Equilibrium Of Game Research Articles

A Game Model Analysis of the International Digital Service Tax in an Asymmetric Market Duopoly

Against the backdrop of the rapidly expanding digital economy, multinational corporations are increasingly exploiting information asymmetry in the market to employ covert and diverse methods of tax avoidance. This poses a significant challenge to tax collection and administration. The proposal of a digital service tax aims to adapt the international tax system to the digitalization of the economy, ensuring fair and reasonable tax payments. To address these challenges, this article assumes an asymmetric international digital economy market and a domestic market characterized by duopoly. It establishes an asymmetric market duopoly game, calculates the digital gains of each country's market using the logistic function, determines the Nash equilibrium of the data service tax game, and analyzes the relationship between the digital service tax rates of strong and weak countries in the digital economy.

Evolutionary game analysis of multiple subjects in the management of major public health emergencies

The frequent occurrence of major public health emergencies (MPHEs) significantly challenges national security, economic stability, social operation and the safety of people's lives and property worldwide. Consequently, enhancing the emergency management of MPHEs is critically urgent. This paper constructs a game model involving local government, social organisations, and the public for MPHE management, exploring strategy combinations and influencing factors across various scenarios. Several results were obtained. (1) Local government, social organisations, and the public each have positive and negative strategy choices based on cost–benefit analysis, leading to eight different strategy combinations. Furthermore, all three take positive strategies as the optimal way to achieve the game equilibrium. (2) The transformation of strategy combinations is primarily influenced by the cost-benefit gap and the strategic decisions of local government. (3) Altering a subject's initial strategy value doesn't change its final choice but impacts the time to achieve a stable strategy equilibrium. The severity of local government punishments on social organisations influences their strategic choices and the time to optimal strategy, whereas rewards to the public or social organisations only affect the time to achieve this strategy. The findings of this study can not only help improve the collaborative governance system of MPHEs but also provide scientific guidance on how governments can manage MPHEs.

Strategic Fertility, Education Choices, and Conflicts in Deeply Divided Societies

Abstract Fertility becomes a strategic choice for minorities when having a larger share of the population helps to increase power. If parents invest resources to educate their children, then raising fertility for strategic reasons might be at the cost of future human capital. We dispel this view using census data from several developing countries. We show that religious and ethnic minorities in Indonesia, China, and Malaysia tend to invest more in both education and fertility compared to larger groups. Solving for the Nash equilibrium of an appropriation game between two groups with education and fertility being prescribed as group-specific behavioral norms, we offer a rationale for the observed patterns provided that human capital is an important input to appropriation.

Computing perfect stationary equilibria in stochastic games

The notion of stationary equilibrium is one of the most crucial solution concepts in stochastic games. However, a stochastic game can have multiple stationary equilibria, some of which may be unstable or counterintuitive. As a refinement of stationary equilibrium, we extend the concept of perfect equilibrium in strategic games to stochastic games and formulate the notion of perfect stationary equilibrium (PeSE). To further promote its applications, we develop a differentiable homotopy method to compute such an equilibrium. We incorporate vanishing logarithmic barrier terms into the payoff functions, thereby constituting a logarithmic-barrier stochastic game. As a result of this barrier game, we attain a continuously differentiable homotopy system. To reduce the number of variables in the homotopy system, we eliminate the Bellman equations through a replacement of variables and derive an equivalent system. We use the equivalent system to establish the existence of a smooth path, which starts from an arbitrary total mixed strategy profile and ends at a PeSE. Extensive numerical experiments, including relevant applications like dynamic oligopoly models and dynamic legislative voting, further affirm the effectiveness and efficiency of the method.

The choice of suppliers and consumers: the evolutionary game equilibrium of platforms

ABSTRACT Platforms competition affects both the structure of the whole industry and the fate of the individual companies within it. The extant studies either simply divide the participants of a platforms war into suppliers and consumers or adopt unbounded rationality assumptions. This paper constructs a game model that includes platforms, suppliers, and the final consumers. We analyze the condition and stability of each equilibrium and the impacts of the characteristics of the core and complementary products on the equilibrium by using the method of evolutionary game. Our results show the following: (1) If the two platforms initially have separate advantages in terms of the number of suppliers or consumers, there will be an equilibrium of the coexistence; otherwise, there will be an equilibrium of single platform dominating. (2) The equilibrium of a single platform dominating is more stable than that of the coexistence of multiple platforms. (3) Incidental shocks to the number of suppliers are more likely to break the coexistence equilibrium than consumers. (4) Platforms with poor core product quality and a high price for core and complementary products can coexist with other superior platforms only if the former has more consumers and suppliers at the beginning of the competition.

On Construction of Subgame Perfect Nash Equilibria in Stackelberg Games

Identifying a Subgame Perfect Nash Equilibrium (SPNE) of a two-player Stackelberg game could be not a manageable task, especially when the players have a continuum of actions and the follower’s best reply correspondence is not single-valued. Aim of the paper is to investigate the issue of construction of SPNEs in Stackelberg games by exploiting perturbations of both the action sets and the payoff functions of the leader and the follower. To achieve the goal, since the limit of SPNEs of perturbed games is not necessarily an SPNE of the original game even for classical perturbations, we prove under nonrestrictive convergence conditions how to produce an SPNE starting from a sequence of SPNEs of general perturbed games. This result allows to describe a procedure to find SPNEs that can accommodate various types of perturbations. More precisely, under mild assumptions on the data of the original game, we show that a large class of perturbed games (including, for example, perturbation approaches relying on the Tikhonov and entropic regularizations or motivated by altruistic and antagonistic behaviors) satisfies the convergence conditions for constructing an SPNE. The specific SPNE selections associated to such a class, together with their possible behavioral interpretations, are discussed and an illustrative example is provided.

Mean-field ranking games with diffusion control

We consider a stochastic differential game, where each player continuously controls the diffusion intensity of her own state process. The players must all choose from the same diffusion rate interval [σ1,σ2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\\sigma _1, \\sigma _2]$$\\end{document}, and have individual random time horizons that are independently drawn from the same distribution. The players whose states at their respective time horizons are among the best p∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (0,1)$$\\end{document} of all terminal states receive a fixed prize. We show that in the mean field version of the game there exists an equilibrium, where the representative player chooses the maximal diffusion rate when the state is below a given threshold, and the minimal rate else. The symmetric n-fold tuple of this threshold strategy is an approximate Nash equilibrium of the n-player game. Finally, we show that the more time a player has at her disposal, the higher her chances of winning.

Game-Theoretic Unlearnable Example Generator

Unlearnable example attacks are data poisoning attacks aiming to degrade the clean test accuracy of deep learning by adding imperceptible perturbations to the training samples, which can be formulated as a bi-level optimization problem. However, directly solving this optimization problem is intractable for deep neural networks. In this paper, we investigate unlearnable example attacks from a game-theoretic perspective, by formulating the attack as a nonzero sum Stackelberg game. First, the existence of game equilibria is proved under the normal setting and the adversarial training setting. It is shown that the game equilibrium gives the most powerful poison attack in that the victim has the lowest test accuracy among all networks within the same hypothesis space when certain loss functions are used. Second, we propose a novel attack method, called the Game Unlearnable Example (GUE), which has three main gradients. (1) The poisons are obtained by directly solving the equilibrium of the Stackelberg game with a first-order algorithm. (2) We employ an autoencoder-like generative network model as the poison attacker. (3) A novel payoff function is introduced to evaluate the performance of the poison. Comprehensive experiments demonstrate that GUE can effectively poison the model in various scenarios. Furthermore, the GUE still works by using a relatively small percentage of the training data to train the generator, and the poison generator can generalize to unseen data well. Our implementation code can be found at https://github.com/hong-xian/gue.

On the Outcome Equivalence of Extensive-Form and Behavioral Correlated Equilibria

We investigate two notions of correlated equilibrium for extensive-form games: the extensive-form correlated equilibrium (EFCE) and the behavioral correlated equilibrium (BCE). We show that the two are outcome-equivalent, in the sense that every outcome distribution achievable under one notion is achievable under the other. Our result implies, to our knowledge, the first polynomial-time algorithm for computing a BCE.

Accelerate Multi-Agent Reinforcement Learning in Zero-Sum Games with Subgame Curriculum Learning

Learning Nash equilibrium (NE) in complex zero-sum games with multi-agent reinforcement learning (MARL) can be extremely computationally expensive. Curriculum learning is an effective way to accelerate learning, but an under-explored dimension for generating a curriculum is the difficulty-to-learn of the subgames –games induced by starting from a specific state. In this work, we present a novel subgame curriculum learning framework for zero-sum games. It adopts an adaptive initial state distribution by resetting agents to some previously visited states where they can quickly learn to improve performance. Building upon this framework, we derive a subgame selection metric that approximates the squared distance to NE values and further adopt a particle-based state sampler for subgame generation. Integrating these techniques leads to our new algorithm, Subgame Automatic Curriculum Learning (SACL), which is a realization of the subgame curriculum learning framework. SACL can be combined with any MARL algorithm such as MAPPO. Experiments in the particle-world environment and Google Research Football environment show SACL produces much stronger policies than baselines. In the challenging hide-and-seek quadrant environment, SACL produces all four emergent stages and uses only half the samples of MAPPO with self-play. The project website is at https://sites.google.com/view/sacl-neurips.

Contests on networks

We develop a model of contests on networks. Each player is connected to a set of contests and exerts a single effort to increase the probability of winning each contest to which she is connected. We explore how behavior is shaped by the pattern of interactions and characterize the networks that tend to induce greater effort; in particular, we show that the complete bipartite network is the unique structure that maximizes aggregate player effort. We also obtain a new exclusion result—akin to the Exclusion Principle of Baye et al. (Am Econ Rev 83(1):289-294, 1993)—which holds under the lottery CSF, and contrasts prior work in contests. Finally, new insight into uniqueness of equilibrium for network contest games is provided. Our framework has a broad range of applications, including research and development, advertising, and research funding.

A paradox of coalition building in public good provision

Does partial coalition among players help mitigate the issue of underprovision of public goods? This study considers endogenous coalition formation and endogenous technology choices in a model of the private provision of global public goods. We show that the possibility of future interstate (partial) coordination may hinder the current adoption of better technology by a country outside the cooperation, which may exacerbate an existing underprovision problem. Specifically, in the subgame perfect equilibrium of a three-stage game, we find two paradoxical results: the inability to form future partial coalitions encourages a country outside the cooperation to adopt better technology, which could lead to an increase in the total public good supply and an improvement of global welfare. These results have important policy implications. For example, in the context of the Paris Agreement, if a major country announces reduced nationally determined contributions in a strategic move to adopt less advanced technology to motivate other countries to form coalitions, it may ultimately lead to a decrease in the overall supply of public goods and global welfare.

A mean field game approach to optimal investment and risk control for competitive insurers

We consider an insurance market consisting of multiple competitive insurers with a mean field interaction via their terminal wealth under the exponential utility with relative performance. It is assumed that each insurer regulates her risk by controlling the number of policies. We respectively establish the constant Nash equilibrium (independent of time) on the investment and risk control strategy for the finite n-insurer game and the constant mean field equilibrium for the corresponding mean field game (MFG) problem (when the number of insurers tends to infinity). Furthermore, we examine the convergence relationship between the constant Nash equilibrium of finite n-insurer game and the mean field equilibrium of the corresponding MFG problem. Our numerical analysis reveals that, for a highly competitive insurance market consisting of many insurers, every insurer will invest more in risky assets and increase the total number of outstanding liabilities to maximize her exponential utility with relative performance.

Scalable Learning for Spatiotemporal Mean Field Games Using Physics-Informed Neural Operator

This paper proposes a scalable learning framework to solve a system of coupled forward–backward partial differential equations (PDEs) arising from mean field games (MFGs). The MFG system incorporates a forward PDE to model the propagation of population dynamics and a backward PDE for a representative agent’s optimal control. Existing work mainly focus on solving the mean field game equilibrium (MFE) of the MFG system when given fixed boundary conditions, including the initial population state and terminal cost. To obtain MFE efficiently, particularly when the initial population density and terminal cost vary, we utilize a physics-informed neural operator (PINO) to tackle the forward–backward PDEs. A learning algorithm is devised and its performance is evaluated on one application domain, which is the autonomous driving velocity control. Numerical experiments show that our method can obtain the MFE accurately when given different initial distributions of vehicles. The PINO exhibits both memory efficiency and generalization capabilities compared to physics-informed neural networks (PINNs).

International environmental agreements when countries behave morally

In the game-theoretical literature on forming international environmental agreements (IEAs) countries use to be self-interested materialists and stable coalitions are small. This paper analyzes IEA games with identical countries that exhibit Kantian moral behavior. Kantians are concerned with doing the right thing which means that they take those actions and only those actions that they advocate all others take as well. Countries may behave morally with respect to both emissions (reduction) and membership in an IEA. If countries are emissions Kantians or membership Kantians the equilibrium of the IEA games is socially optimal. To model more realistic Kantian behavior, we define an emissions [membership] moralist as a country whose welfare is a weighted average of the welfare of an emissions [membership] Kantian and a materialist. The game with emissions moralists produces stable coalitions not larger than those in the standard game with materialists. The game with membership moralists yields stable coalitions that are increasing in the membership morality. The aggregate emissions decline if the degree of morality of either type of moralists increases. Finally, we characterize the equilibrium of an IEA game with moderate moralists with respect to both emissions and membership.

Emergence of trust in the trust game under best experienced payoff dynamics

The subgame perfect equilibrium (SPE) of the trust game is Pareto inferior. In that equilibrium, player 2 will not honor player 1’s trust and hence, player 1 will not trust player 2. We consider a large population version of the trust game. We apply a population game dynamic called the ‘best experienced payoff’ dynamic to this game. We establish conditions under which the SPE outcome in the trust game is unstable under this dynamic. We also show that an outcome where a substantial fraction of agents in population 1 displaying trust and a substantial fraction of agents in population 2 honoring that trust can be globally stable.

MADRL-based UAV swarm non-cooperative game under incomplete information

Unmanned Aerial Vehicles (UAVs) play increasing important role in modern battlefield. In this paper, considering the incomplete observation information of individual UAV in complex combat environment, we put forward an UAV swarm non-cooperative game model based on Multi-Agent Deep Reinforcement Learning (MADRL), where the state space and action space are constructed to adapt the real features of UAV swarm air-to-air combat. The multi-agent particle environment is employed to generate an UAV combat scene with continuous observation space. Some recently popular MADRL methods are compared extensively in the UAV swarm non-cooperative game model, the results indicate that the performance of Multi-Agent Soft Actor-Critic (MASAC) is better than that of other MADRL methods by a large margin. UAV swarm employing MASAC can learn more effective policies, and obtain much higher hit rate and win rate. Simulations under different swarm sizes and UAV physical parameters are also performed, which implies that MASAC owns a well generalization effect. Furthermore, the practicability and convergence of MASAC are addressed by investigating the loss value of Q-value networks with respect to individual UAV, the results demonstrate that MASAC is of good practicability and the Nash equilibrium of the UAV swarm non-cooperative game under incomplete information can be reached.

Quantal Response Equilibrium and Rationalizability: Inside the Black Box

This paper aims to connect epistemic and behavioral game theory by examining the epistemic foundations of quantal response equilibrium (QRE) in static games. We focus on how much information agents possess about the probability distributions of idiosyncratic payoff shocks, in addition to the standard assumptions of rationality and common belief in rationality. When these distributions are transparent, we obtain a solution concept called Δp-rationalizability, which includes action distributions derived from QRE; we also give a condition under which this relationship holds true in reverse. When agents only have common belief in the monotonicity of these distributions (for example, extreme value distributions), we obtain another solution concept called ΔM-rationalizability, which includes action distributions derived from rank-dependent choice equilibrium, a parameter-free variant of QRE. Our solution concepts also provide insights for interpreting experimental and empirical data.

Repeated game behavior between bidder and regulatory agency of construction engineering with intertemporal choice

The traditional theory of bidder and regulatory agency of construction engineering does not take into account the repeated periodicity of the game between the regulator and regulated party, so that the mathematical point of game equilibrium deviates from actual behavioral expression. According to the intertemporal nature of bidder and regulatory agency, this paper analyzed the payoff matrix of the subject of bidder and regulatory agency, constructed the repeated game behavior model of bidder and regulatory agency, and explored the game conditions of the behavioral expression (steady state and unsteady state) between the two game parties of construction engineering. The results shows that: (1) The administrative triggers are adopted in the normalized regulation, which could make both parties between bidder and regulatory agency reach Pareto Optimality; (2) The intertemporal choice behavior of the bidder is related to the economic punishments, extraneous benefits and legitimate benefits. The increase of economic punishments and legitimate benefits could reduce the illegal behaviors; (3) The larger the discounted function, the easier it is for the bidder to choose long-term legal behavior. Our work indicated that the key to establishing a long-term market mechanism between bidder and regulatory agency is to increase the future impact on the present, and construct the administrative trigger measures of infinitely repeated game.

Fast or stable: Research on dynamic behaviors of duopoly automakers under the background of energy transformation in the automobile industry

New energy vehicles (NEVs) are crucial way to solving environmental and energy dilemmas. However, the transition from traditional energy vehicles (TEVs) to NEVs is extremely intricate. If the process for TEVs switching to NEVs falls into out order, the automobile industry will be shaken. This paper takes electric vehicles (EVs), representing NEVs, and internal combustion engine vehicles (ICEVs), representing TEVs, as research objects, and constructs a dynamic Gounod duopoly model for the production decisions of two oligopolistic automakers under the dual credit policy. Combining the nonlinear dynamic theory and Cournot game, the influence of policy, market, technology, and cost on the stability of duopoly automakers’ dynamic production game equilibrium and the nonlinear evolution characteristics of Cournot-Nash equilibrium are analyzed systematically. Results show that: (1) In the transition from ICEVs to EVs, if the speed of production adjustment is too fast, the automobile system would trap into chaos and shakeouts, and the social welfare would show volatility. (2) The revised dual credit policy increases the credit ratio and decreases the credit coefficient which enhances the stability of automobile system. This revision may accelerate the transformation from ICEVs to EVs in the automobile industry. Especially, the credit coefficient plays a more significant role. (3) The consumer acceptance for vehicle prices and the range of EV are positively correlated with the production of EVs and the social welfare, and negatively correlated with the production of ICEVs. Nevertheless, the cost of the two types of vehicle had the opposite effect compared price acceptance and range of EVs.