Mixed Equilibrium Research Articles

Nash at Wimbledon: Evidence from Half a Million Serves

Minimax and its generalization to mixed strategy Nash equilibrium is the cornerstone of our understanding of strategic situations that require decision makers to be unpredictable. Using a dataset of nearly half a million serves from over 3000 matches, we examine whether the behavior of professional tennis players is consistent with the Minimax Hypothesis. The large number of matches in our dataset requires the development of a novel statistical test, which we show is more powerful than the tests used in prior related studies. We find that win rates conform remarkably closely to the theory for men, but conform somewhat less neatly for women. We show that the behavior in the field of more highly ranked (i.e., better) players conforms more closely to theory.

Price Dispersion with Ex Ante Homogeneity: A Reassessment of the Diamond Paradox

If identical firms set prices in a first stage and identical consumers search sequentially in a second stage, then price dispersion arises in the form of a mixed strategy subgame perfect Nash Equilibrium when the search cost is not prohibitively high. The result does not depend on heterogeneity and bridges the gap between monopoly pricing (Diamond, 1971) and marginal cost pricing. Thus, price dispersion is solely supported by information frictions.

Iterative Algorithm for extended mixed equilibrium problem

In this paper, we introduce and study an extended mixed equilibrium problem by using auxiliary principle technique. A generalized predictor-corrector iterative algorithm is defined for solving extended mixed equilibrium problem. The convergence of the method mentioned requires some condition (∗), g-relatively relaxed Lipschitz continuity and relatively g-relaxed monotonicity of the mappings.

GRAPHICAL METHOD FOR INTERVAL BIMATRIX GAMES

This paper deals with two-person non-zero sum games with interval payoffs. Graphical method to find a mixed strategy equilibrium is adapted to interval bimatrix games. In addition, interval bimatrix games Nash equilibrium is attained by graphical method. Nu- merical examples are also illustrated.

System of generalized mixed equilibrium problems, variational inequality, and fixed point problems

AbstractThe purpose of this paper is to introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, the set of common fixed points of a finite family of pseudo contraction mappings, and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a real Hilbert space. We establish results on the strong convergence of the sequence by the proposed scheme to a common element of the above three solution sets. These results extend and improve some corresponding results in this area. Finally, we give a numerical example which supports our main theorem.

Distributionally robust chance-constrained games: existence and characterization of Nash equilibrium

We consider an n-player finite strategic game. The payoff vector of each player is a random vector whose distribution is not completely known. We assume that the distribution of a random payoff vector of each player belongs to a distributional uncertainty set. We define a distributionally robust chance-constrained game using worst-case chance constraint. We consider two types of distributional uncertainty sets. We show the existence of a mixed strategy Nash equilibrium of a distributionally robust chance-constrained game corresponding to both types of distributional uncertainty sets. For each case, we show a one-to-one correspondence between a Nash equilibrium of a game and a global maximum of a certain mathematical program.

Minimax Play at Wimbledon Redux

Initial tests of the predictive power of the mixed-strategy Nash equilibrium in the laboratory pointed to significant deviations. In response to the flat maximum critique, that incentives in the lab were not high enough to induce Nash behavior, researchers turned to the field, particularly sporting environments, where the monetary incentives were high. Field experiments overall, found predictions closer to the marginal distribution of actions predicted by the mixed strategy Nash equilibrium; however, evidence of deviations from the prediction of independent behavior across rounds have been mixed. I use a new dataset of tennis match serves that is two orders of magnitude greater than those used in previous tennis studies. Similarly to previous findings, the marginal distributions of professionals’ serves are not significantly different from the prediction of the normative solution, but there exists significant serial dependence across serves. This behavior is found even among players who at some point in their careers were ranked Number 1 in the world. The deviations of these top players cannot be explained as a strategic best response to expectations of lower-ranked players’ deviations from the Nash equilibrium.

Joint implementation of tradable credit and road pricing in public-private partnership networks considering mixed equilibrium behaviors

This paper investigates joint road charging schemes in a public-private partnership (PPP) network by simultaneously taking into account Cournot-Nash (CN) players and user equilibrium (UE) players. Each joint scheme comprises a tradable credit plan for public roads and a regular tolling plan for private roads. We show that, under UE-CN mixed equilibrium, there exist anonymous nonnegative joint schemes that can support a system optimum link flow pattern. By using preemptive approach, we further design three bi-objective optimization models with hybrid implementation of tradable credit and road pricing. Numerical examples demonstrate that the proposed methods are effective in managing PPP networks.

Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

ABSTRACTIn this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \\hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \\hboxbifunctions are, respectively, maximal monotone and \\hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.

Analysis of information feedback and selfconfirming equilibrium

Recent research emphasizes the importance of information feedback in situations of recurrent decisions and strategic interaction, showing how it affects the uncertainty that underlies selfconfirming equilibrium (e.g., Battigalli et al., 2015, Fudenberg and Kamada, 2015). Here, we discuss in detail several properties of this key feature of recurrent interaction and derive relationships. This allows us to elucidate different notions of selfconfirming equilibrium, showing how they are related to each other given the properties of information feedback. In particular, we focus on Maxmin selfconfirming equilibrium, which assumes extreme ambiguity aversion, and we compare it with the partially-specified-probabilities (PSP) equilibrium of Lehrer (2012). Assuming that players can implement any randomization, symmetric Maxmin selfconfirming equilibrium exists under either “observable payoffs,” or “separable feedback.” The latter assumption makes this equilibrium concept essentially equivalent to PSP-equilibrium. If observability of payoffs holds as well, then these equilibrium concepts collapse to mixed Nash equilibrium.

On the learning and stability of mixed strategy Nash equilibria in games of strategic substitutes

This paper analyzes the learning and stability of mixed strategy Nash equilibria in games of strategic substitutes (GSS), complementing recent work done in the case of strategic complements (GSC). Mixed strategies in GSS are of particular interest because it is well known that such games need not exhibit pure strategy Nash equilibria. First, we establish bounds on the strategy space which indicate where randomizing behavior may occur in equilibrium. Second, we show that mixed strategy Nash equilibria are generally unstable under a wide variety of learning rules. Multiple examples are given.

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

A game-theoretic method for designing distortion function in spatial steganography

Most modern secure image steganographic schemes define distortion functions for constraining the embedding changes to those parts of the image that are difficult to model such as textured or noisy regions. However, thus arguments disregard Kerckhoffs' principle: the warden also knows the adaptivity criterion as well and may be able reproduce or estimate it. This paper proposes a new idea that the embedding distortion is designed based on game theory. We present a two-player zero-sum game between steganographer and attacker related to the security of practical steganography, and there exists a unique mixed strategy Nash equilibrium. The distortion function is first achieved from the original method S-UNIWARD (Spatial-UNIversal Wavelet Relative Distortion), and then we readjust the distortion function according to the above Nash equilibrium. The new distortion not only ensures the embedding changes focus on textured region but also correlates to the Kerckhoffs' principle. Experiment results show that the statistical security of the proposed method is improved by approximately 6.2 % with the features of modern rich models.

Impact of risk attitudes and perception on game theoretic driving interactions and safety

This study employs game theory to investigate behavioural norms of interaction between drivers at a signalised intersection. The choice framework incorporates drivers’ risk perception as well as their risk attitudes. A laboratory experiment is conducted to study the impact of risk attitudes and perception in crossing behaviour at a signalised intersection. The laboratory experiment uses methods from experimental economics to induce incentives and study revealed behaviour. Conflicting drivers are considered to have symmetric disincentives for crashing, to represent a no-fault car insurance environment. The study is novel as it uses experimental data collection methods to investigate perceived risk. Further, it directly integrates perceived risk of crashing with other active drivers into the modelling structure. A theoretical model of intersection crossing behaviour is also developed in this paper. This study shows that right-of-way entitlements assigned without authoritative penalties to at-fault drivers may still improve perceptions of safety. Further, risk aversion amongst drivers attributes to manoeuvring strategies at or below Nash mixed strategy equilibrium. These findings offer a theoretical explanation for interactive manoeuvres that lead to crashes, as opposed to purely statistical methods which provide correlation but not necessarily explanation.

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Braess Paradox of traffic networks with mixed equilibrium behaviors

Under the user equilibrium (UE) behavior assumption, the Braess Paradox (BP) and its variations have been well investigated. However, users do not always follow the UE behavior. In reality, there are likely quiet a few non-collaborative Cournot–Nash (CN) players coexisting with UE players in the common traffic network. Users in a CN player are completely collaborative to minimize their total travel cost and users subordinating to different players are perfectly competitive. Considering both UE and CN players in the congested network, it remains unclear that under what conditions the BP will occur. In this paper, the BP occurrence conditions under the UE–CN mixed equilibrium are firstly investigated using the classical Braess network with linear link cost function. Then, the BP conditions are studied to the ordinary grid network with nonlinear link cost function. It is shown that the BP occurrence in the conventional Braess network depends upon the link travel time function parameters and the demand level of users controlled by UE/CN players, and the BP occurs in the grid network only for certain demand combinations of users under one UE player and two CN players.

Combating Smuggling: What Games We Are Playing: An Indonesian Case Study

Intuitively, a person’s behavioral tendency to corrupt seems to follow several incentives that bound with the outcomes. But changes in outcomes’ payoffs do not always directly affect to person behavior. In games with mixed equilibrium presences, probability of actions taken by other parties (in this paper, “to inspect” and “do not inspect”) will alter a person tendency whether “to comply” or “to cheat”, as shown in garment smuggling case in Indonesia. Game theoretic concepts were employed in this paper to perform framework for analysis in describing the actions of interdependent agents. When the game has mixed equilibrium, probability of one player to take one strategic action does not depend on the opponent’s payoffs (i.e. maximum penalty). What does change is the probability of the opponent’s strategic actions.

The complexity of equilibria for risk-modeling valuations

Following the direction pioneered by Fiat and Papadimitriou in their 2010 paper [12], we study the complexity of deciding the existence of mixed equilibria for minimization games where players use valuations other than expectation to evaluate their costs. We consider risk-averse players seeking to minimize the sum V=E+R of expectationE and a risk valuationR of their costs; R is non-negative and vanishes exactly when the cost incurred to a player is constant over all choices of strategies by the other players. In a V-equilibrium, no player could unilaterally reduce her cost.Say that V has the Weak-Equilibrium-for-Expectation property if all strategies supported in a player's best-response mixed strategy incur the same conditional expectation of her cost. We introduce E-strict concavity and observe that every E-strictly concave valuation has the Weak-Equilibrium-for-Expectation property. We focus on a broad class of valuations shown to have the Weak-Equilibrium-for-Expectation property, which we exploit to prove two main complexity results, the first of their kind, for the two simplest cases of the problem:•Two strategies: Deciding the existence of a V-equilibrium is strongly NP-hard for the restricted class of player-specific scheduling games on two ordered links[22], when choosing R as (1)Var (variance), or (2)SD (standard deviation), or (3) a concave linear sum of even moments of small order.•Two players: Deciding the existence of a V-equilibrium is strongly NP-hard when choosing R as (1)γ⋅Var, or (2)γ⋅SD, where γ>0 is the risk-coefficient, or choosing V as (3) a convex combination of E+γ⋅Var and the concave ν-valuationν−1(E(ν(⋅))), where ν(x)=xr, with r≥2. This is a concrete consequence of a general strong NP-hardness result that only needs the Weak-Equilibrium-for-Expectation property and a few additional properties for V; its proof involves a reduction with a single parameter, which can be chosen efficiently so that each valuation satisfies the additional properties.

Hybrid Projection Methods for Bregman Totally Quasi-D-Asymptotically Nonexpansive Mappings

In this paper, a new iterative scheme by hybrid projection method is proposed for a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings. Conditions ensuring strong convergence are imposed to common elements of set of common fixed points of the mappings and set of common solutions to a system of generalized mixed equilibrium problems in a reflexive Banach space. These results extend many important recent ones in the literature.

Game theory based emotional evolution analysis for chinese online reviews

Sentiment analysis has become one of the mainstream researches in social network analysis. Its impact can be seen in many practical applications, ranging from public opinion analysis to marketing of public praise and information prediction. However, most of the existing research has been performed in the sentiment classification for subjective text, the emotional evolution analysis for complex interactive text (e.g., online reviews) has not yet been thoroughly targeted by the research community. This paper is concerned on short-text Chinese online reviews collected from Tianya forum. First, an efficient affective computing framework is proposed to capture the underlying emotions of Chinese online reviews. It can accurately calculate the semantic orientation of the entire review, without requiring expensive manual labeling of seed words. As users’ attitudes might influence with each other, predicting their future emotional behaviors that only relying on the emotional values of historical reviews is very one-sided. Therefore, we propose a game theory based emotional evolution prediction algorithm combining the affective computing, in which the mixed nash equilibrium strategies are calculated as the future emotional behavior of interactive users. Then, experimental results on the large-scaled review dataset are provided to demonstrate the effectiveness and accurateness of our approaches. Finally, by applying the research results on the pairwise happiness-popularity coordination evaluation, we have revealed some interesting phenomenon on the “World View” board in Tianya forum.