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Achieving Correlated Equilibrium by Studying Opponent's Behavior

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Achieving Correlated Equilibrium by Studying Opponent's Behavior 1 Achieving Correlated Equilibrium by Studying Opponent’s Behavior Through Policy-Based Deep Reinforcement Learning Kuo Chun Tsai, Student Member, IEEE and Zhu Han, Fellow, IEEE Abstract—Game theory is a very profound study on distributed another Nobel winning solution concept called correlated decision-making behavior and has been extensively developed by equilibrium [6]. The idea in the correlated equilibrium concept many scholars. However, many existing works rely on certain is that all players will choose an action according to a public strict assumptions such as knowing the opponent’s private behaviors, which might not be practical. In this work, we focused signal. If all player is satisfied by the recommended strategy, on two Nobel winning concepts, the Nash equilibrium and the the distribution is called a correlated equilibrium distribution correlated equilibrium. Specifically, we successfully reached the [7]. With this concept, players can achieve higher utility in the correlated equilibrium outside the convex hull of the Nash equi- game since they consider the joint distribution instead of the libria with our proposed deep reinforcement learning algorithm. marginal distribution as the Nash equilibrium. However, there With the correlated equilibrium probability distribution, we also propose a mathematical model to inverse the calculation of the does not exist a distributed strategy on how to reach correlated correlated equilibrium probability distribution to estimate the equilibrium outside the Nash equilibrium convex hull since opponent’s payoff vector. With those payoffs, deep reinforcement there does not exist a clear way on how to design the public learning learns why and how the rational opponent plays, instead signal for the player to obtain. Some existing works using of just learning the regions for corresponding strategies and non-regret learning [8] can only achieve the Nash equilibrium actions. Through simulations, we showed that our proposed method can achieve the optimal correlated equilibrium and convex hull but cannot learn the better correlated equilibrium. outside the convex hull of the Nash equilibrium with limited The reason that the players in a non-cooperative game are interaction among players. unable to achieve the correlated equilibrium is due to the Index Terms—correlated equilibrium, deep learning, game fact that they are unable to mine some of the information theory, joint distribution, machine learning, neural network, out from the public signal provided to them. Fortunately, a reinforcement learning. technique called machine learning was developed for studying the underlying factor of the data. Many works, such as [9]– [12] and more, showed the robustness of the machine learning I. INTRODUCTION technique for data mining tasks. However, even today, the link Game theory is the study of mathematical models regarding of machine learning with game theory is seldom studied. the rationality decision making strategic interaction. Origi- Motivated by the above facts, we proposed a policy-based nally, game theory addressed the two-player zero-sum non- deep reinforcement learning model to determine the strategy cooperative games where each participant’s gains or losses are to reach equilibrium under limited information given to the exactly balanced by those of the other participants [1]. Today, player. We then estimate the payoffs of other players via game theory applies to a wide range of behavioral relations our proposed mathematical model. Once we have all the arXiv:2004.08523v1 [cs.GT] 18 Apr 2020 such as cooperative game [2], contract theory, auction theory, payoffs from the other players, we can determine the correlated matching game, dynamic game and more [3], [4]. equilibrium between other players without them interacting The main idea for game theory is for each rational player to with each other. Thus, the contribution we made in this work maximize his or her utility. Take the Nobel winning Nash equi- can be summarized as follows: librium solution concept for example. The Nash equilibrium • We first define the public signal in the system for the solution concept, named after the mathematician John Forbes player to obtain that contains limited information of the Nash Jr., is a solution of a non-cooperative game involving two players. or more players. A Nash equilibrium in the game is where no • With limited information, players will start to interact player has utility increment by changing only their strategy with the environment. By applying our proposed deep [5]. However, the solution under the Nash equilibrium might reinforcement learning model, the player not only can be far from the centralized solution. Hence, in 1974, there understand the structure of the environment but also learn is another mathematician named Robert Aumann discussed the joint distribution among all of the players when exploring the environment. In the end, the players can Kuo Chun Tsai is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX, USA. E-mail: kevint- reach a correlated equilibrium where no one wants to [email protected]. deviate. Zhu Han is with the Department of Electrical and Computer Engineering, • With the correlated equilibrium probability distribution University of Houston, Houston, TX, USA, and also with the Department of Computer Science and Engineering, Kyung Hee University, Seoul, South that all players satisfied with, we proposed a mathemat- Korea. E-mail: [email protected]. ical model that combines the concept of the correlated 2 equilibrium and the force of tension to estimate the payoff vectors of other players. • By knowing the payoff vector of other players in the system, we could compute the correlated equilibrium. In other words, with those payoff vectors, deep rein- forcement learning learns why and how the rational opponent plays, instead of just learning the regions for corresponding strategies and actions. • This paper combines the game theory with machine learning in the sense that the proposed machine learning learns what is the game player’s payoff, instead of just categorizing the strategies of actions according to the current situation. The rest of the paper is organized as follows. In Section II, we discuss the basic concept in the Nash equilibrium Figure 1: Game of Chicken Decision Set Layout and correlated equilibrium concepts along with our system model and problem formulation. Next, in Section III, we the distribution is called a correlated equilibrium. A formal show how the player interacts with the environment with our definition is as follows. proposed policy-based deep reinforcement learning model and learn the joint distribution among the players. In the same Definition II.2. An I-player strategic game (I, Φi,ui) is section, we also proposed a mathematical model to estimate characterized by an action set Φi and utility function ui for the information of the opponent players. Next, we show the each player i. when player i chooses strategy ςi ∈ Φi and the numerical results for our proposed methods in Section IV. remaining players choose a strategy profile ς−i described by Finally, we conclude our work in Section VI. the I − 1 tuple. Then the player i’s utility is ui (ςi, ς−i). A strategy modification for player i is a function φi :Φi → Φi. II. SYSTEM MODEL AND BASIC EQUILIBRIUM CONCEPT That is φi tells player i to modify his or her behavior by playing action φi (ςi) when instructed by play ςi. Let (Ω, Ψ) In this section, we will first go through the basic concept of be a countable probability space. For each player i, let Fi the Nash equilibrium and the correlated equilibrium in Section be his or her information partition, qi be i’s posterior and let II-A. In Section II-B, we will study the relationship between si : Ω → Φi, assigning the same value to states in the same the environment and the players in our system model. Finally, cell of i’s information partition. Then ((Ω, Ψ) , Fi,si) is the in Section II-C, we will discuss the problem formulation. correlated equilibrium of the strategic game (I, Φi,ui) if it satisfies the condition A. Nash Equilibrium and Correlated Equilibrium Basics qi (ω)ui (si (ω) ,s−i (ω)) ≥ ∈ The Nash equilibrium is a solution concept for a non- ωPΩ (1) cooperative game for two or more players [5]. The equilibrium qi (ω)ui (φi (si (ω)) ,s−i (ω)) ω∈Ω outcome of a non-cooperative game is one where no player P wants to deviate from his or her chosen strategy after consid- for every player i and for every strategy modification φi. ering the opponent’s decision. In other words, an individual cannot increment his or her utility from unilaterally changing B. System Model his or her strategies, assuming the other players remain their There are two major components in our system model the strategies. There might be none or multiple Nash equilibrium players and the environment. We can consider the environment in a non-cooperative game depends on the setup of the game is a set of states S where each state sk = (ρ1,ρ2,...,ρH ) and the strategies used by each player. A formal definition of is a unique tuple which contains the probabilities for each Nash equilibrium is as follows. decision set Dh ∈ D for h = 1, 2,...,H, where D is a set of permutation of all player’s decisions. In other words, state Definition II.1. An I-player game is characterized by an sk contains a probability distribution of the decisions in D. action set Φi. Let Bi (ς−i) ⊂ Φi be the set of player pi’s best ∗ ∗ ∗ The order of the players’ decision in decision set Dh will response strategy against ς− ∈ Φ− . ς = (ς ,...,ς ) ∈ Φ i 1 1 I always starts from player p to player p . Take the game of is a Nash equilibrium if ς∗ ∈ B ς∗ for every i ∈ I.
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