Two more classes of games with the continuous-time fictitious play property

Fictitious Play is the oldest and most studied learning process for games. Since the already classical result for zero-sum games, convergence of beliefs to the set of Nash equilibria has been established for several classes of games, including weighted potential games, supermodular games with diminishing returns, and 3×3 supermodular games. Extending these results, we establish convergence of Continuous-time Fictitious Play for ordinal potential games and quasi-supermodular games with diminishing returns. As a by-product we obtain convergence for 3×m and 4×4 quasi-supermodular games.     

The value of information in oligopoly with demand uncertainty

This paper discusses the value of information in supermodular and submodular games, using a simple duopoly model where the level of demand is uncertain. It is shown that the value of information issuperadditive (resp.,subadditive) between players if the game issupermodular (resp.,submodular). For example, in a Bertrand (resp., Cournot) market with (possibly imperfect) substitute products, one firm's information acquisition increases (resp., decreases) the other firm's incentive to acquire the same information. Furthermore, when the game is either supermodular or submodular, the value of information is higher when the player isexpected to be informed according to the opponent's belief than when the player is expected to be uninformed; this result is reversed when the game has asymmetric modularity (i.e., one player's action is substitutional to the other's, and the latter's action is complemental to the former's). These qualitative observations have a potential to be applied to a larger class of games with uncertainty where payoffs are smooth (e.g., twice continuously differentiable) in actions and states.     

Strategic correlativity and network games

We consider the strategic correlativity principle in strategic form games and potential games, which indicate the relationship between an arbitrary pair of players under correlated equilibria. Importantly, the strategies of a pair of players are positively correlated when their expected payoff functions are supermodular, while negatively correlated when their expected payoff functions are submodular. Furthermore, we extend the strategic correlativity principle to strategic form games and potential games in social networks, and investigate the monotonicity of correlated equilibrium in each player's own degree.     

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.     

A Simple Supermodular Mechanism that Implements Lindahl Allocations

This paper introduces a new incentive‐compatible mechanism which, for general preference environments, implements Lindahl allocations as Nash equilibria. We provide an example that having a mechanism induce a supermodular game is not typically sufficient to guarantee dynamic stability of equilibrium. However, for the new mechanism, inducing a supermodular game guarantees that the best reply mapping is a contraction. Thus, this new mechanism provides a connection between the desirable welfare properties of Lindahl allocations and the theoretical/convergence properties of games whose best reply mappings are a contraction.     

Cournot Oligopoly and the Theory of Supermodular Games

We reconsider the Cournot oligopoly problem in light of the theory of supermodular games. Invoking the recent ordinal version of this theory proposed by Milgrom and Shannon, we generalize Novshek's existence result, derive the associated uniqueness result, give an extension of a classical existence result under symmetry, and provide conditions making a Cournot oligopoly into a log-supermodular game (with the natural order on the action sets). We also provide extensive and precise insight as to why decreasing best-responses are widely regarded as being “typical” for the Cournot model with production costs. Several illustrative examples are provided.Journal of Economic LiteratureClassification Numbers: C72, L10, L13.     

Characterising equilibrium selection in global games with strategic complementarities

Global games are widely used to predict behaviour in games with strategic complementarities and multiple equilibria. We establish two results on the global game selection. First, we show that, for any supermodular complete information game, the global game selection is independent of the payoff functions chosen for the game?s global game embedding. Second, we give a simple sufficient criterion to derive the selection and establish noise independence in many-action games by decomposing them into games with smaller action sets, to which we may often apply simple criteria. We also report in which small games noise independence may be established by counting the number of players or actions.     

A family of supermodular Nash mechanisms implementing Lindahl allocations

Summary. We present a family of mechanisms which implement Lindahl allocations in Nash equilibrium. With quasilinear utility functions this family of mechanisms are supermodular games, which implies that they converge to Nash equilibrium under a wide class of learning dynamics. Received: April 27, 2000; revised version: January 16, 2001     

Characterizing solutions of supermodular games; intuitive comparative statics,and unique equilibria

Summary We present a technique for locating both an upper and a lower bound on equilibrium points of supermodular games by looking only at the first derivatives of the payoff functions at points of disequilibrium. This technique is useful for characterizing equilibria of games when the closed form solution is difficult to calculate, for performing comparative statics analysis, and for determining the uniqueness of equilibria.I would like to thank David Easley, James Friedman, Ken Kasa, Rich McLean and Michael Waldman for their helpful comments.     

Interim Bayesian Nash equilibrium on universal type spaces for supermodular games

We prove the existence of a greatest and a least interim Bayesian Nash equilibrium for supermodular games of incomplete information. There are two main differences from the earlier proofs and from general existence results for non-supermodular Bayesian games: (a) we use the interim formulation of a Bayesian game, in which each player's beliefs are part of his or her type rather than being derived from a prior; (b) we use the interim formulation of a Bayesian Nash equilibrium, in which each player and every type (rather than almost every type) chooses a best response to the strategy profile of the other players. There are no restrictions on type spaces and action sets may be any compact metric lattices.     

Beliefs and rationalizability in games with complementarities

We propose two characteristics of beliefs and study their role in shaping the set of rationalizable strategy profiles in games with incomplete information. The first characteristic, type-sensitivity, is related to how informative a player thinks his type is. The second characteristic, optimism, is related to how “favorable” a player expects the outcome of the game to be. The paper has two main results: the first result provides an upper bound on the size of the set of rationalizable strategy profiles; the second gives a lower bound on the change of location of this set. These bounds are explicit expressions that involve type-sensitivity, optimism, and payoff characteristics. Our results generalize and clarify the well-known uniqueness result of global games (Carlsson and van Damme, 1993). They also imply new uniqueness results and allow us to study rationalizability in new environments. We provide applications to supermodular mechanism design (Mathevet, 2010b) and information processing errors.     

Risk-dominance and perfect foresight dynamics in N-player games

In perfect foresight dynamics, an action is linearly stable if expectation that people will always choose the action is self-fulfilling. A symmetric game is a PIM game if an opponent's particular action maximizes the incentive of an action, independently of the rest of the players. This class includes supermodular games, games with linear incentives and so forth. We show that, in PIM games, linear stability is equivalent to u-dominance, a generalization of risk-dominance, and that there is no path escaping a u-dominant equilibrium. Existing results on N-player coordination games, games with linear incentives and two-player games are obtained as corollaries.     

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.     

Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis

This paper is an attempt to develop a unified approach to symmetry-breaking in strategic models arising in industrial organization by constructing two general classes of two-player symmetric games that always possess only asymmetric pure-strategy Nash equilibria. These classes of games are characterized in some abstract sense by two general properties: payoff nonconcavities and some form of strategic substitutability. Our framework relies on easily verified assumptions on the primitives of the game, and relies on the theory of supermodular games. The underlying natural assumptions are satisfied in a number of two-stage models with an investment decision preceding product market competition. To illustrate the generality and wide scope for application of our approach, we present some existing models dealing with R&D, capacity expansion and information provision, which motivated this study.     

Stackelberg versus Cournot Equilibrium

We reconsider Stackelberg's classical critique of the Cournot duopoly, in the framework of endogenous timing for two-player games. For quantity duopoly we provide different sets of minimal conditions, directly on the demand and cost functions, yielding respectively the simultaneous and the two sequential modes of play. While our findings essentially confirm the predominance of the former, they also indicate that the latter is natural under some robust but restrictive conditions. No extraneous assumptions (such as concavity, existence, or uniqueness of equilibria…) are needed, and the analysis makes crucial use of the basic results from the theory of supermodular games.Journal of Economic LiteratureClassification Numbers: B21, C72, D43, L13.     

Eliciting the core of a supermodular capacity

Summary. The paper utilizes duality theory to derive an exact representation of the core of a supermodular capacity for finite-state-space Choquet expected utility preferences. Using the dual representation we develop an algorithm that uses information on willingness to pay and willingness to sell to elicit a supermodular capacity in a finite number of iterations.Received: 21 February 2003, Revised: 26 May 2004, JEL Classification Numbers: D81. Correspondence to: Robert G. ChambersThe authors thank J. Quiggin and an anonymous referee for comments that improved the paper.     

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.     

Finite supermodular design with interdependent valuations

This paper studies supermodular mechanism design in environments with arbitrary (finite) type spaces and interdependent valuations. In these environments, the designer may have to use Bayesian equilibrium as a solution concept, because ex-post implementation may not be possible. We propose direct (Bayesian) mechanisms that are robust to certain forms of bounded rationality while controlling for equilibrium multiplicity. In quasi-linear environments with informational and allocative externalities, we show that any Bayesian mechanism that implements a social choice function can be converted into a supermodular mechanism that also implements the original decision rule. The proposed supermodular mechanism can be chosen in a way that minimizes the size of the equilibrium set, and we provide two sets of sufficient conditions to this effect. This is followed by conditions for supermodular implementation in unique equilibrium.     

Markov perfect equilibria in industries with complementarities

Summary.   This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed. Received: November 9, 1999; revised version: February 10, 2000     

Average Monotonic Cooperative Games

A subclass of monotonic transferable utility (T.U.) games is studied: average monotonic games. These games are totally balanced. We prove that the core coincides with both the bargaining set à la Davis and Maschler and the bargaining set à la Mas-Colell. To obtain this result a technique based on reduced games is used. Journal of Economic Literature Classification Number: C71