Unique Nash Equilibrium Research Articles

Controlling Segregation in Social Network Dynamics as an Edge Formation Game

This paper studies controlling <i>segregation</i> in social networks via exogenous incentives. We construct an edge formation game on a directed graph. A user (node) chooses the probability with which it forms an inter- or intra- community edge based on a utility function that reflects the tradeoff between homophily (preference to connect with individuals that belong to the same group) and the preference to obtain an exogenous incentive. Decisions made by the users to connect with each other determine the evolution of the social network. We explore an <i>algorithmic recommendation mechanism</i> where the exogenous incentive in the utility function is based on <i>weak ties</i> which incentivizes users to connect across communities and mitigates the segregation. This setting leads to a submodular game with a unique Nash equilibrium. In numerical simulations, we explore how the proposed model can be useful in controlling segregation and echo chambers in social networks under various settings.

A General Framework for Learning Mean-Field Games

This paper presents a general mean-field game (GMFG) framework for simultaneous learning and decision making in stochastic games with a large population. It first establishes the existence of a unique Nash equilibrium to this GMFG, and it demonstrates that naively combining reinforcement learning with the fixed-point approach in classical mean-field games yields unstable algorithms. It then proposes value-based and policy-based reinforcement learning algorithms (GMF-V and GMF-P, respectively) with smoothed policies, with analysis of their convergence properties and computational complexities. Experiments on an equilibrium product pricing problem demonstrate that two specific instantiations of GMF-V with Q-learning and GMF-P with trust region policy optimization—GMF-V-Q and GMF-P-TRPO, respectively—are both efficient and robust in the GMFG setting. Moreover, their performance is superior in convergence speed, accuracy, and stability when compared with existing algorithms for multiagent reinforcement learning in the N-player setting. Funding: X. Guo acknowledges the generous support from Coleman Fung Chair endowment fund and the Tsinghua-Berkeley-Shenzhen-Institute.

Predictable Effects of Visual Salience in Experimental Decisions and Games

AbstractBottom-up stimulus-driven visual salience is largely automatic, effortless, and independent of a person’s “top-down” perceptual goals; it depends only on features of a visual stimulus. Algorithms have been carefully trained to predict stimulus-driven salience values for each pixel in any image. The economic question we address is whether these salience values help explain economic decisions. Our first experimental analysis shows that when people pick between sets of fruits that have artificially induced value, predicted salience (which is uncorrelated with value by design) leads to mistakes. Our second analysis uses evidence from games in which choices are locations in images. When players are trying to cooperatively match locations, predicted salience is highly correlated with the success of matching (r = .57). In competitive hider-seeker location games, players choose salient locations more often than predicted by the unique Nash equilibrium. This tendency creates a disequilibrium “seeker’s advantage” (seekers win more often than predicted in equilibrium). The result can be explained by level-k models in which predicted stimulus-driven salience influences level-0 choices and thereby influences overall perceptions, beliefs, and choices of higher-level players. The third analysis shows that there is an effect of visual salience in matrix games, but it is small and statistically weak. Applications to behavioral IO, price and tax salience, nudges and design, and visually influenced beliefs are suggested.

Search for a unique Nash equilibrium in two public goods games: mixed integer programming technique

ABSTRACT We provide the methods for searching for the unique Nash equilibrium using mixed integer programming techniques. We also simulate the model parameters using this technique and derive the numerical solution to show the characteristics of a key player providing both public goods as suggested by Bergstrom et al. (1986). We present two key results. First, when the number of players decreases, the appearance rate of the unique equilibrium accompanying a key player increases. Second, when the variance of the preference or price parameters decreases, the appearance rate of the unique equilibrium accompanying a key player increases.

Which Is the Best Supply Chain Policy: Carbon Tax, or a Low-Carbon Subsidy?

The low-carbon supply chain is key to promoting sustainable development and solving environmental pollution. Government policies related to lowering carbon emissions deeply affect supply chains. This paper builds a supply chain decision-making model under three different regulatory policies: a pure carbon tax, a pure low-carbon subsidy, and a mixed policy with both a carbon tax and a low-carbon subsidy, then compares and analyzes the impacts of these three different regulatory policies on carbon emissions, manufacturer and retailer income, and marginal profit in order to determine the best course of action with respect to supply chain decision-making. Our results indicate that the supply chain decision-making model under the mixed carbon tax and low-carbon subsidy policy results in a unique Nash equilibrium solution between the retailer subsidy rate and the manufacturing carbon reduction rate in a non-cooperative game. Although a carbon tax is beneficial to the ecological environment, retailer income increases slightly as the carbon tax coefficient increases before declining rapidly. Manufacturer income has a negative linear relationship with carbon tax, and an excessive amount of carbon tax increases the burden on companies. Therefore, the government must establish reasonable standards for carbon tax collection while offering moderate low-carbon subsidies at the same time as a means of optimizing social welfare.

The dual of Bertrand with homogenous products is Cournot with perfect complements

The quantity-setting (Cournot) oligopoly with perfect complements is dual to the price-setting (Betrand) oligopoly with homogeneous goods. Under mild technical conditions the former setting has a unique (pure strategy) Nash equilibrium with null quantities. As an implication, the provision of perfectly complementary goods might actually be impossible, if the market is not either perfectly competitive or monopolized.

Accelerate or hinder it? Manufacturer transformation under competition and carbon emission trading

Strengthening carbon regulations and competitive pressures are forcing supply chains to provide environmental (e-) products to replace traditional (t-) products. However, cap-and-trade policy allows manufacturers, the main carbon emission emitters, to trade their emission permits freely. Considering manufacturers' power and key role in producing and carbon trading, we develop a duopoly model consisting of two competing manufacturers. Despite asymmetrical carbon emission reduction efficiencies, the two manufacturers can independently and simultaneously decide their respective t-/e-products outputs, emission reduction levels, and permits trade quotas. The unique Nash equilibrium is derived with the overall cap. We find that the low-cost manufacturer uses its cost advantage to invest more in emission reduction, then sells the excess emission permits to its rival, but makes its rival more competitive in product market. Interestingly, our results show that cap-and-trade policy triggers an industry collusion in which manufacturers reduce competition and improve the power and profits by reducing overall output and adopting differentiated product strategy. Furthermore, the cost advantage and high product substitution can accelerate low-cost manufacturer's transformation but hinder high-cost manufacturers. The impacts of cap-and-trade policy on consumer surplus and social welfare are also discussed to provide guidance for policy makers.

The interaction of emotions and cost-shifting rules in civil litigation

We model civil litigation as a simultaneous contest between a plaintiff and a defendant who have monetary and emotional preferences. The litigants’ emotional variables capture a non-monetary joy of winning and relational emotions toward each other. A contest success function (CSF) describes the litigants’ respective probabilities of success based on their endogenous litigation expenses and exogenous relative advantages. The model does not specify a functional form for the CSF. Instead, it accommodates any CSF that satisfies general and intuitive assumptions, which capture frequently-used functional forms. A cost-shifting rule allows the winner to recover an exogenous proportion of her litigation expenses from the loser. There exists a unique Nash equilibrium with positive expenses. In equilibrium, negative relational emotions (but not a positive joy of winning) amplify the effects of cost shifting, and vice versa. Thus negative relational emotions and positive cost shifting have a similar strategic role, and one can be a substitute for the other. If the litigants’ relative advantages are sufficiently balanced, then more cost shifting (or more negative relational emotions) increases total expenses in equilibrium.

Consensus Game: An Extension of Battle of the Sexes Game

This paper introduces “Consensus Game” as a multiplayer extension of the well-known “Battle of the Sexes” game to model situations where multiple parties (e.g., buyers–sellers) need to come to an agreement on a common issue in order for all players to be better off. We develop closed-form analytical expressions for the mixed-strategy Nash Equilibrium of a Consensus Game and thereafter, under a mildly restrictive assumption of completely mixed strategies, demonstrate the uniqueness of this equilibrium. We also show that in the constant-sum version of the Consensus Game, this unique Nash Equilibrium guarantees equal payoffs to all players. This result pertaining to equal payoffs as the only Nash Equilibrium suggests that in applications where multiple parties need to arrive at a consensus in order to share a common and fixed pool of a resource, an equal-utility distribution of the pool is the only stable solution that does not provide any party a unilateral incentive to deviate.

Location Game of Multiple Unmanned Surface Vessels With Quantized Communications

A placement problem of multiple unmanned surface vessels (USVs) is investigated from a game perspective in this brief. The group of USVs is required to form a desired formation shape, while each USV has its individual objective of being at a particular place relative to the static reference target according to its own benefit. Note that the individual goals of USVs may conflict with the global formation goal of the group. In this context, the location game of multiple USVs is formulated, and a gamed-based controller is designed for each USV to drive it to the equilibrium position that balances the formation goal and the individual goal. Moreover, considering practical limited communication bandwidth, the transmitted information of the USVs is quantized through a uniform quantizer. It is proved that there exists a unique Nash equilibrium of the location game of multiple USVs and the designed controller can drive the USV systems with quantized communication to a small neighborhood of the Nash equilibrium.

A game-theoretic perspective to study a nonlinear stochastic parabolic model of population competition

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players.

Real Time Congestion Management Using Plug in Electric Vehicles (PEV’s): A Game Theoretic Approach

This paper presents a game-theoretic approach for solving network congestion in the Smart Grid. A smart grid environment with high penetration of Electric Vehicles (EVs) for a bidirectional power trade provides an opportunity to use the EVs for congestion management. Optimal power injected or drawn by the EV charging stations can be obtained to mitigate the congestion problem by solving a constrained optimization problem to minimize the change in the power injected at every bus while satisfying several constraints. EV charging stations will be requested to inject or draw optimal power as obtained from the optimization problem. This paper proposes a game-theoretic approach (Greed Game) to develop a smart pricing model and design a mechanism to remove congestion in the network. A unique Nash equilibrium is achieved in the game, which optimally fulfills the line power congestion management problem and yields maximum benefit for the EVs. The algorithm was applied to 16-machine 68-bus test systems, and it has been found that the proposed mechanism maximizes the gain of the EVs and solves the congestion problem in the Grid.

Optimal pricing policy design for selling cost-reducing innovation in Cournot games

In a marketplace where a number of firms produce and sell a homogeneous product, an innovator develops cost-cutting manufacturing technology and decides to sell it to various firms in the form of a license for profit. Given the innovator's license pricing policy, each firm independently decides whether to purchase the innovation license and how many products to produce. To put it simply, the firms are then in a Cournot market in which the product price is a decreasing function of the total amount of the product on the market. Both the innovator and the firms are acting out of self-interest and look to maximize their utilities. We consider the problem of designing optimal pricing policies for the innovator.A pricing policy could be in the form of a one-off upfront fee, a per-unit royalty fee, or a hybrid of both. Building upon the results of Segal [1], we first show that in a properly designed pricing policy, it is a strictly dominant strategy for the firms to accept the pricing policy, and that this constitutes the unique Nash equilibrium of the game. For the hybrid-fee policy, we devise an algorithm that computes the optimal price in time O(n3), where n is the number of firms. For the royalty-fee policy, we show that the problem is captured by convex quadratic programming and can be solved in time O(n6L2), where L is the number of input bits. For the upfront-fee policy, we show the optimal policy problem is NP-complete and we devise an FPTAS algorithm. Moreover, we compare the revenue achievable through the above three pricing policies when all firms are identical.

Distributed Nash Equilibrium Seeking Under Event-Triggered Mechanism

This brief studies distributed event-triggered law design of the Nash equilibrium seeking for noncooperative games under strongly connected graphs. Each player is desired to maximize its payoff function by updating its own action based on other players’ actions. An event-triggered law is proposed, and each player estimates all other players’ actions and exchanges the estimated information with their neighbors only when the triggering conditions are satisfied. Theoretical demonstration is presented to verify that the actions and estimates of players all converge to the unique Nash equilibrium under the proposed event-triggered law and Zeno behavior is excluded. Simulations are provided to show the effectiveness of the main results.

A Green Stackelberg-game Incentive Mechanism for Multi-service Exchange in Mobile Crowdsensing

Although mobile crowdsensing (MCS) has become a green paradigm of collecting, analyzing, and exploiting massive amounts of sensory data, existing incentive mechanisms are not effective to stimulate users’s active participation and service contribution in multi-service exchange in MCS due to its specific features: a large number of heterogeneous users have asymmetric service requirements, workers have the freedom to choose sensing tasks as well as participation levels, and multiple sensing tasks have heterogeneous values which may be untruthful declared by the corresponding requesters. To address this issue, this article develops a green Stackelberg-game incentive mechanism to achieve selective fairness, truthfulness, and bounded efficiency while reducing the burden on the platform. First, we model the multi-service exchange problem as a Stackelberg multi-service exchange game consisting of multi-leader and multi-follower, in which each requester as a leader first chooses the reward declaration strategy and thus the payment for each sensing task, each worker as a follower then chooses the sensing plan strategy to maximize her own utility. We next introduce the concept of virtual currency to maintain the selective fairness to balance service request and service provision between users, in which a user earns/consumes virtual currency for providing/receiving services, and thus no one can always get services without providing services. Then, we present two novel algorithms to compute the unique Nash equilibrium for the sensing plan determination game and the reward declaration determination game, respectively, which together forms a unique Stackelberg equilibrium for the proposed game. Afterwards, we theoretically prove that the proposed green Stackelberg-game incentive mechanism achieves the desirable properties of selective fairness, truthfulness, bounded efficiency. Finally, extensive evaluation results are provided to support the validity and effectiveness of our mechanism compared with both baseline and theoretical optimal approaches.

The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

Poisson voting games under proportional rule

We analyze strategic voting under proportional rule and two parties, embedding the basic spatial model into the Poisson framework of population uncertainty. We prove that there exists a unique Nash equilibrium. We show that it is characterized by a cutpoint in the policy space that is always located between the average of the two parties’ positions and the median of the distribution of voters’ types. We also show that, as the expected number of voters goes to infinity, the equilibrium converges to that of the case with deterministic population size.

Ethical marketing strategies: the unique Nash equilibrium

PurposeCan industrial marketers afford to choose unethical strategies? To answer this question, this study aims to use game theory to analyze whether an industrial marketer choosing and implementing an unethical strategy is successful in maximizing her market share across her strategies.Design/methodology/approachThe competition between two industrial marketers is modeled as a strategic game for the market share of a product that is identical in all attributes except the production process. Each industrial marketer’s objective is to choose to implement either the ethical or the unethical production process to maximize her market share.FindingsThe study finds that both industrial marketers choosing to implement ethical strategies is the unique Nash equilibrium of the game. That is, an industrial marketer choosing to implement an unethical strategy in the production process will be unsuccessful in maximizing her market share when both the industrial marketers are rational.Research limitations/implicationsThe study contributes to the literature on industrial marketing ethics, particularly that on product ethics, by showing that industrial marketers gain market share if they choose ethical strategies.Practical implicationsThe study has implications for industrial marketing executives, as organizational consumers are increasingly aware of the strategies of industrial marketers. Failure to implement ethical strategies will cause industrial marketers to forgo their best possible market shares.Originality/valueThis study’s novelty lies in using a game theoretic approach to demonstrate the positive implications of ethical strategies for industrial marketers.

Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a veri?cation approach and provide su?cient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to in?nity, that is, a suitable mean-?eld game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

A renewable energy-aware power allocation for cloud data centers: A game theory approach

With the rapid emerging of Internet of Things (IoT) devices and the proliferation of cloud-based applications, the cloud computing industry is becoming a vital element for ensuring our daily services. However, cloud computing uses large scale data centers equipped with energy-hungry servers and huge power facilities that massively consume power. This presents a real challenge which can negatively influence the power grid, while exposing the environment to global warming issues. Therefore, minimizing cloud data center power consumption is a challenging problem and has to be addressed. In this paper, we look at renewable energy in the context of a smart grid–cloud architecture and investigate the issue of grid power dispatching to cloud data centers. Since cloud data centers have a non-cooperative nature regarding power demand from the power stations, we model our power allocation problem as a non-cooperative game. Afterwards, we prove the existence and the uniqueness of Nash equilibrium. Moreover, we formulate the payoff function of our game as a non-linear optimization problem before resolving it using Lagrange multipliers and Karush–Kuhn–Tucker (KKT) conditions. Thus, we determine the assigned optimal quantity to each data center based on three main criteria : renewable energy usage, number of critical running applications and workload charge. Extensive simulations are performed by comparing our scheme with an existing work. Results show that our scheme outperforms the comparing approach with a percentage of 31.2% in terms of power load rate and significantly reduces emissions of carbon dioxide.