Sergiu Hart

Potential, Value, and Consistency

Let P be a real-valued function defined on the space of cooperative games with transferable utility, satisfying the following condition: In every game, the marginal contributions of all players (according to P) are efficient (i.e., add up to the worth of the grand coalition). It is proved that there exists just one such function P--called the potential--and moreover that the resulting payoff vector coincides with the Shapley value. The potential approach yields other characterizations for the value; in particular, in terms of a new internal consistency property. Further results deal with weighted values and with the nontransferable utility case. Copyright 1989 by The Econometric Society.

Bargaining and Value

The authors present and analyze a model of noncooperative bargaining among n participants, applied to situations describable as games in coalitional form. This leads to a unified solution theory for such games that have as special cases the Shapley value in the transferable utility case, the Nash bargaining solution in the pure bargaining case, and the recently introduced Maschler-Owen consistent value in the general nontransferable utility case. Moreover, the authors show that any variation (in a certain class) of their bargaining procedure which generates the Shapley value in the transferable utility setup must yield the consistent value in the general nontransferable utility setup. Copyright 1996 by The Econometric Society.

Nonlinearity of Daveport:80Schinzel sequences and of generalized path compression schemes

Davenport—Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport—Schinzel sequence composed ofn symbols is Θ (nα(n)), where α(n) is the functional inverse of Ackermann’s function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.

Uncoupled Dynamics Do Not Lead to Nash Equilibrium

We call a dynamical system uncoupled if the dynamic for each player does not depend on the payoffs of the other players. We show that there are no uncoupled dynamics that are guaranteed to converge to Nash equilibrium, even when the Nash equilibrium is unique.

A General Class of Adaptive Strategies

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: in the long-run, the player is guaranteed an average payoff as large as the best-reply payoff to the empirical distribution of play of the other players; i.e., there is no “regret.” Smooth fictitious play (Fudenberg and Levine [1995, J. Econ. Dynam. Control19, 1065–1090]) and regret-matching (Hart and Mas-Colell [2000, Econometrica68, 1127–1150]) are particular cases. The motivation and application of the current paper come from the study of procedures whose empirical distribution of play is, in the long run, (almost) a correlated equilibrium. For the analysis we first develop a generalization of Blackwells (1956, Pacific J. Math.6, 1–8) approachability strategy for games with vector payoffs. Journal of Economic Literature Classification Numbers: C7, D7, C6.

Termination of Probabilistic Concurrent Program

The asynchronous execution behavior of several concurrent processes, which may use randomization, is studied. Viewing each process as a discrete Markov chain over the set of common execution states, necessary and sufficient conditions are given for the processes to converge almost surely to a given set of goal states under any fair, but otherwise arbitrary, schedule, provided that the state space is finite. (These conditions can be checked mechanically.) An interesting feature of the proof method is that it depends only on the topology of the transitions and not on the actual values of the probabilities. It is also shown that in this model synchronization protocols that use randomization are in certain cases no more powerful than deterministic protocols. This is demonstrated by (1) establishing lower bounds, similar to those known for deterministic protocols, on the size of a shared variable necessary to ensure mutual exclusion and lockout-free behavior of a randomized protocol and (2) showing that no fully symmetric randomized protocol can ensure mutual exclusion and freedom from lockout. 12 references.

Existence of Correlated Equilibria

An elementary proof, based on linear duality, is provided for the existence of correlated equilibria in finite games. The existence result is then extended to infinite games, including some that possess no Nash equilibria.

Nonzero-Sum Two-Person Repeated Games with Incomplete Information

Characterization of all equilibria of nonzero-sum two-person repeated games with incomplete information, in the standard one-sided information case. Informally, each such equilibrium is described by a sequence of communications between the players (consisting of information transmission and coordination), leading to some individually rational agreement. Formally, the concept of a bi-martingale is introduced.

A note on the edges of the n-cube

The following combinatorial problem, which arose in game theory, is solved here: To find a set of vertices of a given size (in the n-cube) which has a maximal number of interconnecting edges.

A Reinforcement Procedure Leading to Correlated Equilibrium

We consider repeated games where at any period each player knows only his set of actions and the stream of payoffs that he has received in the past. He knows neither his own payoff function, nor the characteristics of the other players (how many there are, their strategies and payoffs). In this context, we present an adaptive procedure for play called “modified-regret-matching” — which is interpretable as a stimulus-response or reinforcement procedure, and which has the property that any limit point of the empirical distribution of play is a correlated equilibrium of the stage game.