A characterization of strategic complementarities

I characterize games for which there is an order on strategies such that the game has strategic complementarities. I prove that, with some qualifications, games with a unique equilibrium have complementarities if and only if Cournot best-response dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show that: (1) generic 2×2 games either have no pure-strategy equilibria, or have complementarities; (2) generic two-player finite ordinal potential games have complementarities.     

On the endogeneity of Cournot-Nash and Stackelberg equilibria: games of accumulation

We characterize equilibria of games with two properties: (i) Agents have the opportunity to adjust their strategic variable after their initial choices and before payoffs occur; but (ii) they can only add to their initial amounts. The equilibrium set consists of just the Cournot-Nash outcome, one or both Stackelberg outcomes, or a continuum of points including the Cournot-Nash outcome and one or both Stackelberg outcomes. A simple theorem that uses agents’ standard one-period reaction functions and the one-period Cournot-Nash and Stackelberg equilibria delineates the equilibrium set. Applications include contribution, oligopoly, and rent-seeking games.     

Strategic complements and substitutes,and potential games

We show that games of strategic complements, or substitutes, with aggregation are “pseudo-potential” games. The upshot is that they possess Nash equilibria in pure strategies (NE), even if the strategy sets are not convex; and that various dynamic processes converge to NE. In particular, NE exist in Cournot oligopoly with indivisibilities in production.Our notion of aggregation is quite general and enables us to take a unified view of several disparate models.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

Generalized potentials and robust sets of equilibria

This paper introduces generalized potential functions of complete information games and studies the robustness of sets of equilibria to incomplete information. A set of equilibria of a complete information game is robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to some equilibrium in the set. This paper provides sufficient conditions for the robustness of sets of equilibria in terms of argmax sets of generalized potential functions. These sufficient conditions unify and generalize existing sufficient conditions. Our generalization of potential games is useful in other game theoretic problems where potential methods have been applied.     

Strong comparative statics of equilibria

Some results in the monotone comparative statics literature tell us that if a parameter increases, some old equilibria are smaller than some new equilibria. We give a sufficient condition such that at a new parameter value every old equilibrium is smaller than every new equilibrium. We also adapt a standard algorithm to compute a minimal such newer parameter value and apply this algorithm to a game of network externalities. Our results are independent of a theory of equilibrium selection and are valid for games of strategic complementarities.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Heterogeneous network games: Conflicting preferences

In many economic situations, a player pursues coordination or anti-coordination with her neighbors on a network, but she also has intrinsic preferences among the available options. We here introduce a model which allows to analyze this issue by means of a simple framework in which players endowed with an idiosyncratic identity interact on a social network through strategic complements or substitutes. We classify the possible types of Nash equilibria under complete information, finding two thresholds for switching action that relate to the two-player setup of the games. This structure of equilibria is considerably reduced when turning to incomplete information, in a setup in which players only know the distribution of the number of neighbors of the network. For high degrees of heterogeneity in the population the equilibria is such that every player can choose her preferred action, whereas if one of the identities is in the minority frustration ensues.     

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Bayesian games with unawareness and unawareness perfection

Applying unawareness belief structures introduced in Heifetz et al. (Games Econ Behav 77:100–121, 2013a), we develop Bayesian games with unawareness, define equilibrium, and prove existence. We show how equilibria are extended naturally from lower to higher awareness levels and restricted from higher to lower awareness levels. We apply Bayesian games with unawareness to investigate the robustness of equilibria to uncertainty about opponents’ awareness of actions. We show that a Nash equilibrium of a strategic game is robust to unawareness of actions if and only if it is not weakly dominated. Finally, we discuss the relationship between standard Bayesian games and Bayesian games with unawareness.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

Complements and substitutes in common agency

In this paper we analyse the problem of the rent obtained by the agent in private common agency games. The key features for answering this question are the properties of the cost function of the agent. We prove that if this cost function is submodular (costs complements) then there is no equilibrium in which the agent makes a rent and if the cost function is supermodular (costs substitutes) then in all equilibria the agent makes a rent. We also examine the problem in some intermediate cases.     

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.     

Comparative statics in a simple class of strategic market games

This paper investigates the effects of entry in two-sided markets where buyers and sellers act strategically. Applying new tools from supermodular optimization/games, sufficient conditions for different comparative statics results are obtained. While normality of one good is sufficient for the equilibrium price to be increasing in the number of buyers, normality of both goods is required for equilibrium bids and sellers' equilibrium utilities to be increasing in the number of buyers. When the economy is replicated, normality of both goods and gross substitutes guarantee that the equilibrium of the strategic market game converges monotonically (in quantities) to the competitive equilibrium. Simple counter-examples are provided to settle other potential conjectures of interest.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

A global game with strategic substitutes and complements: Comment

In Karp et al. (2007), an argument is made to show existence of Bayesian–Nash equilibrium in global games that may include both strategic substitutes and complements. This note documents a gap in the proof of that statement.     

Robustness of equilibria in anonymous local games

This paper studies the robustness of symmetric equilibria in anonymous local games to perturbations of prior beliefs. Two priors are strategically close on a class of games if players receive similar expected payoffs in equilibrium under the priors, for any game in that class. I show that if the structure of payoff interdependencies is sparse in a well-defined sense, the conditions for strategic proximity in anonymous local games are strictly weaker than the conditions for general Bayesian games of Kajii and Morris (1998) [11] when attention is restricted to symmetric equilibria. Hence, by exploiting the properties of anonymous local games, it is possible to obtain stronger robustness results for this class.     

Strategic knowledge sharing in Bayesian games

This paper provides a model for the study of direct, public and strategic knowledge sharing in Bayesian games. We propose an equilibrium concept which takes into account communication possibilities of exogenously certifiable statements and in which beliefs off the equilibrium path are explicitly deduced from consistent possibility correspondences, without making reference to perturbed games. Properties of such an equilibrium and of revised knowledge are examined. In particular, it is shown that our equilibrium is always a sequential equilibrium of the associated extensive form game with communication. Finally, sufficient conditions for the existence of perfectly revealing or non-revealing equilibria are characterized in some classes of games. Several examples and economic applications are investigated.