Mixed-strategy Equilibrium Research Articles

Tax Competition in Presence of Profit Shifting

The popular view is that governments should crack down on tax avoidance by multinational firms. In this paper, we analyze how anti-profit-shifting policies influence fiscal competition. Governments commit to profit shifting control effort and then set taxes on capital. Equilibrium tax rates are determined by the elasticities of the two components: profit shifting and capital mobility. Anti-profit-shifting policies decrease the elasticity of the first but increase the elasticity of the second, so that the impact of these policies on the equilibrium of the tax game is ambiguous. We show that there are cases in which laxer policies increase all equilibrium tax rates and that the country announcing laxer profit shifting policies may gain. It appears that there is not always a pure strategy equilibrium in such a fiscal competition game. We construct a mixed strategy equilibrium when the pure strategy equilibrium does not exist.

The n-Player Hirshleifer Contest

We characterize the equilibrium set of the n-player Hirshleifer contest with homogeneous valuations. A symmetric equilibrium always exists. It necessarily corresponds to multilateral peace for sufficient noise and uses finite-support randomized strategies otherwise. Asymmetric equilibria are feasible for n≥3 contestants only, and only for sufficiently small noise. In pure strategies, any asymmetric equilibrium corresponds to one-sided dominance, but there is also a variety of payoff-inequivalent mixed-strategy equilibria for small noise. For arbitrarily small noise, at least two contestants engage in cut-throat competition, while any others become ultimately inactive. Of some conceptual interest is the observation that, for n sufficiently large, the unique equilibrium is multilateral peace.

Refinements of Nash equilibrium in a Pentagonal Fuzzy Bimatrix Game

In this paper, we introduce the concept of symmetric pentagonal fuzzy bimatrix and constant pentagonal fuzzy matrix for which we define the existence of Nash equilibrium in both pure and mixed strategies along with some numerical illustrations. By applying Ranking to the pay-offs, we convert the fuzzy valued game problem to crisp valued game problem. Therefore, Solution concept of perfect equilibrium for normal form game is reviewed and a new concept of proper equilibrium of pentagonal fuzzy bimatrix game is discussed.

On Submodular Search and Machine Scheduling

Suppose that some objects are hidden in a finite set S of hiding places that must be examined one by one. The cost of searching subsets of S is given by a submodular function, and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of S that finds all the objects with minimal expected cost. This problem is NP-hard, and we give an efficient combinatorial 2-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function of the completion times of subsets of jobs. We go on to give better approximations for submodular functions with low total curvature, and we give a full solution when the problem is what we call series-parallel decomposable. Next, we consider a zero-sum game between a cost-maximizing hider and a cost-minimizing searcher. We prove that the equilibrium mixed strategies for the hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.

Supplier Competition and Cost Reduction with Endogenous Information Asymmetry

Problem definition: We consider a buyer sourcing from multiple competing suppliers who exert cost-reduction efforts before procurement contracts are awarded. Academic/practical relevance: The supply chain is subject to the classic hold-up problem—as the lack of a contract commitment hinders suppliers’ incentives to make investment upfront—complicated with supplier competition. Methodology: With deterministic cost-reduction outcomes, suppliers will not exert any effort if this effort is observable, and a pure-strategy equilibrium does not exist if the effort is unobservable. We analyze the mixed-strategy equilibrium with unobservable supplier effort, in which suppliers randomize their efforts and the buyer designs an optimal procurement mechanism. Results: We show that the optimal procurement mechanism can be implemented by a conventional single-price reverse auction with a random reserve price. The mixed strategy of supplier effort generates endogenous information asymmetry on supplier costs that provides suppliers with information rent, which sustains their efforts. The endogenous information asymmetry improves effort efficiency (by inducing positive supplier effort), yet introduces trade inefficiency (by causing the possible failure of trade between the parties). Although increasing supplier competition (measured by the number of suppliers) hurts the effort efficiency, it improves trade efficiency. As a result, the buyer is always better off introducing supplier competition by including more than one supplier in the supply base. However, the desired supply base size (number of suppliers) depends on the product revenue: For high-margin goods, the optimal size is achieved with two suppliers, whereas for low-margin goods, a larger supply base is better for the buyer. We show that the result based on deterministic cost reduction can be established as a limit of the case when uncertainty in cost reduction exists and shrinks to null. Managerial implications: Our study helps to understand the impact of supplier competition when supply-chain parties deliberately make their actions unpredictable to avoid being held up. The findings provide managerial guidance on procurement auction and supply base designs.

Random Access Games With Cost of Waiting for Uplink NOMA Systems

In power domain non-orthogonal multiple access (NOMA) used for uplink random access (RA) systems, users need to control transmit power such that the received power at the base station can be one of the predetermined values. Thus, selecting one of the predetermined values incurs different transmission costs. Furthermore, users can often withdraw their (re)transmission, which involves a waiting (or regret) cost. This letter analyzes ${N}$ -user non-cooperative game of uplink NOMA RA systems, where waiting cost is taken into account. As results, we show the condition, under which a unique mixed-strategy Nash equilibrium (MNE) exists and its efficiency with respect to social welfare maximization (SWM).

When will party whips succeed? Evidence from almost symmetric voting games

The true preferences of some members of parliament sometimes imply voting against the party line, but these members fear sanctions. In the mostly unique pure strategy equilibrium of such a voting game, voters always follow the party line, even if their true preferences are strong and (opportunity) costs from sanctions are arbitrarily small. I qualify this disturbing result by applying the equilibrium selection theory of Harsanyi and Selten (1988). Pure strategy equilibria are selected only for large deviation costs. Otherwise, mixed strategy equilibria are selected which support majorities in true preferences (true majorities), for negligible costs almost with certainty.

Can the US and Europe contain Russian power in the European energy market? A game theoretic approach

The Russian history of politicization of natural gas supplies led Europe to seek alternatives. US LNG is considered to be a viable alternative for Europe, which positioned the US as a strong competitor against Russia. In a game theoretic approach, we model this strategic decision-making process in simultaneous-move and sequential games. Our findings indicate a mixed strategy equilibrium: Europe cannot commit to diversify in the face of Russian advantages, and Russia cannot fully resist the temptation to politicize its supplies to Europe. The US might attempt to intervene in the European commitment problem through implementing sanctions on Russian gas sales. We find that in any case episodic Russian politicization remains likely in equilibrium.

Dynamic pricing under competition with data-driven price anticipations and endogenous reference price effects

Online markets have become highly dynamic and competitive. Many sellers use automated data-driven strategies to estimate demand and to update prices frequently. Further, notification services offered by marketplaces allow to continuously track markets and to react to competitors’ price adjustments instantaneously. To derive successful automated repricing strategies is challenging as competitors’ strategies are typically not known. In this paper, we analyze automated repricing strategies with data-driven price anticipations under duopoly competition. In addition, we account for reference price effects in demand, which are affected by the price adjustments of both competitors. We show how to derive optimized self-adaptive pricing strategies that anticipate price reactions of the competitor and take the evolution of the reference price into account. We verify that the results of our adaptive learning strategy tend to optimal solutions, which can be derived for scenarios with full information. Finally, we analyze the case in which our learning strategy is played against itself. We find that our self-adaptive strategies can be used to approximate equilibria in mixed strategies.

Optimal dynamic auditing based on game theory

A dynamic model based on game-theory is proposed to address the problem of fraud detection in auditing under non-linear payoff functions. Non-linearity is introduced by incorporating learning and sympathy effects in the audit process. It is proven that the audit/fraud detection game between two new engagement parties has a unique mixed strategy equilibrium, between an experienced auditor and a client has a unique pure strategy equilibrium, whereas in the long-run the game converges to a pure strategy equilibrium. In addition to this, to ensure an acceptable level of quality in the audit process, a closed form formula used to estimate the optimal auditor’s replacement time is extracted. The validity of the proposed scheme is tested on empirical data and modeling results comply with the International Standard of Auditing that requires the key audit partner to be rotated after a predefined period.

A complete-information partnership contest with negative productivity spillovers

This note examines a complete-information contest with non-linear rank-order effects. The players’ payoffs incorporate a multiplicative term that allows for negative productivity spillovers. Under certain conditions, our game has a symmetric mixed-strategy equilibrium. To characterize this equilibrium, we adapt a method originally developed for linear settings. We obtain closed-form solutions for the mixing distribution, as well as for the expected efforts and payoffs. The note also explores the implications of negative productivity spillovers for optimal contest design. We show that a planner who seeks to maximize the participants’ expected payoffs may choose interior values for her instruments. We derive simple expressions for these values.

EQUILIBRIUM BALKING STRATEGIES IN THE REPAIRABLE M/M/1 G-RETRIAL QUEUE WITH COMPLETE REMOVALS

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

A strategic approach for the crime of tax evasion

Purpose Crime games cannot be simply read with mixed strategies. These strategies are inconclusive of how the players act rationally. This is undeniably true for the crime of tax evasion, where dishonest taxpayers are rational agents, motivated by the comparison of payoffs, when considering the risk of non-compliance. The purpose of this paper is to illustrate that in the presence of a small “private disturbance” of the players’ payoff, the Nash equilibrium in mixed strategies provides us with the necessary information on equilibria in pure strategies that will be played. Design/methodology/approach In tax-evasion games, an equilibrium must necessarily be interpreted in pure strategies, and the only way to do this is to insert some private information into the game and reinterpret it in a Bayesian scheme. We show that taxpayers’ private,subjective considerations on the effective implementation of the penalty and the revenue agency’s private information on the cost of monitoring and conviction can lead to Bayesian equilibria in pure strategies. The present paper takes issue with this Bayesian equilibrium and the implications for comparative-statics results. Findings In this context, tougher sentencing deters crime, although, as the Italian experience teaches, the necessary condition required is the certainty of punishment and the ability of the government to enforce it. The equilibrium strategies with incomplete information reveal whether it is convenient for the two agents to maintain their “private disturbance” as private information or, on the contrary, it is convenient to expect it to be “common knowledge.” Originality/value A distinct set of studies has adopted a game theoretic approach and shows that the standard economic approach to crime deterrence inspired by Gary Beker’s seminal paper might be flawed. See, among others, Saha and Poole (2000), Tsebelis (1989) and Andreozzi (2010). This paper shows that a greater severity of the penalty and a higher certainty of punishment (a lower possibility of appealing against sanctions and no discounts on due penalties) necessarily lead to a unique Bayesian equilibrium without evasion.

A Data-Driven Game Theoretic Strategy for Developers in Software Crowdsourcing: A Case Study

Crowdsourcing has the advantages of being cost-effective and saving time, which is a typical embodiment of collective wisdom and community workers’ collaborative development. However, this development paradigm of software crowdsourcing has not been used widely. A very important reason is that requesters have limited knowledge about crowd workers’ professional skills and qualities. Another reason is that the crowd workers in the competition cannot get the appropriate reward, which affects their motivation. To solve this problem, this paper proposes a method of maximizing reward based on the crowdsourcing ability of workers, they can choose tasks according to their own abilities to obtain appropriate bonuses. Our method includes two steps: Firstly, it puts forward a method to evaluate the crowd workers’ ability, then it analyzes the intensity of competition for tasks at Topcoder.com—an open community crowdsourcing platform—on the basis of the workers’ crowdsourcing ability; secondly, it follows dynamic programming ideas and builds game models under complete information in different cases, offering a strategy of reward maximization for workers by solving a mixed-strategy Nash equilibrium. This paper employs crowdsourcing data from Topcoder.com to carry out experiments. The experimental results show that the distribution of workers’ crowdsourcing ability is uneven, and to some extent it can show the activity degree of crowdsourcing tasks. Meanwhile, according to the strategy of reward maximization, a crowd worker can get the theoretically maximum reward.

A second-order cone programming formulation for two player zero-sum games with chance constraints

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium in mixed strategies if the row vectors of the random matrices defining the stochastic constraints are elliptically symmetric distributed random vectors. We further show that a saddle point equilibrium can be obtained from the optimal solutions of a primal-dual pair of second-order cone programs.

Equilibrium Market Volatility in Imperfect Competition of General Equilibrium

We develop a theory of equilibrium market volatility in a general equilibrium duopoly with complete information. The resulting economic system possesses a property, which can be described as ‘natural volatility’ of markets, even if players have complete information. Economy is described as a strategic market game where every player has market power as a buyer and a seller. Players are uncertain about each other’s actions, and due to the properties of expected utility maximization there is no convergence of common beliefs of common knowledge to unique mixed strategies equilibrium, hence, rational expectations approach cannot be applied. Volatility at the market appears from a multiplicity of equilibrium mixed strategies; they cannot be computed precisely, but only approximated. We identify several Pareto-efficient strategies and study induced prices, allocations and payoffs, which are all volatile.

Fiat Money Oligopolistic Economy With Labor/Leisure Trade-Off and Equilibrium Default

We are constructing an imperfect competition general equilibrium model, with non-consumable money and labor market; our toolkit is an equilibrium default model of Shubik-Wilson (1978). Our result has an ‘equilibrium volatility’ simultaneously occurring at all three markets: labor, goods, and credit market, with a fixed money supply from a bank. A worker and an entrepreneur strategically and simultaneously trade at the three markets. Players are uncertain about each other’s actions, and do not have a convergence of common beliefs of common knowledge. It is impossible to calculate equilibrium mixed strategies exactly, but it is possible to identify some Pareto-efficient strategies and study induced prices, wages, interest rates, default, allocations and payoffs, which are all volatile, and their fluctuations are not independent. We present an equilibrium result, when a total value of default is bigger than total money supply.

Hotelling Competition With Discrete Consumers: Comparing Equilibrium Concepts in Spatial Competition

Combining the Hotelling line with the realistic assumption that markets contain finitely many consumers, produces a model of spatial competition that does not possess a Nash equilibrium in pure strategies. This paper characterises both the ex-ante stable mixed strategy Nash equilibrium and the ex-post stable undercut-proof equilibrium for the model, and compares them with the equilibrium of the standard Hotelling model with a continuum of consumers. The mixed strategy equilibrium is unique and converges to the equilibrium of the standard model from above, indicating that the use of a continuum is a reasonable approximation of the mixed-strategy equilibrium for a market with a large number of buyers. By contrast, the undercut-proof equilibrium converges to half this level as marginal competition is more intense when the location of the point of indifference relative to the marginal consumers can be observed ex-post. Incorporating location choice into the model shows that the greater intensity of competition in the undercut-proof equilibrium, leads firms to locate closer to the centre of the line. These results provide a novel explanation for the tendency of product characteristics to converge in mature markets.

On competitive nonlinear pricing

We study a discriminatory limit-order book in which uninformed market makers compete in nonlinear tariffs to serve an informed insider. Our model allows for general nonparametric specifications of preferences and for arbitrary discrete distributions for the insider's private information. We show that adverse selection severely restricts possible equilibrium outcomes: in any pure-strategy equilibrium, tariffs must be linear and at most one type may trade, leading to an extreme form of market breakdown. As a result, such equilibria only exist under exceptional circumstances. The Bertrandlike logic underlying these results markedly differs from Cournot-like analyses of the limit-order book that postulate a continuum of types. We argue that these contrasting outcomes can be reconciled when one considers -equilibria of either the game with a large number of market makers or the game with a large number of insider types. Mixed-strategy equilibria, by contrast, lead to a new class of equilibrium predictions that calls for further analysis.

Strategic Joining and Pricing Policies in a Retrial Queue With Orbital Search and Its Application to Call Centers

This paper treats strategic joining and pricing policies in an M/M/1 retrial queue with orbital search which is motivated by the application in call centers, where the server will make orbital search or remain idle whenever he completes a service, the orbital search time follows exponential distribution. Given a natural reward-cost structure and imposed on an admission fee, all arriving customers decide to whether to join the orbit or balk when they find the server busy. Using queueing theory and game theory, we first analyze the Nash equilibrium mixed joining strategy for individual customer. Further we investigate the optimal joining probabilities and corresponding admission pricing problems that maximize the administrator's revenue and social profit, respectively. Finally, we present some numerical examples to demonstrate the effect of some system parameters on the sensitivity of the solutions of the individual maximization, administrator's maximization and social optimization.