Coordination Games and Local Interactions: A Survey of the Game Theoretic Literature
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paper_title: A multinomial probit model of stochastic evolution

paper_content:

A strategy revision process in symmetric normal form games is proposed. Following Kandori, Mailath, and Rob (1993), members of a population periodically revise their strategy choice, and choose a myopic best response to currently observed play. Their payoffs are perturbed by normally distributed Harsanyian (1973) trembles, so that strategies are chosen according to multinomial probit probabilities. As the variance of payoffs is allowed to vanish, the graph theoretic methods of the earlier literature continue to apply. The distributional assumption enables a convenient closed form characterisation for the weights of the rooted trees. An illustration of the approach is offered, via a consideration of the role of dominated strategies in equilibrium selection.
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paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
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paper_title: LEARNING, MUTATION, AND LONG RUN EQUILIBRIA in Games

paper_content:

An evolutionary model with a finite number of players and with stochastic mutations is analyzed. The expansion and contraction of strategies is linked to their current relative success, but mutuation, perturbing the system from its deterministic evolution, are present as well. The focus is on the long run implications of ongoing mutations, which drastically reduce the set of equilibria. For 2 by 2 symmetric games with two symmetric strict Nash equilibria the risk dominant equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium. Copyright 1993 by The Econometric Society.
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paper_title: Altruists, Egoists and Hooligans in a Local Interaction Model

paper_content:

This paper studies a population of agents, each of whom can be either an Altruist or an Egoist. Altruists confer benefits on others at a cost to themselves. Altruism is thus a strictly dominated strategy and cannot survive if agents are rational best-responders. We assume that agents choose their actions by imitating others who earn high payoffs. We also assume that interactions between agents are local, so that each agent affects (and is affected by) only his neighbors. Altruists can survive in such a world if they are grouped together, so that the benefits of altruism are enjoyed primarily by other Altruists, who then earn relatively high payoffs and are imitated.
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paper_title: Random Perturbations of Dynamical Systems

paper_content:

1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.
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paper_title: Complex Social Networks

paper_content:

This 2007 book provides a systematic and self-contained account of the fast-developing theory of complex social networks. Social networks are central to the understanding of most socio-economic phenomena in the modern world. The classical approach to studying them relies on a methodology that abstracts from their size and complexity. In contrast, the approach taken in this book keeps complexity at the core, whilst integrating it with the incentive considerations that are preeminent in traditional economic analysis. The treatment starts with a detailed discussion of the basic models that act as 'benchmarks' for the complex-network literature: random networks, small worlds, and scale-free networks, before studying three different forces that underlie almost all network phenomena in social contexts: diffusion, search, and play. Finally, these forces are combined into a unified framework that is brought to bear on the issue of network formation and the coevolution of agents' behaviour and their pattern of interaction.
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paper_title: Endogenous Networks, Social Games, and Evolution

paper_content:

Abstract This paper studies a social game where agents choose their partners as well as their actions. Players interact with direct and indirect neighbors in the endogenous network. We show that the architecture of any nontrivial Nash equilibrium is minimally connected, and equilibrium actions approximate a symmetric equilibrium of the underlying game. We apply the model to analyze stochastic stability in 2 × 2 coordination games. We find that long-run equilibrium selection depends on a trade-off between efficiency and risk dominance due to the presence of scale effects arising from network externalities. Our results suggest a general pattern of equilibrium selection.
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paper_title: On the Formation of Interaction Networks in Social Coordination Games

paper_content:

There are many situations where two interacting individuals can benefit from coordinating their actions. We examine the endogenous choice of partners in such social coordination games and the implications for resulting play. We model the interaction pattern as a network where individuals periodically have the discretion to add or sever links to other players. With such endogenous interaction patterns we see multiple stochastically stable states of play, including some that involve play of equilibria in the coordination game that are neither efficient nor risk-dominant. Thus the endogenous network structure not only has implications for the interaction pattern that emerges, but it also has a significant impact on the play in the coordination game relative to what would arise if the same interaction network were exogenous.
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paper_title: Economists ' Models of Learning 1

paper_content:

Economic theorists have intensively studied learning in games and decisions over the last decade. This essay puts some of the work in perspective and offers opinions about what still needs to be learned. Journal of Economic Literature Classification Numbers: C73, D83.
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paper_title: On the Co-existence of Conventions

paper_content:

Recent stochastic evolutionary models have shown that the most likely convention when the probability of a mutation is sufficiently small is coordination on the risk-dominant strategy rather than the payoff-dominant one. This paper looks at the consequences of player movement between locations when there are constraints which limit the number of agents who can reside at each location. If the constraints are strong then the risk-dominance result continues to hold. However, we show that when sufficient movement is possible, the most likely outcome involves a mixed state in which agents at different locations coordinate on different strategies. In the asymmetric case, it is the location with the stronger constraint, limiting movement, that coordinates on the payoff-dominant strategy.
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paper_title: Evolution with State-Dependent Mutations

paper_content:

Recent evolutionary models have introduced "small mutation rates" as a way of refining predictions of long-run behavior. We show that this refinement effect can only be obtained by restrictions on how the magnitude of the effect of mutation on evolution varies across states of the system. In particular, given any model of the effect of mutations, any invariant distribution of the "mutationless" process is close to an invariant distribution of the process with appropriately chosen small mutation rates.
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paper_title: Imitation, local interactions, and efficiency

paper_content:

Abstract We consider a circular city model as in Ellison [Ellison, G., 1993. Learning, local interaction, and coordination, Econometrica 61, 1047–1071], where agents follow imitation rules rather than myopic best-response. If interactions are neither global nor limited to the immediate neighbors, payoff-efficient equilibria, and not risk-dominant ones, can be uniquely selected.
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paper_title: Adaptive play with spatial sampling

paper_content:

Abstract We consider a 2×2 coordination game where each agent interacts with his neighbors on a ring. Ellison (1993, Econometrica 61, 1047–1071) shows that the discrete dynamical system generated by the myopic best-reply rule converges to a Nash equilibrium or to a two-period limit cycle. Following Young (1993, Econometrica 61, 57–84), we consider a best-reply process with a sampling procedure. Particularly, we introduce a spatial sampling procedure: each agent observes a sample of information in his neighborhood and plays a best reply to it. We show that if the size of the sample of information is not too large, the best-reply process converges almost surely to a Nash equilibrium. If in addition agents experiment with small probabilities, we show that, in most cases, the risk-dominant equilibrium prevails in the long run. Furthermore, it turns out that the convergence is faster than in Ellison.
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paper_title: Connections: An Introduction to the Economics of Networks

paper_content:

Networks pervade social and economic life, and they play a prominent role in explaining a huge variety of social and economic phenomena. Standard economic theory did not give much credit to the role of networks until the early 1990s, but since then the study of the theory of networks has blossomed. At the heart of this research is the idea that the pattern of connections between individual rational agents shapes their actions and determines their rewards. The importance of connections has in turn motivated the study of the very processes by which networks are formed. In Connections , Sanjeev Goyal puts contemporary thinking about networks and economic activity into context. He develops a general framework within which this body of research can be located. In the first part of the book he demonstrates that location in a network has significant effects on individual rewards and that, given this, it is natural that individuals will seek to form connections to move the network in their favor. This idea motivates the second part of the book, which develops a general theory of network formation founded on individual incentives. Goyal assesses the robustness of current research findings and identifies the substantive open questions. Written in a style that combines simple examples with formal models and complete mathematical proofs, Connections is a concise and self-contained treatment of the economic theory of networks, one that should become the natural source of reference for graduate students in economics and related disciplines.
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paper_title: Efficient Equilibrium Selection in Evolutionary Games with Random Matching

paper_content:

Abstract This paper investigates the sensitivity of recent evolutionary models of learning to the specification of the matching mechanism. We study a literally random matching mechanism, combined with a process of strategy adjustment based on the realized average performance of each strategy. In the key class of symmetric 2×2 coordination games, the Pareto-efficient equilibrium, per se , is selected, rather than the risk-dominant equilibrium, as the probability of experimentation (or “mutation”) goes to zero. Furthermore, convergence to the equilibrium is relatively fast. We extend these results, for example, to games of common interest. Journal of Economic Literature Classification Numbers: C72, C73.
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paper_title: Network Formation and Social Coordination

paper_content:

This paper develops a simple model to examine the interaction between partner choice and individual behavior in games of coordination. An important ingredient of our approach is the way we model partner choice: we suppose that a player can establish ties with other players by unilaterally investing in costly pairwise links. In this context, individual efforts to balance the costs and benefits of links are shown to lead to a unique equilibrium interaction architecture. The dynamics of network formation, however, has powerful effects on individual behavior: if costs of forming links are below a certain threshold then players coordinate on the risk-dominant action, while if costs are above this threshold then they coordinate on the efficient action. These findings are robust to modifications in the link formation process, different specifications of link formation costs, alternative models of mutations as well as the possibility of interaction among indirectly connected players.
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paper_title: Contagion and State Dependent Mutations

paper_content:

Early results of evolutionary game theory showed that the risk dominant equilibrium is uniquely selected in the long run under the best-response dynamics with mutation. Bergin and Lipman (1996) qualified this result by showing that for a given population size, the evolutionary process can select any strict Nash equilibrium if the probability of choosing a nonbest response is state-dependent. This paper shows that the unique selection of the risk dominant equilibrium is robust with respect to state dependent mutation in local interaction games. More precisely, for any given mutation structure there exists a minimum population size beyond which the risk dominant equilibrium is uniquely selected. Our result is driven by contagion and cohesion among players, which exist only in local interaction settings and favor the risk dominant strategy. Our result strengthens the equilibrium selection result of evolutionary game theory.
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paper_title: The Statistical Mechanics of Best-Response Strategy Revision

paper_content:

I continue the study, begun in Blume (1993), of stochastic strategy revision processes in large player populations where the range of interaction between players is small. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. The purpose of this paper is to compare local strategic interaction with global strategic interaction when players update their choice according to the (myopic) best-response rule. I show that randomizing the order in which players update their strategic choice is sufficient to achieve coordination on the risk-dominant strategy in symmetric $2\times 2$ coordination games. The ``persistant randomness'' which is necessary to achieve similar coordination when the range of interaction is global is replaced by spatial variation in choice in the initial condition when strategic interactions are local. An extension of the risk-dominance idea gives the same convergence result for $K\times K$ games with strategic complementarities. Similar results for $K\times K$ pure coordination games and potential games are also presented.
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paper_title: Bandwagon Effects and Long Run Technology Choice

paper_content:

Abstract We introduce a class of “bandwagon games” which generalizes the notion of network externalities when there are more than two competing technologies and various degrees of partial compatibilities among them. We then apply the stochastic evolutionary approach to this class of games. We determine the conditions under which a globally pairwise risk-dominant technology is the long-run equilibrium (LRE). We also show the conditions under which one technology takes over another technology by directly switching consumers from the latter to the former (also called direct domination ). Parametric examples illustrate the properties of the technology which is the LRE. Journal of Economic Literature Classification Numbers: C72, C73.
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paper_title: The Statistical Mechanics of Strategic Interaction

paper_content:

I study strategic interaction among players who live on a lattice. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. I examine various stochastic strategy revision processes, including (myopic) best response and stochastic choice. I discuss both stationary distributions and the limit behavior of these Markov processes. Stationary distributions are partially characterized, and the asymptotic behavior of stochastic choice for those processes whose choice rule is nearly best-response is related to equilibrium selection in symmetric 2 × 2 and n × n coordination games. Journal of Economic Literature Classification Number: C78.
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paper_title: The logit-response dynamics

paper_content:

We develop a characterization of stochastically stable states for the logit-response learning dynamics in games, with arbitrary specification of revision opportunities. The result allows us to show convergence to the set of Nash equilibria in the class of best-response potential games and the failure of the dynamics to select potential maximizers beyond the class of exact potential games. We also study to which extent equilibrium selection is robust to the specification of revision opportunities. Our techniques can be extended and applied to a wide class of learning dynamics in games.
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paper_title: DECENTRALIZATION AND THE COORDINATION PROBLEM

paper_content:

This paper addresses the relationship between the degree of decentralization in a population and the probability of coordinating on an efficient outcome rather than the risk-dominant outcome in a coordination game. The model here extends the literature on evolutionary games by supposing that players interact in groups but that occasionally they may change groups. The main innovation is the group-changing assumption which allows players to ‘vote with their feet.’ In contrast to most of the recent literature (e.g. Kandori et al., 1993) in which the risk-dominant equilibrium is selected in the long run, the model here predicts convergence to the efficient equilibrium if interaction is decentralized enough. Furthermore, it is shown how the model can be applied to the problem of product standardization.
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paper_title: Local Conventions

paper_content:

It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
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paper_title: Long-run equilibria with dominated strategies

paper_content:

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.
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paper_title: COMPETITION AMONG CONVENTIONS

paper_content:

A convention can be seen as a way of resolving a coordination problem. If different conventions exist in various geographical, social or other entities (called &201C;groups&201D;) and if there is some mobility between these groups, which conventions, if any, will emerge as the successful ones? A simple evolutionary process is suggested and it is shown that the process converges to a Nash equilibrium for all games satisfying weak acyclity. Further, if the process converges, it converges to an efficient convention for all games in which the Pareto optimal symmetric equilibria are strict. Hence, the paper presents an explanation for the endogenous evolution of efficiency. In contrast to most recent studies in evolutionary game theory, the conclusions do not rely on random &201C;mutations&201D;. Instead, the driving force is the tendency of players to have increased interaction with member of their own group (viscosity).
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paper_title: Partial bandwagon effects and local interactions

paper_content:

We consider partial bandwagon properties in the context of coordination games to capture the idea of weak network externalities. We then study a local interactions model where agents play a coordination game following a noisy best-reply process. We show that globally pairwise risk dominant strategies are selected in arbitrary 3 × 3 coordination games, but not necessarily in larger games. A comparison with the global interactions benchmark shows that the nature of interactions might alter the long-run results themselves, and not only the speed of convergence. We also illustrate that the simultaneous coexistence of conventions is possible for games with at least 5 strategies. © 2007 Elsevier Inc. All rights reserved.
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paper_title: Evolution of Equilibria in the Long Run: A General Theory and Applications

paper_content:

We extend the evolutionary process studied in Kandori et al., Econometrica 61 (1933), 29-56 to n × n games. The evolutionary process is driven by two forces: players switching to the best response against the present strategy configuration, and players experimenting with new strategies. We show that a unique behavior pattern, called the long-run equilibrium, arises even if the underlying game has multiple (static) equilibria. The paper gives a general algorithm for computing the LRE, and then applies it to two classes of economic games. For games of pure coordination, the LRE is the Pareto-efficient equilibrium. For games with strategic complementarities, the geometry of the best-response correspondence helps identify the LRE. Journal of Economic Literature Classification Numbers: C63, C72, D43.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: LEARNING, MUTATION, AND LONG RUN EQUILIBRIA in Games

paper_content:

An evolutionary model with a finite number of players and with stochastic mutations is analyzed. The expansion and contraction of strategies is linked to their current relative success, but mutuation, perturbing the system from its deterministic evolution, are present as well. The focus is on the long run implications of ongoing mutations, which drastically reduce the set of equilibria. For 2 by 2 symmetric games with two symmetric strict Nash equilibria the risk dominant equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium. Copyright 1993 by The Econometric Society.
---
paper_title: LEARNING, MUTATION, AND LONG RUN EQUILIBRIA in Games

paper_content:

An evolutionary model with a finite number of players and with stochastic mutations is analyzed. The expansion and contraction of strategies is linked to their current relative success, but mutuation, perturbing the system from its deterministic evolution, are present as well. The focus is on the long run implications of ongoing mutations, which drastically reduce the set of equilibria. For 2 by 2 symmetric games with two symmetric strict Nash equilibria the risk dominant equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium. Copyright 1993 by The Econometric Society.
---
paper_title: Economists ' Models of Learning 1

paper_content:

Economic theorists have intensively studied learning in games and decisions over the last decade. This essay puts some of the work in perspective and offers opinions about what still needs to be learned. Journal of Economic Literature Classification Numbers: C73, D83.
---
paper_title: Simple and clever decision rules for a model of evolution

paper_content:

Abstract Under the decision rule specified by Kandori, Mailath, and Rob (1993) [Kandori, M., Mailath, G.J., Rob, R., 1993. Learning, mutation, and long run equilibria in games. Econometrica 61, 29–56], myopic adjustment can lead to surprising results, including coordination on strictly dominated strategies. We show that under an alternative decision rule, convergence to Nash equilibrium is guaranteed. Moreover, if rare mutations are introduced, risk dominant equilibria always correspond to long run equilibria.
---
paper_title: Bandwagon Effects and Long Run Technology Choice

paper_content:

Abstract We introduce a class of “bandwagon games” which generalizes the notion of network externalities when there are more than two competing technologies and various degrees of partial compatibilities among them. We then apply the stochastic evolutionary approach to this class of games. We determine the conditions under which a globally pairwise risk-dominant technology is the long-run equilibrium (LRE). We also show the conditions under which one technology takes over another technology by directly switching consumers from the latter to the former (also called direct domination ). Parametric examples illustrate the properties of the technology which is the LRE. Journal of Economic Literature Classification Numbers: C72, C73.
---
paper_title: Evolution of Equilibria in the Long Run: A General Theory and Applications

paper_content:

We extend the evolutionary process studied in Kandori et al., Econometrica 61 (1933), 29-56 to n × n games. The evolutionary process is driven by two forces: players switching to the best response against the present strategy configuration, and players experimenting with new strategies. We show that a unique behavior pattern, called the long-run equilibrium, arises even if the underlying game has multiple (static) equilibria. The paper gives a general algorithm for computing the LRE, and then applies it to two classes of economic games. For games of pure coordination, the LRE is the Pareto-efficient equilibrium. For games with strategic complementarities, the geometry of the best-response correspondence helps identify the LRE. Journal of Economic Literature Classification Numbers: C63, C72, D43.
---
paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Evolution of Equilibria in the Long Run: A General Theory and Applications

paper_content:

We extend the evolutionary process studied in Kandori et al., Econometrica 61 (1933), 29-56 to n × n games. The evolutionary process is driven by two forces: players switching to the best response against the present strategy configuration, and players experimenting with new strategies. We show that a unique behavior pattern, called the long-run equilibrium, arises even if the underlying game has multiple (static) equilibria. The paper gives a general algorithm for computing the LRE, and then applies it to two classes of economic games. For games of pure coordination, the LRE is the Pareto-efficient equilibrium. For games with strategic complementarities, the geometry of the best-response correspondence helps identify the LRE. Journal of Economic Literature Classification Numbers: C63, C72, D43.
---
paper_title: Bandwagon Effects and Long Run Technology Choice

paper_content:

Abstract We introduce a class of “bandwagon games” which generalizes the notion of network externalities when there are more than two competing technologies and various degrees of partial compatibilities among them. We then apply the stochastic evolutionary approach to this class of games. We determine the conditions under which a globally pairwise risk-dominant technology is the long-run equilibrium (LRE). We also show the conditions under which one technology takes over another technology by directly switching consumers from the latter to the former (also called direct domination ). Parametric examples illustrate the properties of the technology which is the LRE. Journal of Economic Literature Classification Numbers: C72, C73.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Efficient Equilibrium Selection in Evolutionary Games with Random Matching

paper_content:

Abstract This paper investigates the sensitivity of recent evolutionary models of learning to the specification of the matching mechanism. We study a literally random matching mechanism, combined with a process of strategy adjustment based on the realized average performance of each strategy. In the key class of symmetric 2×2 coordination games, the Pareto-efficient equilibrium, per se , is selected, rather than the risk-dominant equilibrium, as the probability of experimentation (or “mutation”) goes to zero. Furthermore, convergence to the equilibrium is relatively fast. We extend these results, for example, to games of common interest. Journal of Economic Literature Classification Numbers: C72, C73.
---
paper_title: Evolution with State-Dependent Mutations

paper_content:

Recent evolutionary models have introduced "small mutation rates" as a way of refining predictions of long-run behavior. We show that this refinement effect can only be obtained by restrictions on how the magnitude of the effect of mutation on evolution varies across states of the system. In particular, given any model of the effect of mutations, any invariant distribution of the "mutationless" process is close to an invariant distribution of the process with appropriately chosen small mutation rates.
---
paper_title: Long-run equilibria with dominated strategies

paper_content:

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
---
paper_title: Altruists, Egoists and Hooligans in a Local Interaction Model

paper_content:

This paper studies a population of agents, each of whom can be either an Altruist or an Egoist. Altruists confer benefits on others at a cost to themselves. Altruism is thus a strictly dominated strategy and cannot survive if agents are rational best-responders. We assume that agents choose their actions by imitating others who earn high payoffs. We also assume that interactions between agents are local, so that each agent affects (and is affected by) only his neighbors. Altruists can survive in such a world if they are grouped together, so that the benefits of altruism are enjoyed primarily by other Altruists, who then earn relatively high payoffs and are imitated.
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paper_title: Dynamic models of segregation†

paper_content:

Some segregation results from the practices of organizations, some from specialized communication systems, some from correlation with a variable that is non‐random; and some results from the interplay of individual choices. This is an abstract study of the interactive dynamics of discriminatory individual choices. One model is a simulation in which individual members of two recognizable groups distribute themselves in neighborhoods defined by reference to their own locations. A second model is analytic and deals with compartmented space. A final section applies the analytics to ‘neighborhood tipping.’ The systemic effects are found to be overwhelming: there is no simple correspondence of individual incentive to collective results. Exaggerated separation and patterning result from the dynamics of movement. Inferences about individual motives can usually not be drawn from aggregate patterns. Some unexpected phenomena, like density and vacancy, are generated. A general theory of ‘tipping’ begins to emerge.
---
paper_title: The Formation of Ghettos as a Local Interaction Phenomenon,” mimeo

paper_content:

A moving yoke comprises a first yoke member, a second yoke member, swivellable relative to the first yoke member about a first axis, and a third yoke member having an arcuate portion. The second yoke member has an arcuate support portion, and the arcuate portion of the third yoke member is supported by the arcuate portion of the second yoke member to enable slidable movement relative thereto along an arcuate path.
---
paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
---
paper_title: Adaptive play with spatial sampling

paper_content:

Abstract We consider a 2×2 coordination game where each agent interacts with his neighbors on a ring. Ellison (1993, Econometrica 61, 1047–1071) shows that the discrete dynamical system generated by the myopic best-reply rule converges to a Nash equilibrium or to a two-period limit cycle. Following Young (1993, Econometrica 61, 57–84), we consider a best-reply process with a sampling procedure. Particularly, we introduce a spatial sampling procedure: each agent observes a sample of information in his neighborhood and plays a best reply to it. We show that if the size of the sample of information is not too large, the best-reply process converges almost surely to a Nash equilibrium. If in addition agents experiment with small probabilities, we show that, in most cases, the risk-dominant equilibrium prevails in the long run. Furthermore, it turns out that the convergence is faster than in Ellison.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Random Perturbations of Dynamical Systems

paper_content:

1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.
---
paper_title: Evolution with State-Dependent Mutations

paper_content:

Recent evolutionary models have introduced "small mutation rates" as a way of refining predictions of long-run behavior. We show that this refinement effect can only be obtained by restrictions on how the magnitude of the effect of mutation on evolution varies across states of the system. In particular, given any model of the effect of mutations, any invariant distribution of the "mutationless" process is close to an invariant distribution of the process with appropriately chosen small mutation rates.
---
paper_title: Contagion and State Dependent Mutations

paper_content:

Early results of evolutionary game theory showed that the risk dominant equilibrium is uniquely selected in the long run under the best-response dynamics with mutation. Bergin and Lipman (1996) qualified this result by showing that for a given population size, the evolutionary process can select any strict Nash equilibrium if the probability of choosing a nonbest response is state-dependent. This paper shows that the unique selection of the risk dominant equilibrium is robust with respect to state dependent mutation in local interaction games. More precisely, for any given mutation structure there exists a minimum population size beyond which the risk dominant equilibrium is uniquely selected. Our result is driven by contagion and cohesion among players, which exist only in local interaction settings and favor the risk dominant strategy. Our result strengthens the equilibrium selection result of evolutionary game theory.
---
paper_title: Bandwagon Effects and Long Run Technology Choice

paper_content:

Abstract We introduce a class of “bandwagon games” which generalizes the notion of network externalities when there are more than two competing technologies and various degrees of partial compatibilities among them. We then apply the stochastic evolutionary approach to this class of games. We determine the conditions under which a globally pairwise risk-dominant technology is the long-run equilibrium (LRE). We also show the conditions under which one technology takes over another technology by directly switching consumers from the latter to the former (also called direct domination ). Parametric examples illustrate the properties of the technology which is the LRE. Journal of Economic Literature Classification Numbers: C72, C73.
---
paper_title: Partial bandwagon effects and local interactions

paper_content:

We consider partial bandwagon properties in the context of coordination games to capture the idea of weak network externalities. We then study a local interactions model where agents play a coordination game following a noisy best-reply process. We show that globally pairwise risk dominant strategies are selected in arbitrary 3 × 3 coordination games, but not necessarily in larger games. A comparison with the global interactions benchmark shows that the nature of interactions might alter the long-run results themselves, and not only the speed of convergence. We also illustrate that the simultaneous coexistence of conventions is possible for games with at least 5 strategies. © 2007 Elsevier Inc. All rights reserved.
---
paper_title: Random Perturbations of Dynamical Systems

paper_content:

1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.
---
paper_title: The Statistical Mechanics of Best-Response Strategy Revision

paper_content:

I continue the study, begun in Blume (1993), of stochastic strategy revision processes in large player populations where the range of interaction between players is small. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. The purpose of this paper is to compare local strategic interaction with global strategic interaction when players update their choice according to the (myopic) best-response rule. I show that randomizing the order in which players update their strategic choice is sufficient to achieve coordination on the risk-dominant strategy in symmetric $2\times 2$ coordination games. The ``persistant randomness'' which is necessary to achieve similar coordination when the range of interaction is global is replaced by spatial variation in choice in the initial condition when strategic interactions are local. An extension of the risk-dominance idea gives the same convergence result for $K\times K$ games with strategic complementarities. Similar results for $K\times K$ pure coordination games and potential games are also presented.
---
paper_title: The Statistical Mechanics of Strategic Interaction

paper_content:

I study strategic interaction among players who live on a lattice. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. I examine various stochastic strategy revision processes, including (myopic) best response and stochastic choice. I discuss both stationary distributions and the limit behavior of these Markov processes. Stationary distributions are partially characterized, and the asymptotic behavior of stochastic choice for those processes whose choice rule is nearly best-response is related to equilibrium selection in symmetric 2 × 2 and n × n coordination games. Journal of Economic Literature Classification Number: C78.
---
paper_title: Path Dependence and Learning from Neighbors

paper_content:

We study the long-run properties of a class of locally interactive learning systems. A finite set of players at fixed locations play a two-by-two symmetric normal form game with strategic complementarities, with one of their “neighbors” selected at random. Because of the endogenous nature of experimentation, or “noise,” the systems we study exhibit a high degree of path dependence. Different actions of a pure coordination game may survive in the long-run at different locations of the system. A reinterpretation of our results shows that the local nature of search may be a robust reason for price dispersion in a search model,Journal of Economic LiteratureClassification Numbers: C72, D83.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
---
paper_title: Local Conventions

paper_content:

It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
---
paper_title: Evolution with State-Dependent Mutations

paper_content:

Recent evolutionary models have introduced "small mutation rates" as a way of refining predictions of long-run behavior. We show that this refinement effect can only be obtained by restrictions on how the magnitude of the effect of mutation on evolution varies across states of the system. In particular, given any model of the effect of mutations, any invariant distribution of the "mutationless" process is close to an invariant distribution of the process with appropriately chosen small mutation rates.
---
paper_title: DECENTRALIZATION AND THE COORDINATION PROBLEM

paper_content:

This paper addresses the relationship between the degree of decentralization in a population and the probability of coordinating on an efficient outcome rather than the risk-dominant outcome in a coordination game. The model here extends the literature on evolutionary games by supposing that players interact in groups but that occasionally they may change groups. The main innovation is the group-changing assumption which allows players to ‘vote with their feet.’ In contrast to most of the recent literature (e.g. Kandori et al., 1993) in which the risk-dominant equilibrium is selected in the long run, the model here predicts convergence to the efficient equilibrium if interaction is decentralized enough. Furthermore, it is shown how the model can be applied to the problem of product standardization.
---
paper_title: Local Conventions

paper_content:

It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
---
paper_title: Long-run equilibria with dominated strategies

paper_content:

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.
---
paper_title: COMPETITION AMONG CONVENTIONS

paper_content:

A convention can be seen as a way of resolving a coordination problem. If different conventions exist in various geographical, social or other entities (called &201C;groups&201D;) and if there is some mobility between these groups, which conventions, if any, will emerge as the successful ones? A simple evolutionary process is suggested and it is shown that the process converges to a Nash equilibrium for all games satisfying weak acyclity. Further, if the process converges, it converges to an efficient convention for all games in which the Pareto optimal symmetric equilibria are strict. Hence, the paper presents an explanation for the endogenous evolution of efficiency. In contrast to most recent studies in evolutionary game theory, the conclusions do not rely on random &201C;mutations&201D;. Instead, the driving force is the tendency of players to have increased interaction with member of their own group (viscosity).
---
paper_title: On the Co-existence of Conventions

paper_content:

Recent stochastic evolutionary models have shown that the most likely convention when the probability of a mutation is sufficiently small is coordination on the risk-dominant strategy rather than the payoff-dominant one. This paper looks at the consequences of player movement between locations when there are constraints which limit the number of agents who can reside at each location. If the constraints are strong then the risk-dominance result continues to hold. However, we show that when sufficient movement is possible, the most likely outcome involves a mixed state in which agents at different locations coordinate on different strategies. In the asymmetric case, it is the location with the stronger constraint, limiting movement, that coordinates on the payoff-dominant strategy.
---
paper_title: The Evolution of Conventions with Mobile 3 Players

paper_content:

This paper analyzes the evolution of conventions in a society with local interaction and mobile players. Three innovative aspects are introduced: Imperfect observability of play outside a player's home location, friction in the strategy adjustment process, and restriced mobility. It is shown that, if mobility is unrestriced, only efficient conventions are stochastichally stable. If there are barriers on mobility, the coexistence of different conventions can be observed. While imperfect observability and friction alone cannot revent society from reaching an overall efficient outcome, restricted mobility can.
---
paper_title: ON THE GEOGRAPHY OF CONVENTIONS

paper_content:

We study a model in which heterogenous boundedly rational agents interact locally in order to play a coordination game. Agents differ in their mobility with mobile agents being able to relocate within a country. The model yields the following predictions: (1) mobile agents always benefit from increased mobility, (2) immobile agents benefit from increased mobility at low levels of mobility, (3) immobile agents lose from increased mobility at high levels of mobility, (4) there is an optimal ``country size,'' (5) ``income inequality'' is weakly increasing in mobility, (6) if there are arbitrarily small payoff differences between two countries, opening borders causes a ``brain drain'' effect; in the long run, all mobile agents reside in the favored (former) country and efficiency is attained only in that country.
---
paper_title: The Statistical Mechanics of Best-Response Strategy Revision

paper_content:

I continue the study, begun in Blume (1993), of stochastic strategy revision processes in large player populations where the range of interaction between players is small. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. The purpose of this paper is to compare local strategic interaction with global strategic interaction when players update their choice according to the (myopic) best-response rule. I show that randomizing the order in which players update their strategic choice is sufficient to achieve coordination on the risk-dominant strategy in symmetric $2\times 2$ coordination games. The ``persistant randomness'' which is necessary to achieve similar coordination when the range of interaction is global is replaced by spatial variation in choice in the initial condition when strategic interactions are local. An extension of the risk-dominance idea gives the same convergence result for $K\times K$ games with strategic complementarities. Similar results for $K\times K$ pure coordination games and potential games are also presented.
---
paper_title: The Statistical Mechanics of Strategic Interaction

paper_content:

I study strategic interaction among players who live on a lattice. Each player interacts directly with only a finite set of neighbors, but any two players indirectly interact through a finite chain of direct interactions. I examine various stochastic strategy revision processes, including (myopic) best response and stochastic choice. I discuss both stationary distributions and the limit behavior of these Markov processes. Stationary distributions are partially characterized, and the asymptotic behavior of stochastic choice for those processes whose choice rule is nearly best-response is related to equilibrium selection in symmetric 2 × 2 and n × n coordination games. Journal of Economic Literature Classification Number: C78.
---
paper_title: Local Conventions

paper_content:

It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
---
paper_title: Long-run equilibria with dominated strategies

paper_content:

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.
---
paper_title: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution

paper_content:

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.
---
paper_title: A multinomial probit model of stochastic evolution

paper_content:

A strategy revision process in symmetric normal form games is proposed. Following Kandori, Mailath, and Rob (1993), members of a population periodically revise their strategy choice, and choose a myopic best response to currently observed play. Their payoffs are perturbed by normally distributed Harsanyian (1973) trembles, so that strategies are chosen according to multinomial probit probabilities. As the variance of payoffs is allowed to vanish, the graph theoretic methods of the earlier literature continue to apply. The distributional assumption enables a convenient closed form characterisation for the weights of the rooted trees. An illustration of the approach is offered, via a consideration of the role of dominated strategies in equilibrium selection.
---
paper_title: Endogenous Networks, Social Games, and Evolution

paper_content:

Abstract This paper studies a social game where agents choose their partners as well as their actions. Players interact with direct and indirect neighbors in the endogenous network. We show that the architecture of any nontrivial Nash equilibrium is minimally connected, and equilibrium actions approximate a symmetric equilibrium of the underlying game. We apply the model to analyze stochastic stability in 2 × 2 coordination games. We find that long-run equilibrium selection depends on a trade-off between efficiency and risk dominance due to the presence of scale effects arising from network externalities. Our results suggest a general pattern of equilibrium selection.
---
paper_title: On the Formation of Interaction Networks in Social Coordination Games

paper_content:

There are many situations where two interacting individuals can benefit from coordinating their actions. We examine the endogenous choice of partners in such social coordination games and the implications for resulting play. We model the interaction pattern as a network where individuals periodically have the discretion to add or sever links to other players. With such endogenous interaction patterns we see multiple stochastically stable states of play, including some that involve play of equilibria in the coordination game that are neither efficient nor risk-dominant. Thus the endogenous network structure not only has implications for the interaction pattern that emerges, but it also has a significant impact on the play in the coordination game relative to what would arise if the same interaction network were exogenous.
---
paper_title: Network Formation and Social Coordination

paper_content:

This paper develops a simple model to examine the interaction between partner choice and individual behavior in games of coordination. An important ingredient of our approach is the way we model partner choice: we suppose that a player can establish ties with other players by unilaterally investing in costly pairwise links. In this context, individual efforts to balance the costs and benefits of links are shown to lead to a unique equilibrium interaction architecture. The dynamics of network formation, however, has powerful effects on individual behavior: if costs of forming links are below a certain threshold then players coordinate on the risk-dominant action, while if costs are above this threshold then they coordinate on the efficient action. These findings are robust to modifications in the link formation process, different specifications of link formation costs, alternative models of mutations as well as the possibility of interaction among indirectly connected players.
---
paper_title: A Noncooperative Model of Network Formation

paper_content:

We present an approach to network formation based on the notion that social networks are formed by individual decisions that trade off the costs of forming and maintaining links against the potential rewards from doing so. We suppose that a link with another agent allows access, in part and in due course, to the benefits available to the latter via his own links. Thus individual links generate externalities whose value depends on the level of decay/delay associated with indirect links. A distinctive aspect of our approach is that the costs of link formation are incurred only by the person who initiates the link. This allows us to formulate the network formation process as a noncooperative game.
---
paper_title: Contagion and State Dependent Mutations

paper_content:

Early results of evolutionary game theory showed that the risk dominant equilibrium is uniquely selected in the long run under the best-response dynamics with mutation. Bergin and Lipman (1996) qualified this result by showing that for a given population size, the evolutionary process can select any strict Nash equilibrium if the probability of choosing a nonbest response is state-dependent. This paper shows that the unique selection of the risk dominant equilibrium is robust with respect to state dependent mutation in local interaction games. More precisely, for any given mutation structure there exists a minimum population size beyond which the risk dominant equilibrium is uniquely selected. Our result is driven by contagion and cohesion among players, which exist only in local interaction settings and favor the risk dominant strategy. Our result strengthens the equilibrium selection result of evolutionary game theory.
---
paper_title: A Strategic Model of Social and Economic Networks

paper_content:

We study the stability and efficiency of social and economic networks, when self-interested individuals have the discretion to form or sever links. First, in the context of two stylized models, we characterize the sets of stable networkds (immune to incentives to form or sever links) and the sets of efficient networks (those which maximize total production or utility). The sets of stable networks and efficients networks do not always intersect. Next, we show that this tension is not unique to these models, but persists generally. In order to assure that there is always at least one efficient graph which is stable, one is forced to allocate resources to nodes (players) who are not responsible for any of the production. We characterize one such allocations rule: the equal split rule. We characterize another rule which fails to assure that efficient graphs are stable, but arises naturally if the allocations result from the bargaining of players.
---
paper_title: Local Conventions

paper_content:

It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
---
paper_title: Potential games in volatile environments ∗

paper_content:

This papers studies the co-evolution of networks and play in the context of finite population potential games. Action revision, link creation and link destruction are combined in a continuous-time Markov process. I derive the unique invariant distribution of this process in closed form, as well as the marginal distribution over action profiles and the conditional distribution over networks. It is shown that the equilibrium interaction topology is an inhomogeneous random graph. Furthermore, a characterization of the set of stochastically stable states is provided, generalizing existing results to models with endogenous interaction structures.
---
paper_title: Partial bandwagon effects and local interactions

paper_content:

We consider partial bandwagon properties in the context of coordination games to capture the idea of weak network externalities. We then study a local interactions model where agents play a coordination game following a noisy best-reply process. We show that globally pairwise risk dominant strategies are selected in arbitrary 3 × 3 coordination games, but not necessarily in larger games. A comparison with the global interactions benchmark shows that the nature of interactions might alter the long-run results themselves, and not only the speed of convergence. We also illustrate that the simultaneous coexistence of conventions is possible for games with at least 5 strategies. © 2007 Elsevier Inc. All rights reserved.
---
paper_title: Small Feedback-based Decisions and Their Limited Correspondence to Description-based Decisions

paper_content:

The present paper explores situations in which the information available to decision makers is limited to feedback concerning the outcomes of their previous decisions. The results reveal that experience in these situations can lead to deviations from maximization in the opposite direction of the deviations observed when the decisions are made based on a description of the choice problem. Experience was found to lead to a reversed common ratio/certainty effect, more risk seeking in the gain than in the loss domain, and to an underweighting of small probabilities. Only one of the examined properties of description-based decisions, loss aversion, seems to emerge robustly in these ‘feedback-based’ decisions. These results are summarized with a simple model that illustrates that all the unique properties of feedback-based decisions can be a product of a tendency to rely on recent outcomes. Copyright # 2003 John Wiley & Sons, Ltd.
---
paper_title: Imitation - Theory and Experimental Evidence

paper_content:

We introduce a generalized theoretical approach to study imitation models and subject the models to rigorous experimental testing. In our theoretical analysis we find that the different predictions of previous imitation models are due to different informational assumptions, not to different behavioral rules. It is more important whom one imitates rather than how. In a laboratory experiment we test the different theories by systematically varying information conditions. We find that the generalized imitation model predicts the differences between treatments well. The data also provide support for imitation on the individual level, both in terms of choice and in terms of perception. But imitation is not unconditional. Rather individuals' propensity to imitate more successful actions is increasing in payoff differences.
---
paper_title: Learning, Local Interaction, and Coordination

paper_content:

This paper discusses the dynamic implications of learning in a large population coordination game, focusing on the structure of the matching process that describes how players meet. As in M. Kandori, G. Mailath, and R. Rob (1992), experimentation and myopia create 'evolutionary' forces that lead players to coordinate on the risk dominant equilibrium. To describe play with finite time horizons, it is necessary to consider the rates at which the dynamic systems converge. In large populations with uniform matching, play is determined largely by historical factors. When players interact with small sets of neighbors, evolutionary forces may determine the outcome. Copyright 1993 by The Econometric Society.
---
paper_title: Altruists, Egoists and Hooligans in a Local Interaction Model

paper_content:

This paper studies a population of agents, each of whom can be either an Altruist or an Egoist. Altruists confer benefits on others at a cost to themselves. Altruism is thus a strictly dominated strategy and cannot survive if agents are rational best-responders. We assume that agents choose their actions by imitating others who earn high payoffs. We also assume that interactions between agents are local, so that each agent affects (and is affected by) only his neighbors. Altruists can survive in such a world if they are grouped together, so that the benefits of altruism are enjoyed primarily by other Altruists, who then earn relatively high payoffs and are imitated.
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paper_title: On the Co-existence of Conventions

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Recent stochastic evolutionary models have shown that the most likely convention when the probability of a mutation is sufficiently small is coordination on the risk-dominant strategy rather than the payoff-dominant one. This paper looks at the consequences of player movement between locations when there are constraints which limit the number of agents who can reside at each location. If the constraints are strong then the risk-dominance result continues to hold. However, we show that when sufficient movement is possible, the most likely outcome involves a mixed state in which agents at different locations coordinate on different strategies. In the asymmetric case, it is the location with the stronger constraint, limiting movement, that coordinates on the payoff-dominant strategy.
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paper_title: Imitation, local interactions, and efficiency

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Abstract We consider a circular city model as in Ellison [Ellison, G., 1993. Learning, local interaction, and coordination, Econometrica 61, 1047–1071], where agents follow imitation rules rather than myopic best-response. If interactions are neither global nor limited to the immediate neighbors, payoff-efficient equilibria, and not risk-dominant ones, can be uniquely selected.
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paper_title: Adaptive play with spatial sampling

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Abstract We consider a 2×2 coordination game where each agent interacts with his neighbors on a ring. Ellison (1993, Econometrica 61, 1047–1071) shows that the discrete dynamical system generated by the myopic best-reply rule converges to a Nash equilibrium or to a two-period limit cycle. Following Young (1993, Econometrica 61, 57–84), we consider a best-reply process with a sampling procedure. Particularly, we introduce a spatial sampling procedure: each agent observes a sample of information in his neighborhood and plays a best reply to it. We show that if the size of the sample of information is not too large, the best-reply process converges almost surely to a Nash equilibrium. If in addition agents experiment with small probabilities, we show that, in most cases, the risk-dominant equilibrium prevails in the long run. Furthermore, it turns out that the convergence is faster than in Ellison.
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paper_title: Efficient Equilibrium Selection in Evolutionary Games with Random Matching

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Abstract This paper investigates the sensitivity of recent evolutionary models of learning to the specification of the matching mechanism. We study a literally random matching mechanism, combined with a process of strategy adjustment based on the realized average performance of each strategy. In the key class of symmetric 2×2 coordination games, the Pareto-efficient equilibrium, per se , is selected, rather than the risk-dominant equilibrium, as the probability of experimentation (or “mutation”) goes to zero. Furthermore, convergence to the equilibrium is relatively fast. We extend these results, for example, to games of common interest. Journal of Economic Literature Classification Numbers: C72, C73.
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paper_title: On Adaptation, Maximization, and Reinforcement Learning Among Cognitive Strategies

paper_content:

Analysis of binary choice behavior in iterated tasks with immediate feedback reveals robust deviations from maximization that can be described as indications of 3 effects: (a) a payoff variability effect, in which high payoff variability seems to move choice behavior toward random choice; (b) underweighting of rare events, in which alternatives that yield the best payoffs most of the time are attractive even when they are associated with a lower expected return: and (c) loss aversion, in which alternatives that minimize the probability of losses can be more attractive than those that maximize expected payoffs. The results are closer to probability matching than to maximization. Best approximation is provided with a model of reinforcement learning among cognitive strategies (RELACS). This model captures the 3 deviations, the learning curves, and the effect of information on uncertainty avoidance. It outperforms other models in fitting the data and in predicting behavior in other experiments.
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paper_title: Imitation and the Role of Information in Overcoming Coordination Failures

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We model the structure of a firm or an organization as a network and consider minimum-effort games played on this network as a metaphor for cooperations failing due to coordination failures. For a family of behavioral rules, including Imitate the Best and the Proportional Imitation Rule, we show that inefficient conventions arise independently of the interaction structure, if information is limited to the interaction neighborhoods. However, in the presence of informational spillovers, a minimal condition on the network guarantees that efficient conventions will eventually dominate. An analogous result is established for average opinion games.
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paper_title: DECENTRALIZATION AND THE COORDINATION PROBLEM

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This paper addresses the relationship between the degree of decentralization in a population and the probability of coordinating on an efficient outcome rather than the risk-dominant outcome in a coordination game. The model here extends the literature on evolutionary games by supposing that players interact in groups but that occasionally they may change groups. The main innovation is the group-changing assumption which allows players to ‘vote with their feet.’ In contrast to most of the recent literature (e.g. Kandori et al., 1993) in which the risk-dominant equilibrium is selected in the long run, the model here predicts convergence to the efficient equilibrium if interaction is decentralized enough. Furthermore, it is shown how the model can be applied to the problem of product standardization.
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paper_title: Local Conventions

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It is shown that player mobility has important consequences for the long-run equilibrium distribution in dynamic evolutionary models of strategy adjustment, when updating is prone to small probability perturbations, i.e. “mistakes" or “mutations." Ellison (1993) concluded that the effect on the matching process of localized “neighborhoods" was to strengthen the stability of risk-dominant outcomes, originally demonstrated by Kandori, Mailath, and Rob (1993) (KMR) and Young (1993). I consider a model in which players can choose the neighborhoods to which they belong. When strategies and locations are updated simultaneously, only efficient strategies survive. The robustness of this conclusion is emphasized in a general locational model in which strategy revision opportunities are allowed to arrive at a faster rate than opportunities to change locations. The efficient strategy persists in all cases in which the locational structure is non-trivial. Moreover, even as the relative frequency of player mobility approaches zero, the efficient strategy occurs with boundedly positive relative frequency. This result is in stark contrast to the conclusions of the previous models.
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paper_title: COMPETITION AMONG CONVENTIONS

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A convention can be seen as a way of resolving a coordination problem. If different conventions exist in various geographical, social or other entities (called &201C;groups&201D;) and if there is some mobility between these groups, which conventions, if any, will emerge as the successful ones? A simple evolutionary process is suggested and it is shown that the process converges to a Nash equilibrium for all games satisfying weak acyclity. Further, if the process converges, it converges to an efficient convention for all games in which the Pareto optimal symmetric equilibria are strict. Hence, the paper presents an explanation for the endogenous evolution of efficiency. In contrast to most recent studies in evolutionary game theory, the conclusions do not rely on random &201C;mutations&201D;. Instead, the driving force is the tendency of players to have increased interaction with member of their own group (viscosity).
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paper_title: Partial bandwagon effects and local interactions

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We consider partial bandwagon properties in the context of coordination games to capture the idea of weak network externalities. We then study a local interactions model where agents play a coordination game following a noisy best-reply process. We show that globally pairwise risk dominant strategies are selected in arbitrary 3 × 3 coordination games, but not necessarily in larger games. A comparison with the global interactions benchmark shows that the nature of interactions might alter the long-run results themselves, and not only the speed of convergence. We also illustrate that the simultaneous coexistence of conventions is possible for games with at least 5 strategies. © 2007 Elsevier Inc. All rights reserved.
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paper_title: Evolutionary dynamics on graphs

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Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.
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paper_title: On the Formation of Interaction Networks in Social Coordination Games

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There are many situations where two interacting individuals can benefit from coordinating their actions. We examine the endogenous choice of partners in such social coordination games and the implications for resulting play. We model the interaction pattern as a network where individuals periodically have the discretion to add or sever links to other players. With such endogenous interaction patterns we see multiple stochastically stable states of play, including some that involve play of equilibria in the coordination game that are neither efficient nor risk-dominant. Thus the endogenous network structure not only has implications for the interaction pattern that emerges, but it also has a significant impact on the play in the coordination game relative to what would arise if the same interaction network were exogenous.
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paper_title: Network Formation and Social Coordination

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This paper develops a simple model to examine the interaction between partner choice and individual behavior in games of coordination. An important ingredient of our approach is the way we model partner choice: we suppose that a player can establish ties with other players by unilaterally investing in costly pairwise links. In this context, individual efforts to balance the costs and benefits of links are shown to lead to a unique equilibrium interaction architecture. The dynamics of network formation, however, has powerful effects on individual behavior: if costs of forming links are below a certain threshold then players coordinate on the risk-dominant action, while if costs are above this threshold then they coordinate on the efficient action. These findings are robust to modifications in the link formation process, different specifications of link formation costs, alternative models of mutations as well as the possibility of interaction among indirectly connected players.
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paper_title: Endogenous Networks, Social Games, and Evolution

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Abstract This paper studies a social game where agents choose their partners as well as their actions. Players interact with direct and indirect neighbors in the endogenous network. We show that the architecture of any nontrivial Nash equilibrium is minimally connected, and equilibrium actions approximate a symmetric equilibrium of the underlying game. We apply the model to analyze stochastic stability in 2 × 2 coordination games. We find that long-run equilibrium selection depends on a trade-off between efficiency and risk dominance due to the presence of scale effects arising from network externalities. Our results suggest a general pattern of equilibrium selection.
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paper_title: Simple and clever decision rules for a model of evolution

paper_content:

Abstract Under the decision rule specified by Kandori, Mailath, and Rob (1993) [Kandori, M., Mailath, G.J., Rob, R., 1993. Learning, mutation, and long run equilibria in games. Econometrica 61, 29–56], myopic adjustment can lead to surprising results, including coordination on strictly dominated strategies. We show that under an alternative decision rule, convergence to Nash equilibrium is guaranteed. Moreover, if rare mutations are introduced, risk dominant equilibria always correspond to long run equilibria.
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paper_title: Network Competition: II. Price Discrimination

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Our companion article developed a clear conceptual framework of negotiated or regulated interconnection agreements between rival operators and studied competition between interconnected networks, under the assumption of nondiscriminatory pricing. This article relaxes this assumption and allows networks to charge different prices for calls terminating on the subscriber's network and those terminating on a rivals network. This creates a price differential between services that are identical for the consumer and generates network externalities despite network interconnection. We show that in both the mature and entry phases of the industry, the nature of competition is substantially affected by such price discrimination.
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paper_title: Bandwagon Effects and Long Run Technology Choice

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Abstract We introduce a class of “bandwagon games” which generalizes the notion of network externalities when there are more than two competing technologies and various degrees of partial compatibilities among them. We then apply the stochastic evolutionary approach to this class of games. We determine the conditions under which a globally pairwise risk-dominant technology is the long-run equilibrium (LRE). We also show the conditions under which one technology takes over another technology by directly switching consumers from the latter to the former (also called direct domination ). Parametric examples illustrate the properties of the technology which is the LRE. Journal of Economic Literature Classification Numbers: C72, C73.
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Title: Coordination Games and Local Interactions: A Survey of the Game Theoretic Literature

Section 1: Introduction
Description 1: Introduce the motivation, main goals, and the significance of studying coordination games and local interactions within the framework of game theory.

Section 2: Global Interactions and Review of Techniques
Description 2: Describe the basic framework and techniques employed in global interaction models, specifically focusing on the uniform matching rule and myopic best response adjustment process.

Section 3: The Global Interactions Model
Description 3: Revisit the global interactions model, discussing specific state conditions and long-run equilibrium criteria under global interactions and best reply learning.

Section 4: Shortcomings of the Global Model
Description 4: Discuss the limitations and potential weaknesses of the global interactions model, including issues related to speed of convergence, noise specification, and robustness to dominated strategies.

Section 5: Local Interactions
Description 5: Explore models where players interact with a small subset of the population, covering settings like the circular city and grid-based interactions.

Section 6: The Circular City
Description 6: Detail the structure and outcomes of local interactions in a circular city model, especially focusing on the spread of risk dominant strategies and the robustness of the model.

Section 7: On the Robustness of the Local Interactions Model
Description 7: Examine the robustness of the local interactions model to various critiques such as speed of convergence, noise independence, and dominated strategies in large populations.

Section 8: Interaction on the Lattice
Description 8: Analyze models where players are situated on a grid or lattice, and discuss how spatial configuration affects the spread of strategies and long-run equilibria.

Section 9: Multiple Locations
Description 9: Investigate models where players can choose locations or islands to interact in, and how mobility influences equilibrium selection and the achievement of efficient outcomes.

Section 10: Restricted Mobility
Description 10: Discuss the effects of constraints on mobility within multiple locations models, and how these constraints can lead to coexistence of conventions or other specific long-run outcomes.

Section 11: Network Formation
Description 11: Present models where players form costly links to decide with whom to interact, focusing on how network formation dynamics influence strategy choices and long-run equilibria.

Section 12: Imitation
Description 12: Explore the role of imitation learning rules in both global and local interactions and how these rules can lead to the adoption of either risk dominant or payoff dominant conventions.

Section 13: Conclusions
Description 13: Summarize the main insights gathered from the survey, noting the robustness and variety of outcomes in coordination games under different interaction and learning structures, and suggest avenues for future research.