Existence Of Nash Equilibrium Research Articles

Expectations-based loss aversion in contests

This paper studies a multi-player lottery contest in which heterogeneous contestants exhibit reference-dependent loss aversion à la Kőszegi and Rabin (2006, 2007). We verify the existence and uniqueness of pure-strategy choice-acclimating personal Nash equilibrium (CPNE) under moderate loss aversion and fully characterize the equilibrium. The equilibrium sharply contrasts with that in the two-player risk-neutral symmetric case. Loss aversion can lead contestants' individual efforts to change nonmonotonically, while the total effort strictly decreases. Further, it always leads to a more elitist distributional outcome, in the sense that a smaller set of contestants remain active in the competition and stronger contestants' equilibrium winning probabilities increase. We demonstrate that loss aversion generates a fundamentally different decision problem than risk aversion and develop a rationale that explains the contrasting predictions from the two frameworks. Finally, our results are robust under the alternative equilibrium concept of preferred personal Nash equilibrium (PPNE).

Ordinal status games on networks

We consider a modification of ordinal status games of Haagsma and von Mouche (2010). A number of agents make scalar choices, e.g., their levels of conspicuous consumption. The wellbeing of each agent depends on her own choice and on her social status determined by comparisons with the choices of others. In contrast to the original model, as well as its modifications considered so far, we allow for some players not caring about comparisons with some others. Assuming that the status of each player may only be “high” or “low,” the existence of a strong Nash equilibrium is shown; for a particular subclass of such games, the convergence of Cournot tatonnement is established. If more than two possible status levels are allowed into consideration, then even Nash equilibrium may fail to exist in very simple examples.

Game-Based Channel Selection for UAV Services in Mobile Edge Computing

Computation offloading is a hot research topic in mobile edge computing (MEC). Computation offloading among multiedge nodes in heterogeneous networks can help reduce offloading cost. In addition, the unmanned aerial vehicles (UAVs) play a key role in MEC, where UAVs in the air communicate with ground base stations to improve the network performance. However, limited channel resources can lead to the increase of transmission delay and the decline of communication quality. Effective channel selection mechanisms can help address those issues by improving transmission rate and ensuring communication quality. In this paper, we study channel selection during communication between multiple UAVs and base stations in an MEC system with heterogeneous networks. To maximize the transmission rate of each UAV user, we formulate a channel selection problem and model it as a noncooperative game. Then, we prove the existence of Nash equilibrium (NE). In addition, we design a multiple UAV-enabled transmission channel selection (UTCS) algorithm to obtain the equilibrium strategy profile of all the UAV users. Experimental results validate that UTCS algorithm can converge after a finite number of iterations and it outperforms random transmission algorithm (RTA) and sequential transmission algorithm (STA).

Invariant equilibria and classes of equivalent games

We consider classes of games for a fixed set of players with fixed strategy sets. For such classes, we analyze and develop the concept of invariance, which is satisfied when the set of Nash equilibria and corresponding equilibrium payoffs are identical for each payoff function within the class. We introduce the condition superior payoff matching, which requires that at any given strategy profile, each player can match her highest payoff near that strategy profile across all games within that class. If a specific game satisfies superior payoff matching, then its equilibria are invariant within a class of games with smaller sets of discontinuities. This condition can be used to prove existence of Nash equilibrium in games that are not quasiconcave or better reply secure.

A unified approach to the Nash equilibrium existence in large games from finitely many players to infinitely many players

A unified approach to the Nash equilibrium existence in large games from finitely many players to infinitely many players

Existence, uniqueness, and comparative statics of Nash equilibrium in a game of voluntary public good provision with two public goods

AbstractThis paper shows the uniqueness of a Nash equilibrium in the presence of many heterogeneous players with two public goods under Cobb–Douglas preferences. It provides a sufficient condition for a unique equilibrium including contributors providing both public goods, and shows that contributors providing both public goods appear frequently. This uniqueness property allows us to conduct a comparative statics analysis with a contributor and a free rider of both public goods, showing the role of the contributor providing both public goods.

Online-Offline Competition with Heterogeneous Consumers: An Example for No Existence of Pure Strategy Nash Equilibrium

Existing literature on competition between online and offline firms has focused on market conditions that guarantee the existence of a pure strategy Nash equilibrium. In this note, by constructing a concrete example, we provide a first attempt to show that the equilibrium existence result does not necessarily hold when consumers’ preferences are heterogeneous. Specifically, we consider the competition between one online firm and several offline firms in a market organized as a Salop model, where consumers’ preferences have a binary distribution. We identify a boundary scenario where the type distribution is binary with one type of consumer loyal to online shopping and the other type loyal to offline shopping. We show that there is no pure strategy Nash equilibrium for this boundary scenario, which indicates that the market may not be stable under such conditions. Our study contributes to a better understanding of the equilibrium existence conditions for the online versus offline retail competition.

Game Theoretical Task Offloading for Profit Maximization in Mobile Edge Computing

In this paper, a novel task offloading architecture called Flex-MEC is proposed, which achieves efficient task allocation and scheduling (TAS) between MEC servers. By adding metadata before task data, we redesign the offloading process in Flex-MEC, the TAS planning can be conducted without finishing the task data receiving. Once planning is done the task data can be directly forwarded to the allocated server and executed. This reduces latency compared to the traditional way of transmitting, planning, forwarding and executing sequentially. For TAS planning, a multi-server multi-task allocation and scheduling (MMAS) problem is formulated to maximize the MEC system profit. The MMAS problem is proven as an NP-complete problem, thus is challenging to solve. Then, a distributed scheme and a centralized scheme are proposed to solve the MMAS problem with low complexity. In the distributed scheme, the MMAS problem is converted into a non-cooperative game and the existence of Nash Equilibrium (NE) is proven and a low complexity response update algorithm is proposed to converge to NE. And the centralized scheme is based on a greedy idea and runs on a MEC controller in a centralized way. Verified by experiments, these two schemes can achieve better performance than compared schemes.

Marketing Resource Allocation Strategy Optimization Based on Dynamic Game Model

Game theory has become an important tool to study the competition between oligopolistic enterprises. After combing the existing literature, it is found that there is no research combining two‐stage game and nonlinear dynamics to analyze the competition between enterprises for advertising. Therefore, this paper establishes a two‐stage game model to discuss the effect of the degree of firms’ advertising input on their profits. And the complexity of the system is analyzed using nonlinear dynamics. This paper analyzes and studies the dynamic game for two types of application network models: data transmission model and transportation network model. Under the time‐gap ALOHA protocol, the noncooperative behavior of the insiders in the dynamic data transmission stochastic game is examined as well as the cooperative behavior. In this paper, the existence of Nash equilibrium and its solution algorithm are proved in the noncooperative case, and the “subgame consistency” of the cooperative solution (Shapley value) is discussed in the cooperative case, and the cooperative solution satisfying the subgame consistency is obtained by constructing the “allocation compensation procedure.” The cooperative solution is obtained by constructing the “allocation compensation procedure” to satisfy the subgame consistency. In this paper, we propose to classify the packets transmitted by the source nodes, and by changing the strategy of the source nodes at the states with different kinds of packets, we find that the equilibrium payment of the insider increases in the noncooperative game with the addition of the “wait” strategy. In the transportation dynamic network model, the problem of passenger flow distribution and the selection of service parameters of transportation companies are also studied, and a two‐stage game theoretical model is proposed to solve the equilibrium price and optimal parameters under Wardrop’s criterion.

The Monostatic Radar and Jammer Games Based on Signal-to-Jamming-Plus-Noise Ratio

In modern electronic warfare, both the radar and the jammer are capable of changing their waveforms to achieve better performance. The conflict relationship between the radar and jammer can be modeled as a non-cooperative game. In this paper, the optimal strategies of a monostatic radar and a jammer are investigated in terms of both Stackelberg game and symmetric game. Instead of using the widely used mutual information as the utility function, signal-to-jamming-plus-noise ratio (SJNR) is used in formulating the utility functions which is directly related to the target detection performance. In Stackelberg game, the Stackelberg equilibria solutions are derived when the leader is the radar and the jammer, respectively. In symmetric game, the existence of Nash equilibrium (NE) is investigated and the corresponding strategies are obtained when Nash equilibrium exists. If Nash equilibrium does not exist, the Stackelberg equilibria strategies are still useful as they can be regarded as the robust strategies which could optimize the worst case performance. The relationship between the mutual information utility function and the proposed SJNR utility function is investigated as well. Simulation results show the effectiveness of the radar and jammer strategies. The investigation in this paper can provide optimal waveform solutions for the monostatic radar and the jammer in the confrontation scenario.

Bio-inspired multi-agent coordination games with Poisson observations

Global games are a class of incomplete information games where the payoffs exhibit strategic complementarity leading to an incentive for the agents to coordinate their actions. Such games have been used to model scenarios in many socioeconomic phenomena, where the private signals available to the agents are typically assumed to be Gaussian. We study an instance of a global game where the agents observe Poisson random variables, which are inspired by applications in microbiology where information signals are disseminated via discrete molecular signals rather than continuous signals. Although this observation model violates the essential technical assumptions present in the Gaussian case, we present preliminary results on the existence of Nash equilibria in pure threshold policies in two variants of the underlying random state-of-the-world: an arbitrarily distributed discrete binary state and a continuous state with an (improper) uniform distribution.

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.

Utility Design for Infinite Distributed Welfare Games

A distributed welfare game is a non-cooperative game-theoretic model for a resource allocation problem that maximizes welfare. In this paper, two utility function designs for resource allocation problems with infinite resources are considered. One is called wonderful life utility, which was proposed for the problem. The second one employs a solution concept of the cooperative game, Egalitarian Non-Separable Contribution (ENSC). The main contributions of the paper are two holds. The first contribution is to clarify the existence of a pure strategy Nash equilibrium in the games defined via the above utility functions when the welfare function is monotone, continuously twice differentiable, and diminishing return (DR) submodular. The second is to derive lower bounds of the Price of Anarchy (PoA) for both games. The PoA is the worst-case ratio between the optimal value and values at the pure strategy Nash equilibria.

Existence of Pure Strategy Nash Equilibrium in Contests: Arbitrary Grouping and Asymmetric Valuations

Existence of Pure Strategy Nash Equilibrium in Contests: Arbitrary Grouping and Asymmetric Valuations

Complex Dynamics of Pricing Game Model in a Dual-Channel Closed-Loop Supply Chain with Delay Decision

In this paper, we study a dual-channel closed-loop supply chain (CLSC) consisting of one manufacturer, one retailer and one third-party firm or platform (3P). The manufacturer wholesales new products through the traditional retail channel and distributes remanufactured products via 3P. We focus on establishing the dynamic Stackelberg game models for nondelayed and delayed cases, respectively. The existence and local stability of Nash equilibrium are examined as well as the complex dynamical behaviors of each model under various scenarios are investigated by numerical simulations, such as stability region, bifurcations, chaos, strange attractors, and so on. Moreover, the impacts of some key parameters on the performance of chain members are analyzed. In addition, the variable feedback control method is utilized to eliminate the system chaos. The results reveal that the high value of the consumer discount perception for remanufactured products and excessively fast price adjustment speed have a destabilization effect on the Nash equilibrium point. In addition, adopting delay decisions by manufacturer does not always make the system more stable because it can exert either positive or negative effect on the system’s stability, while an intermediate delay weight is conducive to the system have a higher chance to stay stable. Furthermore, the manufacturer’s profits will be declined significantly while the profits of retailer and 3P will be elevated to some extent when the system falls into periodic and chaotic motions, so chaos is not always necessarily detrimental to all the decision makers in the dual-channel CLSC.

QoS Optimization for Mobile Ad Hoc Cloud: A Multi-Agent Independent Learning Approach

In the era of ubiquitous computing, offloading certain computing tasks to the mobile ad hoc cloud (MAHC) could help mobile devices reduce execution time. However, if multiple resource demanders (RDs) offload tasks to the MAHC without a proper scheduling policy, it may cause unbalanced load distribution among resource providers (RPs), which will affect the overall quality of service (QoS). To this end, in this paper, we propose a multi-agent independent learning approach aiming to optimize the QoS of MAHC. Firstly, for the distributed MAHC, we formulate the QoS optimization model as a non-cooperative game, where each RD competes for maximizing its own utility. Secondly, based on the potential game theory, we prove the existence of Nash equilibrium. A multi-agent independent learning algorithm is then proposed to obtain the equilibrium points, and the convergence of this algorithm is analyzed. Simulation results confirm that the proposed approach helps balance the load distribution and enhances the QoS of MAHC.

Dynamic two-layer game for striking the balance of interest in multi-agent electricity market considering bilateral contracts and reward-punishment mechanism

• The interactive decision-making behavior of multi-agent in the electricity market is studied by game theory. • Based on the bilateral contract mechanism, a master-slave game model of the wholesale market is established. • A nonlinear reward and punishment price mechanism is proposed to guide users’ behavior, and an evolutionary game is used to simulate the dynamic selection process of users. • A distributed algorithm that can protect users’ privacy is developed to solve the model. To strike the balance of interest in multi-agent electricity market, this paper proposes a dynamic two-layer game in wholesale and retail market environment. First, at the higher layer, a bilateral contract-based master-slave game of wholesale market is developed to investigate the relationship between generators and suppliers. While at the lower layer, a nonlinear reward and punishment mechanism is presented for suppliers to guide users’ electricity consumption. Besides, we take the evolutionary game to model the dynamic selection process of users. Further, this paper proves the existence of Nash equilibrium in multi-agent electricity market and then establishes a distributed algorithm to reach the equilibrium solution. The effectiveness of the dynamic two-layer game strategy proposed in this paper is verified by simulation results, which demonstrates the strategy is conducive to the establishment of a friendly electricity market. Bilateral contracts, and the nonlinear dynamic reward and punishment mechanism can achieve the balance of power supply and demand in the market and realize the balance of interest in multi-agent market. The distributed algorithm performs a fast convergence speed and scalability.

Balanced Computing Offloading for Selfish IoT Devices in Fog Computing

Fog computing, which provides low-latency computing services at the network edge, is an enabler for the emerging Internet of Things (IoT) systems. Offloading tasks to the fog that is closer to IoT users for processing has become a means to ensure that tasks are completed quickly. Fog computing cannot only reduce the congestion of the backbone network but also ensure that the task is completed within the specified time. Since fog resources are limited, there will be resource competition among IoT devices. How to quickly and efficiently make an optimal computation offloading decision for individual selfish IoT devices is a fundamental research issue. This article regards the process of multiple IoT devices competing for fog devices as a game and proposes a distributed computation offloading algorithm. The goal is to optimize the balance of computation delay, energy consumption, and cost for fog nodes. The competition between IoT nodes eventually reaches an equilibrium point, that is the Nash equilibrium point. We prove the existence of Nash equilibrium by Weighted Potential Game. In addition, if a large number of IoT devices select the same node for offloading, which will cause the fog node to run out of power and make some networks unable to work normally. Further, causing part of the network to be paralyzed. Therefore, the paper considers the fairness of offloading to extend the network life cycle. A calculation rate adjustment algorithm is designed for the fairness of offloading to ensure that fog nodes do not run out of power and fail. This paper not only fully considers the performance of the IoT device, but also considers the fairness of the fog. Numerous experiments proved the effectiveness of the proposed algorithm.

Potential Game-Based Radio Resource Allocation in Uplink Multibeam Satellite IoT Networks

In this article, we introduce a potential game-based approach to implement collaborative user scheduling and power allocation in the uplink multibeam satellite-based Internet of things (S-IoT) networks. First of all, a framework of the multibeam uplink S-IoT system is proposed, where full frequency reuse and co-channel interference among different spot beams are considered. To provide broadband S-IoT service effectively, we formulate an initial optimization problem of maximizing uplink sum transmission capacity and then transform it into an interference avoidance problem to address the mathematical intractability. Specifically, a game-theoretic model is implemented to solve the transformed optimization problem, which is proved to be a potential game and existence of Nash equilibriums (NEs). Moreover, an iterative algorithm with low computational complexity, motivated by the finite improvement property, is designed to implement collaborative user scheduling and power allocation to NE point. Finally, the simulation results prove the convergence and effectiveness of the proposed potential game-based approach.

A Game Theoretic Approach to Channel Switching in the Presence of Jamming

In this letter, a channel switching problem is investigated in the presence of jamming based on a game theoretic approach. First, a convex formulation of the optimal channel switching problem is proposed for a given jamming strategy. Then, considering a fixed channel switching strategy, an explicit solution of the optimal jammer power allocation problem is obtained. Consequently, a game theoretic formulation is proposed and the existence of a pure-strategy Nash equilibrium is shown for the proposed channel switching game between the transmitter and the jammer.