Pareto Optimal Nash Equilibrium Research Articles

Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.

Use of Nash equilibrium in finding game theoretic robust security bound on quantum bit error rate

Nash equilibrium is employed to find a game theoretic robust security bound on quantum bit error rate (QBER) for DL04 protocol which is a scheme for quantum secure direct communication that has been experimentally realized recently. The receiver, sender and eavesdropper (Eve) are considered to be quantum players (players having the capability to perform quantum operations). Specifically, Eve is considered to have the capability of performing quantum attacks (e.g., Wójcik’s original attack, Wójcik’s symmetrized attack and Pavičić attack) and classical intercept and resend attack. Game theoretic analysis of the security of DL04 protocol in the above scenario is performed by considering several game scenarios. The analysis revealed the absence of a Pareto optimal Nash equilibrium point within these game scenarios. Consequently, mixed strategy Nash equilibrium points are identified and employed to establish both upper and lower bounds for QBER. Further, the vulnerability of the DL04 protocol to Pavičić attack in the message mode is established. In addition, it is observed that the quantum attacks performed by Eve are more powerful than the classical attack, as the QBER value and the probability of detecting Eve’s presence are found to be lower in quantum attacks compared to classical ones.

Competitive behaviour of major GSM firms’ internet data pricing in Nigeria: A game theoretic model approach

The study investigated the competitive behaviour of major GSM service providers’ Internet data pricing in Nigeria. Two null hypotheses were tested: The major GSM service providers in Nigeria do not have a dominant pricing strategy; and there is no Nash Equilibrium for the pricing strategies of the major GSM service providers in Nigeria. The design was a longitudinal study of the active GSM subscribers and Internet data prices of major GSM service providers in Nigeria (MTN and the others). Two pricing strategies were employed, “charge N1000 per data plan” and “charge N1200 per data plan”. A zero-sum payoff table was obtained from the data on GSM subscribers and Internet bundle prices in Nigeria. The payoff table was analysed using mixed strategy approach. The results revealed that each player has a dominant strategy. The game has a Pareto optimal Nash equilibrium with disparate dominant strategies for both players. The Nash equilibrium is that the other GSM firms charge N1000 per internet data plan while MTN charges N1200 per Internet data plan. At a price of N1000 and N1200 per Internet data plan, the earnings of the other GSM service providers and MTN are maximised respectively irrespective of what the other player does. The study concludes that major GSM firms can use game theory to model the pricing strategy of competitors and predict the behaviour of Internet data subscribers.

On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria

We study the computational complexity of decision problems about Nash equilibria in m-player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or stated in terms of complexity classes, \({\exists {\mathbb {R}}}\)-complete, when m ≥ 3. We show that, unless they turn into trivial problems, they are \({\exists {\mathbb {R}}}\)-hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when m ≥ 3 the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are \({\exists {\mathbb {R}}}\)-complete. The latter result rectifies a previous claim of NP-completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is \({\exists {\mathbb {R}}}\)-hard, answering a question of Bilò and Mavronicolas, and address also the computational complexity of deciding if a game has a rational valued Nash equilibrium. These results also hold for 3-player zero-sum games. Our proof methodology applies to corresponding decision problems about symmetric Nash equilibria in symmetric games as well, and in particular our new results carry over to the symmetric setting. Finally we show that deciding whether a symmetric m-player game has a non-symmetric Nash equilibrium is \({\exists {\mathbb {R}}}\)-complete when m ≥ 3, answering a question of Garg, Mehta, Vazirani, and Yazdanbod.

Learning in two-dimensional beauty contest games: Theory and experimental evidence

We extend the beauty contest game to two dimensions: each player chooses two numbers to be as close as possible to certain target values, which are linear functions of the averages of the two number choices. One of the targets depends on the averages of both numbers, making the choices interrelated. We report on an experiment where we vary the eigenvalues of the associated two-dimensional linear system and find that subjects can learn the Pareto-optimal Nash Equilibrium of the system if both eigenvalues are stable and cannot learn it if both eigenvalues are unstable. Interestingly, subjects can also learn it if the system has the saddlepath property – with one stable and one unstable eigenvalue — but only if the one unstable eigenvalue is negative. We show theoretically that our results cannot be explained by homogeneous level-k models where all agents apply the same level k depth of reasoning to their choices, including the naïve learning model. However, our results can be explained by a mixed cognitive-levels model, including the adaptive learning model. We also run a horserace between many models used in the literature with the winner being a simple mixed model with levels 0, 1, and equilibrium reasoning.

Efficient Multiagent Policy Optimization Based on Weighted Estimators in Stochastic Cooperative Environments

Multiagent deep reinforcement learning (MA-DRL) has received increasingly wide attention. Most of the existing MA-DRL algorithms, however, are still inefficient when faced with the non-stationarity due to agents changing behavior consistently in stochastic environments. This paper extends the weighted double estimator to multiagent domains and proposes an MA-DRL framework, named Weighted Double Deep Q-Network (WDDQN). By leveraging the weighted double estimator and the deep neural network, WDDQN can not only reduce the bias effectively but also handle scenarios with raw visual inputs. To achieve efficient cooperation in multiagent domains, we introduce a lenient reward network and scheduled replay strategy. Empirical results show that WDDQN outperforms an existing DRL algorithm (double DQN) and an MA-DRL algorithm (lenient Q-learning) regarding the averaged reward and the convergence speed and is more likely to converge to the Pareto-optimal Nash equilibrium in stochastic cooperative environments.

Z-equilibria in Bi-matrix games with uncertain payoffs

The concept of Z-equilibrium has been introduced by Zhuk-ovskii (Mathematical Methods in Operations Research. Bulgarian Academy of Sciences, Sofia (1985) 103–195) for games in normal form. This concept is always Pareto optimal and individually rational for the players. Moreover, Pareto optimal Nash equilibria are Z-equilibria. We consider a bi-matrix game whose payoffs are uncertain variables. By appropriate ranking criteria of Liu uncertainty theory, we introduce some concepts of equilibrium based on Z-equilibrium for such games. We provide sufficient conditions for the existence of the introduced concepts. Moreover, using mathematical programming, we present a procedure for their computation. A numerical example is provided for illustration.

Stochastic differential games: the potential approach

A noncooperative stochastic differential game (SDG) is said to be a potential game if there exists an associated stochastic control problem whose optimal solutions are Nash equilibria for the given game. In this paper, we give sufficient conditions for an SDG to be a potential game. To this end, our main tools are suitable versions of the stochastic maximum principle. Our results are illustrated with several examples, including team games and separable games, which have Pareto-optimal Nash equilibria.

Quantum-computing with AI & blockchain: modelling, fault tolerance and capacity scheduling

ABSTRACTWe model the hardware and software architecture for generalized Internet of Things (IoT) by quantum cloud-computing and blockchain. To reduce the measurement error and increase the efficiency of quantum entanglement (i.e. the capability of fault tolerance) in the current quantum computers and communications, we design a quantum-computing chip by modelling it as a multi-input multi-output (MIMO) quantum channel and obtain its channel capacity via our recently derived mutual information formula. To capture the internal qubit data flow dynamics of the channel, we model it via a deep convolutional neural network (DCNN) with generalized stochastic pooling in terms of resource-competition among different quantum eigenmodes or users. The pooling is corresponding to a resource allocation policy with two levels of competitions as in cognitive radio: the first one is on users’ selection in a ‘win–lose’ manner; the second one is on resourcesharing among selected users in a ‘win–win’ manner. To wit, our scheduling policy is the one by mixing a saddle point to a zero-sum game problem and a Pareto optimal Nash equilibrium point to a nonzero- sum game problem. The effectiveness of our policy is proved by diffusion modelling with theory and numerical examples.

A Game Theoretic Approach for Distributed and Coordinated Channel Access Control in Cooperative Vehicle Safety Systems

Fairness and efficiency are two key requirements that have to be guaranteed in channel resource allocation in cooperative vehicle safety systems. Existing channel access control strategies, however, rely on each individual node to adjust networking parameters independently according to its locally measured state information, thus leading to unfairness. Although some coordinated strategies have been proposed to resolve this issue, they pay little attention to the efficiency. In order to achieve the tradeoff between fairness and efficiency, in this paper, we propose a utility function in terms of inter-packet reception time required to capture the performance of consecutive successful packets' reception under various vehicle densities. A channel access control problem among vehicles is then formulated as a non-cooperation game model. This model utilizes a punishment function to penalize a node that monopolizes the channel resources and hence can enable nodes to coordinate with each other to achieve desired fairness and efficiency. Next, a distributed decision-making scheme for channel access control is designed. It adjusts transmission rate in a coordinated manner and can guide each node to reach a Pareto-optimal Nash equilibrium point. The experimental results validate that the proposed strategy can result in fair channel resource allocation and ensure high-tracking accuracy for each vehicle under dynamic traffic conditions.

Competitive Energy Trading Framework for Demand-Side Management in Neighborhood Area Networks

This paper, by comparing three potential energy trading systems, studies the feasibility of integrating a community energy storage (CES) device with consumer-owned photovoltaic (PV) systems for demand-side management of a residential neighborhood area network. We consider a fully competitive CES operator in a non-cooperative Stackelberg game, a benevolent CES operator that has socially favorable regulations with competitive users, and a centralized cooperative CES operator that minimizes the total community energy cost. The former two game-theoretic systems consider that the CES operator first maximizes their revenue by setting a price signal and trading energy with the grid. Then the users with PV panels play a non-cooperative repeated game following the actions of the CES operator to trade energy with the CES device and the grid to minimize energy costs. The centralized CES operator cooperates with the users to minimize the total community energy cost without appropriate incentives. The non-cooperative Stackelberg game with the fully competitive CES operator has a unique Stackelberg equilibrium at which the CES operator maximizes revenue and users obtain unique Pareto-optimal Nash equilibrium CES energy trading strategies. Extensive simulations show that the fully competitive CES model gives the best tradeoff of operating environment between the CES operator and the users.

Representations of Political Power Structures by Strategically Stable Game Forms: A Survey

We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players and alternatives. No prior knowledge of social choice is assumed: As far as definitions are concerned, the paper is self-contained. Section 2 supplies the necessary general tools for the rest of the paper. Each definition is followed by a simple (but nontrivial) example. In Section 3 we give a complete account of representations of committees (proper and monotonic simple games), by exactly and strongly consistent social choice functions. We start with Peleg’s representations of weak games, and then provide a complete and detailed account of Holzman’s solution of the representation problem for simple games without veto players. In Section 4 we deal with representations of constitutions by game forms. Following Gärdenfors we model a constitution by a monotonic and superadditive effectivity function. We fully characterize the representations for three kinds of equilibrium: Nash equilibrium; acceptable equilibrium (Pareto optimal Nash equilibrium); and strong Nash equilibrium. We conclude in Section 5 with a report on two recent works on representations of constitutions under incomplete information.

Strong Nash equilibrium in games with common and complementary local utilities

A rather general class of strategic games is described where the coalitional improvements are acyclic and hence strong Nash equilibria exist: The players derive their utilities from the use of certain facilities; all players using a facility extract the same amount of local utility therefrom, which amount depends both on the set of users and on their actions, and is decreasing in the set of users; the ultimate utility of each player is the minimum of the local utilities at all relevant facilities. Two important subclasses are “games with structured utilities,” basic properties of which were discovered in 1970s and 1980s, and “bottleneck congestion games,” which attracted researchers’ attention quite recently. The former games are representative in the sense that every game from the whole class is isomorphic to one of them. The necessity of the minimum aggregation for the existence of strong Nash equilibria, actually, just Pareto optimal Nash equilibria, in all games of this type is established.

Service composition based on multi-agent in the cooperative game

The principle of service composition based on multi-agent is that multi-agent can coordinate to reach Pareto-optimal Nash equilibrium. Reinforcement learning algorithms can be used to deal with the coordination problem in cooperative games. In this paper, the multi-agent coordination problems in cooperative games for different user preference is investigated. In our case, each agent can represent a user’s preference, and it finally learns a policy that is best fit for that user. Most previous works study the deterministic gain of a state. However, in practical service environments, the gain may be nondeterministic due to unstable Quality of Service (QoS). In addition, user preference should be considered. To avoid local optimal solution, we let each agent randomly change interacting partners in each iteration. Thus, an agent can learn its optimal strategy by interacting repeatedly with the rest of agents representing different user preference. The experimental results show that our reinforcement learning algorithm outperforms other learning methods.

Computing the strong Nash equilibrium for Markov chains games

In this paper, we present a novel method for finding the strong Nash equilibrium. The approach consists on determining a scalar λ* and the corresponding strategies d*(λ*) fixing specific bounds (min and max) that belong to the Pareto front. Bounds correspond to restrictions imposed by the player over the Pareto front that establish a specific decision area where the strategies can be selected. We first exemplify the Pareto front of the game in terms of a nonlinear programming problem adding a set of linear constraints for the Markov chain game based on the c-variable method. For solving the strong Nash equilibrium problem we propose to employ the Euler method and a penalty function with regularization. The Tikhonov’s regularization method is used to guarantee the convergence to a single (strong) equilibrium point. Then, we established a nonlinear programming method to solve the successive single-objective constrained problems that arise from taking the regularized functional of the game. To achieve the goal, we implement the gradient method to solve the first-order optimality conditions. Starting from an utopia point (Pareto optimal point) given an initial λ of the individual objectives the method solves an optimization problem adding linear constraints required to find the optimal strong strategy d*(λ*). We show that in the regularized problem the functional of the game decrease and finally converges, proving the existence and uniqueness of strong Nash equilibrium (Pareto-optimal Nash equilibrium). In addition, we present the convergence conditions and compute the estimatedrate of convergence of variables γ and δ corresponding to the step size parameter of the gradient method and the Tikhonov’s regularization respectively. Moreover, we provide all the details needed to implement the method in an efficient and numerically stable way. The usefulness of the method is successfully demonstrated by a numerical example.

Strategic bidding for multiple price-maker hydroelectric producers

In a market comprised of multiple price-maker firms, the payoff each firm receives depends not only on one’s own actions but also on the actions of the other firms. This is the defining characteristic of a non-cooperative economic game. In this article, we ask: What is the revenue-maximizing production schedule for multiple price-maker hydroelectric producers competing in a deregulated, bid-based market? In every time stage, we seek a set of bids such that, given all other price-maker producers’ bids, no price-maker can improve (increase) their revenue by changing their bid; i.e., a pure-strategy Nash–Cournot equilibrium. From a theoretical game theory perspective, the analysis on the underlying non-cooperative game is lacking. Specifically, existing approaches are not able to detect when multiple equilibria exist and consider any equilibrium found optimal. In our approach, we create interpolations for each price-maker’s best response function using mixed-integer linear programming formulations within a dynamic programming framework. In the presence of multiple Nash equilibria, when one exists, our approach finds the equilibrium that is Pareto optimal. If a Pareto-optimal Nash equilibrium does not exist, we use a tailored bargaining algorithm to determine a unique solution. To illustrate some of the finer details of our method, we present three examples and a case study on an electricity market in Honduras.

Multiagent Reinforcement Social Learning toward Coordination in Cooperative Multiagent Systems

Most previous works on coordination in cooperative multiagent systems study the problem of how two (or more) players can coordinate on Pareto-optimal Nash equilibrium(s) through fixed and repeated interactions in the context of cooperative games. However, in practical complex environments, the interactions between agents can be sparse, and each agent's interacting partners may change frequently and randomly. To this end, we investigate the multiagent coordination problems in cooperative environments under a social learning framework. We consider a large population of agents where each agent interacts with another agent randomly chosen from the population in each round. Each agent learns its policy through repeated interactions with the rest of the agents via social learning. It is not clear a priori if all agents can learn a consistent optimal coordination policy in such a situation. We distinguish two different types of learners depending on the amount of information each agent can perceive: individual action learner and joint action learner . The learning performance of both types of learners is evaluated under a number of challenging deterministic and stochastic cooperative games, and the influence of the information sharing degree on the learning performance also is investigated—a key difference from the learning framework involving repeated interactions among fixed agents.

Some dynamics of signaling games.

Information transfer is a basic feature of life that includes signaling within and between organisms. Owing to its interactive nature, signaling can be investigated by using game theory. Game theoretic models of signaling have a long tradition in biology, economics, and philosophy. For a long time the analyses of these games has mostly relied on using static equilibrium concepts such as Pareto optimal Nash equilibria or evolutionarily stable strategies. More recently signaling games of various types have been investigated with the help of game dynamics, which includes dynamical models of evolution and individual learning. A dynamical analysis leads to more nuanced conclusions as to the outcomes of signaling interactions. Here we explore different kinds of signaling games that range from interactions without conflicts of interest between the players to interactions where their interests are seriously misaligned. We consider these games within the context of evolutionary dynamics (both infinite and finite population models) and learning dynamics (reinforcement learning). Some results are specific features of a particular dynamical model, whereas others turn out to be quite robust across different models. This suggests that there are certain qualitative aspects that are common to many real-world signaling interactions.

Auction-Based Relay Power Allocation: Pareto Optimality, Fairness, and Convergence

It is well known that a cooperative communication technique can offer significant energy saving improvements. In particular, the efficient relay resource allocation makes energy saving practically appealing. In this work, we propose an auction-based relay power allocation scheme over multi-user relay networks from the energy-efficient perspective. In particular, during the relay resource allocation operation, three major design goals are considered: 1) efficient utilization of relay resources, in terms of Pareto optimal relay power allocation; 2) insurance of competitive fairness among competing users; and 3) guarantee of distributed implementation with relaxation of restrictions on complete private knowledge and accurate assessments of convergence. Specifically, we take full advantage of the auction mechanism, i.e., competitive fairness with the incomplete private information of other nodes, to model the interaction among the users as a multi-winner auction based on the optimal bidding decision. By treating the proposed auction mechanism as a non-cooperative game, we obtain the unique and Pareto optimal Nash equilibrium (NE), which yields the optimal bidding decision and allocation of the relay power. Moreover, we design a distributed algorithm based on best-response functions to reach the NE allocation. In particular, the convergence and the convergent rate of the algorithm are analyzed quantitatively to clarify the application scenarios.

Time-Based Competition with Benchmark Effects

We consider a duopoly where firms compete on waiting times in the presence of an industry benchmark. The demand captured by a firm depends on the gap between the firm's offer and the benchmark. We refer to the benchmark effect as the impact of this gap on demand. The formation of the benchmark is endogenous and depends on both firms' choices. When the benchmark is equal to the shorter of the two offered delays, we characterize the unique Pareto optimal Nash equilibrium. Our analysis reveals a stickiness effect in which firms equate their delays at the equilibrium when the benchmark effect is sufficiently strong. When the benchmark corresponds to a weighted average of the two offered delays, we show the existence of a pure Nash equilibrium. In this case, we reveal a reversal effect, in which the market leader, i.e., the firm that offers a shorter delay, becomes the follower when the benchmark effect is sufficiently strong. In both cases, we show that customers' equilibrium waiting times are shorter with the benchmark effect than without it. Our models also capture customers' loss aversion, which, in our setting, states that demand is more sensitive to the gap between the delay and the benchmark when the delay is longer than the benchmark (loss) than when it is shorter (gain). We characterize the impact of this loss aversion on the equilibrium in both settings. Finally, we show numerically that the stickiness and reversal effects still exist when firms also compete on price.