Non-cooperative game theory

This is the first draft of the entry “Game Theory” to appear in the Sage Handbook of the Philosophy of Social Science (edited by Ian Jarvie & Jesús Zamora Bonilla), Part III, Chapter 16.game theory, epstemic foundations, incomplete information,epstemic foundations, incomplete information

Uniform Topologies on Types

We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet (1989)). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris (2006)). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, Incomplete information, Higher-order beliefs, Strategic topology, Electronic mail game

Uniform topologies on types

We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet 1989). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris 2006). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, incomplete information, higher-order beliefs, strategic topology, electronic mail game

On the Strategic Impact of an Event under Non-Common Priors

This paper studies the impact of a small probability event on strategic behavior in incomplete information games with non-common priors. It is shown that the global impact of a small probability event (i.e., its propensity to affect strategic behavior at all states in the state space) has an upper bound that is an increasing function of a measure of discrepancy from the common prior assumption. In particular, its global impact can be arbitrarily large under non-common priors, but is bounded from above under common priors. These results quantify the different implications common prior and non-common prior models have on the (infinite) hierarchies of beliefs.common prior assumption; higher order belief; rationalizability; contagion; belief potential

Uniform continuity of the value of zero-sum games with differential information

We establish uniform continuity of the value for zero-sum games with differential information, when the distance between changing information fields of each player is measured by the Boylan (1971) pseudo-metric. We also show that the optimal strategy correspondence is upper semicontinuous when the information fields of players change, even with the weak topology on players' strategy sets

THE ROBUSTNESS OF EQUILIBRIUM ANALYSIS: THE CASE OF UNDOMINATED NASH EQUILIBRIUM

I consider a strategic game form with a finite set of payoff states and employ undominated Nash equilibrium (UNE) as a solution concept under complete information. I propose notions of the proximity of information according to which the continuity of UNE concept is considered as the robustness criterion. I identify a topology (induced by what I call d?) with respect to which the undominated Bayesian Nash equilibrium (UBNE) correspondence associated with any game form is upper hemi-continuous at any complete information prior. I also identify a slightly coarser topology (induced by what I call d??) with respect to which the UBNE correspondence associated with some game form exhibits a failure of the upper hemi-continuity at any complete information prior. In this sense, the topology induced by d? is the coarsest one. The topology induced by d?? is also used in both Kajii and Morris (1998) and Monderer and Samet (1989, 1996) with some additional restriction. I apply this robustness analysis to the UNE implementation. Appealing to Palfrey and Srivastava’s (1991) canonical game form, I show, as a corollary, that almost any social choice function is robustly UNE implementable relative to d?. I show, on the other hand, that only monotonic social choice functions can be robustly UNE implementable relative to d??. This clarifies when Chung and Ely’s Theorem 1 2003) applies.

Collective intelligence: aggregation of information from neighbors in a guessing game

Complex systems show the capacity to aggregate information and to display coordinated activity. In the case of social systems the interaction of different individuals leads to the emergence of norms, trends in political positions, opinions, cultural traits, and even scientific progress. Examples of collective behavior can be observed in activities like the Wikipedia and Linux, where individuals aggregate their knowledge for the benefit of the community, and citizen science, where the potential of collectives to solve complex problems is exploited. Here, we conducted an online experiment to investigate the performance of a collective when solving a guessing problem in which each actor is endowed with partial information and placed as the nodes of an interaction network. We measure the performance of the collective in terms of the temporal evolution of the accuracy, finding no statistical difference in the performance for two classes of networks, regular lattices and random networks. We also determine that a Bayesian description captures the behavior pattern the individuals follow in aggregating information from neighbors to make decisions. In comparison with other simple decision models, the strategy followed by the players reveals a suboptimal performance of the collective. Our contribution provides the basis for the micro-macro connection between individual based descriptions and collective phenomena.Comment: 9 pages, 9 figure

Strategic Communication Networks

We consider situations in which individuals would like to choose an action which is close to that of others, as well as close to a state of nature, with the ideal proximity to the state varying across agents. Before this coordination game is played, a cheap-talk communication stage is offered to the individuals who decide to whom they reveal their private information about the state. The information transmission occurring in the communication stage is characterized by a strategic communication network. We provide an explicit link between players' preferences and the equilibrium strategic communication networks. A key feature of our equilibrium characterization is that whether communication takes place between two agents not only depends on the conflict of interest between these agents, but also on the number and preferences of the other agents with whom they communicate. Apart from some specific cases, the equilibrium communication networks are quite complex despite our simple one-dimensional description of preference heterogeneity. In general, strategic communication networks cannot be completely Pareto-ranked, but expected social welfare always increases as the communication network expands.Cheap talk ; coordination ; incomplete information ; networks