A nonstandard characterization of sequential equilibrium,perfect equilibrium,and proper equilibrium

New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal and a completely mixed behavioral strategy profile σ′ (so that assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from such that at each information set I for player i, σ _i is an -best response to conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ _i must be a best response to conditional on having reached I. Yet another variant is used to characterize proper equilibrium. This work was supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, and by AFOSR under grant FA9550-05-1-0055.     

Rotation, entropy, and equilibrium states

For a dynamical system and function we consider the corresponding generalised rotation set. This is the convex subset of consisting of all integrals of with respect to -invariant probability measures. We study the entropy of rotation vectors , and relate this to the directional entropy of Geller & Misiurewicz. For a mixing subshift of finite type, and of summable variation, we prove that if the rotation set is strictly convex then the functions and are in fact one and the same. For those rotation sets which are not strictly convex we prove that and can differ only at non-exposed boundary points .

     

Bounded rationality, strategy simplification, and equilibrium

It is frequently suggested that predictions made by game theory could be improved by considering computational restrictions when modeling agents. Under the supposition that players in a game may desire to balance maximization of payoff with minimization of strategy complexity, Rubinstein and co-authors studied forms of Nash equilibrium where strategies are maximally simplified in that no strategy can be further simplified without sacrificing payoff. Inspired by this line of work, we introduce a notion of equilibrium whereby strategies are also maximally simplified, but with respect to a simplification procedure that is more careful in that a player will not simplify if the simplification incents other players to deviate. We study such equilibria in two-player machine games in which players choose finite automata that succinctly represent strategies for repeated games; in this context, we present techniques for establishing that an outcome is at equilibrium and present results on the structure of equilibria.     

Directed graphical structure,Nash equilibrium,and potential games

This paper considers the directed graphical structure of a game, called influence structure, where a directed edge from player $i$ to player $j$ indicates that player $i$ may be able to affect $j$’s payoff via his unilateral change of strategies. We give a necessary and sufficient condition for the existence of pure-strategy Nash equilibrium of games having a directed graph in terms of the structure of that graph. We also discuss the relationship between the structure of graphs and potential games.     

Logit equilibrium as an approximation of Nash equilibrium

We prove that the graph of the logit equilibrium correspondence is a smooth manifold, which is homeomorphic to the space of payoff functions and uniformly approximates the graph of the Nash equilibrium manifold.     

Generalized gradients,bid–ask spreads,and market equilibrium

Abstract

This paper considers economies in which each agent valuates various goods by own generalized gradients. Taken together and appropriately scaled, the latter determine bid–ask spreads. When all such spreads are nil, market equilibrium prevails. Crucial for the arguments is a money commodity which denominates agents' rates of exchange or substitution. Equilibrium obtains when rates coincide across agents. The results may facilitate detailed modelling of market micro-structure, direct deals, and agent-based computations.     

Trimmed equilibrium

Subgame perfect equilibrium (SPE) imposes stronger rationality conditions than necessary to ensure equilibrium outcomes are consistent with its concept. This is demonstrated by characterizing the maximal collection of information sets at which play is irrelevant for an outcome’s consistency with SPE. It is shown that, without affecting the set of equilibrium outcomes, equilibrium conditions can be trimmed by relaxing all conditions on this maximal collection. Therefore a trimmed SPE is a tight version of SPE. However, because its conditions are weaker, trimmed SPE might exist even if SPE does not. This is demonstrated in an application.     

On securitization,market completion and equilibrium risk transfer

We propose an equilibrium framework within which to price financial securities written on non-tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agents are exposed to financial and non-financial risk factors. They can hedge their financial risk in the stock market and trade a structured derivative whose payoff depends on both financial and external risk factors. We prove an existence and uniqueness of equilibrium result for derivative prices and characterize the equilibrium market price of risk in terms of a solution to a non-linear BSDE.     

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.