Dov Samet

Approximating common knowledge with common beliefs

Abstract Strict common knowledge seems almost impossible; we can never be sure what others know. It is shown that common knowledge can be approximated by the weaker and more easily obtained condition of common belief. This approximation justifies the standard assumption in game theory that the description of the game is common knowledge. Aumanns result on the impossibility of agreeing to disagree can also be approximated when common knowledge is replaced by common belief.

On weighted shapley values

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a probabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.

Dissipation of contestable rents by small numbers of contenders

The theory of rent seeking with its origins in the observations of Gordon Tullock (1967) - or to use Jagdish Bhagwatis (1982) proposed term, the theory of directly unproductive profit-seeking activities - is concerned with the potentially adverse effects on resource allocation of incentives to cap- ture and defend artificially-contrived rents and transfers. The scope for so- cial loss proposed by the theory derives from the relation between the value of a contestable prize and the value of the resources attracted into the con- test to determine the beneficiary of the prize. Underlying this social loss is a specification of how rational behavior by optimizing agents links the value of the prize sought to the resources expended. It has been traditional to assume competitive behavior in describing the activities of lobbying and influence seeking. Then, if some further condi- tions are satisfied, l the total value of the resources expended precisely equals the value of the prize sought, so dissipation is complete. 2 Conse- quently, the social cost associated with contestability of a rent can be in- ferred from the value of the rent itself, and the detailed and hard-to-come- by information on individual outlays made in the course of the contest be- comes unnecessary. By basing their analyses on competitive dissipation, contributors to the rent seeking literature (see the review by Robert Tollison, 1982) have been able to presume that the observed value of a contested rent is an exact measure of the associated social cost of monopoly power or regu- lation. Similarly, in the trade-theoretic literature where the rights contested are to quota premia or revenues from trade taxes (Krueger, 1974; Bhagwati

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.

Ignoring ignorance and agreeing to disagree

Abstract A model of information structure and common knowledge is presented which does not take states of the world as primitive. Rather, these states are constructed as sets of propositions, including propositions which describe knowledge. In this model information structure and measurability structure are endogenously defined in terms of the relation between the propositions. In particular, when agents are ignorant of their own ignorance, the information structure is not a partition of the state space. We show that Aumanns (Ann. Statist. 4 (1976), 1236–1239) famous result on the impossibility of agreeing to disagree, which was proved for partitions, can be extended to such information structures.

Utilitarian Aggregation of Beliefs and Tastes.

Harsanyi’s utilitarianism is extended here to Savage’s framework. We formulate a Pareto condition that implies that both society’s utility function and its probability measure are linear combinations of those of the individuals. An indiscriminate Pareto condition has been shown to contradict linear aggregation of beliefs and tastes. We argue that such a condition is not compelling: Society should not necessarily endorse a unanimous choice when it is based on contradictory beliefs. Our Pareto condition is restricted to choices that involve identical beliefs only.

On the Core and Dual Set of Linear Programming Games

We study the relation between the core of a given LP-game and the set of payoff vectors generated by optimal dual solutions to the corresponding linear program. It is well known that the set of dual payoffs is contained in the core, and that cores of games in which players are replicated converge to the set of dual payoffs when the number of replications tends to infinity. We give a necessary and sufficient condition for finite convergence. As corollaries we strengthen a sufficient condition due to Owen and obtain new conditions as well. We also study conditions in which the core and the set of dual payoffs coincide even without replication. We give a necessary and sufficient condition for this phenomenon and present two classes of LP-games with this property which properly subsumes all examples of this type discussed in the literature.

THE DETERMINATION OF MARGINAL COST PRICES UNDER A SET OF AXIOMS

THE MAIN PURPOSE of this paper is to provide an axiomatic approach to marginal cost (MC) pricing and to point out its similarity with Aumann-Shapley (A-S) pricing. The latter is a cost-sharing price mechanism discussed in [3 and 6] that is derived from a set of five natural axioms. In this paper we consider models in which there is one producer with a given technology who faces fixed input prices and produces a finite number of consumption goods. Thus, we can uniquely derive the cost function that describes the minimal cost of producing a given vector of consumption goods. By a price mechanism P(., ) we mean a rule or a function that associates with each cost function F and vector a of quantities, a vector of prices:

An Application of the Aumann-Shapley Prices for Cost Allocation in Transportation Problems

The Aumann-Shapley A-S prices are axiomatically determined on certain classes of piecewise continuously differentiable cost functions. One of these classes consists of all cost functions derived from the transportation problems and some of their generalizations. These prices are used here to allocate costs among destinations in a way that each destination will pay its “real part” in the total transportation costs. An economic transportation model is presented in which the A-S prices are compatible with consumer demands. Finally an algorithm is provided to calculate both the optimal solution and the associated A-S prices for transportation problems.

Bounded versus unbounded rationality: The tyranny of the weak

Abstract We examine the case of a two-person repeated game played by a boundedly rational player versus an unboundedly rational opponent. The former is restricted to strategies which are implementable by connected finite automata. It is shown that the “rational” player has a dominant strategy, and that in some cases the “weaker” (boundedly rational) player may exploit this fact to “blackmail” him. It is also shown that for a repeated zero-sum game, the rational player has a strategy which drives the automaton players limit payoff down to his security (maxmin) level, even if he may choose any finite automaton.