A Madness to Her Method

12 September 2010

During the years when I was wrestling with my paper on indecision, there was at least one paper published that attempts to operationalize incomplete preferences, Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences by Kfir Eliaz and Efe A. Ok in Games and Economic Behavior vol 56 (2006) 61–86. I didn't come across this paper until mine was essentially ready for submission, but I dropped-in a note distinguishing how they sought to operationalize indecision from how I did. What they did was interesting, but I'm distinctly uncomfortable with it.

They propose a scenario in which a Mrs. Watson is selecting one item to be shared by her two children. The children's preferences are thus:

which is to say that A prefers y to z and z to x, while T prefers x to y and y to z. The authors tell us that if y is available then the parent will not select z, as y is preferred to it by both children. Thus, if the choice were only between y and z, then y would be selected. However, should y not be available, and the parent forced to choose between x and z, then there is no clear choice between x and z. Yet if z is not available, there is also no clear choice between x and y. Thus, we have the oddness that y is preferred to z, neither x nor z is preferred to the other, and neither x nor y is preferred to the other. The authors associate this odd confluence with indecision amongst some of the choices. The parent resolves the choice over x and y or over x and z by flipping a coin. (The authors do not recognize coin-flipping as a third option, and what they say elsewhere implies that indifference would also be resolved thus.)

The authors write

It seems quite difficult to argue that there is something clearly wrong in the way Mrs. Watson deals with her two choice problems here.

What the parent has done here is implicitly embraced a sort of democratic process; she knows how the children would vote (if not voting strategically), and casts those votes on the behalf of each, settling ties with the flip of a coin. Well, it's well established that, in the absence of everyone having the same preferences, even if each participant is him- or herself rational, democratic processes don't produce rational choice structures. (For example, if some votes are decided before others, then the order in which things are decided determines the ultimate outcome.) In fact, it was famously shown (with Kenneth Arrow's Impossibility Theorem) that this isn't just a problem for ordinary democracy, but for any collective decision-making process. (I'm not here counting dictatorship as a collective decision-making process.)

So the parent's attempting to meet the desires of her two children is, in this case, formally equivalent to her attempting to meet the desires of one child who is just crazy. (Parents of multiple children will likely have little difficulty in grasping that point intuïtively.)

I'll make more plain the insanity specifically here. Under a decision-making process in which one would simply trade each unit of every xn for a unit of xn+1, by m-stage trades, one can be got to trade all of one's x0 for xm, even if one prefers x0 to xm. If whether each trade is effected depends upon the flip of a coin, then it might take more offers to get one to trade-away all of one's x0 for xm, but that's still where one can be led. (A patient exploiter could get one to surrender all of one's x0 for xm, and then monetize that by selling the x0, with credit for trade-in of the xm.)

The authors acknowledge that their scenario involves social choice theory, but assert that the problem can be generalized to other problems of choice with multiple criteria. However, the example that they give is one of an agent attempting to follow given rules (for awarding a fellowship); what we should see is that whatever rules might produce the same sort of structure are, again, loopy (in at least two senses of that word).

Now, in some case, perhaps in a great many cases, the sane chose to humor the insane; they even bend to the will of the lunatic; but the choice made in yielding to regulation by another is not, properly, the same choice as that of the regulator. Nor would I otherwise be comfortable with a model that could not find indecision except where one can find this sort of irrationality lurking in the background if not in the foreground.


4 Responses to A Madness to Her Method

  • Interestingly enough, I think it becomes political after the first option is ruled out. If y is not available, then x wins by default. Simply rate (or more appropriately slot) the 1st, 2nd, and 3rd choices and assign them values. Be it 3-2-1 or 5-3-1 or some other fair system of importance, and there you go. Obviously not great in terms of predicting economic demand, but it wouldn't surprise me in the least if - at some point - an economist proves the validity of such a system.

    • Daniel says:

      Quite the contrary; the presumption that there is any fair system of importance is without foundation.

      A while back, I began an entry on the problem of interpersonal comparison, which entry is to explain the very strong assumptions required. It's going to take a while before I have it written to my satisfaction. But in the case where, somehow, each individual really can go so far as to impute a degree of desirability to each outcome, there is no basis for thinking that those degrees have any meaning except relative to the degrees within that same person. Even if they did have meaning, it would only be observable to some presumed G_d; if they all doubled for some person, there would be no change observable to the person ascribing own-values or to us. (Calls to G_d to take over decision processes have produced dubious outcomes.)

      Even if, perhaps, A's 6 is truly twice as large as her 3, we don't know that it's larger than T's 5; we can't know.

      • I figured as much. Obviously I'm not an economist, but I do realize and appreciate the process of not making assumptions. However, here is an example, dumbed down for myself.

        Platinum is worth more than gold which is worth more than silver, pretending that the mass of each item is exactly the same.

        P>G>S if you will.

        However, given a choice, I prefer silver, then gold, and then platinum. In a pool of, say, 100 people, or even twenty, my preference is no more important than is that of anyone should the items be valued exactly the same, but since we know they are not, then I would say that the relative values of my choices only increase when the pool is reduced. One platinum-loving advocate might humor me; twenty will not.

        However, should we be talking about exact values, then the odds are entirely the same. Regardless of my irrational love of silver.

        • Daniel says:

          I'm not certain in what in what sense you claim that your preference is not more important than that of anyone else. Where the assignment of market prices is concerned, the preferences of some people are more significant than those of others. If you're talking about some other sort of significance, I'd might want to know why the preferences of one person weren't more important than those of another, and I'd certainly want to know how one went from the proposition that those of one weren't more important than those of another to the proposition that they were equally important, rather than incommensurable.

          Pricing by market or by auctioning process is often advanced as a way of settling competing claims to resources. But any market or auctioning process begins with an initial allocation, implicit or explicit, and those allocations need to have a sound normative foundation if an endorsement of the outcome is to be reasonable; and, perhaps more germain to this discussion, different paths of negotiation will result in different prices and final allocations. In the ideal, limiting case of zero transactions costs, any of these outcomes will be economically efficient, but that doesn't make them fair (nor, of course, does it make them unfair).

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