On Backward Induction

A Guest Post

by

William Carrington

(Note from the proprietor—I am delighted to present this guest post from my correspondent William Carrington, who might or might not have been inspired by the puzzles in Chapter Nine of Can You Outsmart an Economist?. — SL)

Like zombies and Russian spies, there are more economists among us than you might think. This can be dangerous because studies show that economists are more likely than normal people to graze their goats too long on the town commons, to rat out their co-conspirators in jailhouse interrogations, and to show up drunk on their last day at a job. This appears to be both because unethical people are drawn to economics and because economics itself teaches people to be both untrusting and untrustworthy. This feedback loop has led to the creation of famously difficult economists like John Stuart Mill and….well, it’s a long list. Like halitosis and comb-overs, the problem is worse in Washington.

Can you protect yourself against this unseen risk? Sadly, no, as economists often look all too normal and are hard to pick out from the maladjusted crowds that attend us. This is known as the identification problem in economics, and Norway’s Trygve Haavelmo was awarded a Nobel Prize for his work on this issue. Related work by Ken Arrow, also a Nobelist, proved that an infinitesimal group of economists will bollix up the welfare of an arbitrarily large population of otherwise normal people. It’s most disheartening, but I’m here to offer you a failsafe method for identifying economists. You’ll need an old refrigerator.

Patty and I spent the first four married years living in an apartment in Hyde Park, not far from the University of Chicago. The building was lovely, made of wood and brick in the early part of the 20th century, back when they made beautiful buildings, and the dogwood trees in the courtyard flowered during both non-winter months. The apartment had a tiny bathroom and a small 1950s kitchen that included a gas stove with two working burners and a creaky, small refrigerator. The refrigerator had a small interior icebox and, like other refrigerators of its vintage, it had no icemaker. This meant that ice had to be created by filling plastic trays with water, and we had eight such trays stacked one on top of another inside the icebox.

Our young household was divided on when someone had the responsibility to fill an empty or partially empty ice cube tray with water and to then place it on the bottom of the stack of trays in the icebox. My position was that you had to fill up the tray if and only if you took the last cube. I liked this policy because every ice cube was used, thereby minimizing how often we filled the tray with water. But Patty was offended that I would sometimes leave the top tray with only one or two cubes left. It was rude, she said, to leave the tray so that the next ice user would have to fill up the tray with certainty. And she thought that I would game the system by leaving just one cube for the next user, perpetually avoiding the hassle of filling the tray myself.

I certainly did leave the tray with one cube some times, but I thought she had not followed through the logic of her position. If you can’t leave the tray so the next user has to fill up the tray, then it’s true that you can’t leave just one cube in the tray. But if you can’t leave one cube, then you cannot leave two cubes either. After all, the next user will use at least one of the two remaining cubes, and since that next user can’t leave just one cube, then that next user will have to fill up the tray just as surely as if there had been one cube left. And if you can’t leave two cubes, then you can’t leave three cubes, either. And if you can’t leave three cubes, you can’t leave four cubes. And so forth. If you should never leave the tray so that the next person has to fill up the tray, then logic dictates you can never use the ice cube trays at all. The trays would sit mutely in the corner of the icebox while we drank lukewarm mojitos in the hot Chicago summer. The economic logic was airtight.

Patty was unimpressed by my reasoning. “Will, you’re an idiot,” she said. Thus, we looked at the world in vexingly different ways, and this same problem, in slightly different clothes, cropped up around doing the laundry, taking out the garbage and other household chores. And so we sought out third, fourth and fifth opinions from our friends, mostly either medical or economics students. The economists immediately saw the logic of mathematical induction in my position and nodded their heads in agreement. The medical students agreed with Patty, however, most not even acknowledging that my argument made any sense at all. Quite to the contrary, in fact, they thought that, yes, I was an idiot, and a noisome one at that.

I won the rhetorical battle but I lost the ice policy war after spending a few nights on the couch, and I subsequently sought to never leave fewer than two cubes in the tray. We moved a few years later from our Chicago apartment to a Baltimore townhouse that had a refrigerator with an automatic icemaker, so this roadblock to our relationship was removed through the natural course of events. But if you are about to engage in a high-stakes venture with someone you suspect might be an economist – robbing a bank, for example – I suggest asking them about the proper ice cube policy in a world without automatic icemakers. It could save you from marching off to a long prison sentence while your accomplice turns state’s witness.

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29 Responses to “On Backward Induction”


  1. 1 1 Henri Hein

    That was awesome.

    Incidentally, do you take comments and questions about ‘Outsmart,’ and if so, where do we send it? I don’t get the ‘one troll is better than two’ rationale.

  2. 2 2 Steve Landsburg

    Henri: Give me a few days, and I’ll make a post about trolls, just so you can raise your points!

  3. 3 3 James Roberts

    Wonderful!
    This reminds me of the question why I tend to run up stairs but not escalators.

  4. 4 4 Biopolitical

    William is describing naive, socially-awkward economists. There exist sophisticated, socially-gifted economists who would tell Patty she is right, even though they know she is wrong. I am thinking, for example, of Joseph Stiglitz.

  5. 5 5 David R Henderson

    Hilarious! And really well written.

  6. 6 6 Henri Hein

    Steve: Cool, thank you!

  7. 7 7 Enrique

    Alas, the joke is on William (not Patti). Although the logic of backwards induction is sound in a world of automotons, in the real world of social norms (and common courtesy), people cooperate instead of defecting, especially in long-term relational situations where there are going to be many repeat interactions. (As an aside, I just started reading the puzzle book, so I am looking forward to the one troll versus two trolls post.)

  8. 8 8 Jonathan Kariv

    @Enrique #7
    Patty wants the “refill if 0 or 1 cube” rule because she’s worried William will defect under the “refill if 0” rule. The trouble is that William can just as easily defect under either rule. He makes sure he’s leaving exactly 2 instead of exactly 1, thereby essentially shrinking the ice cube tray by the size of a cube.

    The backwards induction comes in when Patty realises this and wants a “0,1 or 2” rule, then a “0-3” rule etc. I think the interesting bit here is that in practice this doesn’t happen but the point isn’t that the “0 or 1” rule doesn’t actually make William cooperate instead of defect.

  9. 9 9 Jonathan Kariv

    Edit to comment 8:
    but the point is (rather than isn’t) that the “0 or 1″ rule doesn’t actually make William cooperate instead of defect.

  10. 10 10 Harold

    Careful, or you might find yourself unexpectedly (perhaps metaphorically?) hanged on Thursday.

  11. 11 11 Advo

    Alas, the automatic ice maker does not solve the problem, because you have to refill it with water.

  12. 12 12 Bennett Haselton

    I suppose this is missing the point, but you could avoid this problem by having a rule that both parties have to refill the tray the same number of times in the long run. If you get too far behind, you have to refill the tray a few times to catch up, regardless of whether you are leaving it with 0 or 1 cubes left.

    To avoid scoring frivolous points by “refilling” the tray when there are 7 out of 8 ice cubes left in the tray, let’s say you can only score a point by refilling it when there are 0, 1, or 2 ice cubes left in the tray.

    The disadvantage of course is that now someone has to keep score to track who is doing too much or too little, but I’ve heard that’s a big part of married life anyway.

    (I’m also assuming nobody is going to nit-pick the difference in effort between refilling 6 empty compartments in the tray versus 8 empty compartments, but then again I’m not married.)

  13. 13 13 James Kirby

    Can you next offer a solution to the “leave toilet seat up or down problem?

  14. 14 14 Enrique

    @Kariz #8, #9
    Quick clarification: I though Patty wants William to refill the tray after each use (regardless of the number of ice cubes that are left in the tray). Isn’t a “refill after every use” a good example of a simple (and fair) rule — simple because it’s easy to tell if one has defected or not (since there are only two players), and fair because it allocated the burden after every use?

  15. 15 15 Bennett Haselton

    @Enrique:

    1) I think it is assumed there is some fix cost associated with taking the tray out and taking it over to the sink, and then replacing it in the freezer and stacking the other seven trays on top of it. In other words, it’s inefficient and everyone loses if you have to refill the tray after every use (even if you only took one cube out), rather than waiting until the tray was nearly empty.

    2) Also, depending on how frequently they withdraw ice and how long it takes to freeze an ice cube, if you re-fill the tray after every use, it might not stay in long enough to form an ice cube next time you need it.
    Suppose between the two of them, they use one ice cube per hour. And suppose it takes 10 hours to freeze a cube. With 8 trays, after 8 hours, the original tray will be back on the top of the pile, and the water that was poured into the empty spot won’t be frozen yet. (You could try taking out one of the other cubes, but it’s difficult to get one cube out of an ice cube tray when there is still unfrozen water in another compartment of the same tray.)

  16. 16 16 Zazooba

    Extra point for use of the word “noisome”.

  17. 17 17 Zazooba

    Often economic results change if you allow some real-world imperfections or larger considerations. Do the following produce a different result?

    1. Include a cost to getting yelled at for acting overly selfish. You can’t repeatedly leave one cube each time because you are obviously being selfish and you will get yelled at, which has imposes a significant cost. Therefore, the optimum selfish strategy is probably to randomize enough that you don’t get yelled at. Does induction fall apart then?

    2. Can the problem be solved by acknowledging the larger context where there are gains to trade? Perhaps one person dislikes refilling the trays more than the other. The efficient solution, therefore, may be to assign refilling to the one who cares less. while the one who cares more does something else. This typically happens since men and women tend to have different preferences and so tasks are naturally divided between them, i.e., specialization of labor naturally flows from differences in tastes or abilities.

    3. Maybe you could split the cost of refilling the tray evenly so there is no benefit to avoiding refilling. One person takes the tray out of the fridge and fills it with water, and the other has to put it in the freezer.

    4. You had eight ice trays. Why did you refill when only one is empty? Shouldn’t you wait for six or seven to be empty to realize economies of scale? (Plus, lifting up seven ice trays to place the refilled one sounds like a lot of work.) The induction problem remains, but I suspect you had way too many ice trays. Were all those trays a sign that there were more serious problems here that needed to be addressed? Were all those trays a cry for help?

  18. 18 18 Henri Hein

    Note to author: apparently, blog comments can also be used to identify economists.

  19. 19 19 Jonathan Kariv

    @Enrique #14
    Enrique, Patty wanted Will to refill if there is 1 (or less) cube left. As Bennett (#15) says it might be simple (easily stated) and fair (symmetric) but it’s inefficient because it involves a lot more refilling than would otherwise occur.

    On a different point the data doesn’t actually point to a test to identify economists. We only sampled on economists and medics. Maybe it is a test for medics and everyone else likes the backwards induction argument. Seems like a great way to figure out who should operate on one…

  20. 20 20 Enrique

    @Kariv (#19): I get that William’s preferred approach is more “efficient” in that it involves less tray transfers (from fridge to sink) but my question boils down to this: why should the efficiency gains of William’s preferred rule trump the values of simplicity and fairness as embodied by Patty’s preferred rule?

  21. 21 21 Jonathan Kariv

    @Enrique There are 3 rules here.
    1. William’s refill if you finish it.
    2. Patty’s refill if you finish it or leave a cube.
    3. Enrique’s refill if you use anything at all.

    In terms of simplicity : Enrique=William>Patty
    In terms of fairness: Enrique=William=Patty (all 3 are symmettric)
    In terms of efficiency: William>Patty>Enrique

    So William’s rule is at least equal to both Patty’s rule and yours on all three criteria.

  22. 22 22 Enrique

    Ok, I get it now! (But I would revise the first ordering above re: simplicity as E > W > P; maybe there is some Condorcet cycling going on!)

  23. 23 23 Zazooba

    @Kariv #21

    Things get confusing for me when I try to spell out the implicit assumptions here.

    Are you assuming that William does not try to game the system by taking fewer cubes than he would like to avoid refilling? Because if he does game the system, then it isn’t fair (at least from Patty’s perspective) because she does all the refilling. It also isn’t efficient because William gets fewer cubes than he would like. In another sense, though, it is more efficient because there are fewer refills. (Curiously, the most efficient outcome in this sense is if no one ever takes the last cube and there are zero refills.)

    The more I think about this, the more confused I get. What is the efficient outcome? Should the cost of refilling be balanced against getting as many cubes as you want? Perhaps everyone should reduce cube consumption to reduce refilling costs. If the marginal value of cube consumption is a constant, then there will be a corner solution.

  24. 24 24 iceman

    Hope I never reach this type of impasse over who changes the TP roll

    15 – “depending on…how long it takes to freeze an ice cube”
    Pro tip: fill the tray with hot water

    4 Biolopolitical — “like”

  25. 25 25 Jonathan Kariv

    @Zabooza #23 The idea is that IF William is going to try to game the system he can do so as easily with rule P as with rule W. Therefore rule P is strictly worse than rule W.

    As for the orderings I posted in #21. I agree it depends on exactly how you define “fairness”, “efficiency” and “simplicity” and we haven’t explicitly done so.

    For efficiency I mean number of cubes gotten per refill.

    For fairness I am assuming no one is gaming the system intentionally. Maybe that’s a bad idea but if Will is going to game it then why wouldn’t Patty also game it?

    For simplicity. Well that’s the trickiest one. Maybe something like “smallest number of accidental rule violations”. Which is probably zero in all 3 of them.

  26. 26 26 Harold

    “For fairness I am assuming no one is gaming the system intentionally.”

    It was one of the explicit conditions that gaming was suspected. Without gaming there is no problem to be solved.

    For efficiency I would use the hypothetical situation where there was only one person. This has to be speculative to some extent, but gives us a framework. We can make some assumptions about Will – he would prefer ice in his drink, he thinks in general the effort of filling the trays is worth it for the benefit of the ice.

    If there were genuinely only one person we would get the efficient outcome. Will would use exactly the number of cubes and fill only when empty, but always do so.

    However, we don’t have just one person – we have present Will and future Will. Present Will could game the system against his future self by failing to fill a tray and putting it off until tomorrow, or by using slightly fewer cubes tonight so Tired Will does not have to fill the tray and fresh Tomorrow Will has to use the last cube.

  27. 27 27 Zazooba

    Maybe Patty has a point and William’s statement formulation of the problem is … um … not as intelligent as it could be.

    Maybe William has not accurately modeled the problem because he has left out Patty’s ability to impose a penalty on William for gaming the system. It is specifically stated that William had to spend a few nights on the couch, and since it was only for a few nights, we can infer that William capitulated. Patty’s penalty is therefore capable of discouraging gaming … at least to some extent.

    If Patty had had the benefit of an economic education, she might have been able to articulate her case more articulately. But her amateur explanation, that William was an “idiot”, is not that bad a rejoinder.

  28. 28 28 Eric

    Maybe Steve needs to do a post about how to factor nights spent on the couch into economic arguments.

    It seems the perfect job for the Armchair Economist.

  29. 29 29 Richard D.

    WC: “The medical students agreed with Patty, however, most not
    even acknowledging that my argument made any sense at all.”

    Tell them, the same reasoning applies to tonsillectomy – whoever
    removes the last tonsil, has to wash away the blood.

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