Nash Equilibrium Outcomes Research Articles

The implementability of liberalism

AbstractThis note shows that under the unrestricted domain, there exists a choice liberal and Nash implementable social choice rule if and only if there are at least three players and the outcome set is at least twice as large as the player set. A social choice rule is choice liberal if and only if for every player, there exists at least one pair of outcomes such that if this player strictly prefers one over the other, the one he prefers is socially desirable and the other one is not. A social choice rule is Nash implementable if and only if there exists a mechanism such that at every preference profile, the set of Nash equilibrium outcomes coincides with the set of socially desirable ones. The proof constructs an intuitive Nash implementing mechanism.

Cardinal Sins? Conspicuous Consumption, Cardinal Status and Inequality

Abstract This paper analyzes the social dilemma arising when a large population of individuals with differing incomes have concerns over relative deprivation in terms of visible or conspicuous consumption. These relative concerns are cardinal—people care about the size of the gap between own and others’ consumption—and include inequity aversion, where negative comparisons are more important than positive, rivalrous preferences, and comparison with mean consumption. The resulting Nash equilibrium is inefficient, with consumption generally exceeding the socially efficient level. In this model, the income distribution has a direct effect on behavior and under rivalrous preferences, an increase in incomes for the rich can raise consumption at all income levels and make almost everyone worse off.

Replicator-mutator dynamics of the rock-paper-scissors game: Learning through mistakes.

We generalize the Bush-Mosteller learning, the Roth-Erev learning, and the social learning to include mistakes, such that the nonlinear replicator-mutator equationwith either additive or multiplicative mutation is generated in an asymptotic limit. Subsequently, we exhaustively investigate the ubiquitous rock-paper-scissors game for some analytically tractable motifs of mutation pattern for which the replicator-mutator flow is seen to exhibit rich dynamics that include limit cycles and chaotic orbits. The main result of this paper is that in both symmetric and asymmetric game interactions, mistakes can sometimes help the players learn; in fact, mistakes can even control chaos to lead to rational Nash-equilibrium outcomes. Furthermore, we report a hitherto-unknown Hamiltonian structure of the replicator-mutator equation.

Open-loop and closed-loop Nash equilibria for the LQ stochastic difference game

This paper investigates the conditions for the existence of open-loop and closed-loop Nash equilibria for the linear quadratic (LQ) two-player stochastic difference game. It is shown that the former is characterized by the solvability of a forward–backward stochastic difference equation (FBSΔE), while the latter is described by the solvability of a class of coupled symmetric difference Riccati equations (ΔREs). Different from the existing works, the relationship between the closed-loop representation of open-loop Nash equilibrium and the closed-loop Nash equilibrium has been studied. It is found that the closed-loop representation of open-loop Nash equilibrium is different from the outcome of closed-loop Nash equilibrium for the general case. However, for the LQ zero-sum stochastic difference game, they are consistent.

Early Adopter or Follower? The Strategic Equilibrium of Blockchain Technology Adoption Strategy for Competing Agri-Food Supply Chains

Despite the touting value of blockchain technology (BT), many agri-food supply chains are still struggling to the adoption of it. In this article, incorporating the benefit and cost associated with BT, we seek to answer whether and under what conditions can two competing agri-food supply chains benefit from the adoption of BT, and how government choose optimal subsidy scheme to promote the adoption of BT. The findings suggest that the Nash equilibrium outcome toward BT adoption strategy will be greatly affected by key parameters, such as competitive intensity, the growth rate of the market size and the investment cost with the adoption of BT, and the planting cost of the agri-food. Specifically, early adopter can always snatch more benefit from the adoption of the BT than the follower, and the gap between them will be increased in the competition intensity. In addition, the decisions selected by agri-food supply chains does not always benefit the consumer surplus and the social welfare, which call for further government subsidy scheme to promote the adoption of BT. The findings provide important implications for both the industrial managers and policy makers on how to benefit from BT in the digital transformation era.

Game-theoretic Flux Balance Analysis Model for Predicting Stable Community Composition.

Models for microbial interactions attempt to understand and predict the steady state network of inter-species relationships in a community, e.g. competition for shared metabolites, and cooperation through cross-feeding. Flux balance analysis (FBA) is an approach that was introduced to model the interaction of a particular microbial species with its environment. This approach has been extended to analyzing interactions in a community of microbes; however, these approaches have two important drawbacks: first, one has to numerically solve a differential equationto identify the steady state, and second, there are no methods available to analyze the stability of the steady state. We propose a game theory based community FBA model wherein species compete to maximize their individual growth rate, and the state of the community is given by the resulting Nash equilibrium. We develop a computationally efficient method for directly computing the steady state biomasses and fluxes without solving a differential equation. We also develop a method to determine the stability of a steady state to perturbations in the biomasses and to invasion by new species. We report the results of applying our proposed framework to a small community of four E. coli mutants that compete for externally supplied glucose, as well as cooperate since the mutants are auxotrophic for metabolites exported by other mutants, and a more realistic model for a gut microbiome consisting of nine species.

Optimal capital structure with supplier market power

AbstractWe use a real‐option model to study the effect of input supplier's market power on a firm's capital structure, and identify the Nash equilibrium outcome (firm's investment and financing policies and its supplier's pricing policy). When its supplier has market power, the firm will reduce leverage ratio and delay investment. This can help explain why observed leverage ratios are lower than in traditional capital‐structure models (without supplier market power). Firm value can be increased by the vertical acquisition of the supplier, which would also result in a higher leverage ratio. This helps explain the observed increase in leverage ratios after acquisitions.

Corporate social responsibility and network externalities: a game-theoretic approach

This research revisits the pioneering work by Katz and Shapiro (Am Econom Rev 75:424–440, 1985) with network (consumption) externalities in a twofold way: first, it considers Corporate Socially Responsible (CSR), instead of profit-maximising, firms; second, it uses a game-theoretic approach and analyses the commitment decision game in which firms face the binary choice to credibly commit (C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C$$\\end{document}) or not to commit (NC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NC$$\\end{document}) themselves to an announced output level in the first decision-making stage. Competition at the market stage occurs à la Cournot. Results show a rich spectrum of sub-game perfect Nash equilibrium (SPNE) outcomes, ranging from the prisoner’s dilemma (self-interest and mutual benefit of output commitment conflict) to the anti-prisoner’s dilemma or deadlock (self-interest and mutual benefit of output commitment do not conflict), passing from the coordination to the anti-coordination game. These outcomes depend on the intensity of the social concern in the firm’s objective and the network size. The article also pinpoints the welfare outcomes corresponding to the SPNE and extends the analysis to a Stackelberg rivalry setting.

Strategic driver’s acceptance-or-rejection behavior and cognitive hierarchy in on-demand platforms

It is critical for an on-demand platform to understand its users’ strategic-decision behaviors and thereby design a good policy to match the supply with the demand. Through behavioral game-theoretical modeling, we investigate the effects of drivers’ strategic acceptance-or-rejection behavior on their incomes and the overall platform users’ welfare, in an on-demand service platform with two service locations (i.e., a hot area and a cold area), wherein the competition between drivers arises due to overall supply exceeding overall demand and the location uncertainty of drivers. We first construct a full-rationality model to investigate drivers’ decision strategy in a competitive environment and attain the Nash Equilibrium outcomes in the stationary state of the system in the long run. We then employ the Cognitive Hierarchy theory to capture the heterogeneity effect in drivers’ strategic acceptance-or-rejection behaviors, which is more realistic than the full-rationality model. We find that the predictions of the Cognitive Hierarchy model can approach the Nash Equilibrium outcomes under an extreme condition. We can conclude that (i) the optimal acceptance probability of the drivers does not decrease with the rejection penalty or with the relative popularity of the cold area under our research settings. (ii) From the profit/welfare perspective, increasing the popularity of the cold area is beneficial for the platform users; however, a higher rejection penalty may not be always better for the platform users as a whole. Managerial implications are discussed.

Asymmetric information, capacity constraint and segmentation in credit markets

PurposeThe purpose of this study is to propose a model of competition between a formal lender (bank) and an informal lender (moneylender) with informational asymmetry between these two lenders. Further, the authors introduce capacity constraint on the lending capacity of the moneylender and assume that borrowers differ in risk and wealth.Design/methodology/approachThe solution concept of Nash equilibrium has been used to derive the optimal strategies of the lenders.FindingsThe equilibrium strategies in most of the results depend on the difference between the expected returns from risky and safe projects where the risky project has higher expected returns. The credit market is segmented in terms of risk and wealth levels. Rationing of poor safe borrowers from the credit market is inevitable when the moneylender's capacity is sufficiently small, suggesting a low-income trap for them. Further, when moneylender has capacity constraint of some form, a zero-profit outcome is never a Nash equilibrium outcome.Research limitations/implicationsThere is a possibility of collusion between the lenders. However, the authors do not derive all possible outcomes under capacity constraintPractical implicationsWhen the informal lender has limited capacity, competition between formal and informal lenders may not alleviate credit rationing, instead aggravate the problem. Thus, the government should devise policies to ensure credit availability to resource poor householdsOriginality/valueWhile the literature models strategic interaction between lenders under the assumption of zero-profit (Bertrand Paradox) condition, this study shows that zero profit is not the only outcome under such a set-up. Also, in presence of capacity constraint of the moneylender, a zero-profit outcome is never a Nash equilibrium outcome for the lenders. There is an optimal contract at which the lenders differentiate in terms of repayment and collateral and earn positive profits under certain conditions.

Green innovation downturn: The role of imperfect competition

We use strategic interactions to analyze the role of China’s state-subsidized production expansion in the recent downturn in solar photovoltaics innovation. To that end, we develop a dynamic game model in which N solar panel manufacturers compete in price and invest in cost-reducing research. The resulting Nash equilibrium displays an inverted U relationship between a manufacturer’s market share and its research effort. In the case where a local firm is competing against a foreign state-subsidized one, we obtain analytical and numerical results that are consistent with three stylized facts: (i) the foreign manufacturer progressively expands to become the dominant player, (ii) competition and innovation bring marginal costs down to zero, and (iii) each manufacturer’s research effort follows an increasing-then-decreasing curve. At a policy level, our model shows that national technology-push policies, such as R&D subsidies, can affect foreign innovation by changing the structure of global competition.

Geo-Fencing or Geo-Conquesting? a strategic analysis of Location-Based coupon under different market structures

Location-based technology enables firms to target consumers with personalized coupons based on their real-time locations, making location-based coupons (LBC) an innovative marketing tool. In this paper, we consider two types of LBC strategies, namely defensive geo-fencing versus offensive geo-conquesting. With a defensive LBC strategy, a company sets a virtual geo-fence by offering coupons with deeper discounts to consumers located closer to the focal firm. Using a spatial model, we examine how two competing companies choose between defensive and offensive LBC strategies, as well as the impact of LBC strategies on company profits, consumer surplus, and social welfare. The results show that the defensive LBC strategy lowers revenue in a monopoly market but increases it under duopoly conditions. In a duopoly market, both firms adopting the defensive LBC strategy is the Nash equilibrium outcome, leading to the highest profits but lowest consumer surplus. The misalignment of interests among firms and consumers requires policymakers to regulate firms’ LBC strategies for the sake of customers.

Resource-Aware Cost-Sharing Methods for Scheduling Games

In large distributed systems, ensuring the efficient utilization of the available resources is a very challenging task. Given limited information regarding the state of the system and no centralized control over the outcome, decentralized scheduling mechanisms are unable to enforce optimal utilization. To better understand such systems, some classic papers that introduced game theoretic models used the “price of anarchy” measure to evaluate the system’s performance. The paper “Resource-Aware Cost-Sharing Methods for Scheduling Games” by Christodoulou, Gkatzelis, and Sgouritsa overcomes some of the overly pessimistic results shown in this prior work by enhancing the scheduling mechanisms with access to some additional information regarding the state of the system: a “resource-aware” mechanism knows what machines are available in the system and uses this information to carefully incentivize the users toward more efficient Nash equilibrium outcomes.

When does an additional stage improve welfare in centralized assignment?

We study multistage centralized assignment systems to allocate scarce resources based on priorities in the context of school choice. We characterize schools’ capacity-priority profiles under which an additional stage of assignment may improve student welfare when the deferred acceptance algorithm is used at each stage. If the capacity-priority profile is acyclic, then no student prefers any subgame-perfect Nash equilibrium (SPNE) outcome of the 2-stage system to the truthful dominant-strategy equilibrium outcome of the 1-stage system. If the capacity-priority profile is not acyclic, then an SPNE outcome of the 2-stage system may Pareto dominate the truthful dominant-strategy equilibrium outcome of the 1-stage system. If students are restricted to playing truncation strategies, an additional stage unambiguously improves student welfare: no student prefers the truthful dominant-strategy equilibrium outcome of the 1-stage system to any SPNE outcome of the 2-stage system.

Generic catastrophic poverty when selfish investors exploit a degradable common resource.

The productivity of a common pool of resources may degrade when overly exploited by a number of selfish investors, a situation known as the tragedy of the commons. Without regulations, agents optimize the size of their individual investments into the commons by balancing incurring costs with the returns received. The resulting Nash equilibrium involves a self-consistency loop between individual investment decisions and the state of the commons. As a consequence, several non-trivial properties emerge. For N investing actors we prove rigorously that typical payoffs do not scale as 1/N, the expected result for cooperating agents, but as (1/N)2. Payoffs are hence reduced with regard to the functional dependence on N, a situation denoted catastrophic poverty. We show that catastrophic poverty results from a fine-tuned balance between returns and costs. Additionally, a finite number of oligarchs may be present. Oligarchs are characterized by payoffs that are finite and not decreasing when N increases. Our results hold for generic classes of models, including convex and moderately concave cost functions. For strongly concave cost functions the Nash equilibrium undergoes a collective reorganization, being characterized instead by entry barriers and sudden death forced market exits.

Influence in social networks with stubborn agents: From competition to bargaining

The literature on game-theoretic models of opinion dynamics in social networks mainly focuses on the Nash equilibrium, which reflects a competitive situation between influencing agents called players. In some real-world situations, however, players negotiate over a game; thus, a different type of solution needs to be considered to account for possible outcomes. In this paper, we examine an opinion dynamics game based on the Friedkin–Johnsen model for which we characterize the Pareto frontier, including the Nash bargaining solution. Next, we analyze this solution when there are changes in the susceptibility of noninfluencing agents with respect to their initial opinions. We also quantify how the Nash equilibrium outcome differs from the outcome prescribed by the Nash bargaining solution.

Instabilities in multi-asset and multi-agent market impact games

We consider the general problem of a set of agents trading a portfolio of assets in the presence of transient price impact and additional quadratic transaction costs and we study, with analytical and numerical methods, the resulting Nash equilibria. Extending significantly the framework of Schied and Zhang (2019) and Luo and Schied (2020), who considered the single asset case, we prove the existence and uniqueness of the corresponding Nash equilibria for the related mean-variance optimization problem. We then focus our attention on the conditions on the model parameters making the trading profile of the agents at equilibrium, and as a consequence the price trajectory, wildly oscillating and the market unstable. While Schied and Zhang (2019) and Luo and Schied (2020) highlighted the importance of the value of transaction cost in determining the transition between a stable and an unstable phase, we show that also the scaling of market impact with the number of agents J and the number of assets M determines the asymptotic stability (in J and M) of markets.

Competition with indivisibilities and few traders

We study minimal conditions for competitive behavior with few agents. We adapt a price-quantity strategic market game to the indivisible commodity environment commonly used in double auction experiments, and show that all Nash equilibrium outcomes with active trading are competitive if and only if there are at least two buyers and two sellers willing to trade at every competitive price. Unlike previous formulations, this condition can be verified directly by checking the set of competitive equilibria. In laboratory experiments, the condition we provide turns out to be enough to induce competitive results, and the Nash equilibrium appears to be a good approximation for market outcomes. Subjects, although possessing limited information, are able to act as if complete information were available in the market.

Better Approximation for Distributed Weighted Vertex Cover via Game-Theoretic Learning

Toward better approximation for the minimum-weighted vertex cover (MWVC) problem in multiagent systems, we present a distributed algorithm from the perspective of learning in games. For self-organized coordination and optimization, we see each vertex as a potential game player who makes decisions using local information of its own and the immediate neighbors. The resulting Nash equilibrium is classified into two categories, i.e., the inferior Nash equilibrium (INE) and the dominant Nash equilibrium (DNE). We show that the optimal solution must be a DNE. To achieve better approximation ratios, local rules of perturbation and weighted memory are designed, with the former destroying the stability of an INE and the latter facilitating the refinement of a DNE. By showing the existence of an improvement path from any INE to a DNE, we prove that when the memory length is larger than 1, our algorithm converges in finite time to DNEs, which could not be improved by exchanging the action of a selected node with all its unselected neighbors. Moreover, additional freedom for solution efficiency refinement is provided by increasing the memory length. Finally, intensive comparison experiments demonstrate the superiority of the presented methodology to the state of the art, both in solution efficiency and computation speed.

Effects of Defensive and Proactive Measures on Competition Between Terrorist Groups

A two-stage game investigates how counterterrorism measures affect within-country competition between two rival terrorist groups. Although such competition is commonplace (e.g., al-Nusra Front and Free Syria Army; Revolutionary Armed Forces of Colombia and the National Liberation Army; and al-Fatah and Hamas), there is no theoretical treatment of how proactive and defensive measures influence this interaction. Previous studies on rival terrorist groups are solely empirical concerning group survival, outbidding, and terrorism level, while ignoring the role that government countermeasures exert on the rival groups’ terrorism. In a theoretical framework, alternative counterterrorism actions have diverse impacts on the level of terrorism depending on relative group sizes and government-targeting decisions. In the two-stage game, optimal counterterrorism policy rules are displayed in terms of how governments target symmetric and asymmetric terrorist groups. Comparative statics show how parameter changes affect Nash or subgame perfect equilibrium outcomes.