A General Structure Theorem for the Nash Equilibrium Correspondence

We consider n--person normal form games where the strategy set of each player is a non--empty compact convex subset of a Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in player i''s strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. We demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well--known structure theorem in Kohlberg and Mertens (On the Strategic Stability of Equilibria, Econometrica, 54, 1003--1037, 1986). It is supplemented by an extension analogous to the unknottedness theorems in Demichelis and Germano (On (Un)knots and Dynamics in Games, Games and Economic Behavior, 41, 46--60, 2002): the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.mathematical economics;

Social choice theory, game theory, and positive political theory

We consider the relationships between the collective preference and non-cooperative game theory approaches to positive political theory. In particular, we show that an apparently decisive difference between the two approachesthat in sufficiently complex environments (e.g. high-dimensional choice spaces) direct preference aggregation models are incapable of generating any prediction at all, whereas non-cooperative game-theoretic models almost always generate predictionis indeed only an apparent difference. More generally, we argue that when modeling collective decisions there is a fundamental tension between insuring existence of well-defined predictions, a criterion of minimal democracy, and general applicability to complex environments; while any two of the three are compatible under either approach, neither collective preference nor non-cooperative game theory can support models that simultaneously satisfy all three desiderata

Asymptotic equivalence between Cournot–Nash and Walras equilibria in exchange economies with atoms and an atomless part

In this paper, we consider an exchange economy à la Shitovitz (Econometrica 41:467–501, 1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley, known as the Shapley window model. We analyze the relationship between the set of the Cournot–Nash allocations of the strategic market game and the Walras allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot–Nash allocation of the game is not a Walras allocation of the underlying exchange economy. Accordingly, in the original spirit of Cournot (Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris, 1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. Our main theorem shows that any sequence of Cournot–Nash allocations of the strategic market games associated with the partial replications of the exchange economy has a limit point for each trader and that the assignment determined by these limit points is a Walrasian allocation of the original economy

Robust implementation under alternative information structures

In this paper we consider a model in which agents have complete information about their neighbours and, possibly, incomplete information about the rest of the economy. We consider two different informational frameworks. In the first, agents do not have priors about what is going on in the rest of the economy. In the second, agents are supposed to have priors about the unknown characteristics. We present a mechanism which any social choice correspondence satisfying monotonicity and no veto powet in both informational settings for every possible prior thus requiring little knowledge from the point of view of the designer of the information possesed by agents about the economy

Extremal Choice Equilibrium: Existence and Purification with Infinite-Dimensional Externalities

We prove existence and purification results for equilibria in which players choose extreme points of their feasible actions in a class of strategic environments exhibiting a product structure. We assume finite-dimensional action sets and allow for infinite-dimensional externalities. Applied to large games, we obtain existence of Nash equilibrium in pure strategies while allowing a continuum of groups and general dependence of payoffs on average actions across groups, without resorting to saturated measure spaces. Applied to games of incomplete information, we obtain a new purification result for Bayes-Nash equilibria that permits substantial correlation across types, without assuming conditional independence given the realization of a finite environmental state. We highlight our results in examples of industrial organization, auctions, and voting.

Morphisms of open games

We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the more intuitively obvious definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category in which the horizontal 1-cells are open games, vertical 1-morphisms are lenses, and 2-cells are morphisms of open games. States (morphisms out of the monoidal unit) in the vertical category give a flexible solution concept that includes both Nash and subgame perfect equilibria. Products in the vertical category give an external choice operator that is reminiscent of products in game semantics, and is useful in practical examples. We illustrate the above two features with a simple worked example from microeconomics, the market entry game