Pure Strategy Nash Equilibrium in a Group Formation Game with Positive Externalities

This paper identifies a domain of payoff functions inno spillovernoncooperative games withPositive externalitywhich admit a pure strategy Nash equilibrium. Since in general a Nash equilibrium may fail to exist, in order to guarantee the existence of an equilibrium, we impose two additional assumptions,AnonymityandOrder preservation. The proof of our main result is carried out by constructing, for a given gameG, a potential function Ψ over the set of strategy profiles in such a way that the maximum of Ψ yields a Nash equilibrium in pure strategies ofG.Journal of Economics LiteratureClassification Numbers: C72, D62, H73.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

Minimizing Regret: The General Case

In repeated games with differential information on one side, the labelling “general case” refers to games in which the action of the informed player is not known to the uninformed, who can only observe a signal which is the random outcome of his and his opponent's action. Here we consider the problem of minimizing regret (in the sense first formulated by J. Hannan (1956, in Contributions to the Theory of Games, Vol. III, Annals of Mathematics Studies, Vol. 39, Princeton, NJ: Princeton Univ. Press.) when the information available is of this type. We give a simple condition describing the approachable set. Journal of Economic Literature Classification Number: D81, D82, D83.     

Strong equilibrium in cost sharing connection games

We study network games in which each player wishes to connect his source and sink, and the cost of each edge is shared among its users either equally (in Fair Connection Games—FCG's) or arbitrarily (in General Connection Games—GCG's). We study the existence and quality of strong equilibria (SE)—strategy profiles from which no coalition can improve the cost of each of its members—in these settings. We show that SE always exist in the following games: (1) Single source and sink FCG's and GCG's. (2) Single source multiple sinks FCG's and GCG's on series parallel graphs. (3) Multi source and sink FCG's on extension parallel graphs. As for the quality of the SE, in any FCG with n players, the cost of any SE is bounded by H(n) (i.e., the harmonic sum), contrasted with the Θ(n) price of anarchy. For any GCG, any SE is optimal.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Unfolding Social Hierarchies

Consider a large population of finitely-lived agents organized into n different hierarchical levels. Every period, all those placed at each level are randomly matched to play a given symmetric game. Based on the resulting outcome, a ρ-fraction of agents who (within their own level) attain the highest payoffs are promoted upwards. On the other hand, newcomers replacing those who die every period enter at the lowest level and choose irreversibly the strategy to be played for the rest of their life. This choice is made, with some noise, by imitating one of the strategies adopted at the highest level.  In this setup, the unique long-run behavior of the system is fully characterized for the whole class of 2×2 coordination games and two alternative variations of the model. The results crucially depend on the key “institutional” parameters ρ and n. In particular, it is shown that inefficient behavior prevails in the long run (even when risk-dominated) if promotion is only mildly selective—high ρ—and the social system is quite hierarchical—large n. In a stylized manner, these parameter conditions may be viewed as reflecting a sort of institutional deficiency that impairs economic performance. Journal of Economic Literature Classification Numbers: C70, C72, C73, D72.     

Simple search methods for finding a Nash equilibrium

We present two simple search methods for computing a sample Nash equilibrium in a normal-form game: one for 2-player games and one for n-player games. Both algorithms bias the search towards supports that are small and balanced, and employ a backtracking procedure to efficiently explore these supports. Making use of a new comprehensive testbed, we test these algorithms on many classes of games, and show that they perform well against the state of the art—the Lemke–Howson algorithm for 2-player games, and Simplicial Subdivision and Govindan–Wilson for n-player games.     

Belief-Based Refinements in Signalling Games

There are many economic problems which, when modelled as games of incomplete information, give rise to many sequential equilibria, severely limiting the usefulness of the model. There has recently been a large literature devoted to "refining" the set of equilibria in order to reduce this multiplicity by restricting the set of admissible disequilibrium beliefs. This paper argues that the logical foundations of some refinements and the equilibria they focus on are problematic and, further, proposes an alternative refinement that avoids the difficulties. We also provide an existence theorem covering a broad class of signalling games often studied in economics. Journal of Economic Literature Classification Numbers: C72, C70, D82.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Rapid evolution under inertia

This paper demonstrates that inertia driven by switching costs leads to more rapid evolution in a class of games that includes m×m pure coordination games. Under the best-response dynamic and a fixed rate of mutation, the expected waiting time to reach long-run equilibrium is of lower order in the presence of switching costs, due to the creation of new absorbing states that allow Ellison's [Ellison, G., 2000. Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud. 67, 17–45] “step-by-step” evolution to occur.     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.     

Repeated Games with Bounded Entropy

We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer's strategies while the other player is unrestricted. We will show that if the bound η(n), a function of the number of repetitions n, satisfies the condition η(n)/n → γ (n → ∞), then the maxmin value Wⁿ(η(n)) converges to (cav U)(γ), the concavification of the maxmin value of the stage game in which the maximizer's actions are restricted to those with entropy at most γ. A similar result is obtained for the infinitely repeated games. Journal of Economic Literature Classification Numbers: C73, C72.     

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.     

Concatenate coordination and mutual coordination

We tell of the evolving meaning of the term coordination as used by economists. The paper is based on systematic electronic searches (on “coord,” etc.) of major works and leading journals. The term coordination first emerged in professional economics around 1880, to describe the directed productive concatenation of factors or activities within a firm. Also, transportation economists used the term to describe the concatenation of routes and trips of a transportation system. These usages represent what we term concatenate coordination. The next major development came in the 1930s from several LSE economists (Hayek, Plant, Hutt, and Coase), who extended that concept beyond the eye of any actual coordinator. That is, they wrote of the concatenate coordination of a system of polycentric or spontaneous activities. These various applications of concatenate coordination prevailed until the next major development, namely, Thomas Schelling and game models. Here coordination referred to a mutual meshing of actions. Game theorists developed crisp ideas of coordination games (like “battle of the sexes”), coordination equilibria, convention, and path dependence. This “coordination” was not a refashioning, but rather a distinct concept, one we distinguish as mutual coordination. As game models became more familiar to economists, it was mutual coordination that economists increasingly had in mind when they spoke of “coordination.” Economists switched, so to speak, to a new semantic equilibrium. Now, mutual coordination overshadows the older notion of concatenate coordination. The two senses of coordination are conceptually distinct and correspond neatly to the two dictionary definitions of the verb to coordinate. Both are crucial to economics. We suggest that distinguishing between the two senses can help to clarify “coordination” talk. Also, compared to talk of “efficiency” and “optimality,” concatenate coordination allows for a richer, more humanistic, and more openly aesthetic discussion of social affairs. The narrative is backed up by Excel worksheets that report on systematic content searches of the writings of economics using the worldwide web and, using JSTOR, of Quarterly Journal of Economics, Economic Journal, Journal of Political Economy, American Economic Review, and Economica.     

The Core of Large Differentiable TU Games

For suitable non-atomic TU games ν, the core can be determined by computing appropriate derivatives of ν, yielding one of two stark conclusions: either core(ν) is empty or it consists of a single measure that can be expressed explicitly in terms of derivatives of ν. In this sense, core theory for a class of games may be reduced to calculus. Journal of Economic Literature Classification Number: C71.     

Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283-314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games. Journal of Economic Literature Classification Numbers: C72, D72.