Pure-strategy Bayesian Nash Equilibrium Research Articles

On monotone pure-strategy Bayesian-Nash equilibria of a generalized contest

We introduce a new approach to studying the existence of a monotone pure-strategy Bayesian-Nash equilibrium in an n-player single-prize contest model that covers both perfectly and imperfectly discriminating contests. The contestants have continua of possible types and bids, atomless type distributions, and their valuations and costs might depend not only on their own bids and types but also on other bidders' bids and types. Many, quite different contests are covered by our generalized contest model and equilibrium existence in monotone pure strategies in them follows from this paper's results.

Pure-strategy equilibrium in Bayesian potential games with absolutely continuous information

We prove the existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games with absolutely continuous information and a Bayesian potential that is upper semi-continuous in actions for any realization of the players' types. In particular, all Bayesian potential games with finitely many actions and absolutely continuous information possess a pure-strategy Bayesian Nash equilibrium.

Bayesian Nash equilibrium existence in (almost continuous) contests

We prove the existence of a behavioral-strategy Bayesian Nash equilibrium in contests where each contestant’s probability to win is continuous in efforts outside the zero-effort profile, monotone in his own effort, and greater that 1/2 if that contestant is the only one exerting positive effort. General type spaces, and in particular a continuum of information types, are allowed. As a corollary, the existence of a pure-strategy Bayesian Nash equilibrium is established in generalized Tullock contests, where the probability to win is strictly concave in one’s own effort for any non-zero effort profile of other players.

On the existence of Pareto undominated mixed-strategy Nash equilibrium in normal-form games with infinite actions

We show that each mixed strategy of a normal-form game can be reformulated as a pure strategy of an induced Bayesian game. Moreover, a normal-form game has a Pareto undominated mixed-strategy Nash equilibrium if and only if its induced Bayesian game has a pure-strategy Bayesian Nash equilibrium. By relying on the existence result of Pareto undominated pure-strategy Nash equilibrium in Bayesian games in Fu and Yu (2015), we also show that every normal-form game has a Pareto undominated mixed-strategy Nash equilibrium.

Equilibrium existence in games with a concave Bayesian potential

We establish existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games that have a continuous and concave potential at all states of nature, without assuming absolute continuity of information. As an application, we show that Bayesian Nash equilibrium exists in many well-known games that have semi-quadratic payoffs (including Bertrand and Cournot oligopolies with linear demand), for general information structures.

War of Attrition with Incomplete Information and Fuzzy Players' Types

The result on existence of a pure-strategy symmetric Bayesian Nash equilibrium in the war of attrition is generalized for fuzzy players’ actions and types.

Information Correlation in a 2 × 2 Game and an Extension of Purification Rationale

In this paper, we study a 2 × 2 Bayesian entry game with correlated private information. The distribution of private information is modelled by a symmetric joint normal distribution. Therefore, the correlation coefficient of the private information distribution reflects the degree of dependence of players' private information. Under such specification, players' private information can be correlated flexibly, which is not confined to the typical additive specification of private payoff shocks or private information by Carlson and van Damme (1993), where the private information is correlated due to the common payoff shock. In our game, if the private information is correlated, we find that given the variances of the private information, there exists a restriction on the degree of correlation of players' private information that allows the game can be solved by cutoff strategies. Specifically, given the variances of the private information, if players' private information in strategic substitutes (strategic complements) Bayesian games are positively (negatively) correlated, the range of correlation coefficient that allows the game can be solved by cutoff strategies is restricted so that if the correlation is out of the range, the game cannot be solved by cutoff strategies. Alternatively, given positive (negative) correlation of private information, the value of variances that allows a strategic substitutes (strategic complements) Bayesian games can be solved by cutoff strategies are restricted within certain range. If the value of variances fall out of the range, the Bayesian game cannot be solved by cutoff strategies. However, given negative (positive) correlation of players' private information in strategic substitutes (strategic complements) Bayesian games, in which the Bayesian games can always be solved by cutoff strategies, we prove that as the variances converge to zero, all pure strategy Bayesian Nash equilibria of the perturbed games converge to the respective Nash equilibria of the corresponding strategic substitutes (strategic complements) complete information games. Based on the result, we conclude that the purification rationale proposed by Harsanyi (1973) can be extended to games with dependent perturbation errors that follow a symmetric joint normal distribution if the correlation coefficient is positive for the strategic complements games or negative for the strategic substitutes games.

On monotone approximate and exact equilibria of an asymmetric first-price auction with affiliated private information

This paper develops a new approach to investigating equilibrium existence in first-price auctions with many asymmetric bidders whose types are affiliated and valuations are interdependent and not necessarily strictly increasing in own type. We begin with studying a number of continuity-related properties of the model, which are used, in conjunction with tieless single crossing and H-convexity, to establish the existence of monotone approximate interim equilibria. Then we provide two sets of sufficient conditions for the game to have a sequence of monotone approximate equilibria whose limit points are pure-strategy Bayesian-Nash equilibria.

Non-Cooperative Game of Throughput and Hash Length for Adaptive Merkle Tree in Mobile Wireless Networks

The restricted resources of mobile wireless networks such as limited bandwidth have resulted in stringent requirements on the available quality of service (QoS). To enhance the communication performance, mobile nodes attempt to optimize the available bandwidth of wireless networks, which, however, makes them vulnerable to the malicious attacks of adversaries owing to the open and mobile context. Furthermore, improving the security level of the communication consumes bandwidth and thus degrades the communication quality. In order to harmonize the QoS and security, in this paper, the two conflicting terms “communication player” and “security player” are considered as two individual players and are balanced using game theory. Furthermore, we construct a non-cooperative game model, wherein the utilities of the communication player and security player are maximized by controlling the number of nodes transmitted and the length of hash, respectively. We formulate the utility functions of the communication player with respect to the throughput as well as the security level of the Merkle tree for the security player. This paper considers complete information game and incomplete information game, for which we theoretically determine the pure strategy Nash equilibrium and the pure strategy Bayesian Nash equilibrium, respectively. Extensive numerical calculations are performed to comprehensively investigate the utilities of each player and the Nash equilibrium for various combinations of the concerned parameters. The results show that the presented theoretical game approach enables a mobile node to adaptively perform specific actions based on the derived Nash equilibrium. With the obtained Nash equilibrium values, both communication player's and security player's utilities reach to the maximum so that satisfactory communication quality can be realized without compromising the security.

Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

Emerging short-lived (ephemeral) connections between wireless mobile devices have raised concerns over the security of ephemeral networks. An important security challenge in these networks is to identify misbehaving nodes, especially in places where a centrally managed station is absent. To tackle this problem, a local voting-based scheme (game) in which neighboring nodes quickly decide whether to discredit an accused (target) node in mobile networks has been introduced in the literature. However, nodes’ beliefs and reactions significantly affect the outcome of target node identification in the collaboration. In this paper, a plain Bayesian game between a benign node and a target node in one stage of a local voting-based scheme is proposed in order to capture uncertainties of nodes for target node identification. In this context, the expected utilities (payoffs) of players in the game are defined according to uncertainties of nodes regarding their monitoring systems, the type of target node and participants, and the outcome of the cooperation. Meanwhile, incentives are offered in payoffs in order to promote cooperation in the network. To discourage nodes from abusing incentives, a variable-benefit approach that rewards each player according to the value of their contribution to the game is introduced. Then, possible equilibrium points between a benign node and a malicious node are derived using a pure-strategy Bayesian Nash equilibrium (BNE) and a mixed-strategy BNE, ensuring that no node is able to improve its payoffs by changing its strategy. Finally, the behavior of malicious and benign nodes is studied via simulations. Specifically, it is shown how the aforementioned uncertainties and the designed incentives impact the strategies of the players and, consequently, the correct target-node identification.

Managing and reallocating inventory across two markets with local information

Consider a firm that controls inventory centrally for two separate markets that are managed by regional managers having local demand information. The central planner provides a dedicated inventory level to each market, to ensure a minimum service level, but can reallocate inventory once associated demands are filled. How should such a firm solicit regional managers to share their local information so inventory levels can be set efficiently? In this paper, we study a forecast sharing game in such a firm, where demand information lies within regions and inventory is managed by a central decision maker. We show that incentives are misaligned and truthful information sharing is not an equilibrium. We study two potential ways to improve the system: a) regional managers directly place orders (rather than simply passing demand forecasts), or b) a pricing scheme for inventory reallocations is imposed. We show that under direct ordering a unique pure strategy Bayesian Nash equilibrium exists and makes retailers place orders based on their true forecasts, although the requested order quantities are not system optimal (e.g., some inventory pooling benefits are lost). Under an endogenous transfer pricing mechanism set by the central planner, we find that a first best truth telling equilibrium is possible under certain conditions. We conduct a numerical study to examine the different cases and find that when the critical fractile is high, direct ordering results in lower inventories while the reverse is true when the critical fractile is low. Expected profit comparisons show the value of local information.

Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

Equilibrium existence in Tullock contests with incomplete information

We prove an existence theorem for pure strategy Bayesian Nash equilibrium in Tullock contests where the information endowment of each contender is described by a countable partition.

Estimation of discrete games with correlated types

Summary In this paper, I focus on the identification and estimation of static games of incomplete information with correlated types. Instead of making the independence assumption on players' types in order to simplify the equilibrium set, I propose an approach that allows me to identify subsets of the space of covariates (i.e. publicly observed state variables in payoff functions), for which there exists a unique pure strategy Bayesian Nash equilibrium (BNE) and the equilibrium strategies are monotonic functions. Moreover, I characterize the monotonic pure strategy BNE in a simple manner and propose an estimation procedure that uses observations only from the subset of the covariate space where the game admits a unique monotonic pure strategy BNE. Furthermore, I show that the proposed estimator is -consistent and has a limiting normal distribution.

On the existence of pure-strategy equilibria in games with private information: A complete characterization

This paper reports a definitive resolution to the question of the existence of a pure-strategy Bayesian–Nash equilibrium in games with a finite number of players, each with a compact metric action set and private information. The resolution hinges on saturated spaces. If the individual spaces of information are saturated, there exists a pure-strategy equilibrium in such a game; and if there exists a pure-strategy equilibrium for the class of games under consideration and with uncountable action sets, the spaces of private information must be saturated. As such, the paper offers a complete characterization of a longstanding question, and offers another game-theoretic characterization of the saturation property, one that complements a recent result of Keisler–Sun (2009) on large non-anonymous games with complete information.

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.

Incomplete information in rent-seeking contests

We consider a variant of the Tullock lottery contest. Each player’s constant marginal cost of effort is drawn from a potentially different continuous distribution. In order to study the impact of incomplete information, we compare three informational settings to each other; players are either completely informed, privately informed about their own costs, or ignorant of all cost realizations. For the first and the third setting, we determine the unique pure-strategy Nash equilibrium. Under private information, we prove existence of a pure-strategy Bayesian Nash equilibrium and identify a sufficient condition for uniqueness. Assuming that unit cost distributions all have the same mean, we show that under ignorance of all cost realizations ex ante expected aggregate effort is lower than under both private and complete information. Ex ante expected rent dissipation, however, is higher than in the latter settings if we focus on the standard lottery contest and assume costs are all drawn from the same distribution. Between complete and private information, there is neither a general ranking in terms of effort nor in terms of rent dissipation.

Signaling game based strategy of intrusion detection in wireless sensor networks

As Wireless Sensor Networks (WSNs) become increasingly popular, it is necessary to require Intrusion Detection System (IDS) available to detect internal malicious sensor nodes. Because sensor nodes have limited capabilities in terms of their computation, communication, and energy, selecting the profitable detection strategy for lowering resources consumption determines whether the IDS can be used practically. In this paper, we adopt the distributed-centralized network in which each sensor node has equipped an IDS agent, but only the IDS agent resided in the Cluster Head (CH) with sufficient energy will launch. Then, we apply the signaling game to construct an Intrusion Detection Game modeling the interactions between a malicious sensor node and a CH-IDS agent, and seek its equilibriums for the optimal detection strategy. We illustrate the stage Intrusion Detection Game at an individual time slot in aspects of its player’s utilities, pure-strategy Bayesian–Nash equilibrium (BNE) and mixed-strategy BNE. Under these BNEs the CH-IDS agent is not always on the Defend strategy, as a result, the power of CH can be saved. As the game evolves, we develop the stage Intrusion Detection Game into a multi-stage dynamic Intrusion Detection Game in which, based on Bayesian rules, the beliefs on the malicious sensor node can be updated. Upon the current belief and the Perfect Bayesian equilibrium (PBE), the best response strategy for the CH-IDS agent can be gained. Afterward, we propose an intrusion detection mechanism and corresponding algorithm. We also study the properties of the multi-stage dynamic Intrusion Detection Game by simulations. The simulation results have shown the effectiveness of the proposed game, thus, the CH-IDS agents are able to select their optimal strategies to defend the malicious sensor nodes’ Attack action.

Extensive Robustness of Large Games with a Continuum of Players

'We present a model of Bayesian games with a continuum of players and prove that pure strategy Bayesian Nash equilibria exist and are extensively robust. This extends Kalai's (2004) approximate extensive robustness results to the limiting continuum model and highlights the roles played by key assumptions such as semi-anonymity and independence of type distributions.

Incomplete Information in Rent-seeking Contests

We consider a variant of the Tullock lottery contest. Each player’s constant marginal cost of effort is drawn from a potentially different continuous distribution. In order to study the impact of incomplete information we compare three informational settings to each other: players are either completely informed, privately informed about their own costs, or ignorant of all cost realizations. For the first and the third setting we determine the unique pure-strategy Nash equilibrium. Under private information we prove existence of a pure-strategy Bayesian Nash equilibrium and identify a sufficient condition for uniqueness. Assuming that unit cost distributions all have the same mean, we show that under ignorance of all cost realizations ex ante expected aggregate effort is lower than under both private and complete information. Ex ante expected rent dissipation, however, is higher than in the latter settings if we focus on the standard lottery contest and assume costs are all drawn from the same distribution. Between complete and private information there is neither a general ranking in terms of effort nor in terms of rent dissipation.