Mixed-strategy Equilibrium Research Articles

To Commit or Not to Commit: Revisiting Quantity vs. Price Competition in a Differentiated Industry

Previous studies have shown that quantity commitment by all firms in an industry mitigates price competition. This paper shows that when firms have a choice, asymmetric outcomes can arise, with some firms choosing to commit to a quantity and other firms choosing not to. To study the commitment decision, we analyze a multistage game within a duopoly of differentiated firms à la Hotelling. In the first stage of the game, firms choose whether or not to commit to a quantity. In the second stage, each firm that chose to commit sets a quantity, which represents an upper bound on how much it can sell to consumers. In the third stage, both firms set prices strategically, regardless of whether or not they committed. In the final stage, demand is allocated as consumers maximize their utilities. Firm(s) that chose not to commit to a quantity in the first stage can fulfill any quantity that consumers demand at the equilibrium prices. We find that if product differentiation is sufficiently low, both firms choose to commit to a quantity in equilibrium. This symmetric equilibrium allows both firms to avoid the intense price competition associated with low product differentiation. If the level of differentiation is sufficiently high, the equilibrium in quantity commitment is asymmetric such that one firm chooses to commit and its competitor chooses not to commit. Under this equilibrium, a pricing equilibrium in pure strategies does not exist, only a pricing equilibrium in mixed strategies does, and the equilibrium prices can result in a committed firm not clearing the entire quantity it chose earlier. This paper was accepted by J. Miguel Villas-Boas, marketing.

Static versus Contingent Pricing Under Competition

Should capacitated firms set prices responsively to uncertain market conditions in a competitive environment? We study a duopoly selling differentiated substitutable products with fixed capacities under demand uncertainty, where firms can either commit to a fixed price ex ante, or elect to price contingently ex post, e.g., to charge high prices in booming markets, and low prices in slack markets. Interestingly, we analytically show that even for completely symmetric model primitives, asymmetric equilibria of strategic pricing decisions may arise, in which one firm commits statically and the other firm prices contingently; in this case, there also exists a unique mixed strategy equilibrium. Such equilibrium behavior tends to emerge, when capacity is ampler, and products are less differentiated or demand uncertainty is lower. With asymmetric fixed capacities, if demand uncertainty is low, a unique asymmetric equilibrium emerges, in which the firm with more capacity chooses committed pricing and the firm with less capacity chooses contingent pricing. We identify two countervailing profit effects of contingent pricing under competition: gains from responsively charging high price under high demand, and losses from intensified price competition under low demand. It is the latter detrimental effect that may prevent both firms from choosing a contingent pricing strategy in equilibrium. We caution that responsive price changes under aggressive competition of less differentiated products can result in profit-killing discounting. We show that our results remain valid when capacity decisions are endogenized.

External Economies and International Trade Redux: Comment*

Abstract Recently, Gene Grossman and Esteban Rossi-Hansberg (GRH; “External Economies and International Trade: Redux,” Quarterly Journal of Economics 125 [2010], 829–858) proposed a novel way to think about the implications of international trade in the presence of national external economies at the industry level. Instead of perfect competition and two industries, GRH assume Bertrand competition and a continuum of industries. GRH conclude that the equilibrium is unique if transport costs are low, that there is no trade for high transport costs, and that there is no equilibrium in pure strategies when transport costs are intermediate. In this note we reexamine the equilibrium analysis under different transport costs for a single industry (partial equilibrium) version of GRH’s model. We confirm many of GRH’s results, but also find that there are circumstances under which there are multiple equilibria, including equilibria in which trade patterns run counter to “natural” comparative advantage, and also find that there is a profitable deviation to the mixed-strategy equilibrium postulated by GRH for intermediate trading costs. We propose an alternative set of strategies for this case and establish that they constitute an equilibrium.

Implementability of correlated and communication equilibrium outcomes in incomplete information games

In a correlated equilibrium, the players’ choice of actions is directed by correlated random messages received from an outside source, or mechanism. These messages allow for more equilibrium outcomes than without any messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also reflect the types of the players, either because they are affected by extraneous factors that also affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Mechanisms may be further differentiated by the connections between the messages that the players receive and their own and the other players’ types, by whether the messages are statistically dependent or independent, and by whether they are random or deterministic. Consequently, whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the corresponding number is 14 or 15 for correlated equilibria and even larger—15, 16 or 17—for communication equilibria. For both solution concepts, the implication relations between the different kinds of equilibria form a two-dimensional lattice, which is considerably more intricate than the single-dimensional one of the complete information case.

A generalization of Nash's theorem with higher-order functionals

The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.

An Application of Decision Theoretical Models of Probabilistic Choice to Solving 2x2 Simultaneous-Move Noncooperative Games in the Normal Form

Several models of probabilistic choice under uncertainty allow for deterministic choice if one act state-wise dominates the other. Such models have a natural application in game theory where probabilistic choice corresponds to mixed strategies, deterministic choice — to pure strategies and state-wise dominance — to strategic dominance. This paper considers an application to the simplest possible 2x2 simultaneous-move noncooperative game in the normal form. We derive a new equilibrium solution concept that coincides with the standard Nash equilibrium in pure strategies but resembles a quantal response equilibrium in mixed strategies. In particular, players randomize between strategies not in order to keep their opponent indifferent between his or her strategies (as in the mixed strategy Nash equilibrium) but because they are more likely to (but not always) choose strategies yielding higher utility.

A Note on Purification Theorem

The purification Theorem of Harsanyi (1973) argues that one can understand the mixed strategy Nash equilibrium of a simultaneous move complete information game as the limit of a particular Bayesian Nash equilibrium of the incomplete information version of the same game where the players choose pure strategies and the incompleteness of information almost vanishes. This paper analyzes an incomplete information coordination game and the purification of the pure strategy Bayesian Nash equilibrium (as stipulated by Harsanyi (1973)). Through comparative static analysis, the paper shows that besides inheriting the problematic feature of the mixed strategy equilibrium of the complete information game, the incomplete information game also has new problematic feature of its own.

Markov Game Analysis for Attack-Defense of Power Networks Under Possible Misinformation

Electricity grids are critical infrastructures. They are credible targets of active (e.g., terrorist) attacks since their disruption may lead to sizable losses economically and in human lives. It is thus crucial to develop decision support that can guide administrators in deploying defense resources for system security and reliability. Prior work on the defense of critical infrastructures has typically used static or Stackelberg games. These approaches view network interdictions as one-time events. However, infrastructure protection is also a continual process in which the defender and attacker interact to produce dynamic states affecting their best actions, as witnessed in the continual attack and defense of transmission networks in Colombia and Yemen. In this paper, we use zero-sum Markov games to model these interactions subject to underlying uncertainties of real-world events and actions. We solve equilibrium mixed strategies of the players that maximize their respective minimum payoffs with a time-decayed metric. We also show how the defender can use deception as a defense mechanism. Using results for a 5-bus system, a WECC 9-bus system, and an IEEE standard 14-bus system, we illustrate that our game model can provide useful insights. We also contrast our results with those of static games, and quantify the gain in defender payoff due to misinformation of the attacker.

The Classical Notion of Competition Revisited

This article seeks to fill a lacuna within classical economics concerning the process of market price determination in situations of market disequilibrium. To this aim, first we distinguish the classical notion of free competition from the Walrasian notion of perfect competition and we argue that the latter is beset with some theoretical difficulties alien to the former. Second, we reconstruct in some detail Smith’s and Marx’s views concerning market price determination and show that Marx’s extensive use of metaphors and numerical examples foreshadows the modern taxonomy of buyers’ market, sellers’ market, and mixed strategy equilibrium in the capacity space of a standard Bertrand duopoly model. Finally, we highlight similarities and differences between the classical notion of competition and contemporary Bertrand competition models.

Rebates in a Bertrand game

We study a price competition game in which customers are heterogeneous in the rebates they get from either of two firms. We characterize the transition between competitive pricing (without rebates), mixed strategy equilibrium (for intermediate rebates), and monopoly pricing (for larger rebates).In the mixed equilibrium, a firm’s support consists of two parts: (i) aggressive prices that can steal away customers from the other firm, and (ii) defensive prices that can only attract customers who get the rebate. Both firms earn positive expected profits.We show that, counter-intuitively, for intermediate rebates, an increase in rebates leads to a lower market segmentation.

On the genericity of the finite number of equilibria in multicriteria games: a counterexample

The famous Harsanyi’s (1973) Theorem states that a generic finite game has an odd number of Nash equilibria in mixed strategies. In this paper, counterexamples are given showing that for finite multicriteria games (games with vector-valued payoffs) this kind of result does not hold. In particular, it is shown that it is possible to find balls in the space of games such that every game in this set has uncountably many equilibria. This result then formalises the intuitive idea that games with uncountable sets of equilibria are not non-generic in the multicriteria case.

On the Use of Games Theory in Financial Studies: The Case of Mixed Strategy Equilibrium

Copyright: © 2012 Teng J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Since the 1980s, there has been an increasing use of games theory in the field of finance, including stock and forex trading [1]. The standard assumption of perfect competition was challenged. Analysis with the assumption of strategic interactions which uses games theory implicitly or explicitly becomes more popular. The use of games theory in the field of finance brings many new insights and should definitely be encouraged. But it also means that finance specialists need to be familiar with a new field of knowledge. Furthermore, one should also caution that games theory itself is a relatively new field and many of the game theoretic solution concepts are still being researched, refined and revised, and are not without controversy.

Mixed Strategy Nash Equilibria in Signaling Games

Signaling games are characterized by asymmetric information where the more informed player has a choice about what information to provide to its opponent. In this paper, decision trees are used to derive Nash equilibrium strategies for signaling games. We address the situation where neither player has any pure strategies at Nash equilibrium, i.e. a purely mixed strategy equilibrium. Additionally, we demonstrate that this approach can be used to determine whether certain strategies are part of a Nash equilibrium containing dominated strategies. Analyzing signaling games using a decision-theoretic approach allows the analyst to avoid testing individual strategies for equilibrium conditions and ensures a perfect Bayesian solution.

Equilibrium Mixed Strategies of Customers in an Unobservable Queue with Multiple Vacations

This paper studies and compares the equilibrium mixed strategies of customers with different decision criteria in an unobservable single-server queue with multiple vacations. Arriving customers can’t observe the queue length and the server state, whereas they are informed that the service time is exponentially distributed with unknown parameter, which is either an alternative of two values or belongs to a range. We consider three kinds of customers classified by their different waiting cost functions: linear, quadratic and exponential functions, and find that for the two types of partial information on service time parameter, both the equilibrium threshold expressions of customers’ effective arrival rates based on each criterion are consistent with each other except for the Laplace thresholds.

Towards the role of social connectivity and aspiration level on evolutionary game

Investigating the evolutionary game dynamics in structured populations is challenging due to the complexity of social interactions. There has been a growing interest in evolutionary game on social networks, particularly concerning how a specific network structure affects the evolution of strategies. Here, we consider a social network of interacting individuals playing the anti-coordination games with mixed strategies, and present a deterministic nonlinear equation for the evolution of strategies where the aspiration level is an incentive in the selection of strategies. We find that with an intermediate aspiration level, there exists an evolutionarily-stable mixed-strategy equilibrium if the cost-to-benefit ratio of altruistic is chosen below a threshold, which is determined by the largest Laplacian eigenvalue of the network. We also give extensive numerical simulations on regular and scale-free networks which confirm the validity of our analytical findings.

Continuous-Time Stochastic Games of Fixed Duration

We study nonzero-sum continuous-time stochastic games, also known as continuous-time Markov games, of fixed duration. We concentrate on Markovian strategies. We show by way of example that equilibria need not exist in Markovian strategies, but they always exist in Markovian public-signal correlated strategies. To do so, we develop criteria for a strategy profile to be an equilibrium via differential inclusions, both directly and also by modeling continuous-time stochastic as differential games and using the Hamilton–Jacobi–Bellman equations. We also give an interpretation of equilibria in mixed strategies in continuous time and show that approximate equilibria always exist.

Minimax play by teams

We analyze the behavior of two-person teams and individuals who repeatedly play the game with a unique mixed strategy equilibrium in the laboratory. When teams play OʼNeillʼs 4×4 game against another team, the choice frequencies are consistent with equilibrium of the game at the decision-maker level. In contrast, individuals against another individual play far from equilibrium, as previous experiments have found. The hide-and-seek game experiment reveals that teamsʼ behavior is less heterogeneous than individuals. When teams play OʼNeillʼs game against individuals, teams win at above the equilibrium rate in one treatment, but not in the other.

Strategic defense and attack for series and parallel reliability systems: reply 1 to comment 1

Kovenock and Roberson's [2011] comment provides initial work which has the potential, when suitably extended, to advance the research frontier. Kovenock and Roberson's paper consists of three sections. The first section is an interesting introduction. The second section, titled ‘Model and Main Result,’ provides no contribution beyond Hausken [2008a]. It consists of Equations (1)–(10) which are equivalent to equations developed by Hausken, and Equation (11) which is equivalent to the utility requirements u ⩾ 0 and U ⩾ 0 provided after Equation (17) in Hausken. The third section provides interesting ideas about mixed-strategy equilibria that can be extended in future research.

Equilibrium in a discrete Downsian model given a non-minimal valence advantage and linear loss functions

This note complements Aragonès and Palfrey [Aragonés, E., Palfrey, T., 2002. Mixed strategy equilibrium in a Downsian model with a favored candidate. Journal of Economic Theory 103, 131–161.] and Hummel [Hummel, P., 2010. On the nature of equilibriums in a Downsian model with candidate valence. Games and Economic Behavior 70 (2), 425–445.] by characterizing an essentially unique mixed strategy Nash equilibrium in a two-candidate Downsian model where one candidate enjoys a non-minimal non-policy advantage over the other candidate. The policy space is unidimensional and discrete (even number of equidistant locations), the preferences of the median voter are not known to the candidates and voter’s preferences on the policy space are represented by linear loss functions. We find that if the uncertainty about the median voter’s preferences is sufficiently low, then the mixed strategy σˆA= play the two intermediate locations with probability12 for the advantaged candidate and the mixed strategy σˆD= play the least liberal location that guarantees positive probability of election givenσˆAwith probability12and the least conservative strategy that guarantees positive probability of election givenσˆAwith probability12 for the disadvantaged candidate, constitute a Nash equilibrium of the game for any admissible value of the non-policy advantage.

Independendly‐Friendly Logic: Dependence and Independence of Quantifiers in Logic

AbstractIndependence‐Friendly logic (IF‐logic) introduced by Hintikka and Sandu (1989) studies patterns of dependence and independence of quantifiers which exceed those found in ordinary first‐order logic. The present survey focuses on the game‐theoretical interpretation of IF‐logic, including connections to solution concepts (equilibria in mixed strategies) in classical game theory, but we shall also present its compositional interpretation together with its connections to notions of dependence and dependence between terms.