A Madness to Her Method
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:
|A||y ≻ z ≻ x|
|T||x ≻ y ≻ z|
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.
Tags: decision theory