Mixed Strategy Nash Equilibrium Research Articles

Existence and Uniqueness of Nash Equilibrium in Electoral Competition Games: The Hybrid Case

AbstractThis paper analyzes the traditional unidimensional, two‐party electoral competition game when parties have mixed motivations, in the sense that they are interested in winning the election, but also in the policy implemented after the contest. In spite of having discontinuous payoffs, this game, referred to as the hybrid election game, is shown to be payoff secure and reciprocally upper semi‐continuous. Conditional payoffs, however, are not quasi‐concave. Hence, the existence of a pure strategy Nash equilibrium (psne) is ensured only if parties have homogenous interests in power. In that case, an equilibrium not only exists, but it is also unique. Instead, if parties have heterogeneous motivations, depending upon the relationship between the electoral uncertainty, the aggregate opportunism, and its distribution across parties, a psne may or may not exist. The mixed extension, however, is always better reply secure. Therefore, a mixed strategy Nash equilibrium does indeed exist.

Multiproduct price competition with heterogeneous consumers and nonconvex costs

This paper extends the oligopolistic model of price competition to environments with multiple goods, heterogeneous consumers, and arbitrary continuous cost functions. A Nash equilibrium in mixed strategies with an endogenous sharing rule is proven to exist. It is also shown that, in environments with fixed costs and constant marginal costs, all (symmetric and asymmetric) equilibria exhibit price dispersion across stores. Furthermore, the paper identifies scenarios in which prices will necessarily be random. In these markets, stores keep each other guessing because, given the fixed costs, they would incur a loss if their price strategies were anticipated and beaten by competitors. This is interpreted as an important economic feature that is possibly behind random price promotions such as weekly specials.

Game theoretic approach in routing protocol for wireless ad hoc networks

This paper introduces a game theoretic method, called forwarding dilemma game (FDG), which controls routing overhead in dense multi-hop wireless ad hoc networks. The players of the game are the wireless nodes with set of strategies { Forward, Not forward}. The game is played whenever an arbitrary node in the network receives a flooding packet. In FDG, every player needs to know the number of players of the game. That is why a neighbor discovery protocol (NDP) is introduced. In order for NDP to function, a field is attached to the flooding packets (routing overhead packets). The mixed strategy Nash equilibrium is used as a solution for the FDG. This provides the probability that the flooding packet would be forwarded by the receiver node. FDG with NDP is implemented in AODV protocol in Network Simulator NS-2 to verify its performance with simulations. FDG with NDP improves performance of the AODV compared to the same network with only AODV protocol in moderate and high node densities. FDG can be applied to any routing protocol that uses flooding in the route discovery phase.

Correlated Equilibria, Good and Bad: An Experimental Study

We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of the Nash equilibrium concept. Specifically, we seek to understand the conditions under which subjects playing the game of Chicken will condition their behavior on private, third-party recommendations drawn from known distributions. In a good-recommendations treatment, the distribution we use is a correlated equilibrium with payoffs better than any symmetric payoff in the convex hull of Nash equilibrium payoff vectors. In a bad-recommendations treatment, the distribution is a correlated equilibrium with payoffs worse than any Nash equilibrium payoff vector. In a Nash-recommendations treatment, the distribution is a convex combination of Nash equilibrium outcomes (which is also a correlated equilibrium), and in a fourth very-good-recommendations treatment, the distribution yields high payoffs, but is not a correlated equilibrium. We compare behavior in all of these treatments to the case where subjects do not receive recommendations. We find that when recommendations are not given to subjects, behavior is very close to mixed-strategy Nash equilibrium play. When recommendations are given, behavior does differ from mixed-strategy Nash equilibrium, with the nature of the differences varying according to the treatment. Our main finding is that subjects will follow third-party recommendations only if those recommendations derive from a correlated equilibrium, and further, if that correlated equilibrium is payoff-enhancing relative to the available Nash equilibria.

Teaching Mixed Strategy Nash Equilibrium to Undergraduates

The authors present a simple and effective method for improving student comprehension of mixed strategies. After reviewing available texts,and with some trial and error, we feel that this method eases the teaching of the information and increases the students’ability to understand and retain concepts.Through a simple alteration to the normal form of a game, students can clearly see the results of employing mixed strategies, practise calculating expected payoffs, visualise the accompanying ‘indifference’in the payoffs, and recognise the mixed strategy Nash equilibrium in a familiar way.

Strategic bidding in continuous electricity auctions: an application to the Spanish electricity market

In this paper we introduce an asymmetric model of continuous electricity auctions with limited production capacity and bounded supply functions. The strategic bidding is studied with this model by means of an electricity market game. We prove that for every electricity market game with continuous cost functions a mixed-strategy Nash equilibrium always exists. In particular, we focus on the behavior of producers in the Spanish electricity market. We consider a very simple form for the Spanish electricity market: an oligopoly consisting just of independent hydro-electric power production units in a single wet period. We show that a pure-strategy Nash equilibrium for the Spanish electricity market game always exists.

Strategic games with security and potential level players

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players.

Competition and coordination in experimental minority games

This work presents experimental results on a coordination game in which agents must repeatedly choose between two sides, and a positive fixed payoff is assigned only to agents who pick the minoritarian side. We conduct laboratory experiments in which stationary groups of five players play the game for 100 periods, and manipulate two treatment variables: the amount of information about other players’ past choices and the salience of information regarding the game history (i.e., the length of the string of past outcomes that players can see on the screen while choosing). Our main findings can be summarized as follows: aggregate efficiency in the game is in most cases significantly higher than the level corresponding to the symmetric mixed strategy Nash equilibrium. In addition, providing players with information about individual choices in the group does not improve aggregate efficiency with respect to when such information is absent. Displaying information about more rounds than just the previous one, on the other hand, seems to have a positive effect on aggregate efficiency. At the individual level, we find a stronger statistical relation between players’ current choices and their own past choices than between players’ choices and previous aggregate outcomes. In addition, the depth of the relation between present and past choices seems to be affected by the prompt availability of information about the game history. Finally, we detect evidence of a mutual co-adaptation between players’ choices over time that is partly responsible for the high level of efficiency observed.

Mixed Strategies in Games of Capacity Manipulation in Hospital–Intern Markets

We investigate games of capacity manipulation in hospital-intern markets as proposed by Konishi and Unver (Soc Choice Welfare, in press). While Konishi and Unver (Soc Choice Welfare, in press) show that there may not exist a pure-strategy Nash equilibrium in general, there exists a mixed-strategy Nash equilibrium in such a game. We show that every hospital weakly prefers a Nash equilibrium to any “larger” capacity profiles, whether the equilibrium is in pure or mixed strategies. In particular, a Nash equilibrium is weakly preferred by hospitals to the outcome that results from truthful reporting.

Uniform payoff security and Nash equilibrium in compact games

We introduce a condition, uniform payoff security, for games with compact Hausdorff strategy spaces and payoffs bounded and measurable in players’ strategies. We show that if any such compact game G is uniformly payoff secure, then its mixed extension G ¯ is payoff secure. We also establish that if a uniformly payoff secure compact game G has a mixed extension G ¯ with reciprocally upper semicontinuous payoffs, then G has a Nash equilibrium in mixed strategies. We provide several economic examples of compact games satisfying uniform payoff security.

Decentralised unit commitment in a market structure: problem formulation and solution advancement

Electricity markets based on power pools, in which the owners of generating units submit their marginal-price schedules, implicitly leave the problem of unit commitment and its associated constraints and costs aside. The paper addresses the unit-commitment problem faced by participants if they are to self-commit their units in a market structure. This strategic problem is formulated through a probabilistic-dynamic-programming model, where the probability of a particular generator being online during a specific market period is set by its owner. The solutions adopted for that problem are Nash equilibria in mixed strategies. The paper presents an example with a small group of generating units, where a comparison is made between the results for the integrated utility and for the market structure. This example is solved making use of a direct Monte Carlo method.

A Model of Search Intermediaries and Paid Referrals

In this paper we pursue three main objectives: (1) to develop a model of an intermediated search market in which matching between consumers and firms takes place primarily via paid referrals; (2) to address the important managerial question (for a search intermediary) of finding a suitable mechanism for selling referrals to firms; and (3) to characterize and analyze the firms' resulting equilibrium bidding strategies given consumers' equilibrium search behavior. To achieve these objectives we develop a two-stage model of search intermediaries in a vertically differentiated product market. In the first stage an intermediary chooses a search engine design that specifies to which extent a firm's search rank is determined by its bid and to which extent it is determined by the product offering's overall performance. In the second stage, based on the search engine design, competing firms place their open bids to be paid for each referral by the search engine. We find that the revenue-maximizing search engine design bases rankings on a weighted average of relative product performance and bid amount. The firms' bid amounts are generally nonmonotonic in product performance and a nonzero pure-strategy equilibrium of the underlying discontinuous bidding game does not exist. We derive the unique mixed-strategy Nash equilibrium and show that firms of low product performance tend to fully dissipate their rents. These rents are fully appropriated by the search intermediary and the better firm, and are therefore not socially wasteful. However, the intermediary's design choice, by attributing a positive weight to the firms' bids, generally obfuscates search results and reduces overall consumer surplus compared to the socially optimal design of providing fully transparent results ranked purely on product performance.

Unraveling Ancient Mysteries: Reimagining the Past Using Evolutionary Computation in a Complex Gaming Environment

In this paper, we use principles from game theory, computer gaming, and evolutionary computation to produce a framework for investigating one of the great mysteries of the ancient Americas: why did the pre-Hispanic Pueblo (Anasazi) peoples leave large portions of their territories in the late A.D. 1200s? The gaming concept is overlaid on a large-scale agent-based simulation of the Anasazi. Agents in this game use a cultural algorithm framework to modify their finite-state automata (FSA) controllers following the work of Fogel (1966). In the game, there can be two kinds of active agents: scripted and unscripted. Unscripted agents attempt to maximize their survivability, whereas scripted agents can be used to test the impact that various pure and compound strategies for cooperation and defection have on the social structures produced by the overall system. The goal of our experiments here is to determine the extent to which cooperation and competition need to be present among the agent households in order to produce a population structure and spatial distribution similar to what has been observed archaeologically. We do this by embedding a "trust in networks" game within the simulation. In this game, agents can choose from three pure strategies: defect, trust, and inspect. This game does not have a pure Nash equilibrium but instead has a mixed strategy Nash equilibrium such that a certain proportion of the population uses each at every time step, where the proportion relates to the quality of the signal used by the inspectors to predict defection. We use the cultural algorithm to help us determine what the mix of strategies might have been like in the prehistoric population. The simulation results indeed suggest a mixed strategy consisting of defectors, inspectors, and trustors was necessary to produce results compatible with the archaeological data. It is suggested that the presence of defectors derives from the unreliability of the signal which increases under drought conditions and produced increased stress on Anasazi communities and may have contributed to their departure.

Learning in and about Games

We study learning in finitely repeated 2x2 normal form games, when players have incomplete information about their opponents' payoffs. In a laboratory experiment we investigate whether players (a) learn the game they are playing, (b) learn to predict the behavior of their opponent, and (c) learn to play according to a Nash equilibrium of the repeated game. Our results show that the success in learning the opponent's type depends on the characteristics of the true game. The learning success is much higher for games with pure strategy Nash equilibria than for games with a unique mixed strategy Nash equilibrium, and it is higher for games with symmetric pure strategy Nash equilibria than for games with asymmetric equilibria. Moreover, subjects learn to predict the opponents' behavior very well. However, they rarely play according to a Nash equilibrium and we observe no correlation between equilibrium play and learning about the game.

Uncertainty and resistance to reform in laboratory participation games

This paper presents a participation game experiment to study the impact of uncertainty and costly political participation on the incidence of reform. We introduce intra-group conflict into the framework of Fernandez and Rodrik (1991) [Fernandez, R., Rodrik, D., 1991. Resistance to reform: status quo bias in the presence of individual-specific uncertainty. American Economic Review 81, 1146–1155] by incorporating costly political participation, which creates a natural incentive for free-riding on fellow group members' efforts to influence policy outcomes. Our experimental findings show that uncertainty reduces the incidence of reform even with costly political participation and that an increase in the cost of participation reduces the participation of all agents, regardless of whether they belong to the majority and minority. This second result cannot be reconciled with the standard mixed strategy Nash equilibrium, but is consistent with the quantal response equilibrium.

A Note on the Existence of Nash Equilibrium in Games with Discontinuous Payoffs

This paper generalises the approach taken by Dasgupta & Maskin (1986) and Simon (1989) and provides necessary and sufficient conditions for the existence of pure and mixed strategy Nash equilibrium in games with continuous strategy spaces and discontinuous payoff functions. The conditions can be applied widely, and examples for existence of pure strategy and monotonic equilibria in First-Price auctions are provided. The conditions are also appropriate for ensuring that computer generated equilibrium solutions can be extended to continuous strategy spaces.

Short-Term Electricity Market Auction Game Analysis: Uniform and Pay-as-Bid Pricing

This paper analyzes the competing pricing mechanisms of uniform and pay-as-bid pricing in an electricity market. Game theory and auction theory are adopted to analyze the strategic behavior of a big player and a small player in a short-term auction game. Contrary to what would be expected from the conclusion of the equivalence theorem, we prove that for a two-player static game the Nash Equilibrium (NE) under pay-as-bid pricing will yield less total revenue in expectation than under uniform pricing when demand is inelastic. To confirm this theoretical result we simulated the model using a mixed-strategy NE solver. We extended the model to an elastic demand case and showed that pay-as-bid pricing also led to a larger expected total demand being served when demand is elastic.

A Model of Search Intermediaries and Paid Referrals

In this paper we pursue three main objectives: (1) to develop a model of an intermediated search market in which matching between consumers and firms takes place primarily via paid referrals; (2) to address the question of designing a suitable mechanism for selling referrals to firms; and (3) to characterize and analyze the firms' bidding strategies given consumers' equilibrium search behavior. To achieve these objectives we develop a two-stage model of search intermediaries in a vertically differentiated product market. In the first stage an intermediary chooses a search engine design that specifies to which extent a firm's search rank is determined by its bid and to which extent it is determined by the product offering's performance. In the second stage, based on the search engine design, competing firms place their open bids to be paid for each referral by the search engine. We find that the revenue-maximizing search engine design bases rankings on a weighted average of product performance and bid amount. Nonzero pure-strategy equilibria of the underlying discontinuous bidding game generally exist but are not robust with respect to noisy clicks in the system. We determine a unique nondegenerate mixed-strategy Nash equilibrium that is robust to noisy clicks. In this equilibrium firms of low product performance fully dissipate their rents, which are appropriated by the search intermediary and the firm with the better product. The firms' expected bid amounts are generally nonmonotonic in product performance and depend on the search engine design parameter. The intermediary's profit-maximizing design choice, by attributing a positive weight to the firms' bids, tends to obfuscate search results and reduce overall consumer surplus compared to the socially optimal design of fully transparent results ranked purely on product performance.

Rewarding Policemen Increases Crime. Another Surprising Result from the Inspection Game

Inspection games are 2x2 games in which one playermust decide whether to inspect the other player, who in turnmust decide whether to infringe a norm or a regulation.Inspection games have a single, mixed strategy Nashequilibrium, which has counter-intuitive comparative staticsproperties. This result has been used by Tsebelis (1989) andHoller (1992) to show that the economic approach to lawenforcement is not likely to generate clear-cut predictions.In this paper I discuss a variant of the inspection game inwhich the inspector can act as a Stackelberg leader. I willalso show that this version of the inspection game hascounter-intuitive comparative statics properties. Inparticular, increasing inspector’s incentives to enforce thelaw increases the frequency of law infractions.

Observation, Measurement, and Computation in Finite Games

Limitations in agents’ ability to observe, measure and compute are studied. A formulation in terms of induced changes in the players’ subjective perception of the game makes it possible to study these phenomena on a common footing with objective variation in the game. We study a model where payoffs undergo continuous variation and players reason about the process from limited local information. Specifically, finite games, and statements about them, are interpreted within sheaf models, which incorporate both variation and a logic of limited observation. Nash’s theorem on the existence of equilibrium is not valid unless special observational properties hold, although closely related statements are valid. The possibility of learning to play a mixed strategy Nash equilibrium from observation of past play is also examined within the framework.