). You do need them for U(A)>U(B). It was afterall developed to deal with uncertainty.

Btw, I can adjust the variance, skewness, etc. of payoffs I face by flipping coins or drawing random numbers. But I think people would agree that flipping a coin to play poker or to buy hurricane insurance, would provide a fundamentally different experience eventhogh the payoff distributions are adjusted to my liking.

121. 121 121 Ken B

@117: The amount I am willing to pay is 99% of the cost of flipping the coin myself. Now what?

122. 122 122 Frosty

@110 Steven So we disagree on the following question: I) Given a choice between Urns A, B, and a coin flip between them, it’s irrational to prefer the coin flip.

Correct

So I presume you disagree with this as well

Correct. Due to Harold’s comments, my opinion on the dog and cat scenario is evolving.

I certainly agree with your statements 1), 2) and 3)

This puzzles me…

X) you believe choosing Urn C is irrational,
Y) you agree 3.) Urn C is middle ground;,

but, in #97, when I inferred from X and Y that you were there claiming choosing a middle ground is irrational, you strongly objected (#98).

@116 Ken B My preference is for a particular set of guarantees offered by C that is not offered by either A or B in the original urns problem

Correct

@117 ThomasBayes If not, then what does it mean to prefer Urn C over Urns A and B?

If repeating myself doesn’t help, then see Ken B #116. Preferring Urn C over Urns A and B simply means I prefer a middle ground. That’s it. I don’t prefer A because the chance of winning something is too small. I don’t prefer B because the chance to win big is zero. I think we all agree Urn C can be replaced with a two step process (step 1 – coin flip, step 2 draw from Urn A or B based on result of step 1). If you don’t agree Urn C is a middle ground or you believe choosing a middle ground is irrational, then that’s precisely where we disagree. I think we’re stuck though because we’re coming from two different directions. I’m convinced C is the middle ground and is rational and therefore the two step process must be rational. You’re convinced the two step process is irrational and therefore choosing C must be irrational.

123. 123 123 Steve Landsburg

Frosty:

Y) you agree 3.) Urn C is middle ground;,

but, in #97, when I inferred from X and Y that you were there claiming choosing a middle ground is irrational, you strongly objected (#98).

This is because your definition of “middle ground” keeps changing. By one definition, a labradoodle is middle ground between a labrador and a poodle. By a different definition, a coin flip between a labrador and a poodle is middle ground between a labrador and a poodle. Many of your arguments seem to rely on confusing one sort of “middle ground” with the other.

By one definition, Urn C is middle ground and is irrational. By an entirely different definition, choosing a middle ground is not irrational. This is not a contradiction.

124. 124 124 Frosty

@122 I accept that. The Labradoodle was a half-hearted attempt to use an animal analogy. It didn’t work. You promptly and correctly described the flaw, and I wish I had acknowledged I was abandoning it.

125. 125 125 Frosty

@122 By an entirely different definition, choosing a middle ground is not irrational.

Last night, I was blissfully inferring something and decided to verify…

Is 3.) Urn C is middle ground; better (than B) chance to win big and better (than A) chance to win something one of the definitions that is not irrational?

126. 126 126 iceman

Now if only we could get dogs and cats to inter-breed…

Frosty – I also was initially stuck on the idea that we’re dealing with 3 different distribution profiles, each of which offers some objective advantage over the others (this is what seems to be missing in the “outcome”-based dog-cat analogy), so hard to see how we can rule out a set of preferences that could rationally support any of them (even without invoking surprise). However, I think I get now how Alchian (or Markowitz) cleverly cuts through the comparability issue via deduction. The trick is establishing that one is indifferent between A and something else that clearly *dominates* the profile of C. I’ll see if I can make it work with dogs and cats:

To keep this involving distribution profiles, let’s say you prefer either pet to being petless (i.e. any prize to no prize), so instead of flipping a coin you’re rolling dice.
1) Start with the case where you’re indifferent between a cat and a dog, so also indifferent between a choice of A) rolling 1-2 for a cat (otherwise nothing) or B) 1-2 for a dog. So you’re also indifferent between those options and C) a roll of 1 = cat, 2 = dog. (In this case you have the same odds of getting a pet of some kind, so this part is similar to the coin flip analogy but with the chance of also getting nothing.)
2) Next assume you like cats better, so B is out, and in this case it’s obvious that C is out too, since 2 chances to get a cat is better than 1 chance at a cat and 1 chance at something you like less. That is, A dominates C. More generally you can construct it this way: saying A is better (but the value of B is non-zero too) means there’s some improved chance of getting a dog, say rolling 1-4, where you’re indifferent between the cat odds and the dog odds (if your preferences are really strong, increase the # of dice). Along the lines of step 1, this suggests you’re also indifferent about combining those two options into a roll of 1 = cat, 2-3 = dog (e.g. you’re equally happy if a friend chooses which game to play on your behalf (randomly or otherwise)). And since that is *definitively* better than the existing option C (1 = cat, 2 = dog) — i.e. it provides a “balance” that’s more consistent with your preferences, if you will — this “proves” you in fact prefer profile A to profile C, even though each of the 3 profiles seemingly offers some advantage over the others. I hope that made sense.

127. 127 127 Frosty

@125 iceman

I appreciate your taking the time to help me understand. Your assessment of where I’m stuck is accurate. If I’m following what you’re doing, you’re making the cat more like the \$1000 payoff and the dog more like the \$100 and adjusting probabilities accordingly. Nice. But how in the end have we proven I in fact prefer profile A to profile C? Because A dominates C?

Here’s my final attempt because I’ve already spent more time on this than I can rationally justify. Back to the urns. I like money and I’m indifferent toward the three urns. After all, the urns are rigged so that in expectation, I’ll win exactly \$1 regardless of the urn, so to me they’re equally good. Am I already being irrational before I choose an urn? If so, why? If not, what can I do to avoid being deemed irrational according to the test?

1) I prefer a free chance to win money, therefore I must pick one of the urns, yet
2) The three urns are equally good and therefore I cannot prefer/pick any of them.

See the issue? It isn’t my choosing C that renders me irrational, it’s my indifference.

128. 128 128 Steve Landsburg

Frosty: As has been said many many times, it’s not irrational to be indifferent among all three urns.

129. 129 129 Frosty

Which urn can be rationally picked by an indifferent person?

130. 130 130 Steve Landsburg

Frosty:

Which urn can be rationally picked by an indifferent person?

Allow me gently to suggest that questions like this are the reason some of us suspect you haven’t been paying close attention to this discussion.

131. 131 131 Frosty

I appreciate your being gentle, but won’t you please just answer the question?

132. 132 132 Steve Landsburg

Frosty: The definition of indifference is that you don’t care which urn you pick.

133. 133 133 Frosty

Another condescending non-answer from Steve. Unless someone asks me a direct question, I bid you farewell and leave you with these final thoughts.

My definition of irrationality is choosing something which is inconsistent with your preferences. Therefore, assuming the Von Neumann/Morgenstern axioms, my opinion is that when offered choices A, B, and a coin flip:

1) If choosing either A or B has positive utility, it is irrational to choose none.
2) If you prefer A, it is irrational to choose B or C.
3) If you prefer B, it is irrational to choose A or C.
4) If you prefer neither A nor B, then none of the choices is irrational. You cannot claim to prefer any of the choices, but you can choose any of them. Choosing is different from preferring. At best, choosing the coin flip proves only that you’re indifferent toward A and B, not that you’re irrational.

If my opinions are inconsistent with economists’ definition of irrationality or with revealed preference theory, then so be it.

134. 134 134 Steve Landsburg

Frosty: Your “opinions” 1) through 4) are not matters of opinion; they are all correct statements.

135. 135 135 Ken B

I don’t think Frosty the best advocate of the point he raised. I think Dan in 119 is. I have not seen anything yet to convince me Dan is wrong. Have I missed something, or does Steve agree with 119?

136. 136 136 Steve Landsburg

Dan 119 (and Ken B 134): I do not understand your comment. You say I don’t need all the vN/M axioms to derive U(C)>U(D). What are C and D?

137. 137 137 Harold

Would it be correct to say that if one is indifferent between A and B, is not irrational to *pick* Jar C. It is irrational to *prefer* Jar C – or indeed oxymoronic to both prefer Jar C and be indifferent.

Dan in 119 says “the preferences are over gambles not known payoffs. The cats/dogs example is therefore not a good analogy” Surely it is precisely because the preferences are over gambles that makes the cat / dog analogy a good one? Sorry to repeat myself (again) but for me replacing jar B with 2000 \$1 balls (and jar C with 1 \$1000 ball and 1000 \$1 balls (plus 999 black)) makes this clear.

Jar A: \$1 in 1000 chance of \$1000
Jar B: \$1

We could keep the expected payoffs the same by subtracting \$1 from each. Would you spend \$1 for a lottery ticket with a 1 in 1000 chance of \$1000? You can answer either yes or no – it is rational to prefer either depending on preferences. Jar C now becomes “do you toss a coin to see if you buy a ticket?” I think it is clear that the only way C appears rational is because you cannot decide -not because it offers any genuine middle way.

There are all sorts of reasons why either may be prefered. Lets say you are at a fair and you have just enough for the \$ fare home, but it is a fairly short walk. You also want a \$100 phone. It would seem rational to prefer Jar B – the certainty of a bus home to a 1 in 1000 chance of a big win.

I think our minds interpret the numbers as significant, when they are not. As Dan said, the preferences are for *gambles* not money. I think our minds get mixed up between Jar C and lets say Jar D, which has 20 x \$100 (the old jar B). Jar D now gives us new odds and a new amount -\$100 – which was not offerd by A, B or C. We may rationally prefer Jar D over A and B – it offers us a chance to get that \$100 phone.

138. 138 138 Ken B

@Steve 135: A Cat for certain and a Dog for certain. Frosty’s initial point, expanded upon by Dan and myself, is that the original problem offers lotteries as prizes. A lottery is not a cat or a dog, the uncertainty might matter. So axioms relevant for lotteries are not required for certainties.

139. 139 139 Dan

@Steve 135 and @Harold 136. C = Cat, D = Dog. A and B are the gambles provided by Urn A and B. We need the vN/M assumptions to write U(A) as sum(pi*U(Xi)) where Xi are the gamble payoffs.

All the so called paradoxes relate to choice under uncertainty. And once you allow for uncertainty, you need additional information like how often do I face these gambles, can I combine them, etc. which do not arise when we talk about certain payoffs like cats and dogs.

I am not suggesting that the argument is wrong, it’s just that not enough information is provided, and that the cat/dog analogy is not the right one. This is why so many found the puzzle confusing.

Harold, I think it’s more informative to start with Urn C being optimal and then work backwards. In your fair example, imagine a roulette wheel with half of it painted black with numbers from 1 to 1000, and the other half painted in red with no numbers. You pick a number and place your bet. If you hit red, you get your money back, if it’s black then then if you hit your number you get \$1000. There is another table featuring a roullete with only the black half.

Now you can spend your money to take the bus or play one of the roulette wheels. Say you see someone who really likes playing the two-colored wheel. You can say they are being irrational. You can tell them ‘hey you can replicate your experience by flipping a coin and then depending on the outcome play the the one-colored roulette or go take the bus to go home” They’d most likely say no they would not have the same experience. Again, this could be the definition of being irrational (or alternatively expected utility framework being wrong). At any rate, the cat/dog analogy provides confusion in this situation.

140. 140 140 iceman

Frosty – well they say the best way to make sure you understand something is to try to explain it so someone else.

In the first case I used where the odds were the same for A and B and one is preferred, that directly dominates the profile of C. (Like I said that seems closest to the coin flip but retains some uncertainty.) That extends into the more general idea that if you prefer A to B (or B to A), you can imagine an improved version of B (or A) where you’re indifferent between them, and this also corresponds to an improved version of C – one with a profile that is not just different than, but clearly superior to, the old C (e.g. at least equal to in all respects and better in some). And being indifferent between *that* and A means you must prefer A to the old C. Pretty slick.

As you said, it’s only irrational to choose something when you really prefer something else. As others have said, it’s not irrational to choose any of the 3 randomly if you’re indifferent between A and B.

141. 141 141 iceman

Or you value surprise for its own sake of course

142. 142 142 ThomasBayes

If I understand this alternative theory that several people are proposing for rational decision making, then it should be considered rational to prefer heads over tails when calling the flip of a fair coin to decide on a lottery prize.

Example:

Option A: You win \$1M if the coin shows heads.
Option B: You win \$1M if the coin shows tails.

A rational person could prefer Option A OVER Option B because they will feel better if they win with heads than if they win with tails? Correct?

Prior to this, I might have used this as a simple example of behavior that is irrational. I probably still will, but it is good to know that some people disagree.

143. 143 143 Ken B

@Thomas Bayes 141:
You need to see Rosenkrantz and Guildenstern Are Dead.

144. 144 144 Dan

@Thomas Bayes 141.

You example is not useful because nobody would behave ‘irrationally’ based on your test.

Instead try something like the ellsberg test: http://en.wikipedia.org/wiki/Ellsberg_paradox.

Pretty much everyone here would fail that test. And according to the strict definition of rationality by Steve and you (failure to follow expected utility) everyone would be considered ‘irrational’. Most people I think would agree that calling people irrational based on this alone would be silly.

The question is whether something like the Allais test any different. We have a model and we have the reality of how people actually make decisions. I grant that we have to make a distinction between preferences and rational behavior. But I think you guys are pushing it too far. I am saying that the model is wrong. You guys are saying the model is right and the reality is wrong. This seems strange to me as we have many models to choose from. For instance you can introduce ambiguity aversion and no longer be ‘irrational’ for failing the ellsberg test or introduce dissapointment aversion or rank-dependent expected utility to deal with the allais paradox.

145. 145 145 Frosty

@139 – iceman As others have said, it’s not irrational to choose any of the 3 randomly if you’re indifferent between A and B.

Footnote 23 from the Alchian paper says “Only the reader who chose C should continue, for his choice has revealed irrationality or denial of the axioms.”[emphasis mine]. I inferred the author was supporting the “revealed irrationality” case, not the “denial of the axioms” case and is therefore claiming that choosing C is irrational.

The footnote continues “This can be shown easily. He states he prefers C to A and to B.” Hasn’t the author equated choosing with preferring at this point?

146. 146 146 Steve Landsburg

Frosty: Hasn’t the author equated choosing with preferring at this point?

Preferring \$C\$ to \$A\$ is defined as choosing \$C\$ over \$A\$ (when given the choice).

147. 147 147 Frosty

1) The Alchian paper says “Only the reader who chose C should continue, for his choice has revealed irrationality or denial of the axioms.
2) Steve posted “So Urn C, like the coin flip, is the irrational choice.

3) In 132, I said “4) If you prefer neither A nor B, then none of the choices is irrational. You cannot claim to prefer any of the choices, but you can choose any of them. Choosing is different from preferring. At best, choosing the coin flip proves only that you’re indifferent toward A and B, not that you’re irrational.
4) in 133, Steve said “Your “opinions” 1) through 4) are not matters of opinion; they are all correct statements.
5) In 139, iceman said “As others have said, it’s not irrational to choose any of the 3 randomly if you’re indifferent between A and B.

I think the Alchian paper (1) and Steve (2) are saying choosing C is irrational.
I think I (3), Steve (4), and iceman (5) are saying choosing C is not irrational.

Can anyone help me resolve this apparent contradiction?

148. 148 148 Frosty

@139 – iceman

Feel free to point out if/where I go astray. You said “…the more general idea that if you prefer A to B…“. To verify I’m understanding, I need to begin with a scenario where I actually prefer A to B. I propose urn A has 2 red (\$1000) balls and the rest black balls, urn Y has only 12 blue (\$100) balls and the rest black balls, and urn Z has 1 red ball, only 6 blue balls, and the rest black. Clearly, I prefer urn A to urn Y and to urn Z.

You said “…you can imagine an improved version of B (or A) where you’re indifferent between them…. Increase the appeal of urn Y and urn Z by adjusting the probabilities (replacing black balls with blue balls) in urn Y and the corresponding changes in urn Z to the point at which I become indifferent, i.e.. until Y has transformed into urn B, and Z into urn C.

You said “…this also corresponds to an improved version of C – one with a profile that is not just different than, but clearly superior to, the old C…” and that’s true, I clearly prefer urn C (new C) to urn Z (old C).

Then you say “And being indifferent between *that* and A means you must prefer A to the old C.“, which I translate into being indifferent between urn C (new C) and A means I must prefer A to urn Z (old C), which is also true.

However, I must be missing something because the conclusion (prefer A to Z) is something I presumed must be true by the premise that I prefer A to Y (your B).

149. 149 149 Harold

#139 and #147 – Frosty and Iceman – are you related?
Urn A = 2 \$1000 balls. Expected return = \$1
Urn B = 20 x \$80 balls. Expected return = 60c

Urn C must be 1/2 A and 1/2 B. So it is 1 x \$1000 ball and 10 x \$80 balls.

I prefer A to B.
I increase the value of the balls in B (and C).

When we get to \$100 balls the expected return is the same – the original puzzle. I still prefer A – because I prefer a small chance at a big win over a larger chance of a small win. Expected return for each jar is \$1

With \$120 balls in B, I prefer B. The slightly greater value of the small win means I now prefer this to the small chance of a big win. Expected returns are now A= \$1, B=\$1.20 and C=\$1.10. We now have my point of indifference – \$110 balls in B as close as we can get. This indiffernce point will be different for different people Some may really need \$70, so would prefer it with only \$70 balls in B. Others would realy like the chance at \$1000, so will still pick A even with \$130 balls in B.

The question now becomes – is it rational for me (or anyone) to prefer C for any value of balls in B?

Taking the extreme cases – well away from my indiference point. At \$80 balls in B I prefer A to B. I think we all agree that picking C is like tossing a coin to decide which jar to take. Since I prefer A to B, why would I prefer to toss a coin to decide which jar to take since I know which I prefer?

The arguments put forward are (I think):
1) as twofer said: “picking C as kind of a hybrid between still having some of the thrill of possibility winning \$1000 and while retaining some of the higher odds of winning” This is just as true with \$80 balls as with \$100 balls. It is just as true for \$20 balls. Even if you prefer C to A at your indifferece point, how far away would you have to go to still prefer C? What precisely is it that you are getting for the loss of expected return?

2) I enjoy the thrill of the coin toss. I don’t think this makes much sense either. I am going to get the thrill of drawing the balls, increasing the randomness wouldn’t give me much extra utility.

At my indifference point, I cannot decide. Choosing C then saves me from trying to decide the undecideable, and has a marginal extra utility. Whilst it may not be rational in terms of the vN/M axioms, we would not say that an individual making such a choice was irrational.

At anywhere other than my indiffernce point, is it rational to choose C? If so, where is the extra value coming from? Personally, it makes no sense to me to prefer A then opt for C.

I think the real reasons people prefer C is becaue they are unable to decide A or B.

150. 150 150 Frosty

@148 Harold “Frosty and Iceman – are you related?” made me laugh.

I agree adjusting the value of blue balls works. Personally, my indifference point is precisely when the expected payoff is identical. I am not willing to accept any loss of expected return. If blue balls were \$99, I’d choose urn A. If blue balls were \$101, I’d choose urn B. But at \$100, I’d choose C because, using your words, “At my indifference point, I cannot decide. Choosing C then saves me from trying to decide the undecideable, and has a marginal extra utility.“. TomM pointed this out immediately in #2.

Although avoiding expending any resources trying to decide the undecidable is enough to tip the scale for me, I’m trying to take it one step further. I believe choosing C also has marginal extra utility by most accurately reflecting my indecision and therefore best avoids inaccurately revealing a preference for either A or B. Assume ThomasBayes offers me urn A, urn B, and the coin flip. If I just walk up, choose A and draw from urn A, ThomasBayes understandably adds me to his group of “people who prefer urn A to urn B”. But if I choose the coin flip and then subsequently draw from urn A and ThomasBayes adds me to the group of “people who prefer urn A to urn B”, then ThomasBayes is a fool.

I think the real reasons people prefer C is becaue they are unable to decide A or B.

I agree. I don’t object to “coin flippers” being categorized as indifferent or indecisive. I object to them being categorized as irrational. If deemed irrational due to some definition/axiom, then I steadfastly continue to cast my vote for “too crabbed”. Who’s with me?

Is the horse dead yet?

151. 151 151 iceman

Harold – it’s just that we’re both so cool.

Frosty – At the risk of resuscitating the mare…but what happened to your preference for “balance” if it’s suddenly obvious that A > B (or Y) => A > C (or Z)? Note even your Z is still better than A in one respect and B in another (the original puzzle is a little more interesting because you start with the same EVs so it’s purely about risk preference). It’s the deduction from adding balls to non-preferred options to re-establish indifference that objectively proves if you have a preference it can’t be C. Again I found that method to be kinda slick.

I’m trying to agree with those who said it’s irrational to *prefer* C. I think the famous footnote tries to lay out that precondition, but we’re getting hung up on the word “choose”. Selecting randomly (or in real life “just picking one”) doesn’t indicate a preference, e.g. the footnote says if you’re indifferent you don’t care if your friend flips a coin on your behalf, so why couldn’t you flip the coin yourself? We don’t want to be in a position of saying if you’re indifferent but have to choose something you’re irrational by definition. From there we’ve identified some potential tweaks the axioms exclude, like the value of surprise, or avoiding the effort of determining we’re actually indifferent. Apparently Frosty is also concerned about accurately signaling his indifference to others.

152. 152 152 ThomasBayes

149:

But if I choose the coin flip and then subsequently draw from urn A and ThomasBayes adds me to the group of “people who prefer urn A to urn B”, then ThomasBayes is a fool.

Yes, Frosty, if someone prefers urn B to urn A, but flips a coin and selects from urn A, then, sure, they will have fooled me real good.

By the way, the issue has never been about indifference. The issue has been about a person who prefers urn B to urn A, but would instead draw from urn A because a coin happened to show heads. The too crabbed theory of rational decision making calls this irrational. I suppose we could use a different word, but I think it is valuable to have some name for this type of behavior, and irrational seems fine to me.

Just for the record, I often behave irrationally according to this definition. Some of those behaviors I’d like to change, some I wouldn’t. But I don’t see how using a different definition would matter.

153. 153 153 Frosty

@151 ThomasBayes “Yes, Frosty, if someone prefers urn B to urn A, but flips a coin and selects from urn A, then, sure, they will have fooled me real good

Isn’t it clear to you that I’m not talking about someone who prefers urn B to urn A? I’m talking about people who prefer neither A nor B. For heavens sake, if you read to the bottom of 149, you’ll see I’m agreeing with Harold that the primary motivation for choosing C is indecisiveness. I also made this abundantly clear in 132. I’ve also taken the time to re-read each of my comments. In none of them did I support choosing C when preferring A to B or B to A. If you could tell me which of my comments led you to believe I was supporting such a position, I’d appreciate it.

By the way, the issue has never been about indifference

Well, we couldn’t disagree more on this point and probably explains why our debate is going so poorly. My objection revolves around indifference/indecisiveness. Naturally I think you’re simply wrong to conclude the issue has never been about indifference. The Alchian footnote starts out saying choosing C reveals irrationality and supports the opinion by first presuming indifference toward A and B and concluding there is no reason for choosing C. This is what I’m challenging. Steve’s post doesn’t include a qualifier excluding indifferent people when he says urn C like the coin flip is irrational.

154. 154 154 ThomasBayes

Frosty 152: “This is what I’m challenging. Steve’s post doesn’t include a qualifier excluding indifferent people when he says urn C like the coin flip is irrational.”

Here’s a quote from Steve’s original text on this post:

“You can prefer A, you can prefer B, and you can like all three urns equally, but you can’t prefer C.”

Again, indifference has never been the issue. The issue has always been whether or not it is rational to prefer C over both A and B. I don’t know what else to say.

155. 155 155 Steve Landsburg

Frosty:

“This is what I’m challenging. Steve’s post doesn’t include a qualifier excluding indifferent people when he says urn C like the coin flip is irrational.”

Thomas Bayes:

Here’s a quote from Steve’s original text on this post:

“You can prefer A, you can prefer B, and you can like all three urns equally, but you can’t prefer C.”

I think it’s pretty clear at this point that all of Frosty’s confusion stems from a problem with reading comprehension.

156. 156 156 Harold

Not so fast: SL: “Preferring \$C\$ to \$A\$ is defined as choosing \$C\$ over \$A\$ (when given the choice).”

This does seem a side point, applicable only in the special case where there is indifffernce between A and B. It is irrational to prefer C, but also impossible to prefer A or B (since we have stated indifference). It is therefore not rational or possible to prefer any jar. Since from the quote above choosing is defined as prefering, it is impossible to choose any jar. In the case of having only 2 jars, it is impossible to choose either jar. This does not seem satisfactory.

157. 157 157 Steve Landsburg

Harold:

Since from the quote above choosing is defined as prefering

Well, actually it’s the other way around; preferring is defined as choosing. Preferring \$A\$ to \$C\$ means that you’ll always choose \$A\$ over \$C\$. Choosing \$A\$ over \$C\$ therefore says that you *don’t* prefer \$C\$ over \$A\$, which is not the same as saying that you prefer \$A\$ over \$C\$.

Still, I take your point to be that one ought to say all this more carefully, and I agree.

158. 158 158 ThomasBayes

I think I see where the difference between “choosing” and “preferring” is causing a problem. And though the comment Harold cited does confuse the issue, I think Steve has been clear about this in several other comments (including his opening statement that I cited earlier):

91: “If you are truly indifferent between cats and dogs, you’ll be indifferent between cats, dogs, and the coin flip. There’s no circumstance in which the coin flip is preferred to both the cat and the dog.”

127: “As has been said many many times, it’s not irrational to be indifferent among all three urns.”

131: “The definition of indifference is that you don’t care which urn you pick.”

If the dispute has been about whether or not an indifferent person can “choose” C while not “preferring” C, then I apologize for my role in prolonging the debate. For the record: I believe it is rational to accept option C if you do not have a preference between A and B. I do not believe it is rational (by the standard definition) to prefer option C if you do not have a preference between A and B.

159. 159 159 ThomasBayes

To be more clear: change my last sentence to . . .

I do not believe it is rational (by the standard definition) to prefer option C over both A and B.

160. 160 160 Harold

156: “Well, actually it’s the other way around” Yes, that makes a difference. You can’t just swap ‘em round like I did.

Can we conclude that the cat / dog analogy is a good one, and it is only because of indifference that it appears not to be?

161. 161 161 Frosty

@150 iceman “it’s just that we’re both so cool.” Again, very amusing.

…but what happened to your preference for ‘balance’

Nothing, it’s ultimately just a term used to express or explain my indifference. My desire for a chance to win big is equal to my desire for a chance to win something, i.e. the two desires are in “balance”.

Apparently Frosty is also concerned about accurately signaling his indifference to others.

I actually believe, in #69, ThomasBayes lends support to my proposal to assign value to C for most accurately predicting a preference for neither A nor B. Thomas offers only urns A and B. A secretive coin flipper ends up in the group of “people who prefer A to B”. I infer this bothers Thomas and it bothers me. Most likely, the coin flipper doesn’t actually prefer A to B and is polluting the group. I propose we offer a third choice, a coin flip. Our secretive coin flipper becomes a known coin flipper and we can properly place him in the group of “people who prefer neither A nor B”.

…but we’re getting hung up on the word “choose”

Absolutely, the precise definitions of the words “choose” and “prefer” are causing problems. Neither the Alchian test nor Steve’s test asks which do you “prefer”, Alchian asks which would you “choose” and Steve asks which would you “pick”. I choose/pick C and my response to “You can prefer A, you can prefer B, and you can like all three urns equally, but you can’t prefer C.”” is simply…ok, but if you’re most interested in which I “prefer”, why didn’t you ask which I prefer? If anyone says he derived I prefer C from my picking C, I say…well there’s your mistake. I’m indifferent and therefore my picking C reveals only my preference for choosing an urn relative to choosing none of the urns.

1) I think it can be rational to, as Steve asked, “pick” C. I’m under the impression there is considerable agreement on this. The debate seems to have turned to whether this is even relevant.

2) I go a step further by thinking there are valid reasons to attach marginal extra utility to C such that it is possible to “prefer” C. People who disagree cite the vN/M axioms. But this is unsatisfying and unpersuasive. I’m voting for too crabbed because I disagree with the axioms. In any case, neither side seems likely to persuade the other.

162. 162 162 Frosty

159 : “Can we conclude that the cat / dog analogy is a good one…” I vote yes.

163. 163 163 Ken B

re 159: I vote no, on the basis of 138, where Dan notes the need for different sets of axioms. Since the point at issue the adequacy of that model and those axioms, this seems like an important difference.

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