Perfect Equilibrium In Pure Strategies Research Articles

The Rise of the Superstars: Uncovering the Composition Effect of International Trade That Cements the Dominant Position of Big Businesses

International markets are extremely polarised, with a few big superstar businesses operating alongside numerous small competitors, and globalisation has been highlighted as a powerful force behind the superstars’ increasingly dominant presence. The empirical literature has established that superstars are more efficient compared to their smaller counterparts, and, unlike them, they exhibit strategic behaviour. Building on this evidence, we develop a model to examine how an initial productivity advantage allows a select few firms to expand, via innovation, to the extent that it becomes optimal to adopt strategic behaviour, and show how polarised markets emerge endogenously as the unique subgame perfect equilibrium in pure strategies. We then introduce international trade and show that, in polarised markets, trade liberalisation puts into motion a novel composition effect, reallocating market share from smaller to larger rivals and raising large firms’ profits. This effect suppresses the pro-competitive welfare gains from trade and cements the dominant position of big businesses, who come out as the big winners of globalisation. We find that, although trade increases welfare, by reducing average markup and markup heterogeneity, in the presence of a handful of large powerful firms, welfare gains are severely diminished, and subsidising smaller enterprises may turn out to be welfare-enhancing.

The absence of attrition in a war of attrition under complete information

We consider a two-player game of war of attrition under complete information. It is well-known that this class of games admits equilibria in pure, as well as mixed strategies, and much of the literature has focused on the latter. We show that if the players' payoffs whilst in “war” vary stochastically and their exit payoffs are heterogeneous, then the game admits Markov Perfect equilibria in pure strategies only. This is true irrespective of the degree of randomness and heterogeneity, thus highlighting the fragility of mixed-strategy equilibria to a natural perturbation of the canonical model. In contrast, when the players' flow payoffs are deterministic or their exit payoffs are homogeneous, the game admits equilibria in pure and mixed strategies.

Multi-lateral strategic bargaining without stationarity

This paper establishes existence of subgame perfect equilibrium in pure strategies for a general class of sequential multi-lateral bargaining games, without assuming a stationary setting. The only required hypothesis is that utility functions are continuous on the space of economic outcomes. In particular, no assumption on the space of feasible payoffs is needed. The result covers arbitrary and even time-varying bargaining protocols (acceptance rules), externalities, and other-regarding preferences. As a side result, we clarify the meaning of assumptions on “continuity at infinity.”

Equilibrium Selection in the War of Attrition Under Complete Information

We consider a two-player game of war of attrition under complete information. It is well- known that this class of games admits equilibria in pure, as well as mixed strategies, and much of the literature has focused on the latter. We show that if the players’ payoffs whilst in “war” vary stochastically and their exit payoffs are heterogeneous, then the game admits Markov Perfect equilibria in pure strategies only. This is true irrespective of the degree of randomness and heterogeneity, thus highlighting the fragility of mixed-strategy equilibria to a natural perturbation of the canonical model. In contrast, when the players’ flow payoffs are deterministic or their exit payoffs are homogeneous, the game admits equilibria in pure and mixed strategies.

Extremal Markov Perfect Equilibria of Stochastic Games with Complementarities

Under natural assumptions, we show that the set of Stationary Markov Perfect Equilibria in pure strategies is non-empty for stochastic repeated games with complementarities. We characterize the set of extremal SMPE as unique fixed points of well-chosen contractions. Those extremal equilibria can be approximated exponentially fast, and uniform convergence obtains for any initial guess chosen on a relevant functional set. This characterization also allows to derive a standard result in Monotone Comparative Statics. We finally show how to extend our approach to some other classes of repeated stochastic games.

Sequential lottery contests with multiple participants

We apply aggregative games techniques in a novel fashion in the analysis of sequential lottery contests with n players to show that, there exists a unique subgame perfect equilibrium in pure strategies.

Dynamic Contests With Bankruptcy: The Despair Effect

AbstractWe analyze a two-period contest in which agents may become bankrupt at the end of the first period. A bankrupt agent is excluded from the contest in the second period of the game. We investigate the existence of a subgame perfect equilibrium in pure strategies. We distinguish between a borrowing equilibrium in which at least one agent might be bankrupted and a non borrowing equilibrium in which no agent is bankrupted. We prove that the former occurs when the agent taking loans is relatively poor and the future does not matter very much. This action represents the Despair Effect, in which severely handicapped agents take actions that jeopardize their existence in the long run but are currently helpful. We find conditions under which borrowing and non borrowing equilibria overlap and do not overlap. We provide an example in which no equilibrium exists.

On equilibrium refinements in supermodular games

We show that supermodular games satisfying sequential better-reply security possess a pure strategy perfect equilibrium and a strategically stable set of pure strategy equilibria. We illustrate that in continuous supermodular games, perfect equilibria may contain weakly dominated actions. Moreover, in discontinuous supermodular games satisfying sequential better-reply security, perfect equilibria may involve play of actions in the interior of the set of weakly dominated actions. We show that supermodular games satisfying sequential better-reply security possess pure strategy perfect equilibria outside the interior of the set of weakly dominated action profiles.

Dynamic Contests with Bankruptcy: The Despair Effect

We analyze a two period contest in which agents may become bankrupt at the end of the first period. A bankrupt agent is excluded from the contest in the second period of the game. We investigate the existence of a subgame perfect equilibrium in pure strategies. We distinguish between a borrowing equilibrum where at least an agent might be bankrupt and a non borrowing equilibrium where no agent is bankrupted. We prove that the former occurs when the agent taking loans is relatively poor. This is the despair effect where severely handicapped agents take actions that are risky. We also show conditions under which both kind of equilibria overlap or not. We provide an example in which no equilibrium exists.

Electing a parliament

We present a model where a society elects candidates belonging to two parties to a national parliament. The electoral rule determines the seats distribution between the two parties. The policy outcome is a function of the number of seats the two parties win in the election. We analyze two electoral rules, multidistrict majority and single district proportional. We prove that under both systems there is a unique pure strategy perfect equilibrium outcome. We compare the outcomes under the two systems.

Existence of a Pure Strategy Equilibrium in Markov Games with Strategic Complementarities for Finite Actions and States

In this paper, we provide the sufficient conditions for a Markov perfect equilibrium in pure strategies to exist for a class of stochastic games with finite horizon, in which any stage game has strategic complementarities. In contrast to previous studies (e.g., Curtat (1996) and Amir (2002), we assume herein that the sets of actions and the set of states is finite and do not assume dominant diagonal conditions for payoffs and the transition probability, which yield the uniqueness of equilibria. The greatest equilibrium is monotonically increasing in the state. The main result is applied to a model of a Bertrand competition with investment.

On the existence of pure-strategy perfect equilibrium in discontinuous games

We provide sufficient conditions for a (possibly) discontinuous normal-form game to possess a pure-strategy trembling-hand perfect equilibrium. We first show that compactness, continuity, and quasiconcavity of a game are too weak to warrant the existence of a pure-strategy perfect equilibrium. We then identify two classes of games for which the existence of a pure-strategy perfect equilibrium can be established: (1) the class of compact, metric, concave games satisfying upper semicontinuity of the sum of payoffs and a strengthening of payoff security; and (2) the class of compact, metric games satisfying upper semicontinuity of the sum of payoffs, strengthenings of payoff security and quasiconcavity, and a notion of local concavity and boundedness of payoff differences on certain subdomains of a player's payoff function. Various economic games illustrate our results.

Go for broke or play it safe? Dynamic competition with choice of variance

We consider a differential game in which the joint choices of the two players influence the variance, but not the mean, of the one‐dimensional state variable. We show that a pure strategy perfect equilibrium in stationary Markov strategies (ME) exists and has the property that patient players choose to play it safe when sufficiently ahead and to take risks when sufficiently behind. We also provide a simple condition that implies both players choose risky strategies when neither one is too far ahead, a situation that ensures a dominant player emerges “quickly.”

HOTELLING'S BEACH WITH LINEAR AND QUADRATIC TRANSPORTATION COSTS: EXISTENCE OF PURE STRATEGY EQUILIBRIA*

In Hotelling type models consumers have the same transportation cost function. We deviate from this assumption and introduce two consumer types. Some consumers have linear transportation costs, while the others have quadratic transportation costs. If at most half the consumers have linear transportation costs, a subgame perfect equilibrium in pure strategies exists for all symmetric locations. Furthermore, no general principle of differentiation holds. With two consumer types, the equilibrium pattern ranges from maximum to intermediate differentiation. The degree of product differentiation depends on the fraction of consumer types.

A selling mechanism

For the assignment game, we propose the following selling mechanism: sellers, simultaneously, fix their prices first; then buyers, sequentially, decide which object to buy, if any. The first phase of the game determines the potential prices, while the second phase determines the actual matching. We prove that the set of subgame perfect equilibria in pure strategies in the strong sense of the mechanism coincides with the set of sellers' optimal stable outcomes when buyers use maximal strategies.

Voting for voters: the unanimity case

We present a simplified model of the evolution of a society which is regulated by a formal unanimity voting procedure. We examine several protocols, which depend on whether admission or expulsion are permissible, and on the order with which they are implemented. Conditions which ensure the existence of pure-strategy perfect equilibrium profiles for some voting protocols, and counter examples for the existence of such profiles in other protocols are presented. Finally, we prove that, if the agents insist on perfect equilibrium strategy profiles in a one-stage play, the original founders would prefer a protocol in which expulsion precedes admission to protocols in which either admission precedes expulsion, or the two are treated simultaneously. The paper concludes with an overview and a discussion on the results and suggestions for further research.

Spatial competition with profit‐maximising and labour‐managed firms

Abstract The nature of the equilibria arising under spatial differentiation is investigated here in a duopoly model, where at least one firm maximises value added per worker. The study shows that if firms’objectives differ, there exists a subgame perfect equilibrium in pure strategies, which is possibly characterised by asymmetric locations. If both firms are labour‐managed, there exists a (symmetric) subgame perfect equilibrium in pure strategies with firms located at the first and third quartiles, if and only if the setup cost is low enough. Otherwise, undercutting is profitable.

Spatial competition with profit-maximising and labour-managed firms

The nature of the equilibria arising under spatial differentiation is investigated here in a duopoly model, where at least one firm maximises value added per worker. The study shows that if firms' objectives differ, there exists a subgame perfect equilibrium in pure strategies, which is possibly characterised by asymmetric locations. If both firms are labour-managed, there exists a (symmetric) subgame perfect equilibrium in pure strategies with firms located at the first and third quartiles, if and only if the setup cost is low enough. Otherwise, undercutting is profitable.

Implementation of Bargaining Sets via Simple Mechanisms

We propose two simple mechanisms that implement two bargaining sets in super-additive environments. The first bargaining set is a close variation of the one proposed by L. Zhou (1994, Games Econom. Behav.6, 512–526), and the second is the Pareto optimum payoffs of the A. Mas-Colell (1989, J. Math. Econom.18, 129–139) bargaining set. We adopt a simple framework in which the cooperative outcomes are realized as non-cooperative subgame perfect equilibria in pure strategies of a two-stage game played by an auxiliary set of individuals competing over the cooperative agents. Journal of Economic Literature Classification Numbers: C71, C72.

Unicity of the equilibrium in the unconstrained Hotelling model

A spatial duopoly is described, where firms are free to locate along the real axis, while consumers are distributed along a linear city of finite length. By adding a preplay stage where firms are required to set the timing of moves, one can analyze issues such as the endogenous choice of roles and the nature of differentiation at equilibrium. The distribution of roles is discussed both at the location stage and at the price stage. The game exhibits a unique subgame perfect equilibrium in pure strategies (up to a permutation), where firms play simultaneously in locations and sequentially in prices. Moreover, the results highlight the relevance of endogenous differentiation in altering some of the conclusions reached by previous literature. Finally, it turns out that the nature of differentiation may be heavily affected by the sequence of decisions.