Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.     

Unequal Contributions from Symmetric Agents in a Local Interaction Model

The main findings of the theory on the private provision of public goods under the assumptions of symmetric agents and normality are that (1) there exists a unique Nash equilibrium in which everybody contributes the same; and (2) this pattern is stable. We show that these findings no longer hold in a context characterized by local interaction. In this context, it is always possible to find preferences satisfying the assumption of normality such that the symmetric Nash equilibrium is unstable, and there exist asymmetric Nash equilibria which are locally stable.     

Rationing rules and Bertrand-Edgeworth equilibria in large markets

In a market with concave downward sloping demand and symmetric firms which compete in prices with constant marginal costs and capacity limits it is shown that the supports of the symmetric (mixed strategy) Nash equilibria converge to the unique competitive price provided that unsatisfied demand is allocated according to the surplus-maximizing rationing rule.     

Public Input Competition under Stackelberg Equilibrium: A Note

This paper examines the Stackelberg equilibrium for public input competition and compares it with the noncooperative Nash equilibrium. Given two asymmetric regions, we show that under the Nash equilibrium the more productive region tends to spend more on public input, which results in this region attracting more capital than the less productive region. The comparison of the two equilibria reveals that the leader region obtains a first‐mover advantage under the Stackelberg setting. This suggests that if regions interact with each other sequentially as in the Stackelberg equilibrium, then the regional disparity that is due to the heterogeneity of productivity is likely to be mitigated or enlarged, depending on which region performs the leadership role in the competition process.     

QUANTITY VERSUS PRICE IN A HOMOGENEOUS PRODUCT DUOPOLY*

In a simple homogeneous product setting, the paper looks at the debate on whether firms should choose quantity or price as their strategic variable. It examines a two-stage game between firms with symmetric costs in which the firms choose the strategic mode of operation in the first period and then, in the second period, price or output are chosen simultaneously according to the mode chosen in the first stage. In this game it is possible to have two Nash equilibria where either both play in quantities or both play in prices. One firm choosing price and the other quantity can never be a Nash equilibrium in the two-stage game. Both choosing quantity is always a Nash equilibrium. Both choosing prices may be a Nash equilibrium only in some situations: the structure of the cost functions decides this issue.     

Public input competition

Public investment in infrastructure may help create agglomeration economies, by attracting mobile factors such as skilled labour. Competition among regions in public investment can then be destructive. This paper analyzes the Nash equilibria to a simple model of public input competition. Even though the regions are assumed identical, the equilibrium may not be symmetric. The problem with non-cooperative behaviour is not (only) that regions invest too much, but that too many regions may choose to invest. Depending on the parameter values, the Nash equilibrium may be efficient, may be inefficient or may not exist (at least not in pure strategies). Better mobility among regions leads to more aggressive competition. The analysis suggests that rents from public investment may be dissipated by governments' competition to attract mobile factors.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Diligence and laziness in the Chinese countryside revisited

A game-theoretic model is used to asses the impact of changing disutility of labour and distribution schemes on labour supply in Chinese agricultural product teams. It is argued that a previous analysis of the problem by chinn (1980) overlooks some inherent complications. It is shown that application of the symmetric, pure strategy Nash equilibrium concept yields no equilibria over a significant range of values for relevant variables. Asymmetric, pure strategy equilibria are shown to exist over this range. Allowing production teams continuous choice of labour inputs restores symmetric equilibrium.     

Cyber Technology and the Arms Race

Cyber technology represents digital military capability with the purpose of causing damage to the military strength of a potential enemy. War using conventional weapons may be preceded by a strike using cyber technology. This paper introduces such technology into the theory of conflicts. The cost of war relative to the payoff from victory turns out to be crucial for the results on armament decisions. In the war game, two types of Nash equilibria may arise. One is subject to warfare while the other is not (‘equilibrium of terror’), depending on the perceived cost of war. In a symmetric war game, cyber capabilities are neutral with respect to the investments in conventional weapons, but they make wars more likely. Asymmetric access to cyber technology limits the international arms race with conventional weapons. A low success probability in the cyber programme encourages exercising the cyberattack option as the enemy may not have access to cyber capability. Uncertainty of the success of a cyber programme makes countries cautious when allocating resources not only to these programmes but also in conventional armament.     

Price vs quantity in a duopoly supergame with Nash punishments

We examine the endogenous choice between price and quantity behaviour in a duopoly supergame with product differentiation. We find that (i) if cartel profits are evenly split between firms, then only symmetric equilibria s obtained; (i) if instead the additional profits available through collusion are split according to the Nash bargaining solution, there are parameter regions where all subgame perfect equilibria are asymmetric, with firms colluding in price–quantity supergames.     

Weakest-link public goods: Giving in-kind or transferring money

This paper extends the analysis of weakest-link public goods, whose level equals the smallest of the agents’ provision levels, by permitting an agent either to increase one's own provision or else augment both one's own provision and that of the other agent(s). Nash equilibria may be either symmetric with agents matching one another's provision or else asymmetric with an in-kind transfer by one agent to another. An equilibrium with cash transfers, but no in-kind transfers, may be Pareto superior to one with only in-kind transfers. If agents differ in their efficiency, then in-kind transfers by the low-cost agent may dominate a cash transfer. The possibility of Pareto-improving transfers is enhanced as the number of agents increases.     

Evolution with Delay

This paper analyses evolutionary models with delays in players’ observations. I apply best response dynamics to symmetric 2 × 2 games, in particular the coordination game and the game with no symmetric pure strategy equilibrium. When the dynamics have no delay, their solutions converge to symmetric Nash equilibria monotonically. However when I introduce delays to the dynamics, although these dynamics represented by differential difference equations are seemingly simple, it shows that they have periodic solutions. JEL Classification Number: C79.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.     

EQUILIBRIA IN NON-COOPERATIVE GAMES II: DEVIATIONS BASED REFINEMENTS OF NASH EQUILIBRIUM

In any Nash equilibrium no player will unilaterally deviate. However, many games have multiple Nash equilibria. In this paper, we survey some refinements of Nash equilibria based on the hypothesis that any player may consider a deliberate deviation from a Nash equilibrium vector while expecting other players to respond optimally to this deviation. The concepts studied here differ in the expectations players have about other players' responses to a deviation. This sort of deviations philosophy is predicated on the thought process of players. Therefore, the validity of a particular equilibrium concept to an economic model may depend upon the relevance of the thought process implied by the concept.     

The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games

Envelope theorems are established for a ubiquitous class of finite horizon differential games. The theorems cover open-loop and feedback information patterns in which the corresponding Nash equilibria are locally differentiable with respect to the parameters of the game. Their relationship with extant envelope results is discussed and an application of them to a generalized capital accumulation game is provided. An important implication of the theorems is that, in general, the archetypal economic interpretation of the costate vector, namely, as the shadow value of the state vector along the Nash equilibrium, is valid for feedback Nash equilibria, but not for open-loop Nash equilibria.     

Learning in Elections and Voter Turnout

Voter turnout in game theoretic models of voting has typically been difficult to predict because of the problem of multiple Nash equilibria ( Palfrey and Rosenthal 1983, 1985 ). Many of these equilibria require an extreme precision of beliefs among voters that is unlikely to be reached in real elections. At the same time, mechanisms like pre‐election polls exist to shape the beliefs of voters about expected turnout. We combine these two features in a model of voter learning in elections and characterize the asymptotically stable equilibria of both complete and incomplete information games in a simple symmetric setting with two candidates. We also show how the model can be used to qualitatively explain several phenomena observed in reality: increases in costs of voting affect turnout adversely but there may be persistence of turnout levels between elections even though costs and other parameters change. Increase in uncertainty increases turnout while increases in the size of the electorate decrease it, in line with intuition.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Bertrand Games and Sharing Rules

We consider asymmetric Bertrand games with arbitrary payoffs at ties or sharing rules, and identify sufficient conditions for the zero-profit outcome and the existence of Nash equilibria. Subject to some technical conditions on non-tied payoffs the following hold. If the sharing rule is strictly tie-decreasing all players but one receive zero equilibrium payoffs, while everybody does so if non-tied payoffs are symmetric. Mixed (pure) strategy Nash equilibria exist if the sharing rule is (norm) tie-decreasing and coalition-monotone. I would like to thank Fernando Branco, the audience at Pompeu Fabra (Barcelona), ISEG (Lisbon), University of Mannheim, ESEM 2003 (Venice), EARIE 2005 (Porto), two anonymous referees, and the editor Dan Kovenock for very useful comments. This research received financial support under project POCTI/ECO/37925/2001 of FCT and FEDER.     

Dynamic Interactions in Trade Policy in a Differential Game Model of Tariff Protection

This paper develops a two‐country dynamic game model of tariff protection to reconsider optimal trade policies and their implications for welfare. The authors show that an import subsidy is optimal in the feedback Nash equilibria, which results in a curious possibility that the domestic market is monopolized by the foreign firrm. However, welfare comparisons among Nash equilibria, free trade, and autarky reveal that feedback Nash equilibria involve higher welfare than both autarky and free trade, i.e. dynamic noncooperative choices of policy serve as tacit policy coordination and ensure larger trade gains relative to free trade.     

Essential equilibria of discontinuous games

We introduce spaces of discontinuous games in which games having essential Nash equilibria are the generic case. In order to prove the existence of essential Nash equilibria in such spaces, we provide new results on the Ky Fan minimax inequality. In the setting of potential games, we show that games with essential Nash equilibria are the generic case when their potentials satisfy a condition called weak upper pseudocontinuity that is weaker than upper semicontinuity.