Mixed Equilibria in Games of Strategic Complements are Unstable

In games with strict strategic complementarities, properly mixed Nash equilibria -equilibria that are not in pure strategies- are unstable for a broad class of learning dynamics.Facultad de Ciencias Económica

Mixed Equilibria in Games of Strategic Complements are Unstable

In games with strict strategic complementarities, properly mixed Nash equilibria -equilibria that are not in pure strategies- are unstable for a broad class of learning dynamics.Facultad de Ciencias Económica

Essays on Games of Strategic Substitutes with Incomplete Information

This dissertation consists of three individual chapters. The first chapter applies lattice theoretic techniques in order to establish fundamental properties of Bayesian games of strategic substitutes (GSS) when the underlying type space is ordered either in increasing or decreasing first-order stochastic dominance. Existence and uniqueness of equilibria is considered, as well as the question of when such equilibria can be guaranteed to be monotone in type, a property which is used to guarantee monotone comparative statics. The second chapter uses the techniques of the first and combines them with the existing results for strategic complements (GSC) in order to extend the literature on global games under both GSC and GSS. In particular, the model of Carlsson and Van Damme (1993) is extended from 22 games to GSS or GSC involving a finite amount of players, each having a finite action space. Furthermore, the possibility that groups of players receive the same signal is allowed for, a condition which is new to the literature. It is shown that under this condition, the power of the model to resolve the issue of multiplicity is unambiguously increased. The third chapter considers stability of mixed strategy Nash equilibria in GSS. Chapter 1 analyzes Bayesian games of strategic substitutes under general conditions. In particular, when beliefs are order either increasingly or decreasingly by first order stochastic dominance, the existence and uniqueness, monotonicity, and comparative statics in this broad class of games are addressed. Unlike their supermodular counterpart, where the effect of an increase in type augments the strategic effect between own strategy and opponent’s strategy, submodularity produces competing effects when considering optimal responses. Using adaptive dynamics, conditions are given under which such games can be guaranteed to exhibit Bayesian Nash equilibria, and it is shown that in many applications these equilibria will be a profile of monotone strategies. Comparative statics of parametrized games is also analyzed using results from submodular games which are extended to incorporate incomplete information. Several examples are provided. The framework of Chapter 1 is applied to global games in Chapter 2. Global games methods are aimed at resolving issues of multiplicity of equilibria and coordination failure that arise in game theoretic models by relaxing common knowledge assumptions about an underlying parameter. These methods have recently received a lot of attention when the underlying complete information game is a GSC. Little has been done in this direction concerning GSS, however. This chapter complements the existing literature in both cases by extending the global games method developed by Carlsson and Van Damme (1993) to multiple player, multiple action GSS and GSC, using a p-dominance condition as the selection criterion. This approach helps circumvent recent criticisms to global games by relaxing some possibly unnatural assumptions on payoffs and parameters necessary to conduct analysis under current methods. The second part of this chapter generalizes the model by allowing groups of players to receive homogenous signals, which, under certain conditions, strengthens the model’s power of predictability. Chapter 3 analyzes the learning and stability of mixed strategy Nash equilibria in GSS, complementing recent work done in the case of GSC. Mixed strategies in GSS are of particular interest because it is well known that such games need not exhibit pure strategy Nash equilibria. First, a bound on the strategy space which indicate where randomizing behavior may occur in equilibrium is established. Second, it is shows that mixed strategy Nash equilibria are generally unstable under a wide variety of learning rules

Learning in Perturbed Asymmetric Games

We investigate the stability of mixed strategy equilibria in 2 person (bimatrix) games under perturbed best response dynamics. A mixed equilibrium is asymptotically stable under all such dynamics if and only if the game is linearly equivalent to a zero sum game. In this case, the mixed equilibrium is also globally asymptotically stable. Global convergence to the set of perturbed equilibria is shown also for (rescaled) partnership games (also know as games of identical interest). Some applications of these result to stochastic learning models are given.Games, Learning, Best Response Dynamics, Stochastic Fictitious Play, Mixed Strategy Equilibria, Zero Sum Games

Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

This paper is an attempt to develop a unified approach to endogenous heterogeneity by constructing general class 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: payo? non-concavities and some form of strategic substitutability. We provide a detailed discussion of the relationship of this work with Matsuyama’s symmetry breaking framework and with business strategy literature. Our framework generalizes a number of models dealing with two-stage games, with long term investment decisions in the first stage and product market competition in the second stage. We present the main examples that motivate this study to illustrate the generality of our approach.firm heterogeneity; submodular games; business strategy; innovation strategies.

Supermodular mechanism design

This paper introduces a mechanism design approach that allows dealing with the multiple equilibrium problem, using mechanisms that are robust to bounded rationality. This approach is a tool for constructing supermodular mechanisms, i.e. mechanisms that induce games with strategic complementarities. In quasilinear environments, I prove that if a social choice function can be implemented by a mechanism that generates bounded strategic substitutes - as opposed to strategic complementarities - then this mechanism can be converted into a supermodular mechanism that implements the social choice function. If the social choice function also satisfies some efficiency criterion, then it admits a supermodular mechanism that balances the budget. Building on these results, I address the multiple equilibrium problem. I provide sufficient conditions for a social choice function to be implementable with a supermodular mechanism whose equilibria are contained in the smallest interval among all supermodular mechanisms. This is followed by conditions for supermodular implementability in unique equilibrium. Finally, I provide a revelation principle for supermodular implementation in environments with general preferences.Implementation, mechanisms, learning, strategic complementarities, supermodular games

Differentiated Standards and Patent Pools

We consider patent pool formation by owners of essential patents for differentiated standards that may be complements or substitutes in use. Pooling improves coordination in terms of royalty setting within a standard but provokes a strategic response from licensors in the competing standard. We characterise the incentives to form and defect from pools within standards and show how pool formation and stability depend on competition between standards. We also examine strategic patent pool formation by consortium standards and show that policies promoting compatibility of standards may increase or decrease welfare depending on the effects on the incentives to form pools.Patent pools, competing standards, consortium standards

Evolutionary network games: equilibria from imitation and best-response dynamics

We consider games of strategic substitutes and strategic complements on networks. We introduce two different evolutionary dynamics in order to refine their multiplicity of equilibria, and we analyse the system through a mean field approach. We find that for the best-shot game, taken as a model for substitutes, a replicator-like dynamics does not lead to Nash equilibria, whereas it leads to unique equilibria (full cooperation or full defection, depending on the initial condition and the game parameter) for complements, represented by a coordination game. On the other hand, when the dynamics becomes more cognitively demanding in the form of a best response evolution, predictions are always Nash equilibria (at least when individuals are fully rational): For the best-shot game we find equilibria with a definite value of the fraction of contributors, whereas for the coordination game symmetric equilibria arise only for low or high initial fractions of cooperators. We also extend our study by considering complex heterogeneous topologies, and show that the nature of the selected equilibria does not change for the best-shot game. However for coordination games we reveal an important difference, namely that on infinitely large scale-free networks cooperation arises for any value of the incentive to cooperate