Robert J. Aumann

Subjectivity and correlation in randomized strategies

Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘objective’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘subjective’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.). Even a fairly superficial such examination yields some startling results, as follows :

Game theoretic analysis of a bankruptcy problem from the Talmud

Abstract For three different bankruptcy problems, the 2000-year old Babylonian Talmud prescribes solutions that equal precisely the nucleoli of the corresponding coalitional games. A rationale for these solutions that is independent of game theory is given in terms of the Talmudic principle of equal division of the contested amount; this rationale leads to a unique solution for all bankruptcy problems, which always coincides with the nucleolus. Two other rationales for the same rule are suggested, in terms of other Talmudic principles. (Needless to say, the rule in question is not proportional division).

Cooperative games with coalition structures

Many game-theoretic solution notions have been defined or can be defined not only with reference to the all-player coalition, but also with reference to an arbitrary coalition structure. In this paper, theorems are established that connect a given solution notion, defined for a coalition structure ℬ with the same solution notion applied to appropriately defined games on each of the coalitions in ℬ. This is done for the kernel, nucleolus, bargaining set, value, core, and thevon Neumann-Morgenstern solution. It turns out that there is a single function that plays the central role in five out of the six solution notions in question, though each of these five notions is entirely different. This is an unusual instance of a game theoretic phenomenon that does not depend on a particular solution notion but holds across a wide class of such notions.

Backward induction and common knowledge of rationality

We formulate precisely and prove the proposition that if common knowledge of rationality obtains in a game of perfect information, then the backward induction outcome is reached. Journal of Economic Literatur Classification Numbers: C72 D81.

Endogenous Formation of Links Between Players and of Coalitions: An Application of the Shapley Value

Consider the coalitional game v on the player set (1,2,3) defined by

Cooperation and bounded recall

v(S) = \left\{ \begin{array}{l} 0\quad if{\kern 1pt} \left| S \right| = 1, \\ 60\quad if{\kern 1pt} \left| S \right| = 2, \\ 72\quad if{\kern 1pt} \left| S \right| = 3, \\ \end{array} \right.

Values of Markets with a Continuum of Traders

(1) were |S| denotes the number of players in S. Most cooperative solution concepts “predict” (or assume) that the all-player coalition {1, 2, 3} will form and divide the payoff 72 in some appropriate way. Now suppose that P 1 (player 1) and P 2 happen to meet each other in the absence of P3. There is little doubt that they would quickly seize the opportunity to form the coalition {1, 2} and collect a payoff of 30 each. This would happen in spite of its inefficiency. The reason is that if Pi and P 2 were to invite P 3 to join the negotiations, then the three players would find themselves in effectively symmetric roles, and the expected outcome would be {24,24,24}. P1 and P 2 would not want to risk offering, say, 4 to P 3 (and dividing the remaining 68 among themselves), because they would realize that once P 3 is invited to participate in the negotiations, the situation turns “wide open” — anything can happen.

A variational problem arising in economics

There have been continuing expressions of interest from a variety of quarters in the development of techniques for modelling national behavior in a long-term context of continuing international rivalry — for short, “long term competition”. The most characteristic feature of these models is that they extend over time in a fairly regular or repetitive manner. The underlying structure of possible actions and consequences remains the same, though parameters may vary and balances shift, and the decisions and policies of the national decision makers are by no means constrained to be constant or smoothly-varying, or even “rational” in any precisely identifiable sense. The use of game theory or an extension thereof is obviously indicated, and considerable theoretical progress has been made in this area. But the ability of the theory to handle real applications is still far from satisfactory. The trouble lies less with the descriptive modelling, — i.e., formulating the “rules of the game” in a dynamic setting, than with the choice of a solu- tion concept that will do dynamic justice to the interplay of motivations of the actors. (Game theoreticians, like mathematical economists, have always been more comfortable with static than dynamic models.) Since any predictions, recommendations, etc. that a mathematical analysis can produce will likely be very sensitive to the rationale of the solution that is used, and since the big difficulties are conceptual rather than technical, it seems both possible and worthwhile to discuss salient features of the theory without recourse to heavy mathematical apparatus or overly formal arguments, and thereby perhaps make the issues involved accessible to at least some of the potential customers for the practical analyses that we wish we could carry out in a more satisfactory and convincing manner.