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.     

Pure strategy equilibria in games with countable actions

In this paper, we extend the Radner–Rosenthal theorem with finite action spaces on the existence of a pure strategy equilibrium for a finite game to the case that the action space is countable and complete. We also prove the existence of a pure strategy equilibrium for a game with a continuum of players of finite types and with a countable and complete action space. To work with the countably infinite action spaces, we prove some regularity properties on the set of distributions induced by the measurable selections of a correspondence with a countable range by using the Bollobás–Varopoulos extension of the marriage lemma.     

Subgame-perfect equilibria in stochastic timing games

We develop a notion of subgames and the related notion of subgame-perfect equilibrium – possibly in mixed strategies – for stochastic timing games. To capture all situations that can arise in continuous-time models, it is necessary to consider stopping times as the starting dates of subgames. We generalize Fudenberg and Tirole’s (Rev. Econom. Stud. 52, 383–401, 1985) mixed-strategy extensions to make them applicable to stochastic timing games and thereby provide a sound basis for subgame-perfect equilibria of preemption games. Sufficient conditions for equilibrium existence are presented, and examples illustrate their application as well as the fact that intuitive arguments can break down in the presence of stochastic processes with jumps.     

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Pareto-undominated and socially-maximal equilibria in non-atomic games

This paper makes the observation that a finite Bayesian game with diffused and disparate private information can be conceived of as a large game with a non-atomic continuum of players. By using this observation as its methodological point of departure, it shows that (i) a Bayes–Nash equilibrium (BNE) exists in a finite Bayesian game with private information if and only if a Nash equilibrium exists in the induced large game, and (ii) both Pareto-undominated and socially-maximal BNE exist in finite Bayesian games with private information. In particular, it shows these results to be a direct consequence of results for a version of a large game re-modeled for situations where different players may have different action sets.     

Pure strategy equilibria in games with private and public information

We introduce a new game form which allows the players’ strategies to depend on their strategy-relevant private information as well as on some publicly announced information. The players’ payoffs depend on their own payoff-relevant private information and some payoff-relevant common information. Under the assumption that the players’ strategy-relevant private information is diffuse and their private information is conditionally independent given the public and payoff-relevant common information, we prove the existence of a pure strategy equilibrium for such a game by developing a distribution theory of correspondences via vector measures.     

Real options and contingent convertibles with regime switching

We consider a firm with no assets in place but an option to invest in a project. The investment is irreversible but delayable in a regime-switching economy. The firm issues equity, straight bonds (SBs) and contingent convertibles (CoCos). We provide the closed-form prices for the firm׳s securities and the pricing and timing of the option. Our numerical analyses discover that issuing CoCos instead of SBs induces much less agency cost of debt. The agency cost is higher in a boom economy than in recession but the difference is small. There is a unique CoCos׳ conversion ratio such that the agency cost arrives at the minimum value zero. The inefficiencies arising from asset substitution and debt overhang are much more significant in recession than in boom. Only if the conversion ratio is not too small, the two inefficiencies disappear during boom periods. While the effects of the conversion rate on optimal capital structure and firm value and those of supervision and jump intensity on optimal CoCos׳ coupon are ambiguous and weak, the stricter the supervision or the longer the economy remains in recession, the less the option value and the optimal SBs׳ coupon.     

Markov Perfect equilibria in repeated asynchronous choice games

This paper examines the issue of multiplicity of Markov Perfect equilibria in alternating move repeated games. Such games are canonical models of environments with repeated, asynchronous choices due to inertia or replacement. Our main result is that the number of Markov Perfect equilibria is generically finite with respect to stage game payoffs. This holds despite the fact that the stochastic game representation of the alternating move repeated game is “non-generic” in the larger space of state dependent payoffs. We further obtain that the set of completely mixed Markov Perfect equilibria is generically empty with respect to stage game payoffs.     

Transitions between equilibria in bilingual games under logit choice

We study the effect of introducing a bilingual option on the long run equilibrium outcome in a class of two-strategy coordination games with distinct payoff and risk dominant equilibria under the logit choice rule. Existing results show that in the class of two-strategy games under consideration, the inefficient risk dominant equilibrium is selected in the long run under noisy best response models. We show that if the cost of the bilingual option is sufficiently low then the efficient payoff dominant equilibrium will be selected in the long run under the logit choice rule.     

Perfect and limit admissible perfect equilibria in discontinuous games

We compare the properties of several notions of trembling-hand perfection within classes of compact, metric, and possibly discontinuous games, and show that in the presence of payoff discontinuities, standard notions of trembling-hand perfection fail a weakening of admissibility termed limit admissibility. We also provide conditions ensuring the existence of a limit admissible perfect equilibrium.     

Probabilistic forecasting of industrial electricity load with regime switching behavior

This paper suggests a novel inhomogeneous Markov switching approach for the probabilistic forecasting of industrial companies’ electricity loads, for which the load switches at random times between production and standby regimes. The model that we propose describes the transitions between the regimes using a hidden Markov chain with time-varying transition probabilities that depend on calendar variables. We model the demand during the production regime using an autoregressive moving-average (ARMA) process with seasonal patterns, whereas we use a much simpler model for the standby regime in order to reduce the complexity. The maximum likelihood estimation of the parameters is implemented using a differential evolution algorithm. Using the continuous ranked probability score (CRPS) to evaluate the goodness-of-fit of our model for probabilistic forecasting, it is shown that this model often outperforms classical additive time series models, as well as homogeneous Markov switching models. We also propose a simple procedure for classifying load profiles into those with and without regime-switching behaviors.     

On strategic complementarities in discontinuous games with totally ordered strategies

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.     

Communication complexity and stability of equilibria in economies and games

The theory of decentralized allocation mechanisms introduced by Leonid Hurwicz enabled economists to analyze the implications of requiring an allocation mechanism to perform well over a range of possible economic environments. One of the earliest major results in this theory was the formalization and proof of Hayek’s famous assertion that the competitive market mechanism minimizes the communication needed to achieve Pareto efficient allocations. The informational efficiency theorem spawned a large literature on communication complexity in allocation mechanisms. This paper provides an exposition of some results on the communication required to make equilibria locally stable, and the possibility of eliciting the required communication in allocation mechanisms and games.     

Collusive equilibria with switching costs: The effect of consumer concentration

I study the ability of two competing firms to set collusive prices in markets where consumers have switching costs. In consumer markets (with a small number of consumers), I find an antifolk result in which collusion, in the presence of switching costs, does not arise with patient firms. Patient firms compete aggressively and consumers expect a large utility. A collusive equilibrium is unstable because a deviating firm incorporates the future consumer utility in its deviating price. Also, consumers have a strategic impact so, with the prospects of large utility, they decide to switch to destabilize the firms' collusive agreement. These results do not eventuate in markets with a large number of consumers. In mass markets (a continuum of consumers), a single consumer lacks a strategic impact to destabilize a collusive agreement and a deviating firm cannot appropriate the consumer utility when deviating from collusion. Collusion, then, becomes straightforward to achieve. We show that for any number of consumers, switching costs make collusion easy to sustain.     

Shortfall risk minimization in a discrete regime switching model

Abstract There have been profound ideas on how to measure risk which have influenced the financial market. Shortfall risk minimization is one of the methods which has attracted considerable attention. This problem has been studied for the binomial model in Runggaldier and Zaccaria (2000) and Runggaldier, Trivellato and Vargiolu (2002) and for the trinomial model in Scagnelatto and Vargiolu (2002). In this paper, we investigate shortfall risk minimization in a discrete regime switching model. In the model, we have two possible regimes, which are both binomial. To fix ideas, we can think of the second regime as being the consequence of the presence of inside information, but this can also be due to other factors. Explicit solutions for one-period models are given.     

The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games

The comparative dynamics of locally differentiable feedback Nash equilibria are derived for the ubiquitous class of autonomous and exponentially discounted infinite horizon differential games. The resulting refutable implications are intrinsic to the said class of differential games, and thus form their basic, empirically testable, properties. Their relationship with extant results in the optimal control theory and the static game theory is discussed. Separability conditions are identified on the instantaneous payoff and transition functions under which the intrinsic comparative dynamics collapse, in form, to those in optimal control problems. Applications of the results to capital accumulation and sticky-price games are provided.     

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.     

A note on the symmetry of all Nash equilibria in games with increasing best replies

It is well known that a symmetric game has only symmetric (pure strategy) Nash equilibria if its best-reply correspondences admit only increasing selections and its strategy sets are totally ordered. Several nonexamples of the literature show that this result is generally false when the totality condition of the relation that orders the strategy sets is simply dropped. Making use of the structure of interaction functions, this note provides sufficient conditions for the symmetry of all (pure strategy) Nash equilibria in symmetric games where best-reply correspondences admit only increasing selections, but strategy sets are not necessarily totally ordered.