Unique Pure Nash Equilibrium Research Articles

Incentivizing Collaboration in Heterogeneous Teams via Common-Pool Resource Games

We consider a team of heterogeneous agents, which is collectively responsible for servicing, and subsequently reviewing a stream of homogeneous tasks. Each agent has an associated mean service time and a mean review time for servicing and reviewing the tasks, respectively. Agents receive a reward based on their service and review admission rates. The team objective is to collaboratively maximize the number of “serviced and reviewed” tasks. We formulate a common-pool resource game, and design utility functions to incentivize collaboration among heterogeneous agents in a decentralized manner. We show the existence of a unique pure Nash equilibrium (PNE), and establish convergence of the best response dynamics to this unique PNE. Finally, we establish an analytic upper bound on three inefficiency measures of the PNE, namely the price of anarchy, the ratio of the total review admission rate, and the ratio of latency.

On Evolutionary Vaccination Game in Activity-Driven Networks

Since human face-to-face communications change over time, their perceived probabilities associated with uncertain outcomes usually present a nonlinear fashion in reality. This article incorporates both humans active characteristics and risk perception into epidemic spreading and explores a comprehensive evolutionary vaccination game in activity-driven networks. According to the weighting effect, the individuals’ activity rate thresholds are obtained in different cases under the framework of the pure Nash equilibrium. Furthermore, how the number of connected edges of activated individuals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> and infection rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> affect individuals’ vaccination, decision-making is theoretically analyzed. It is proven that some unvaccinated individuals gradually become vaccinated individuals with the increase of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> . Moreover, there is a unique pure Nash equilibrium social state (Pareto optimization), and the pure Nash equilibrium together with Pareto optimization has the same equilibrium for our proposed evolutionary vaccination game, which provides an alternative condition judging the stability of game system. In the case of complete information and individual complete rationality, the factors, including vaccination cost, individuals’ activity rate, the number of connected edges of activated individuals, and infection rate, have a significant influence on the final average social cost.

A Hashing Power Allocation Game with and without Risk-free Asset

Abstract Miners in various blockchain-backed cryptocurrency networks compete to maintain the validity of the underlying distributed ledgers to earn the bootstrapped cryptocurrencies. With limited hashing power, each miner needs to decide how to allocate their resource to different cryptocurrencies so as to achieve the best overall payoff. Together all the miners form a hashing power allocation game. We consider two settings of the game, depending on whether each miner can allocate their fund to a risk-free asset or not. We show that this game admits unique pure Nash equilibrium in closed-form for both settings.

Shadowing and shielding: Effective heuristics for continuous influence maximisation in the voting dynamics.

Influence maximisation, or how to affect the intrinsic opinion dynamics of a social group, is relevant for many applications, such as information campaigns, political competition, or marketing. Previous literature on influence maximisation has mostly explored discrete allocations of influence, i.e. optimally choosing a finite fixed number of nodes to target. Here, we study the generalised problem of continuous influence maximisation where nodes can be targeted with flexible intensity. We focus on optimal influence allocations against a passive opponent and compare the structure of the solutions in the continuous and discrete regimes. We find that, whereas hub allocations play a central role in explaining optimal allocations in the discrete regime, their explanatory power is strongly reduced in the continuous regime. Instead, we find that optimal continuous strategies are very well described by two other patterns: (i) targeting the same nodes as the opponent (shadowing) and (ii) targeting direct neighbours of the opponent (shielding). Finally, we investigate the game-theoretic scenario of two active opponents and show that the unique pure Nash equilibrium is to target all nodes equally. These results expose fundamental differences in the solutions to discrete and continuous regimes and provide novel effective heuristics for continuous influence maximisation.

The effect of strategic noise in linear regression

We build on an emerging line of work which studies strategic manipulations in training data provided to machine learning algorithms. Specifically, we focus on the ubiquitous task of linear regression. Prior work focused on the design of strategyproof algorithms, which aim to prevent such manipulations altogether by aligning the incentives of data sources. However, algorithms used in practice are often not strategyproof, which induces a strategic game among the agents. We focus on a broad class of non-strategyproof algorithms for linear regression, namely \(\ell _p\) norm minimization (\(p > 1\)) with convex regularization. We show that when manipulations are bounded, every algorithm in this class admits a unique pure Nash equilibrium outcome. We also shed light on the structure of this equilibrium by uncovering a surprising connection between strategyproof algorithms and pure Nash equilibria of non-strategyproof algorithms in a broader setting, which may be of independent interest. Finally, we analyze the quality of equilibria under these algorithms in terms of the price of anarchy.

Stability of dynamic asymmetric contests with endogenous prizes

Repeated asymmetric contest games are examined under conditions which guarantee the existence and uniqueness of pure Nash equilibrium (Hirai and Szidarovszky 2013). Conditions are derived for the local asymptotical stability of the equilibrium under continuous and discrete dynamics with gradient adjustments. In both cases, a crucial assumption is the nonexistence of a dominant player at the equilibrium level. In the case of continuous time scales, this is sufficient for stability, and in the discrete case, the speeds of adjustments have to be sufficiently small. As special cases, symmetric and semi-symmetric games are analyzed.

Nash versus coarse correlation

We run a laboratory experiment to test the concept of coarse correlated equilibrium (Moulin and Vial in Int J Game Theory 7:201–221, 1978), with a two-person game with unique pure Nash equilibrium which is also the solution of iterative elimination of strictly dominated strategies. The subjects are asked to commit to a device that randomly picks one of three symmetric outcomes (including the Nash point) with higher ex-ante expected payoff than the Nash equilibrium payoff. We find that the subjects do not accept this lottery (which is a coarse correlated equilibrium); instead, they choose to play the game and coordinate on the Nash equilibrium. However, given an individual choice between a lottery with equal probabilities of the same outcomes and the sure payoff as in the Nash point, the lottery is chosen by the subjects. This result is robust against a few variations. We explain our result as selecting risk-dominance over payoff dominance in equilibrium.

Indirect Mechanism Design for Efficient and Stable Renewable Energy Aggregation

Mechanism design is studied for aggregating renewable power producers (RPPs) in a two-settlement power market. Employing an indirect mechanism design framework, a payoff allocation mechanism is derived from the competitive equilibrium of an especially formulated market with transferrable payoff. Given the designed mechanism, the strategic behaviors of the participating RPPs entail a non-cooperative game: It is proven that a unique pure Nash equilibrium exists among the RPPs, for which a closed-form expression is found. Moreover, it is proven that the designed mechanism achieves a number of key desirable properties at the NE: these include efficiency (i.e., an ideal “Price of Anarchy” of one), stability (i.e., “in the core” from a coalitional game theoretic perspective), and no collusion. In addition, it is shown that a set of desirable “ex-post” properties are also achieved by the designed mechanism. Extensive simulations are conducted and corroborate the theoretical results.

Adaptive topologies against jamming attacks in wireless networks: A game-theoretic approach

Towards securing wireless networks and ensuring their dependability under jamming attacks, this paper presents and analyzes game-theoretic formulations between an adversary and a defender. The adversary jams a subset of nodes to increase the level of interference in the network, while the defender makes judicious adjustments of the transmission power level of the nodes, thereby continuously adapting the underlying network topology to reduce the impact of the attack. The defender's strategy is based on playing Nash equilibria (NE) strategies securing a worst-case network utility. First, a discrete control set is considered in which the space of strategies of the defender grows exponentially with the network size. Scalable decomposition-based approaches are developed yielding a defense strategy whose performance closely approaches that of the non-decomposed game. Second, we develop a marginal strategy to assign the power levels while satisfying a coverage constraint thereby evading the combinatorial complexity associated with enumerating all pure actions. Third, we generalize the marginal-based strategy by considering a continuous action space for both players. For this setting, we prove the existence of a unique pure Nash equilibrium. The presented numerical results show the effectiveness of the proposed defense approach against various attack policies and demonstrate the assignment strategies on a real wireless network deployed in a three-story building.

Analysis of Competition Fronting the Popularity of Content in Social Networks

&lt;p&gt;In the telecommunications domain they are several providers, but customers seeking those that there are good services. In this paper, a study is seeking on two types of providers: content providers CPs and Internet Service Providers ISPs. In this study, we analyzed the impact of Selfishness of Content Providers and Internet Service Providers on their strategies of Price and QoS on their decision strategies. Yet, we formulate our problem as a non-cooperative game among multiple CPs, multiple ISPs competing for the same market. We prove through a detailed analysis uniqueness of pure Nash Equilibrium (NE). Furthermore, a fully distributed algorithm to converge to the NE point is presented. In order to quantify how efficient is the NE point, a detailed analysis of the Price of Anarchy (PoA) is adopted to ensure the performance of the system at equilibrium. Finally, we provide an extensive numerical study to point out the importance of QoS and credibility in the market and the in-fluence of the existing economic relationship between content providers and Internet service providers.&lt;/p&gt;

Strategic bidding in an accumulating priority queue: equilibrium analysis

We study the strategic purchasing of priorities in a time-dependent accumulating priority M/G/$1$ queue. We formulate a non-cooperative game in which customers purchase priority coefficients with the goal of reducing waiting costs in exchange. The priority of each customer in the queue is a linear function of the individual waiting time, with the purchased coefficient being the slope. The unique pure Nash equilibrium is solved explicitly for the case with homogeneous customers. A general characterisation of the Nash equilibrium is provided for the heterogeneous case. It is shown that both avoid the crowd and follow the crowd behaviours are prevalent, within class types and between them. We further present a pricing mechanism that ensures the order of the accumulating priority rates in equilibrium follows a $C\mu$ type rule and improves overall efficiency.

Electric Vehicle Charging Stations With Renewable Power Generators: A Game Theoretical Analysis

In this paper, we study the price competition among electric vehicle charging stations (EVCSs) with renewable power generators (RPGs). As electric vehicles (EVs) become more popular, a competition among EVCSs to attract EVs is inevitable. Thereby, each EVCS sets its electricity price to maximize its revenue by taking into account the competition with neighboring EVCSs. We analyze the competitive interactions between EVCSs using game theory, where relevant physical constraints such as the transmission line capacity, the distance between EV and EVCS, and the number of charging outlets at the EVCSs are taken into account. We show that the game played by EVCSs is a supermodular game and there exists a unique pure Nash equilibrium for best response algorithms with arbitrary initial policy. The electricity price and the revenue of EVCSs are evaluated via simulations, which reveal the benefits of having RPGs at the EVCSs.

On the Structure of Weakly Acyclic Games

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.

Discovering theorems in game theory: Two-person games with unique pure Nash equilibrium payoffs

In this paper we provide a logical framework for two-person finite games in strategic form, and use it to design a computer program for discovering some classes of games that have unique pure Nash equilibrium payoffs. The classes of games that we consider are those that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisseʼs class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique pure Nash equilibrium payoffs. It also came up with new classes of strict games that have unique pure Nash equilibria, where a game is strict if for both player different profiles have different payoffs.

Pricing and Distributed Power Control in Wireless Relay Networks

In this paper, we consider a wireless amplify-and-forward relay network with one relay node and multiple source-destination pairs/users and propose a pricing framework that enables the relay to set prices to maximize either its revenue or any desirable system utility. Specifically, depending on the quality of the received signals, the relay sets prices and correspondingly charges the users utilizing its resources for their transmissions. The price is determined in such a way that the relay's revenue or system utility is maximized. Given the specified price, the users competitively employ the relay node to forward their signals. We model each user as a rational player, which aims at maximizing its own net utility through power allocation, and analyze the competition among the users within the framework of noncooperative game theory. It is shown that, in the game played by the users, there always exists a unique pure Nash equilibrium point that can be achieved through distributed iterations. Next, subject to the availability of complete information about the users at the relay, we propose a low-complexity uniform pricing algorithm and an optimal differentiated pricing algorithm, in which the relay either charges the users at a suboptimal uniform price or charges different users at different prices. We also show that, by applying the differentiated pricing algorithm that enforces the users to transmit at certain power levels, any system utility can be maximized. Extensive simulations are conducted to quantify the performance of the proposed methods.

A Game-Theoretical Study of Robust Networked Systems

This paper analyses the robustness of networked systems from a game-theoretical perspective. Networked systems often consist of several subsystems sharing resources interdependently based on local preferences. These systems can be modelled by a dependence game, which is a generalisation of stable paths problem. A unique pure Nash equilibrium in a dependence game can characterise the robustness of the represented networked system, precluding oscillations and nondeterminism. We show that the absence of a structure termed a generalised dispute wheel is useful to ensure the existence of a unique pure Nash equilibrium. Furthermore, we consider more sophisticated settings: tie-breaking over non-strict preferences and asynchronous communications among subsystems. We also obtain stronger results that the absence of a generalised dispute wheel can be useful to ensure the consistency of tie-breaking and asynchronous convergence to a pure Nash equilibrium.

Cost-Balancing Tolls for Atomic Network Congestion Games

We investigate the existence of optimal tolls for atomic symmetric network congestion games with unsplittable traffic and arbitrary nondecreasing latency functions. We focus on pure Nash equilibria, and consider a natural toll mechanism, which we call _cost-balancing tolls_. A set of cost-balancing tolls turns every path with positive traffic on its edges into a minimum-cost path. Hence any given configuration is induced as a pure Nash equilibrium of the modified game with the corresponding cost-balancing tolls. We show how to compute in linear time a set of cost-balancing tolls for the optimal solution such that the total amount of tolls paid by any player in any pure Nash equilibrium of the modified game does not exceed the latency on the maximum-latency path in the optimal solution. Our main result is that for congestion games on series-parallel networks with strictly increasing latency functions, the optimal solution is induced as the _unique_ pure Nash equilibrium of the game with the corresponding cost-balancing tolls. To the best of our knowledge, only linear congestion games on parallel links were known to admit optimal tolls prior to this work. To demonstrate the difficulty of computing a better set of optimal tolls, we show that even for two-player linear congestion games on series-parallel networks, it is NP-hard to decide whether the optimal solution is the unique pure Nash equilibrium or there is another pure Nash equilibrium of total cost at least 6/5 times the optimal cost.