Convergence to Walrasian equilibrium with minimal information

We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.

     

Internalized social norms in conflicts: an evolutionary approach

An individual with an internalized norm prefers to comply with the norm as long as the opponents also comply. This paper studies the evolution of such norms in a conflict. Individuals play with the same opponent for a number of rounds, during which they adapt their behavior according to a myopic best response dynamic. The payoffs earned during this equilibration period enter into the cultural evolutionary process that determines which norms that survive. It is found that if agents play against the same opponent for sufficiently many times, then only norms that reduce conflict relative to the Nash equilibrium survive. If instead agents meet the same opponent only once, then only agents that play the Nash equilibrium survive. By way of example it is also shown that in the intermediate case the outcome of evolution can be either that conflict reducing norms or Nash behavior dominates. These results indicate that the common practice of disregarding equilibration in models of preference evolution is not without a cost.     

An evolutionary model with best response and imitative rules

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms.     

Equilibria in a class of aggregative location games

Consider a multimarket oligopoly, where firms have a single license that allows them to supply exactly one market out of a given set of markets. How does the restriction to supply only one market influence the existence of equilibria in the game? To answer this question, we study a general class of aggregative location games where a strategy of a player is to choose simultaneously both a location out of a finite set and a non-negative quantity out of a compact interval. The utility of each player is assumed to depend solely on the chosen location, the chosen quantity, and the aggregated quantity of all other players on the chosen location. We show that each game in this class possesses a pure Nash equilibrium whenever the players’ utility functions satisfy the assumptions negative externality, decreasing marginal utility, continuity, and Location–Symmetry. We also provide examples exhibiting that, if one of the assumptions is violated, a pure Nash equilibrium may fail to exist.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Bounds for best response functions in binary games

This paper studies the identification of best response functions in binary games without making strong parametric assumptions about the payoffs. The best response function gives the utility maximizing response to a decision of the other players. This is analogous to the response function in the treatment-response literature, taking the decision of the other players as the treatment, except that the best response function has additional structure implied by the associated utility maximization problem. Further, the relationship between the data and the best response function is not the same as the relationship between the data and the response function in the treatment-response literature. We focus especially on the case of a complete information entry game with two firms. We also discuss the case of an entry game with many firms, non-entry games, and incomplete information. Our analysis of the entry game is based on the observation of realized entry decisions, which we then link to the best response functions under various assumptions including those concerning the level of rationality of the firms, including the assumption of Nash equilibrium play, the symmetry of the payoffs between firms, and whether mixed strategies are admitted.     

Second-Price Auctions with Asymmetric Payoffs: An Experimental Investigation

A series of two-player, second-price common-value auctions are reported. In symmetric auctions, bidders suffer from a winner's curse. In asymmetric auctions in which one bidder has a private value advantage, the effect on bids and prices is proportional rather than explosive (the prediction of Nash equilibrium bidding theory). Although advantaged bidders are close to making best responses to disadvantaged bidders, the latter bid much more aggressively than in equilibrium, thereby earning negative average profits. Experienced bidders consistently bid closer to the Nash equilibrium than inexperienced bidders, although these adjustments towards equilibrium are small and at times uneven.     

Characterizing neutrality in the voluntary contribution mechanism

When public goods are provided by voluntary contributions, redistribution of initial holdings among an unchanged set of contributors will not alter the Nash equilibrium allocation.Constraint invariance, where a participant faces the same class of constraints even after her endowment is changed, impliesglobal neutrality under a generalized Nash equilibrium concept. Geometrically,path invariance, where the best response path does not move even after the endowment change, is equivalent to global neutrality assuming a sufficiently rich class of economies. We thank Shinji Ohseto, Hiroaki Osana and Stephen Turnbull for their comments. We are also grateful to an anonymous referee and an associate editor for their comments and suggestion. Research was partially supported by the Nomura Foundation for the Social Sciences and the Grant in Aid for General Research 03803008 of the Ministry of Education, Science and Culture of Japan.     

Empirical Analysis of a Dynamic Duopoly Model of Competition

Empirically validating and testing the specification of game theoretic models has received limited attention in the marketing literature. The authors provide an econometric framework for estimating the parameters of response functions when the observed data in the market place are the Nash equilibrium outcomes of an underlying dynamic duopoly game specification. Specifically, the estimation procedure accounts for the joint endogeneity of market shares and marketing efforts of market rivals using a system of simultaneous equations that included the market response function and the Nash equilibrium conditions. A formal statistical test is used to detect model misspecification. The empirical analysis is carried out using data from four product markets: pharmaceutical, soft drink, beer, and detergent. Comparisons are provided with conventional estimation of the response function parameters in which the equilibrium conditions are ignored in the estimation. Managerial implications of the empirical results are discussed.     

Observable implications of Nash and subgame-perfect behavior in extensive games

We provide necessary and sufficient conditions for observed outcomes in extensive game forms, in which preferences are unobserved, to be rationalized first, weakly, as a Nash equilibrium and then as the unique subgame-perfect equilibrium. Thus, one could use these conditions to find that play is (a) consistent with subgame-perfect equilibrium, or (b) not consistent with subgame-perfect behavior but is consistent with Nash equilibrium, or (c) consistent with neither.     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.     

Stable matchings and the small core in Nash equilibrium in the college admissions problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. Received: 30 December 1998 / Accepted: 12 October 2001 This paper is a revision of the paper “Manipulation and Stability in a College Admissions Problem” circulated since 1994. I thank Rich McLean, Abraham Neyman, Mark Satterthwaite, Sang-Chul Suh, and Tetsuji Yamada for helpful discussions. I thank the associate editor and the two anonymous referees for their helpful comments that have greatly improved the paper. I am grateful to the Kellogg G.S.M. at the Northwestern University for the hospitality for my visit. Any errors are mine.     

从博弈论的角度分析我国税务筹划的现状

税务筹划作为一种节税行为应当符合政府的立法意图。但在现实条件下,由于政府征税与纳税人追求最大利益是相违背的,因此形成了一种博弈关系。纳税人和税务机关之间的博弈存在着混合战略纳什均衡,并且我国目前税务筹划在实践中应用滞后的现状与纳税人和税务机关之间的博弈关系存在着紧密联系。     

Is having a unique equilibrium robust?

We investigate whether having a unique equilibrium (or a given number of equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium and correlated equilibrium. We show that the set of n  -player finite games with a unique correlated equilibrium is open, while this is not true of Nash equilibrium for n>2$n>2$. The crucial lemma is that a unique correlated equilibrium is a quasi-strict Nash equilibrium. Related results are studied. For instance, we show that generic two-person zero-sum games have a unique correlated equilibrium and that, while the set of symmetric bimatrix games with a unique symmetric Nash equilibrium is not open, the set of symmetric bimatrix games with a unique and quasi-strict symmetric Nash equilibrium is.     

Inefficiency of credible strategies in oligopolistic resource markets with uncertainty

The implications of different information patterns for firms in oligopolistic resource markets are considered. The traditional open-loop Nash equilibrium with static information sets is one of many possible Nash equilibria and is not suitable for stochastic environments. When shocks to resource growth are serially uncorrelated, there are no gains from conditioning the harvest on past stock levels and the feedback or credible Nash equilibrium is the appropriate Nash equilibrium concept. This credible equilibrium assumes that firms have knowledge of current stocks of reserves, which typically leads to more rapid extraction of the resource and possibly extinction. Since the open-loop Nash equilibrium is efficient when demand is iso-elastic and extraction costs are zero, it is clear that an increase in information can be detrimental to firms in the industry.     

The women of Cairo: Equilibria in large anonymous games

In this article, I characterize Nash equilibria of large anonymous games by providing the following neccessary and sufficient condition for an equilibrium distribution: for no subset K of actions more players play actions in K than have a best response in K to the given distribution. While neccessity is trivial the proof for sufficiency relies on a theorem by [Math. Proc. Camb. Philos. Soc. 78 (1974) 323] which is an extension of Hall’s theorem or the marriage lemma well known from graph theory. The veiling problem for the women of Cairo serves as an illustrating heuristic example explaining the nature of the result.     

Convergence in games with continua of equilibria

In game theory, the question of convergence of dynamical systems to the set of Nash equilibria has often been tackled. When the game admits a continuum of Nash equilibria, however, a natural and challenging question is whether convergence to the set of Nash equilibria implies convergence to a Nash equilibrium. In this paper we introduce a technique developed in Bhat and Bernstein (2003) as a useful way to answer this question. We illustrate it with the best-response dynamics in the local public good game played on a network, where continua of Nash equilibria often appear.     

Full Nash implementation of neutral social functions

This paper characterizes neutral social functions that are fully implementable. A neutral social function must be monotonic and simple to be fully implementable under either the Nash equilibrium concept or the strong Nash equilibrium concept. If a neutral social function is monotonic and simple, then the social function is fully implementable by a set of Nash equilibria. For finite alternative sets a neutral social function will be fully implementable by a set of strong Nash equilibria if and only if it is monotonic, simple and dictatorial.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

OLIGOPOLY GAMES AND THE COURNOT–BERTRAND MODEL: A SURVEY

Traditional oligopoly models hold that firms compete in the same strategic variable, output (Cournot) or price (Bertrand). Alternatively, a hybrid model allows some firms to compete in output and other firms to compete in price, also known as the Cournot–Bertrand model. When the choice of strategic variable is endogenous, the established dominant strategy is output competition. A growing body of work demonstrates, however, that the Cournot–Bertrand outcome can be a subgame‐perfect Nash equilibrium in the presence of market asymmetries. Observations of real‐world markets consistent with Cournot–Bertrand behavior bolster justification for the model and have stimulated an impressive and evolving literature on advances and applications. We lay out the roots of the Cournot–Bertrand model and explore a number of model developments. We categorize 12 primary models in the literature based on alternative assumptions. In particular, some authors consider when the timing of play as well as the choice of strategic variable are endogenous. Altogether, this research identifies when Cournot–Bertrand behavior can emerge in a dynamic setting and under alternative market conditions. We also review the Cournot–Bertrand model applications in the fields of international economics, industrial organization, labor, and public economics. We expect the literature to continue to expand in the future.