Game Theory, Procedure, and Consent: Focusing on Fair Division

ion that are involved in game theory to construct possible ways peoThis content downloaded from 207.46.13.114 on Thu, 26 May 2016 06:43:35 UTC All use subject to http://about.jstor.org/terms 1098 LAW AND SOCIAL INQUIRY pie might agree to resolve their disputes. The "divide a dollar" problem has occupied some of the most powerful minds in game theory (e.g., Harsanyi and Selten 1988). The original game requires two (or more) players to divide a dollar by bidding how much they would like to have, subject to a payoff structure that their independent bids cannot total more than a dollar or they all get nothing. The Brams and Taylor "solution" is ask the players, "Perhaps you would like to play a different game?" One key element of the different game in this case is to pay off the smaller bids first, leaving the largest-bids unpaid if their payment would take the total above a dollar. The change in the incentives then is that the people who make greedy bids pay disproportionately for the collective excess of greed. The risks of having to pay for everyone's excess and so get nothing increases as the amount of a person's bid over equality increases. It is a lot easier to solve this game than the original one (even without the Bayesian statistics used by Harsanyi and Selten). If disputants can understand Brams and Taylor more easily than I can understand Harsanyi and Selten, they may even get consent to the game's outcome (each bidder in equilibrium gets 1/n of the dollar, where n is the number of bidders). Of, course they probably thought that was fair in the first place, game theory to the contrary. If permitted to discuss it honestly in ordinary language, instead of in game-theory moves, they might well choose that outcome directly. There are two main components in the formalization of bargaining procedures itself, besides the agreement on isolation of the situation and the agreement on implementation of the outcome after the game. The first is agreement on the abstraction of the bargaining problem, or building the algorithm of the formalization, which I will call "substantive formalization." The second is specifying the connections between bargainers' acts and outcomes, which I will call "payoff formalization." Thus, in the example about Bob and Carol, the substantive formalization involves identifying issues, giving Bob and Carol equal numbers of points to describe the importances of those issues, describing the method of ordering the issues near the cutting point between bargainers so that Carol can get her equitable points from among those cheapest to Bob (in points). It also involves, in the game described above, assuming away the problem of people misrepresenting their valuation of issues for strategic reasons, and the separation of bargaining about how to split the refrigerator that is on the final dividing point and the like. The payoff formalization describes the actions taken by the parties or a referee that takes the basic algorithm and decides which issues go wholly to Bob, which wholly to Carol, and which one is split by a separate bargain. What is most distinctive about the new game Brams and Taylor ask people dividing the dollar to do, as compared with its original "game theory" verThis content downloaded from 207.46.13.114 on Thu, 26 May 2016 06:43:35 UTC All use subject to http://about.jstor.org/terms Game Theory, Procedure, and Consent 1099 sion, is that the dollar is paid off to the lowest bidders first, and to the highest bidders if any money is left. This turns out to provide very near to an egalitarian optimum bid by all players, and an egalitarian outcome if all optimize.2 Thus, formalization may use formal abstracted procedures, or routines rigidly adhered to that ignore the rest of life, but it can be made a matter of choice what abstractions and procedures one is subject to. The mood is very different from solutions in which the game theorist allows the game to be set by the nature of the world or the nature of the psychological experimenter. Try to imagine the perverse personality it would take to set the conditions of the traditional divide-a-dollar game; one can only imagine its real-life analog in a cognitive psychologist's laboratory. On the other hand, the Brams and Taylor version has considerable poetic justice, in the greedy bidders getting less.