Pure Nash Equilibrium Research Articles

Quantum Games with Strategies Induced by Basis Change Rules

The aim of the paper is to draw attention to a special class of two parameter unitary strategies in the Eisert-Wilkens-Lewenstein scheme for quantum games. We identify the players’ strategies with basis change matrices. Then we prove that the resulting quantum game is invariant with respect to isomorphic transformations of the input game. Moreover, it is shown that the game so obtained may not be trivial with respect to pure Nash equilibria, compared with the model with strategies being the special unitary group SU(2).

Distributed cell selection in heterogeneous wireless networks

A critical issue in many wireless networks is how to establish the best possible quality connections between users to base stations. This is in particular challenging when users are randomly located over a geographical region, and each covered by a number of heterogeneous base stations. To this end, we design a smart and efficient cell selection mechanism to improve the user-base station connection in heterogeneous wireless networks. We formulate the cell selection problem as an asymmetric congestion game with consideration of users’ heterogeneity in their locations and their data rates to various cells. We show the existence of pure Nash equilibria (PNE) and propose a concurrent distributed learning algorithm to converge to them. In the algorithm, we allow users to perform random error-tolerant updates synchronously, and guarantee them to reach one or multiple PNE with the largest utilities. In addition, we do a systematical investigation on the implementation of the algorithm in practical networks. Simulation results show that the algorithm can achieve satisfactory performance with acceptable convergence rate.

Airline Booking Limit Competition Game Under Differentiated Fare Structure

We address a two-firm booking limit competition game in the airline industry. We assume aggregate common demand, and differentiated ticket fare and capacity, to make this study more realistic. A game theoretic approach is used to analyze the competition game. The optimal booking limits and the best response functions are derived. We show the existence of a pure Nash equilibrium and provide the closed-form equilibrium solution. The location of the Nash equilibrium depends on the relative magnitude of the ratios of the full and discount fares. We also show that the sum of the booking limits of the two firms remains the same regardless of the initial allocation proportion of the demand.

A game model of competition between a new good producer and a remanufacturer using negative advertising

In this paper we analyze the strategic interaction between a new good producer and a remanufacturer who use negative advertising on television (TV) to compete for a greater share of the market for a particular good. Government regulations limit the total amount of negative advertising time either firm can buy. The two rival firms choose how much negative advertising time to buy simultaneously. Our analysis of this duopolistic interaction leads to four results. First, we provide the normal form representation of the game between the new good producer and the remanufacturer. Second, we specify the best response functions of the two firms. Third, we determine the pure strategy Nash equilibrium of the game under study and point out that this equilibrium is unique. Finally, we ascertain the amount of negative advertising time the two firms would buy if they could come to a binding agreement to curtail this kind of advertising.

Robust fully distributed file caching for delay-tolerant networks: A reward-based incentive mechanism

This article exhibits a reward-based incentive mechanism for file caching in delay-tolerant networks. In delay-tolerant networks, nodes use relay’s store-carry-forward paradigm to reach the final destination. Thereby, relay nodes may store data in their buffer and carry it till an appropriate contact opportunity with destination arises. However, the relays are not always willing to assist data forwarding due to a limited energy or a low storage capacity. Our proposal suggests a reward mechanism to uphold and to sustain cooperation among relay nodes. We model this distributed network interaction as a non-cooperative game. Namely, the source node offers to the relay nodes a positive reward if they accept to cache and to forward a given file successfully to a target destination, whereas the relay nodes may either accept or reject the source deal, depending on the reward attractiveness and on their battery status (their actual energy level). Next, full characterizations of both pure and mixed Nash equilibria are provided. Then, we propose three fully distributed algorithms to ensure convergence to the Nash equilibria (for both pure equilibrium and mixed equilibrium). Finally, we validate our proposal through extensive numerical examples and many learning simulations and draw some conclusions and insightful remarks.

On Strategic Complementarities in Discontinuous Games

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.

On the Efficiency of All-Pay Mechanisms

We study the inefficiency of mixed Nash equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 (Syrgkanis and Tardos in Proceedings of the 45th symposium on theory of computing (STOC ’13), 2013) to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi-unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mixed Nash equilibria in mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy with respect to social welfare, revenue and maximum bid.

Dynamics and stability of potential hyper-networked evolutionary games

This paper considers the modeling and convergence of hyper-networked evolutionary games (HNEGs). In an HNEG the network graph is a hypergraph, which allows the fundamental network game to be a multi-player one. Using semi-tensor product of matrices and the fundamental evolutionary equation, the dynamics of an HNEG is obtained and we extend the results about the networked evolutionary games to show whether an HNEG is potential and how to calculate the potential. Then we propose a new strategy updating rule, called the cascading myopic best response adjustment rule (MBRAR), and prove that under the cascading MBRAR the strategies of an HNEG will converge to a pure Nash equilibrium. An example is presented and discussed in detail to demonstrate the theoretical and numerical results.

Constrained Pure Nash Equilibria in Polymatrix Games

We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consistent with some pure Nash equilibrium or, alternatively, with all pure Nash equilibria. We identify the most natural tractable cases and show NP or coNP-completness of these problems already for unweighted DAGs.

Fans Economy and All-Pay Auctions with Proportional Allocations

In this paper, we analyze an emerging economic form, called fans economy, in which a fan donates money to the host and gets allocated proportional to the amount of his donation (normalized by the overall amount of donation). Fans economy is the major way live streaming apps monetize and includes a number of popular economic forms ranging from crowdfunding to mutual fund. We propose an auction game, coined all-pay auctions with proportional allocation (APAPA), to model the fans economy and analyze the auction from the perspective of revenue. Comparing to the standard all-pay auction, which normally has no pure Nash-Equilibrium in the complete information setting, we solve the pure Nash-Equilibrium of the APAPA in closed form and prove its uniqueness. Motivated by practical concerns, we then analyze the case where APAPA is equipped with a reserve and show that there might be multiple equilibria in this case. We give an efficient algorithm to compute all equilibria in this case. For either case, with or without reserve, we show that APAPA always extracts revenue that 2-approximates the second-highest valuation. Furthermore, we conduct experiments to show how revenue changes with respect to different reserves.

The Game-Theoretic Formation of Interconnections Between Networks

We introduce a network design game where the objective of the players is to design the interconnections between the nodes of two different networks $G_{1}$ and $G_{2}$ in order to maximize certain local utility functions. In this setting, each player is associated with a node in $G_{1}$ and has functional dependencies on certain nodes in $G_{2}$ . We use a distance-based utility for the players in which the goal of each player is to purchase a set of edges (incident to its associated node) such that the sum of the distances between its associated node and the nodes it depends on $G_{2}$ is minimized. We consider a heterogeneous set of players (i.e., players have their own costs and benefits for constructing edges). We show that finding a best response of a player in this game is NP-hard. Despite this, we characterize some properties of the best response actions, which are helpful in determining a Nash equilibrium for certain instances of this game. In particular, we prove the existence of pure Nash equilibria in this game when $G_{2}$ contains a star subgraph, and provide an algorithm that outputs such an equilibrium for any set of players. Finally, we show that the price of anarchy in this game can be arbitrarily large.

A fully distributed algorithm for user-base station association in HetNets

The dense deployment of small-cell base stations in heterogeneous networks (HetNets) requires efficient resource allocation and interference management techniques. Especially, the problem of associating users to base stations (BSs) and allocating frequency channels must be revised and carefully studied. Finding the optimal solution of such problem is NP-hard. Further, it requires huge amount of information exchange between the BSs. In order to efficiently solve this problem in a distributed manner, we model it using non-cooperative game theory. The proposed game model is proved to not always admit pure Nash equilibria (PNEs), even though simulations show that, for slow fading channels, a PNE exists for most instances of the game. It is shown that, when the game admits PNEs, its prices of anarchy and stability are close to one. By modifying the players’ actions set and hence obtaining a new game model, we guarantee the existence of PNEs at the expense of performance degradation. Next, a fully distributed algorithm, based on a learning mechanism, is proposed. It requires no communication between the BSs and needs only one bit of feedback. Simulations show that the fully distributed algorithm has tight-to-optimal performance and solves efficiently the trade-off between complexity, information exchange and performance. We benchmark the proposed algorithm against the centralized optimal algorithm, the maximum signal to interference-plus-noise ratio algorithm, the best response dynamics algorithm and the randomized weighted majority algorithm.

Network Movement Games

We introduce a new class of games, called Network Movement Games, naturally related to the movement problems of [7], which models the spontaneous creation of multi-hop communication networks by the distributed and uncoordinated movement of k selfish mobile devices placed on the nodes of an underlying graph. Devices are players aiming to find the final positions which achieve a global property of the induced subnetwork. We actually focus on solutions (i.e. Nash equilibria) connecting all the players while minimizing the distances from their home locations. We show that the game always admits a pure Nash equilibrium, and that the convergence after a finite number of improving moves is guaranteed only when players perform their best possible moves; in this case, we provide tight bounds on the speed of convergence to Nash equilibria, both when initial positions are arbitrary and when players start at their home positions. As for the Nash equilibria performances, we first prove that the price of stability is 1 (i.e., an optimal solution is also a Nash equilibrium). Furthermore, we provide tight bounds on the price of anarchy, also showing that better performances are obtained when players start at their home positions. Finally, through extensive experiments, we show that high bounds on the price of anarchy as well as high convergence time of best move dynamics to Nash equilibria are only due to pathological worst cases. Moreover, quite often improving move dynamics (i.e., where players do not necessarily perform a best possible move) converge to Nash equilibria on the tested instances even though the class of instances does not guarantee the convergence.

Distributed ABS-Slot Access in Dense Heterogeneous Networks: A Potential Game Approach With Generalized Interference Model

This paper investigates the problem of almost blank subframe (ABS)-slot access in the heterogeneous networks to optimize network performance under the ABS scheme. Since maximizing network throughput requires global information and is not feasible in large scale or dense networks, the problem is solved from an interference minimization perspective after investigating the inherent relationship between local interference and achievable throughput. Since traditional binary interference model does not consider accumulated interference and thus results in inaccuracy, we take the accumulated interference into account and propose a generalized interference model, in which small cells are classified into different distant neighborhoods. An interference minimization game is formulated and is then proved to be an exact potential game, which has at least one pure Nash equilibrium (NE), and the best pure strategy NE point is a global optimum of minimizing aggregate interference. Combining the relationship between throughput and interference, the throughput maximization problem is proved to be an ordinal potential game, which possesses nice properties as well. A distributed algorithm is proposed to reach the NE, which minimizes the aggregate interference and maximizes the network throughput globally or locally. Furthermore, simulation results show that the proposed generalized interference model outperforms the traditional binary interference model.

Competitive infrastructure up-gradation in federated cloud

In a federated cloud, the virtual network requests are mapped to the resources belonging to multiple CPs. An individual CP gets better utility if the usage of its own part of the network increases. In this paper, we study the problem of connectivity upgradation in the federated cloud. The usage of a CP's network depends on its connectivity with the rest of the network. A CP will be interested in improving the network in such a way that such improvement makes its part of the network more accessible from other clouds. At the same time, being competitive, it will avoid any improvement that facilitates better accessibility among other CPs. In this paper we study how the CPs decide on such improvement by themselves using a non-cooperative game. We study the existence of the pure Nash equilibrium, the complexity to find the pure Nash equilibrium and calculate the price of anarchy.

Energy Aware Offloading for Competing Users on a Shared Communication Channel

This paper considers a set of mobile users that employ cloud-based computation offloading . In order to execute jobs in the cloud, the user uploads must occur over a base station channel that is shared by all of the uploading users. Since the job completion times are subject to hard deadline constraints, this restricts the feasible set of jobs that can be processed. The system is modelled as a competitive game in which each user is interested in minimizing its own energy consumption. The game is subject to the real-time constraints imposed by the job execution deadlines, user specific channel bit rates, and the competition over the shared communication channel. The paper shows that for a wide range of parameters, a game where each user independently sets its offloading decisions always has a pure Nash equilibrium, and a Gauss-Seidel-like method for determining this equilibrium is introduced. Results are presented that illustrate that the system always converges to a Nash equilibrium using the Gauss-Seidel method. Data is also presented that show the number of iterations required, and the quality of the solutions. We find that the solutions perform well compared to a lower bound on total energy performance.

Energy Management Based on Demand-Side Pricing: A Supermodular Game Approach

This paper proposes a pricing strategy for energy management in the electricity market composed of one generation company, multiple competing utility companies, and consumers with heating, ventilation, and air conditionings, which participate in demand response to keep the electricity real-time balance. The utility company plays the role of an intermediary agent between the generation company and consumers. We model the cost of the utility companies from the demand-side regulation and give the condition of a unique optimal regulation price. For the multiple competing utility companies, we demonstrate that the retail pricing game belongs to the supermodular game and the game exists a pure Nash equilibrium. The uniqueness of the Nash equilibrium is also proved, and then a distributed algorithm is developed to search for the optimal equilibrium price. Furthermore, we study the influence of the wholesale price on the equilibrium. Finally, numerical results demonstrate the efficiency of the proposed pricing strategy.

Game Theory Based Correlated Privacy Preserving Analysis in Big Data

Privacy preservation is one of the greatest concerns in big data. As one of extensive applications in big data, privacy preserving data publication (PPDP) has been an important research field. One of the fundamental challenges in PPDP is the trade-off problem between privacy and utility of the single and independent data set. However, recent research has shown that the advanced privacy mechanism, i.e., differential privacy, is vulnerable when multiple data sets are correlated. In this case, the trade-off problem between privacy and utility is evolved into a game problem, in which payoff of each player is dependent on his and his neighbors' privacy parameters. In this paper, we first present the definition of correlated differential privacy to evaluate the real privacy level of a single data set influenced by the other data sets. Then, we construct a game model of multiple players, in which each publishes data set sanitized by differential privacy. Next, we analyze the existence and uniqueness of the pure Nash Equilibrium. We refer to a notion, i.e., the price of anarchy, to evaluate efficiency of the pure Nash Equilibrium. Finally, we show the correctness of our game analysis via simulation experiments.

How Asymmetrically Increasing Joint Strike Costs Need Not Lead to Fewer Strikes

The model states that the incentive to strike is inversely related to the total costs associated with workers' and firms' strike activities. Not only has this model been tested with mixed results, but also the joint costs model is problematic in explaining several stylized facts in the strike literature because higher strike costs do not always yield a lower incidence of strike activity. This paper illustrates how the joint cost model can yield these counterintuitive results. It shows that strike incidence need not decrease when joint strike costs increase. The innovation is to raise union and firm joint strike costs in an asymmetric way. Increasing a particular side's strike costs necessarily decreases its incentive to strike. However, in response, the other side's incentive can increase, since under a number of circumstances it holds out with a higher probability in order to collect the relatively larger expected rents coming about because the other side's implicit threat point decreases. To illustrate this, we model contract negotiations as a simple one-period game. (No need for more complex repeated games such as attrition since our point is only to show as simply as possible why the joint-costs model yields ambiguous results.) We use standard Hicksian concession curves to derive a payoff matrix. The payoff matrix results in contract negotiations following along the lines of a of chicken. The solution to the game yields no one stable pure Nash-equilibrium strategy, but instead a mixed strategy so that choices become probabilistic depending upon union and firm concession curve parameters. The results indicate that increasing either party's strike costs can have ambiguous effects on strike incidence. This ambiguity may explain why higher strike costs need not always lead to fewer strikes, and thus may account for the mixed success observed in studies that empirically test the joint costs model with strike incidence data. Although couched in terms of strikes, the results are equally applicable to other negotiation situations.

Coordination games on graphs

We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between (Formula presented.) and (Formula presented.). The latter upper bound is tight for (Formula presented.) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases.