Robert Samuel Simon

Games of incomplete information, ergodic theory, and the measurability of equilibria

We present an example of a one-stage three-player game of incomplete information played on a sequence space {0, 1}Z such that the players’ locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1}Z, each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g., all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is also a philosophical issue.

Equilibrium existence and topology in some repeated games with incomplete information

This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals. This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping F on a compact convex set P C R n , we give further conditions which imply that every point p 0 ∈ P belongs to the convex hull of a finite subset P 0 of the domain of F satisfying ∩ x ∈ P0 F(x) ¬= O.

A Topological Approach to Quitting Games

This paper presents a question of topological dynamics and demonstrates that its affirmation would establish the existence of approximate equilibria in all quitting games with only normal players. A quitting game is an undiscounted stochastic game with finitely many players where every player has only two moves, to end the game with certainty or to allow the game to continue. If nobody ever acts to end the game, all players receive payoffs of 0. A player is normal if and only if by quitting alone she receives at least her min-max payoff. This proof is based on a version of the Kohlberg–Mertens [Kohlberg, E., J.-F. Mertens. 1986. On the strategic stability of equilibria. Econometrica 54(5) 1003–1037] structure theorem designed specifically for quitting games.

Equilibria in a class of games and topological results implying their existence.

We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.ResumenExponemos resultados relacionados con el problema de la existencia de equilibrios en algunas clases de juegos bipersonales infinitamente repetidos con información incompleta por una de las partes, considerados por primera vez por Aumann, Maschler y Stearns. Generalizamos este marco a uno más amplio de problemas de agentes principales. También discutimos los resultados topológicos necesarios, presentándolos dualmente (usando cohomología en lugar de homología) y de modo más sistemático que en nuestros artículos anteriores.

Value and perfection in stochastic games

A stochastic game isvalued if for every playerk there is a functionrk:S→R from the state spaceS to the real numbers such that for every ε>0 there is an ε equilibrium such that with probability at least 1−ε no states is reached where the future expected payoff for any playerk differs fromrk(s) by more than ε. We call a stochastic gamenormal if the state space is at most countable, there are finitely many players, at every state every player has only finitely many actions, and the payoffs are uniformly bounded and Borel measurable as functions on the histories of play. We demonstrate an example of a recursive two-person non-zero-sum normal stochastic game with only three non-absorbing states and limit average payoffs that is not valued (but does have ε equilibria for every positive ε). In this respect two-person non-zero-sum stochastic games are very different from their zero-sum varieties. N. Vieille proved that all such non-zero-sum games with finitely many states have an ε equilibrium for every positive ε, and our example shows that any proof of this result must be qualitatively different from the existence proofs for zero-sum games. To show that our example is not valued we need that the existence of ε equilibria for all positive ε implies a “perfection” property. Should there exist a normal stochastic game without an ε equilibrium for some ε>0, this perfection property may be useful for demonstrating this fact. Furthermore, our example sews some doubt concerning the existence of ε equilibria for two-person non-zero-sum recursive normal stochastic games with countably many states.

A parametrized version of the Borsuk Ulam theorem

We show that for a ‘continuous’ family of Borsuk–Ulam situations, parametrized by points of a compact manifold W, its solution set also depends ‘continuously’ on the parameter space W. By such a family we understand a compact set Z⊂W×Sm×ℝm, the solution set consists of points (w, x, v)∈Z such that also (w,−x, v)∈Z. Here, ‘continuity’ means that the solution set supports a homology class that maps onto the fundamental class of W. We also show how to construct such a family starting from a ‘continuous’ family Y⊂∂ W×ℝm when W is a compact top-dimensional subset in ℝm+1. This solves a problem related to a conjecture that is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with ℤ/2-coefficients.

The Difference Between Common Knowledge of Formulas and Sets

Abstract. Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.

The challenge of non-zero-sum stochastic games

For a broad definition of time-discrete stochastic games, their zero-sum varieties have values. But the existence of

The generation of formulas held in common knowledge

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