Nash Equilibrium Conditions Research Articles

A Mixed Cooperative Dual to the Nash Equilibrium

A mixed dual to the Nash equilibrium is defined for n-person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other n-1 players’ strategies. Conversely, in the dual equilibrium every n-1 players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each n>2 there exists a game for which no dual equilibrium exists.

Energy Efficient Based Channel Assignment Game Algorithm for Wireless Sensor Network

In wireless sensor network, communication interference problem is serious due to the widespread use of wireless sensor network. Meanwhile, the network may perform abnormally for the low energy node failure. The energy consumption of data retransmission is the main reason for node failure. The data retransmission is caused by communication interference. As we all know, the multi-channel technology is an effective way to alleviate the interference among nodes by using the available channel resource reasonably. The existing algorithms have higher complexity, because they assign channel for link through respectively determining the receiving and sending channel of two nodes on the link. However, the sensor node's energy and the ability of computing access are limited greatly. Thus, it makes the solution of channel allocation problem for WSN a new challenge. To effectively solve the problem, a multiple channels allocation game model is established. It considers the influence of node energy to avoid the extra energy consumption with effect. And, the path gain is introduced to accurately describe the interference. Furthermore, a kind of Energy Efficiency based multiple channels allocation algorithm is proposed in this paper. In addition, the algorithm only assigns receiving channel for sensor node to avoid the high complexity of algorithm. And it combines dynamic channel switching to complete channel adjustment in communication. The analysis results demonstrate that the channel allocation game is a potential game and it can converge to the state of Nash equilibrium. The simulation results show that the algorithm can greatly decrease the convergence round, effectively reduce the interference of low-energy nodes and improve the anti-jamming network performance.

Order-Driven Markets are Almost Competitive

This article studies a market game under uncertainty in which agents may submit multiple limit and market orders. When agents know their preferences at all states, the competitive equilibrium can be supported as a Nash equilibrium of the market game, that is, agents behave as if they were price takers. Therefore, if the associated competitive economy has a fully revealing rational expectations equilibrium, then so does the market game. This resolves the puzzle that agents behave as if prices were given, even though prices aggregate private information, at least for this “private values” case. Necessary conditions for Nash equilibrium show that the resulting allocation cannot deviate too far from a competitive equilibrium. When agents do not know their preferences at some states, though, a characterization result shows that the Nash equilibria of the market game tend to be far from competitive. (author's abstract)

Corrigendum to “Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples”

Levy (2013) presented examples of discounted stochastic games that do not have stationary equilibria. The second named author has pointed out that one of these examples is incorrect. In addition to describing the details of this error, this note presents a new example by the first named author that succeeds in demonstrating that discounted stochastic games with absolutely continuous transitions can fail to have stationary equilibria.

Threshold models of mutual insurance

Game-theoretic models of mutual insurance are considered. Here a player chooses between participating in a mutual insurance fund or not. Player's behavior depends on its risk aversion. Using a scalar coefficient of agent's risk aversion degree, we define utility functions leading to the threshold behavior of agents. Next, the game-theoretic models of anonymous and non-anonymous insurants are studied and Nash equilibrium conditions are established for both models.

A Game Theoretic Model for Smart Grids Demand Management

Demand-side management (DSM) plays a key role in the future of smart grids. Recently, DSM researchers have developed various mathematical models to optimize the demand response. Most of these works ignore the channel impairments' impact on the optimization process. In this paper, we propose a new noncooperative game theoretic model for the management of smart grid's demand considering the packet error rate in our formulation. We set the Nash equilibrium conditions for the proposed model. Under an assumption on the form of the utility functions, we develop a 0-1 mixed linear programming approach to compute nondominated extreme Nash equilibria. Results on a numerical example are provided and some useful insights are presented. Under some assumptions and a fully proven proposition, a feasible nondominated Nash equilibrium solution is found. Finally, we report and comment on computational experiments on randomly generated smart grid DSM game instances with different characteristics.

A Characterization of Stationary Nash Equilibria of Single Controller Constrained Stochastic Games

We consider a two player finite state-action general sum single controller constrained stochastic game with both discounted and average cost criteria. We consider the situation where player 1 has subscription-based constraints and player 2, who controls the transition probabilities, has realization-based constraints which can also depend on the strategies of player 1. It is known that a stationary Nash equilibrium for discounted case exists under strong Slater condition, while, for the average case, stationary Nash equilibrium exists if additionally the Markov chain is unichain. For each case we show that the set of stationary Nash equilibria of this game has one to one correspondence with the set of global minimizers of a certain nonconvex mathematical program. If the constraints of player 2 do not depend on the strategies of player 1, then the mathematical program reduces to a quadratic program. The known linear programs for zero sum games of this class can be obtained as a special case of above quadratic programs.

On Asynchronicity of Moves and Coordination

This paper shows that asynchronicity of moves can lead to a unique prediction in coordination games, in an infinite-horizon setting, under certain conditions on off-equilibrium payoffs. In two-player games we derive necessary and sufficient conditions for play ultimately being absorbed in the Pareto dominant Nash equilibrium of the stage game, for every Markov perfect equilibrium. For players patient enough, the condition is that the Pareto dominant Nash equilibrium is also risk dominant, but for lower levels of patience the condition departs from simple risk-dominance. For general n-player symmetric games with patient players, we show that a necessary and sufficient condition for the Pareto dominant Nash equilibrium to be the unique limit outcome in all symmetric Markov perfect equilibrium is a particular generalization of risk-dominance for more than two players. We provide extensions to the unique selection results to all subgame perfect Nash equilibria, and to coordination games in which different players prefer different Nash equilibria of the stage game.

Algorithmic mechanisms for reliable crowdsourcing computation under collusion.

We consider a computing system where a master processor assigns a task for execution to worker processors that may collude. We model the workers’ decision of whether to comply (compute the task) or not (return a bogus result to save the computation cost) as a game among workers. That is, we assume that workers are rational in a game-theoretic sense. We identify analytically the parameter conditions for a unique Nash Equilibrium where the master obtains the correct result. We also evaluate experimentally mixed equilibria aiming to attain better reliability-profit trade-offs. For a wide range of parameter values that may be used in practice, our simulations show that, in fact, both master and workers are better off using a pure equilibrium where no worker cheats, even under collusion, and even for colluding behaviors that involve deviating from the game.

Decision, Uncertainty and Cooperation: A Behavioral Interpretation Based on Quantum Strategy

This paper presents a new attempt to explore people’s cooperative behavior under the natural uncertainty in decision making process. The most recent development of quantum cognition creatively enriched the exploration to the endogenous uncertainty in human behavior by expanding the strategic space. In this paper we extend the quantum decision-making model to a two-player Prisoner’s dilemma game by bringing in the evolutional decision operators with quantum phases. These quantum phases, following uniform random distribution when the decisions are confined to individual decision-making process, will change according to the different levels of implied cooperative inclination, and lead to possible new Nash Equilibriums that emerge from a “hyper” decision space. We also bring in the Dissimilarity Index from complex network literature, to capture and measure the cooperative inclination between the players. Conditions of Quantum Nash Equilibrium are derived out from dynamic quantum equations under the Heisenberg Picture.

Failure of gradualism under imperfect monitoring

We consider a two-player Prisoner's Dilemma type game with continuous actions, where players choose how much to contribute to a public project. This game is played infinitely many times and actions are irreversible: players cannot decrease their actions over time. While it is strictly dominant for players not to contribute in the stage game, some strictly positive level of contribution is Pareto optimal. It is known that when players perfectly observe each other's actions, cooperation can be achieved through gradual increases in contribution levels. I show that introducing an arbitrarily small amount of smooth noise in the monitoring makes cooperation impossible and players play the static Nash equilibrium of the stage-game forever.

A Game-Theoretic Model of Underpricing and Over-Subscription in Chinese IPO's

In Chinese A-share IPO's the bulk of available stock is allocated to investors via a lottery. Since the probability of an investor winning the lottery depends on the dollar value of deposits submitted by other investors (and vice versa), the payoff structure of participating in an IPO resembles a game. We solve a simple version of the game for the static Nash equilibrium in continuous strategies and derive the optimal IPO deposit for an arbitrary number of investors with common risk aversion within a two-moment decision model. As the number of investors grows large, the over-subscription ratio implied by our model reduces asymptotically to the ratio of expected IPO underpricing to the net cost of funding of deposits. Notably, investor risk aversion does not enter into this asymptotic solution. A data set of 1,121 Chinese A-share IPO's provides empirical support for our results.

Computational Framework for Optimal Carbon Taxes Based on Electric Supply Chain Considering Transmission Constraints and Losses

A modeling and computational framework is presented for the determination of optimal carbon taxes that apply to electric power plants in the context of electric power supply chain with consideration of transmission constraints and losses. In order to achieve this goal, a generalized electric power supply chain network equilibrium model is used. Under deregulation, there are several players in electrical market: generation companies, power suppliers, transmission service providers, and consumers. Each player in this model tries to maximize its own profit and competes with others in a noncooperative manner. The Nash equilibrium conditions of these players in this model form a finite-dimensional variational inequality problem (VIP). By solving this VIP via an extragradient method based on an interior point algorithm, the optimal carbon taxes of power plants can be determined. Numerical examples are provided to analyze the results of the presented modeling.

Cooperative Coevolutionary Algorithm-Based Model Predictive Control Guaranteeing Stability of Multirobot Formation

This paper proposes a novel cooperative coevolutionary algorithm (CCEA)-based distributed model predictive control (MPC) that guarantees asymptotic stability of multiagent systems whose state vectors are coupled and nonseparable in a cost function. While conventional evolutionary algorithm-based MPC approaches cannot guarantee stability, the proposed CCEA-based MPC approach guarantees asymptotic stability regardless of the optimality of the solution that the CCEA-based algorithm generates with a small number of individuals. To guarantee stability, a terminal state constraint is found, and then a repair algorithm is applied to all candidate solutions to meet the constraint. Furthermore, as the proposed CCEA-based algorithm finds the Nash-equilibrium state in a distributed way, robots can quickly move into a desired formation from their locations. A novel dynamic cooperatively coevolving particle swarm optimization (CCPSO), dynamic CCPSO (dCCPSO) in short, is proposed to deal with the formation control problem based on the conventional CCPSO, which was the most recently developed algorithm among CCEAs. Numerical simulations and experimental results demonstrate that the CCEA-based MPC greatly improves the performance of multirobot formation control compared with conventional particle swarm optimization-based MPC.

Constrained stochastic games with the average payoff criteria

In this paper we study two-person nonzero-sum constrained stochastic games under the average payoff criteria. The state space is denumerable and the action spaces of the players are Borel spaces. Under the suitable conditions, we prove the existence of a constrained stationary Nash equilibrium via the vanishing discount approach. Furthermore, we use an example to illustrate our conditions.

Complexity analysis of a Cournot–Bertrand duopoly game with different expectations

In this paper, we consider a Cournot–Bertrand mixed duopoly model with different expectations, where the market has linear demand and the firms have fixed marginal cost functions. Two firms choose output and price as decision variables, respectively, under the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market is occupied by the one that applies a lower price. The two players are considered to have bounded rational and static expectations. The existence and local stability condition of Nash equilibrium is investigated. We find the stability region of Cournot–Bertrand system is bigger than that of Cournot or Bertrand system under the same conditions. Furthermore, there are two different kinds of bifurcations when the parameters pass through the different boundary curves of the stability region, which is different from the Cournot or Bertrand model. Numerical simulation method is used to display the dynamic behaviors of the dynamical system, such as periodic cycles, bifurcation diagrams and strange attractors of the systems. The economic explanations of the complex dynamic behaviors are also given.

Potential game design for a class of distributed optimisation problems

The state-based potential game is discussed and a game-based approach is proposed for distributed optimisation problem in this paper. A continuous-time model is employed to design the state dynamics and learning algorithms of the state-based potential game with Lagrangian multipliers as the states. It is shown that the stationary state Nash equilibrium of the designed game contains the optimal solution of the optimisation problem. Moreover, the convergence and stability of the learning algorithms are obtained for both undirected and directed communication graph. Additionally, the application to plug-in electric vehicle management is also discussed.

The multi-item newsvendor model with cross-selling and the solution when demand is jointly normally distributed

Products are often demanded in tandem because of the cross-selling effect. The demand for an item can increase if sales of its cross-selling-associated items are achieved or decrease when the associated items are out of stock, resulting in lost sales. Therefore, a joint inventory policy should be pursued in a cross-selling system. This paper introduces customer-driven cross-selling into centralized and competitive newsvendor (NV) models by representing an item’s effective demand as a function of other items’ order quantities. We derive first-order optimality conditions for the centralized model in addition to pure-strategy Nash equilibrium conditions and uniqueness conditions of the equilibria for the competitive model. We further develop gradient-based (GB) and iteration-based (IB) algorithms to solve the centralized and competitive models, respectively. A computational study verifies the effectiveness of the proposed algorithms. The computational results show that a larger cross-selling effect leads to a larger order quantity in a centralized NV model but a smaller order quantity in a competitive NV model, and a larger positive correlation between items’ demands leads to higher profits with smaller order quantities in both models. Moreover, NVs will order more items if the demand variance is greater, however resulting in lower profits. In a competitive situation, one will prefer smaller order quantities than in a centralized decision situation.

Cooperation and Game between Producers and Managers Based on the Linear Contract

There is a cooperative game between the manager and the producer in the enterprise. In this paper, we firstly construct the cooperative game model based on the principal-agent theory. Under the conditions of Nash equilibrium and linear contract, the paper calculates the net income of the client, the total risk and welfare of the agents when the agents have the cooperation or not. The result shows that the correlation coefficient between their output has a direct relationship with the cooperation. Secondly, according to the power distribution theory another model is developed. We analyze the game process and critical state. Furthermore, we deduce the share proportion of the profit and the control size when they have the cooperation. Finally, we summarize all the research achievements, which are of universal significance for the practical cooperation game problems.

A Dynamic Model of the Tragedy of the Commons in Marketing-Intensive Industries

This study provides a dynamic model and analyzes its process that may plunge the business ecosystem into ToC (the Tragedy of the Commons). When developing the model, we have in mind some industries where the marketing competition to secure a large installed base is intense. The social commerce industry is a representative example of this type of industries, but the scope of this study is not limited to the industry. We first introduce a previous study focusing on the static Nash equilibrium, and then present an extended version of the basic model in a dynamic perspective. According to our analyses on the dynamic equilibria together with their stability, there may be a unique interior equilibrium, but it is highly likely unstable. In addition, possible (near) boundary equilibria are also unstable for a wide range of parameter values. We also conduct some numerical experiments and discover cycles as solutions to some particular instances. Since those cycles contain the ToC traps, a policy measure or regulation may need to be employed. Our approach and results will help to figure out a clue to escape from the ToC trap, thereby shedding new light on the sustainable growth of the business ecosystem, which is prone to excessive marketing competition.