Multiplicity and Sensitivity of Stochastically Stable Equilibria in Coordination Games

We investigate the equilibrium selection problem in n-person binary coordination games by means of the adaptive play with mistakes (Young 1993). We show that whenever the difference between the deviation losses of respective equilibria is not overwhelming, the stochastic stability exhibits a notable dependence on payoff parameters associated with strategy profiles where the numbers of players for the respective strategies are nearly equal. This feature necessitates the existence of games that possess multiple stochastically stable equilibria.Equilibrium selection, stochastic stability, unanimity game, coordination game

Stochastically Stable Equilibria in Coordination Games with Multiple Populations

We investigate the equilibrium selection problem in n-person binary coordination games by means of adaptive play with mistakes (Young 1993). The size and the depth of a particular type of basins of attraction are found to be the main factors in determining the selection outcome. The main result shows that if a strategy has the larger basin of attraction, and if it is deep enough, then the strategy constitutes a stochastically stable equilibrium. The existence of games with multiple stochastically stable equilibria is an immediate consequence of the result. We explicitly address the qualitative difference between selection results in multi-dimensional stochastic evolution models and those in single dimensional models, and shed some light on the source of the difference.Equilibrium selection, stochastic stability, unanimity game, coordination game

Multiple Stochastically Stable Equilibria in Coordination Games

In an (n,m)-coordination game, each of the n players has two alternative strategies. A strategy generates positive payoff only if there are at least m-1 others who choose the same, where m>n/2. The payoff is nondecreasing in the number of such others so that there are exactly two strict equilibria. Applying the adaptive play with mistakes (Young 1993) to (n,m)-coordination games, we point out potential complications inherent in many-person games. Focusing on games that admit simple analysis, we show that there is a nonempty open set of (n,m)-coordination games that possess multiple stochastically stable equilibria, which may be Pareto ranked, if and only if m>(n+3)/2, which in turn is equivalent to the condition that there is a strategy profile against which every player has alternative best responses.Equilibrium selection, stochastic stability, unanimity game, coordination game, collective decision making

Robust stochastic stability

A strategy profile of a game is called robustly stochastically stable if it is stochastically stable for a given behavioral model independently of the specification of revision opportunities and tie-breaking assumptions in the dynamics. We provide a simple radius-coradius result for robust stochastic stability and examine several applications. For the logit-response dynamics, the selection of potential maximizers is robust for the subclass of supermodular symmetric binary-action games. For the mistakes model, the weaker property of strategic complementarity suffices for robustness in this class of games. We also investigate the robustness of the selection of risk-dominant strategies in coordination games under best-reply and the selection of Walrasian strategies in aggregative games under imitation.Learning in games, stochastic stability, radius-coradius theorems, logit-response dynamics, mutations, imitation

Nonspecific Networking

A new model of strategic network formation is developed and analyzed, where an agent's investment in links is nonspecific. The model comprises a large class of games which are both potential and super- or submodular games. We obtain comparative statics results for Nash equilibria with respect to investment costs for supermodular as well as submodular networking games. We also study logit-perturbed best-response dynamics for supermodular games with potentials. We find that the associated set of stochastically stable states forms a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest stochastically stable state. Finally, we provide a broad spectrum of applications from social interaction to industrial organization. Models of strategic network formation typically assume that each agent selects his direct links to other agents in which to invest. Nonspecific networking means that an agent cannot select a specific subset of feasible links which he wants to establish or strengthen. Rather, each agent chooses an effort level or intensity of networking. In the simplest case, the agent faces a binary choice: to network or not to network. If an agent increases his networking effort, all direct links to other agents are strengthened to various degrees. We assume that benefits accrue only from direct links. The set of agents or players is finite. Each agent has a finite strategy set consisting of the networking levels to choose from. For any pair of agents, their networking levels determine the individual benefits which they obtain from interacting with each other. An agent derives an aggregate benefit from the pairwise interactions with all others. In addition, the agent incurs networking costs, which are a function of the agent's own networking level. The agent's payoff is his aggregate benefit minus his cost.Network Formation, Potential Games, Supermodular Games

Evolution with Mutations Driven by Control Costs

Bergin and Lipman (1996) show that the refinement effect from the random mutations in the adaptive dynamics in Kandori, Mailath and Rob (1993) and Young (1993) is due to restrictions on how these mutation rates vary across population states. We here model these mutation rates as endogenously determined mistake probabilities, by assuming that players at some cost or disutility can control their mistake probability, i.e., the probability of implementing another pure strategy than intended. This is shown to corroborate the result in Kandori-Mailath-Rob and Young that the risk-dominant equilibrium is selected in 2£ 2-coordination games.games;probability

Interaction on Hypergraphs

Interaction on hypergraphs generalizes interaction on graphs, also known as pairwise local interaction. For games played on a hypergraph which are supermodular potential games, logit-perturbed best-response dynamics are studied. We find that the associated stochastically stable states form a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest stochastically stable state. In the special case of networking games, we obtain comparative statics results with respect to investment costs, for Nash equilibria of supermodular games as well as for Nash equilibria of submodular games.

Evolution and Refinement with Endogenous Mistake Probabilities

Bergin and Lipman (1996) show that the refinement effect from the random mutations in the adaptive population dynamics in Kandori, Mailath and Rob (1993) and Young (1993) is due to restrictions on how these mutation rates vary across population states. We here model mutation rates as endogenously determined mistake probabilities, by assuming that players with some effort can control the probability of implementing the intended strategy. This is shown to corroborate the results in Kandori-Mailath-Rob (1993) and, under certain regularity conditions, those in Young (1993). The approach also yields a new refinement of the Nash equilibrium concept that is logically independent of Selten's (1975) perfection concept and Myerson's (1978) properness concept.game theory;probability