Unique Pure Strategy Nash Equilibrium Research Articles

Multi-battle contests over complementary battlefields

AbstractThis paper studies contests with complementary prizes where each agent simultaneously distributes a fixed budget over multiple battlefields. Each battlefield has a single prize which is divided among the competitors in proportion to an arbitrary power function of their investment levels. A unique pure strategy Nash equilibrium is shown to exist under arbitrarily sensitive battlefield success functions if objective functions exhibit constant subunitary elasticity of substitution between prize shares. In contrast, Blotto contests with linear objectives have only mixed strategy Nash equilibria if battlefield success functions are sufficiently sensitive to investment levels. Sufficient complementarity between prize shares allows pure strategy Nash equilibria to exist under arbitrarily sensitive battlefield success functions.

Non-Cooperative Games for Privacy-Preserving and Cost-Efficient Smart Grid Energy Management

In this paper, we design privacy-preserving and cost-efficient energy management strategies for smart grid users that are equipped with renewable energy sources. The adversary is assumed to employ a factorial hidden Markov model based inference for load disaggregation, and the corresponding joint log-likelihood of the model is utilized as the privacy measure. The studied dynamic pricing model is applicable to a commodity-limited market, where the price of unit amount of energy is determined by the users’ aggregated power request. The users’ energy management strategies are designed under a non-cooperative game framework, where each user aims to optimize a weighted sum objective of both privacy measure and energy cost saving. The users’ non-cooperative game is shown to admit a unique pure strategy Nash equilibrium. As an extension, a computational-efficient distributed Nash equilibrium energy management strategy seeking method is proposed, which also avoids the privacy leakage due to the sharing of payoff functions between users. The performance of practical designs of the energy management strategies in the equilibrium is finally illustrated by numerical experiments.

Efficient and Robust Equilibrium Strategies of Utilities in Day-Ahead Market With Load Uncertainty

We consider the scenario where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> utilities strategically bid for electricity in the day-ahead market and balance the mismatch between the committed supply and actual demand in the real-time market, with uncertainty in demand and local renewable generation in consideration. We model the interactions among utilities as a noncooperative game, in which each utility aims atminimizing its per-unit electricity cost. We investigate utilities’ optimal bidding strategies and show that all utilities bidding according to (net load) prediction is a unique pure strategy Nash equilibrium with two salient properties. First, it incurs no loss of efficiency; hence, competition among utilities does not increase the social cost. Second, it is robust and (0, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N-1$</tex-math></inline-formula> ) fault immune. That is, fault behaviors of irrational utilities only help to reduce other rational utilities’ costs. The expected market supply–demand mismatch is minimized simultaneously, which improves the planning and supply-and-demand matching efficiency of the electricity supply chain. We prove the results hold under the settings of correlated prediction errors and a general class of real-time spot pricing models, which capture the relationship between the spot price, the day-ahead clearing price, and the market-level mismatch. Simulations based on real-world traces corroborate our theoretical findings. Our article adds new insights to market mechanism design. In particular, we derive a set of fairly general sufficient conditions for the market operator to design real-time pricing schemes so that the interactions among utilities admit the desired equilibrium.

Competition and equilibrium effort choice

We examine optimal effort choice in a competition model where the agents are averse to low relative status and to exerting excessive effort. The game has a unique pure strategy Nash equilibrium. When the agents are homogeneous, a stronger competition incentive induces higher effort levels and results in lower utilities. When agents are heterogeneous, the model predicts that (i) all else equal, initial losers exert more effort but cannot become final winners, (ii) changes in effort choices of a subgroup of agents have a spillover effect, and (iii) agents with competitive advantages may prefer not to compete if the advantages are not significant enough. Some extensions and variants of the baseline model are also examined. Overall, the model’s theoretical predictions are consistent with some competition-related findings documented in the empirical literature.

M Equilibrium: A Theory of Beliefs and Choices in Games

We introduce a set-valued solution concept, M equilibrium, to capture empirical regularities from over half a century of game theory experiments. We show M equilibrium serves as a meta theory for various models that hitherto were considered unrelated. M equilibrium is empirically robust and, despite being set-valued, falsifiable. Results from a series of experiments that compare M equilibrium to leading behavioral game theory models demonstrate its virtues in predicting observed choices and stated beliefs. Data from experimental games with a unique pure-strategy Nash equilibrium and multiple M equilibria exhibit coordination problems that could not be anticipated through the lens of existing models. (JEL C72, C90, D83)

The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

Non-cooperative queueing games on a network of single server queues

This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2times 2-grid, and (iv) 2-player games on an Atimes B-grid with small differences in the service rates.

M Equilibrium: A Theory of Beliefs and Choices in Games

We introduce a set-valued solution concept, M equilibrium, to capture empirical regularities from over half a century of game-theory experiments. We show M equilibrium serves as a meta theory for various models that hitherto were considered unrelated. M equilibrium is empirically robust and, despite being set-valued, falsifiable. We report results from a series of experiments comparing M equilibrium to leading behavioral-game-theory models and demonstrate its virtues in predicting observed choices and stated beliefs. Data from experimental games with a unique pure-strategy Nash equilibrium and multiple M equilibria exhibit coordination problems that could not be anticipated through the lens of existing models.

Hierarchical Optimal Scheduling for Air Conditioner’s Load Aggregator

In this paper, a scheduling framework based on air-conditioning load aggregator is first proposed. Then the split constant frequency air-conditioning load is optimized by minute level. The step function characteristic between the average air-conditioning load per hour and the temperature difference set at the beginning and the end of the time is obtained and linearized. A temperature difference-power model suitable for air-conditioning hourly scheduling is established. Secondly, aiming at the multiple load aggregators existing in the distribution area, this paper proposes a non-cooperative game model of air conditioning load aggregators. Considering the problem of discretization of control variables in the game revenue function, this paper introduces continuous intermediate variables, establishes a hierarchical optimization model of the game, and gives the proof that the hierarchical optimization of the game has a unique pure strategy Nash equilibrium solution and the solution method. Finally, the feasibility of the proposed method is verified by numerical simulation.

Imposing Choice under Ambiguity: The Case of Dynamic Currency Conversion

It is a common experience for present-day consumers making an international payment via credit or debit card to be invited to choose the currency in which they wish to have the transaction executed. While this choice, made feasible by a technology known as dynamic currency conversion (DCC), seems to foster competition, we show that the opposite is the case. In fact, the unique pure-strategy Nash equilibrium in anatural fee-setting game turns out to be highly asymmetric, entailing fees for the service provider that always exceed the monopoly level. Although losses in welfare may be substantial, a regulatory solution is unlikely to come about due to a global free-rider problem.

A Game-Theoretic Framework for Multiperiod-Multicompany Demand Response Management in the Smart Grid

By utilizing tools from the game theory, we develop a novel multiperiod-multicompany demand response framework considering the interactions between companies (sellers of energy) and their consumers (buyers of energy). We model the interactions in terms of a Stackelberg game, where companies set their prices and consumers respond by choosing their demands. We show that the underlying game has a unique equilibrium at which the companies maximize their revenues, while the consumers maximize their utilities subject to their local constraints. Closed-form expressions are provided for the optimal strategies of all players. Based on these solutions, a power-allocation game has been formulated, which is shown to admit a unique pure-strategy Nash equilibrium, for which closed-form expressions are also provided. This equilibrium is found under the assumption that companies can freely allocate their power across the time horizon, but we also demonstrate that it is possible to relax this assumption. We further provide a fast distributed algorithm for the computation of all optimal strategies using only local information. We also study the effect of variations in the number of periods (subdivisions of the time horizon) and the number of consumers. As a consequence, we are able to find an appropriate company-to-consumer ratio when the number of consumers participating in demand response exceeds some threshold. Furthermore, we show, both analytically and numerically, that the multiperiod scheme provides incentives for energy consumers to participate in demand response compared with the single-period framework studied by Maharjan et al. In our framework, we provide a condition for the minimum budget consumers need and carry out case studies using real-life data to demonstrate the benefits of the approach, which show potential savings of up to 30% and equilibrium prices that have low volatility.

Tullock Contest: A Model of Proof-of-Work Mining in Cryptocurrencies

This paper presents an N-player Tullock contest where players purchases shares in a lottery at a fixed common cost. Once all players have purchased their shares, a common value reward is raffled off. The model has some similarities to all-pay auctions, however the assignment mechanism differs. This difference ensures that, unlike all-pay auctions, the N-player Tullock contest has an unique pure strategy Nash equilibrium. This article presents also a new kind of Tullock contest that integrates the main mechanism behind proof-of-work mining of cryptocurrencies such as Bitcoin. The application of the model is expanded to analyze some topics of proof-of-work protocols, such as concentration of hashing power, double-spending attacks and environmental externalities.

Variational Inequality and Distributed Learning for a Bidding Game in Electricity Supply Markets

This study uses a game-theoretic analysis of bid-based electricity supply market equilibrium. Electricity supply markets are modeled as strategic interactions of bidders that supply electric power to the market and the bidders’ pure strategies are the cost function parameters of power generation. We demonstrate that the resultant bidding game is a convex game and has a unique pure-strategy Nash equilibrium (PNE) when the bid-cost functions are parameterized by marginal costs of power generation. The PNE of the power-supply bidding game is reformulated in terms of a variational inequality and as a fixed-point of a recursive mapping. We propose two distributed learning algorithms and their variations with convergence analysis to compute a PNE. Three types of measures are proposed and analyzed for quantification of inefficiency due to falsified bidding actions corresponding to the marginal cost function parameters of supply-market participative generators. A numerical case study with a 26-bus power network is presented to illustrate and demonstrate our results.

Multi-market Oligopoly of Equal Capacity

We consider a variant of Cournot competition, where multiple firms allocate the same amount of resource across multiple markets. We prove that the game has a unique pure-strategy Nash equilibrium (NE), which is symmetric and is characterized by the maximal point of a "potential function". The NE is globally asymptotically stable under the gradient adjustment process, and is not socially optimal in general. An application is in transportation, where drivers allocate time over a street network.

Unique equilibrium in contests with incomplete information

Considered are imperfectly discriminating contests in which players may possess private information about the primitives of the game, such as the contest technology, valuations of the prize, cost functions, and budget constraints. We find general conditions under which a given contest of incomplete information admits a unique pure-strategy Nash equilibrium. In particular, provided that all players have positive budgets in all states of the world, existence requires only the usual concavity and convexity assumptions. Information structures that satisfy our conditions for uniqueness include independent private valuations, correlated private values, pure common values, and examples of interdependent valuations. The results allow dealing with inactive types, asymmetric equilibria, population uncertainty, and the possibility of resale. It is also shown that any player that is active with positive probability ends up with a positive net rent.

A novel GSP auction mechanism for ranking Bitcoin transactions in blockchain mining

Bitcoin is gaining ground in recent years. In the Bitcoin system, miners provide computing power to confirm transactions and mine blocks in pursuit of transaction fees, while users compete by bidding transaction fees for faster confirmation. This process is in essence analogous to online ad auctions, where advertisers bid for more prominent ad slots. Therefore, inspired by the Generalized Second Price (GSP) mechanism dominantly used in online ad auctions, we propose to adopt the GSP auction model in the Bitcoin transaction confirmation game. Also, we use weighted fees as the new ranking basis, which can be calculated by user-submitted fees, transaction size, quality scores and virtual fees accumulated from the waiting time. We show that the formulated static GSP transaction fee auction with complete information has a unique Pure Strategy Nash Equilibrium. Then, we discuss the impacts of quality scores and virtual fees on users' equilibrium fee decisions and payoffs. Finally, computational experiments are designed to validate our theoretical models and analysis. Our research findings indicate that this novel GSP mechanism is superior to the currently adopted GFP mechanism, and can help users save fees. Besides, quality scores and virtual fees are also proven to be effective on reducing users' paid fees. Moreover, the design of virtual fees allows all transactions to be processed more efficiently in a uniform pipeline, and the interests of transactions with and without associated fees are taken into consideration.

When Are Mixed Equilibria Relevant?

Mixed Nash equilibrium (NE) is a cornerstone of game theory, but its empirical relevance has always been questionable. We study in the laboratory two games whose unique NE is in completely mixed strategies; other treatments include the matching protocol (pairwise random vs population mean matching), whether time is discrete or continuous, and whether players can specify mixtures explicitly or only pure strategy realizations. NE mixes predict observed behavior better than maximin in all treatments, but uniform mixes are better predictors than any equilibrium mixture in many treatments. By contrast, in a control game with a unique NE in pure strategies, the best point prediction is NE. Mixed equilibrium predictions are more useful in population mean matching than in standard pairwise matching. Regret-based sign preserving dynamics capture regularities across all treatments.

Managing Price Uncertainty in Prosumer-Centric Energy Trading: A Prospect-Theoretic Stackelberg Game Approach

In this paper, the problem of energy trading between smart grid prosumers, who can simultaneously consume and produce energy, and a grid power company is studied. The problem is formulated as a single-leader, multiple-follower Stackelberg game between the power company and multiple prosumers. In this game, the power company acts as a leader who determines the pricing strategy that maximizes its profits, while the prosumers act as followers who react by choosing the amount of energy to buy or sell so as to optimize their current and future profits. The proposed game accounts for each prosumer’s subjective decision when faced with the uncertainty of profits, induced by the random future price. In particular, the framing effect, from the framework of prospect theory (PT), is used to account for each prosumer’s valuation of its gains and losses with respect to an individual utility reference point. The reference point changes between prosumers and stems from their past experience and future aspirations of profits. The followers’ noncooperative game is shown to admit a unique pure-strategy Nash equilibrium (NE) under classical game theory which is obtained using a fully distributed algorithm. The results are extended to account for the case of PT using algorithmic solutions that can achieve an NE under certain conditions. Simulation results show that the total grid load varies significantly with the prosumers’ reference point and their loss-aversion level. In addition, it is shown that the power company’s profits considerably decrease when it fails to account for the prosumers’ subjective perceptions under PT.

Simultaneous and sequential voting under general decision rules

In an economic theory of voting, voters have positive or negative costs of voting in favor of a proposal and positive or negative benefits from an accepted proposal. When votes have equal weight then simultaneous voting mostly has a unique pure strategy Nash equilibrium which is independent of benefits. Voting with respect to (arbitrarily small) costs alone, however, often results in voting against the ‘true majority’ . If voting is sequential as in the roll call votes of the US Senate then, in the unique subgame perfect equilibrium, the ‘true majority’ prevails. It is shown that the result for sequential voting holds also with different weights of voters (shareholders), with multiple necessary majorities (European Union decision-making), or even more general rules. Simultaneous voting in the general model has more differentiated results.

Bertrand Oligopoly Competition in Composite Web Service Network

Because of its heterogeneous nature, a web service can be composed of multiple composite web services. To improve profitability in the software-component economy, software as a service (SaaS) service providers compete more at the composite-service level than at the single-service level. Moreover, because of the collaborative environment, composite web service networks determine both the applicability of the web service and its expected economic behavior. Based on the traditional linear demand model, this article presents a congestion-aware demand model that makes several assumptions regarding the SaaS service marketplace. Then, it formulates the SaaS service providers' pricing behaviors as a network Bertrand oligopoly competition. Key game-theoretic analysis includes the existence and uniqueness of the pure strategy Nash equilibrium. Moreover, this article provides one sufficient condition, where if all SaaS service providers follow the best response strategy, the strategy profile converges to the unique pure strategy Nash equilibrium.