Existence Of Pure Nash Equilibrium Research Articles

Pure Nash equilibria in a generalization of congestion games allowing resource failures

We introduce a model for congestion games in which resources can fail with some probability distribution. These games are an extension of classical congestion games, and like these, have exact potential functions that guarantee the existence of pure Nash equilibria (PNE). We prove that the agent's cost functions for these games can be hard to compute by giving an example of a game for which the cost function is hard for Valiant's #P class, even in the case when all failure probabilities coincide. We characterize the complexity of computing PNE in congestion games with failures with an extension of the local search class PLS that allows queries to a #P function, and show examples of games for which the PNE search problem is complete for this class. Furthermore, we provide a variant of the game with the property that a PNE can be constructed in polynomial time if this also holds in the restricted game without failures.

Public goods games in directed networks

Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to prove, finding the equilibrium is PPAD-complete. Finally, when the treewidth of the directed network is appropriately bounded, we prove that polynomial-time algorithms are possible.

Existence of pure Nash equilibria in 2-player information diffusion games with strict public preferences

Existence of pure Nash equilibria in 2-player information diffusion games with strict public preferences

Posetal Games: Efficiency, Existence, and Refinement of Equilibria in Games With Prioritized Metrics

Modern applications require robots to comply with multiple, often conflicting rules and to interact with the other agents. We present Posetal Games as a class of games in which each player expresses a preference over the outcomes via a partially ordered set of metrics. This allows one to combine hierarchical priorities of each player with the interactive nature of the environment. By contextualizing standard game theoretical notions, we provide two sufficient conditions on the preference of the players to prove existence of pure Nash Equilibria in finite action sets. Moreover, we define formal operations on the preference structures and link them to a refinement of the game solutions, showing how the set of equilibria can be systematically shrunk. The presented results are showcased in a driving game where autonomous vehicles select from a finite set of trajectories. The results demonstrate the interpretability of results in terms of minimum-rank-violation for each player.

Equilibria in Multiclass and Multidimensional Atomic Congestion Games

This paper studies the existence of pure Nash equilibria in atomic congestion games with different user classes where the cost of each resource depends on the aggregated demand of each class. A set of cost functions is called consistent for this class if all games with cost functions from the set have a pure Nash equilibrium. We give a complete characterization of consistent sets of cost functions showing that the only consistent sets of cost functions are sets of certain affine functions and sets of certain exponential functions. This characterization is also extended to a larger class of games where each atomic player may control flow that belongs to different classes.

Selfish Routing Game-Based Multi-Resource Allocation and Fair Scheduling of Indivisible Jobs in Edge Environments

Distributed and heterogeneous edge computing environments require efficient allocation and scheduling of multiple users’ applications. This paper presents a game-theoretic solution to model the competition between time-sensitive internet of things (IoT) applications with indivisible loads to be allocated and scheduled to edge servers. We model the allocation problem as a selfish routing game such that no job is unsatisfied with its allocation. Also, the allocated jobs are scheduled using a weighted time-sharing approach, which assigns resources to jobs proportional to their demands and deadlines. We proved the existence of pure Nash equilibrium in the introduced game and presented a greedy and polynomial time algorithm to obtain the solution called SAFSA. Using an extended version of the CloudSim simulator, we assess our approach in different situations by employing a real-world dataset of the Google cluster. Since the proposed algorithm considers jobs’ deadlines beside resource requests in allocation and scheduling decisions, it can simultaneously minimize average response time and maximize the number of jobs completed within their deadlines.The simulation results confirm the ability of SAFSA to efficiently execute the jobs within their deadlines, especially when there are many IoT devices and edge servers in the system, which is the main characteristic of IoT environments.

Pure Nash Equilibria in Resource Graph Games

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively.
 For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium.
 For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load.
 We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

Nash equilibrium and group strategy consensus of networked evolutionary game with coupled social groups

This paper studies the existence of Nash equilibrium and group strategy consensus problem of networked evolutionary game with coupled social groups, which is formulated as a two-layer networked game. First, by the approach of semi-tensor product, the payoff functions of players are converted into the algebraic forms, and two necessary and sufficient conditions are derived for the existence of pure Nash equilibrium. Second, the strict game algebraic equation by the best response adjustment rule with group intelligence is established, by which the group strategy consensus problem is investigated. At last, an example is worked out to verify our conclusions.

On the costly voting model: the mean rule

In this paper, we study a model in which a policy is chosen by a group of people through the formation of a committee. Attending the committee is costly, and each person decides whether to take part in it or not. Our work complements Osborne, Rosenthal and Turner (2000) since we allow various types of costs. We work with the mean compromise function, and we do not restrict the distribution of the favourite policy of the group members. Under these assumptions, we establish the existence of pure Nash equilibrium and show that, in comparison to the case of the median compromise function, the outcome of the committee’s work is less random and is not likely to be extreme. In the case of constant costs of participation, we show that when the costs increase, the size of the equilibrium committee decreases and the spread of the outcomes increases.

NASH EQUILIBRIA FOR INFORMATION DIFFUSION GAMES ON WEIGHTED CYCLES AND PATHS

The information diffusion game, which is a type of non-cooperative game, models the diffusion process of information in networks for several competitive firms that want to spread their information. Recently, the game on weighted graphs was introduced and pure Nash equilibria for the game were discussed. This paper gives a full characterization of the existence of pure Nash equilibria for the game on weighted cycles and paths according to the number of vertices, the number of players and weight classes.

Efficient Black-Box Reductions for Separable Cost Sharing

In cost-sharing games with delays, a set of agents jointly uses a subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a nonshareable player-specific delay for each resource. A separable cost-sharing protocol determines cost shares that are budget-balanced, separable, and guarantee existence of pure Nash equilibria (PNE). We provide black-box reductions reducing the design of such a protocol to the design of an approximation algorithm for the underlying cost-minimization problem. In this way, we obtain separable cost-sharing protocols in matroid games, single-source connection games, and connection games on n-series-parallel graphs. All these reductions are efficiently computable - given an initial allocation profile, we obtain a cheaper profile and separable cost shares turning the profile into a PNE. Hence, in these domains, any approximation algorithm yields a separable cost-sharing protocol with price of stability bounded by the approximation factor.

Multi-tenancy and URLLC on unlicensed spectrum: Performance and design

We study in this paper the transport of Ultra-Reliable Low-Latency Communication (URLLC) in a scenario of Industry 4.0 where transmission in uplink is possible in both licensed and unlicensed spectrum. We propose a transmission policy where the packet first attempts transmission in unlicensed spectrum during a time budget smaller than the delay constraint, if the transmission fails then the packet uses the remaining time budget to attempt transmission in licensed spectrum. The goal of using this policy is to minimize the cost of licensed bandwidth required for URLLC. We first consider the case of one tenant managing the industrial area and quantify the needed licensed bandwidth to attain reliability target within the target delay. We optimize the system by choosing the transmission policy which minimizes this cost. We then study the case of multiple tenants present in the same area and sharing unlicensed spectrum. Each tenant tries to minimize its cost of licensed bandwidth by utilizing unlicensed resources, which may result in the tragedy of the commons like situation. We formulate the problem using a game-theoretic approach to model the non-cooperative multi-tenant scenario. We model the medium access of unlicensed system for this case and derive the strategies that minimize individual cost functions. We then prove the existence of pure Nash equilibria analytically and identify them numerically. Finally, we quantify the so-called price of anarchy, i.e., ratio of the utility yielded by the competitive setting to the outcome of a cooperative scenario.

Existence and Efficiency of Equilibria for Cost-Sharing in Generalized Weighted Congestion Games

This work studies the impact of cost-sharing methods on the existence and efficiency of (pure) Nash equilibria in weighted congestion games. We also study generalized weighted congestion games , where each player may control multiple commodities. Our results are fairly general; we only require that our cost-sharing method and our set of cost functions satisfy certain natural conditions. For general weighted congestion games, we study the existence of pure Nash equilibria in the induced games, and we exhibit a separation from the standard single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees existence of pure Nash equilibria. With respect to efficiency, we present general tight bounds on the price of anarchy, which are robust and apply to general equilibrium concepts. Our analysis provides a tight bound on the price of anarchy, which depends only on the used cost-sharing method and the set of allowable cost functions. Interestingly, the same bound applies to weighted congestion games and generalized weighted congestion games. We then turn to the price of stability and prove an upper bound for the Shapley value cost-sharing method, which holds for general sets of cost functions and which is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we provide a somewhat surprising result, showing that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, for this case, the price of stability becomes as bad as the price of anarchy. Again, our bounds on the price of stability are independent on whether players are single or multi-commodity.

Blockchain Competition Between Miners: A Game Theoretic Perspective

We model the competition over mining resources and over several cryptocurrencies as a non-cooperative game. Leveraging results about congestion games, we establish conditions for the existence of pure Nash equilibria and provide efficient algorithms for finding such equilibria. We account for multiple system models, varying according to the way that mining resources are allocated and shared and according to the granularity at which mining puzzle complexity is adjusted. When constraints on resources are included, the resulting game is a constrained resource allocation game for which we characterize a normalized Nash equilibrium. Under the proposed models, we provide structural properties of the corresponding types of equilibrium, e.g., establishing conditions under which at most two mining infrastructures will be active or under which no miners will have incentives to mine a given cryptocurrency.

Simple Utility Design for Welfare Games under Global Information

Welfare game is a game-theoretic model for resource allocation problem which is to find an allocation to maximize the welfare function. In order to determine it in a distributed way, each agent is assigned to an admissible utility function such that the resulting game possesses desirable properties, for example, scalability, existence and efficiency of pure Nash equilibria, and budget balance. In this paper, supposing that each agent can access the global information, marginal contribution based utility design is proposed. It is shown that utility functions based on the above design have scalability and existence of pure Nash equilibria. Furthermore, efficiency is the same as that of the conventional utility design via Shapley value.

Verification and Design of Zero-Sum Potential Games

Zero-sum game is a class of game where one player’s gain is equivalent to another’s loss, which can be used in competitive situation. But pure Nash equilibrium maybe not exist in general zero-sum games. Potential games have nice properties, such as existence of pure Nash equilibrium. To combine advantages of zero-sum games and potential games, zero-sum potential game is proposed in this paper. Verification for a finite non-cooperative game being a zero-sum potential game is considered. Conversely, how to design a zero-sum potential game is also studied when the potential function is given. We show that verification and design of zero-sum potential game can be realized by solving linear equations. Furthermore, we find that if any two players play the zero-sum potential game in a network, then the networked game is also a zero-sum potential game.

Generalizations of weighted matroid congestion games: pure Nash equilibrium, sensitivity analysis, and discrete convex function

Congestion games provide a model of human’s behavior of choosing an optimal strategy while avoiding congestion. In the past decade, matroid congestion games have been actively studied and their good properties have been revealed. In most of the previous work, the cost functions are assumed to be univariate or bivariate. In this paper, we discuss generalizations of matroid congestion games in which the cost functions are n-variate, where n is the number of players. First, motivated from polymatroid congestion games with $$\mathrm {M}^\natural $$ -convex cost functions, we conduct sensitivity analysis for separable $$\mathrm {M}^\natural $$ -convex optimization, which extends that for separable convex optimization over base polyhedra by Harks et al. (SIAM J Optim 28:2222–2245, 2018. https://doi.org/10.1137/16M1107450 ). Second, we prove the existence of pure Nash equilibria in matroid congestion games with monotone cost functions, which extends that for weighted matroid congestion games by Ackermann et al. (Theor Comput Sci 410(17):1552–1563, 2009. https://doi.org/10.1016/j.tcs.2008.12.035 ). Finally, we prove the existence of pure Nash equilibria in matroid resource buying games with submodular cost functions, which extends that for matroid resource buying games with marginally nonincreasing cost functions by Harks and Peis (Algorithmica 70(3):493–512, 2014. https://doi.org/10.1007/s00453-014-9876-6 ).

Mining competition in a multi-cryptocurrency ecosystem at the network edge

We model the competition over several blockchains characterizing multiple cryptocurrencies as a non-cooperative game. Then, we specialize our results to two instances of the general game, showing properties of the Nash equilibrium. In particular, leveraging results about congestion games, we establish the existence of pure Nash equilibria and provide efficient algorithms for finding such equilibria.

Multistage interval scheduling games

We study a game theoretical model of multistage interval scheduling problems in which each job consists of exactly one task (interval) for each of t stages (machines). In the game theoretical model, the machine of each stage is controlled by a different selfish player who wants to maximize her total profit, where the profit for scheduling the task of a job j is a fraction of the weight of the job that is determined by the set of players that also schedule their corresponding task of job j. We provide criteria for the existence of pure Nash equilibria and prove bounds on the Price of Anarchy and the Price of Stability for different social welfare functions.

Pure-strategy Nash equilibria on competitive diffusion games

This paper treats two types of competitive facility location games on graphs: information diffusion games and discrete Voronoi games. Both of these games can be regarded as models of the rumor spreading processes on the networks, where each player of the game wants to select an influencer who can widely spread information throughout the network. For each game, given a graph and the number of players, we are interested in whether there exist pure Nash equilibria or not. In this paper, we discuss the existence of pure Nash equilibria on graphs with small diameter, path graphs, and cycle graphs. The results include the behavior of the discrete Voronoi games on graphs with diameter two, and the complete characterization of the existence of the pure Nash equilibria in the discrete Voronoi games and information diffusion games on path graphs.