Overview of game theory and examples of integration and implementation with AI technology

Overview of Game Theory

Game theory is a theory for determining optimal strategies when there are multiple decision makers (players) who influence each other, such as in competition or cooperation, by mathematically modeling their strategies and their outcomes. It is used primarily in economics, social sciences, and political science.

In game theory, each player aims to maximize his or her own profit. Players can change their own strategies depending on the strategies of their opponents, and game theory models such a situation as a game.

In game theory, the profits and payoffs (rewards) of each player are expressed mathematically to find a solution to the game. The solution to the game is called the Nash equilibrium, a state in which each player can optimize his or her own strategy when he or she knows the strategy of the opponent.

Game theory has been applied to real-world social and business situations. For example, in competitive markets and auctions, the equilibrium of the market is determined by each player’s choice of his or her own strategy. Also, in negotiations and cooperative relationships, game theory can be used to determine the optimal strategy.

However, as described in “Decision Theory and Mathematical Decision-Making Techniques” in the real world, each player can choose not only the max-max strategy to maximize his or her own profit, as used in game theory, but also the max-min strategy to secure the minimum profit in consideration of risk, the max-min strategy to minimize regret, the max-survage strategy to minimize the expected value of all choices, and the max-min strategy to maximize the value of all choices, as used in game theory. There are various strategies, such as the “serve-as-you-go” strategy, the “expect-value” strategy, which chooses the expected value of all choices, etc., and these outcomes may be more desirable from a societal point of view. Therefore, when applying game theory, it is necessary to take them into account.

On the Application of Game Theory to Case Studies

Game theory has been applied to various social, business, and other situations. Examples of applications are described below.

• Competitive markets: In competitive markets, it is important for firms to set prices when selling their products and services. Game theory can be used to determine the optimal strategy for setting prices. For example, setting prices based on Nash equilibrium can increase the competitiveness of the market.
• Politics: In politics, there are multiple parties vying for power through elections. Each party can attract the support of the electorate depending on the policies that it advocates in the election. Game theory can be used to determine the optimal strategy for policy making.
• Business strategy: In a competitive market, a company needs to change its strategy depending on the behavior of its competitors. Game theory allows companies to optimize their strategies based on the strategies of their competitors.
• Social Problems: Game theory can also be applied to social problems. For example, in the case of traffic congestion, congestion can occur because each driver maximizes his or her own profit. Game theory can be used to determine the optimal strategy to relieve traffic congestion.
• International Relations: Game theory can also be applied to international relations. For example, when several states have nuclear weapons, the behavior of other states may change depending on whether or not each state uses nuclear weapons. Game theory can determine the optimal strategy to avoid using nuclear weapons.

Furthermore, game theory is also used in combination with AI technology.

The Application of AI Technology to Game Theory

Artificial intelligence techniques have become useful tools in game theory research and applications. Some specific examples are listed below.

• Predicting player behavior using machine learning: In game theory, it is important to predict what strategies players will adopt when multiple players interact with each other. Machine learning can be used to make such predictions. By building machine learning models, it is possible to predict future behavior by using players’ past behavior history and in-game information.
• Automatic game play using neural networks: Neural networks can be used to create AI players with the same decision-making ability as humans. Using this technology, AI players can achieve high performance in complex games. Such technology is expected to help improve game design and game balance in the future.
• Developing AI agents based on game theory: AI agents can develop AI agents based on game theory, which can take optimal strategies in situations where multiple players interact with each other. For example, an AI agent can be developed to optimize pricing in a competitive market.
• Learning Game AI with Deep Reinforcement Learning: Using deep reinforcement learning, an AI player can learn the optimal strategy for a game. This technique can be used to learn various strategies in games and is expected to lead to the development of more advanced game AI in the future.

Next, we discuss implementation techniques for game theory.

On game theory implementation methods

Game theory has been widely studied in the field of computer science, and several methods have been proposed for its implementation.

• Minimax method: The minimax method described in “Overview of the minimax method and examples of algorithms and implementations” is one of the most basic methods in game theory, and it is a method for determining the optimal strategy in a zero-sum game in which multiple players are competing against each other. In this method, one determines the optimal action to be taken by oneself against all possible actions taken by one’s opponent, with the goal of maximizing one’s own profit. This method is suitable for implementing simple games of shapes and strings.
• Monte Carlo Tree Search: Monte Carlo Tree Search described in “Overview of Monte Carlo Tree Search and Examples of Algorithms and Implementations” is suitable for implementing very complex games. The method simulates the overall game state by taking random steps and examining how each step affects the score. This allows the optimal procedure to be determined.
• Reinforcement Learning: Reinforcement learning is widely used in the field of artificial intelligence and is well suited for implementing complex games. In this technique, an AI agent plays a game and learns the strategies needed to win. The agent chooses actions with the goal of maximizing the reward. Rewards are set higher for victory and lower for defeat.
• Deep Learning: Deep learning is a widely used technique in image recognition and natural language processing that can be useful in game implementations. This technique uses neural networks to learn the optimal strategy for an AI player to play a game. By using deep learning, the AI player can select different strategies depending on the situation.

Specific implementation examples are described below.

Implementation with R

R is a programming language widely used for statistical and data analysis, and is also used to implement game theory.

The following is an example of a simple game implementation in R. In this example, Player 1 and Player 2 take turns choosing numbers, and the first player to reach 100 is the winner.

library(gameTheory)
library(pracma)

game <- createGame(list(c("Player 1", "Player 2")),
                   list(matrix(c(0, 1, -1, 0), ncol=2)))
result <- nashEquilibrium(game)

p1 <- 0
p2 <- 0

while (p1 < 100 && p2 < 100) {
  move1 <- as.integer(readline("Player 1's move: "))
  move2 <- as.integer(readline("Player 2's move: "))
  
  p1 <- p1 + move1
  p2 <- p2 + move2 cat("Player 1's score:", p1, "n") cat("Player 2's score:", p2, "n") } if (p1 >= 100) {
  cat("Player 1 wins!n")
} else {
  cat("Player 2 wins!n")
}

In this example, the gameTheory package is used to define a cooperative game with two players. The rules of this game are that player 1 can gain one point and player 2 can lose one point. The first player to reach 100 points wins.

Then, using the pracma package, mathematical operations can be performed to find the nesh equilibrium. Finally, a loop is implemented in which player 1 and player 2 take turns choosing numbers to determine the winner.

This example is very simple and may require more R packages and techniques to implement more complex games. However, this example is useful for understanding the basics of how to implement game theory using R.

Implementation with Python

Python can be one of the most useful programming languages for implementing game theory. Below is an example of how to implement game theory in Python.

1. Minimax Method Implementation

def minimax(board, depth, maximizingPlayer):
    if depth == 0 or game_over(board):
        return evaluate(board)

    if maximizingPlayer:
        maxEval = -float("inf")
        for move in get_possible_moves(board):
            eval = minimax(move, depth - 1, False)
            maxEval = max(maxEval, eval)
        return maxEval
    else:
        minEval = float("inf")
        for move in get_possible_moves(board):
            eval = minimax(move, depth - 1, True)
            minEval = min(minEval, eval)
        return minEval

This code defines a function called minimax. This function takes as arguments the board in its current state, a depth that specifies how many moves ahead the player will look (depth), and whether the current player is trying to maximize (maximizingPlayer). Within the function, the board is evaluated recursively and all possibilities are explored to determine the optimal move.

2. Implementation of Monte Carlo tree search

class Node:
    def __init__(self, state, parent=None):
        self.state = state
        self.parent = parent
        self.children = []
        self.wins = 0
        self.visits = 0

    def is_fully_expanded(self):
        return len(self.children) == len(get_possible_moves(self.state))

    def add_child(self, child_state):
        child_node = Node(child_state, self)
        self.children.append(child_node)
        return child_node

def mcts(root_state, itermax):
    root_node = Node(root_state)

    for i in range(itermax):
        node = root_node
        state = root_state.copy()

        # selection phase
        while node.is_fully_expanded() and node.children:
            node = ucb_select_child(node)
            state.play(node.move)

        # expansion phase
        if not node.is_fully_expanded():
            move = get_untried_move(node)
            state.play(move)
            node = node.add_child(state)

        # Simulation Phase
        while not game_over(state):
            state.play(random.choice(get_possible_moves(state)))

        # Backup Phase
        while node:
            node.visits += 1
            if node.state.current_player == state.winner:
                node.wins += 1
            node = node.parent

    return max(root_node.children, key=lambda n: n.visits).state

The code defines a Node class and a function called mcts. the Node class represents a node in a Monte Carlo tree search.

Implementation with Clojure

Clojure is a Lisp-like programming language, suitable for implementing game theory because it is suited for functional programming. Below is an example of how to implement game theory in Clojure.

1. Implementation of the minimax method

(defn minimax [board depth maximizing-player]
  (if (or (= depth 0) (game-over? board))
    (evaluate board)
    (if maximizing-player
      (reduce max (map #(minimax % (- depth 1) false) (get-possible-moves board)))
      (reduce min (map #(minimax % (- depth 1) true) (get-possible-moves board))))))

This code defines a function called minimax. This function takes as arguments the board in its current state, a depth (depth) specifying how many moves ahead the player will look, and whether the current player is trying to maximize (maximizing-player). Within the function, the board is evaluated recursively and all possibilities are explored to determine the optimal move.

2. Implementation of Monte Carlo tree search

(defn mcts [root-state itermax]
  (let [root-node (Node. root-state)]
    (dotimes [_ itermax]
      (let [node (atom root-node)
            state (atom (copy-state root-state))]
        ; selection phase
        (while (and (fully-expanded? @node) (seq (:children @node)))
          (reset! node (ucb-select-child @node))
          (play! state (:move @node)))

        ; expansion phase
        (if-not (fully-expanded? @node)
          (let [move (get-untried-move @node)]
            (play! state move)
            (reset! node (add-child @node state move))))

        ; Simulation Phase
        (while (not (game-over? @state))
          (play! state (rand-nth (get-possible-moves @state))))

        ; Backup Phase
        (while @node
          (swap! node update-visit-win-counts @state)
          (reset! state (copy-state (:parent @node)))
          (reset! node (:parent @node)))))

    (:state (max-key visit-count root-node))))

This code defines a Node class and a function called mcts. the Node class represents a node in a Monte Carlo tree search. the mcts function takes the current state of the board and the maximum number of moves to search (itermax) as arguments and returns the optimal move.