Game Theory Basics in Unity

Game Theory Basics in Unity

Game theory basics can be applied in Unity to create strategic games that involve decision-making and interactions between players. This comprehensive guide will help you understand and implement game theory concepts in Unity, covering more advanced aspects such as payoff matrices, user interactions, Nash Equilibrium analysis, and strategy optimization.

Understanding Game Theory

Game Theory: Game theory is a mathematical framework for modeling scenarios where conflicts of interest exist among players. It’s used in various fields, including economics, political science, psychology, and computer science.

• Types of Games:
  • Cooperative vs. Non-cooperative: Cooperative games allow players to form coalitions and make binding agreements, while non-cooperative games do not.
  • Symmetric vs. Asymmetric: In symmetric games, payoffs depend only on the strategies employed, not on the players. Asymmetric games have payoffs that differ depending on who is playing.
  • Zero-Sum vs. Non-Zero-Sum: In zero-sum games, one player’s gain is another player’s loss. Non-zero-sum games have varying total payoffs.
• Key Concepts:
  • Nash Equilibrium: A set of strategies where no player can benefit by changing their strategy unilaterally.
  • Dominant Strategy: A strategy that is best for a player, regardless of others’ strategies.
  • Pareto Efficiency: A situation where no player can be made better off without making at least one player worse off.

Implementing Game Theory in Unity

Step 1: Setting Up Unity Project

1. Create a New Project: Open Unity Hub, create a new 2D or 3D project.
2. Organize Your Project: Create folders for “Scripts,” “Prefabs,” and “Scenes.”

Step 2: Implementing Players and Strategies

1. Player Script:

csharp

using UnityEngine;

public class Player : MonoBehaviour
{
    public string playerName;
    public int strategy;
    public int payoff;

    public void ChooseStrategy(int chosenStrategy)
    {
        strategy = chosenStrategy;
    }

    public void SetPayoff(int payoffValue)
    {
        payoff = payoffValue;
    }
}

2. GameManager Script:

csharp

using UnityEngine;

public class GameManager : MonoBehaviour
{
    public Player player1;
    public Player player2;

    private int[,] payoffMatrixPlayer1 = { { 3, 0 }, { 5, 1 } }; // Player 1's payoffs
    private int[,] payoffMatrixPlayer2 = { { 3, 5 }, { 0, 1 } }; // Player 2's payoffs

    void Start()
    {
        player1.ChooseStrategy(0);
        player2.ChooseStrategy(1);

        CalculatePayoffs();
    }

    public void CalculatePayoffs()
    {
        int player1Strategy = player1.strategy;
        int player2Strategy = player2.strategy;

        int player1Payoff = payoffMatrixPlayer1[player1Strategy, player2Strategy];
        int player2Payoff = payoffMatrixPlayer2[player1Strategy, player2Strategy];

        player1.SetPayoff(player1Payoff);
        player2.SetPayoff(player2Payoff);

        Debug.Log($"{player1.playerName} payoff: {player1.payoff}");
        Debug.Log($"{player2.playerName} payoff: {player2.payoff}");
    }
}

Step 3: Creating the User Interface

1. Canvas Setup: Add a Canvas to the scene to display player strategies and payoffs.
2. UI Elements: Add Text elements to display current strategy and payoff for each player, and buttons for strategy selection.
3. UI Script:

csharp

using UnityEngine;
using UnityEngine.UI;

public class UIScript : MonoBehaviour
{
    public Text player1StrategyText;
    public Text player2StrategyText;
    public Text player1PayoffText;
    public Text player2PayoffText;
    public GameManager gameManager;

    void Update()
    {
        player1StrategyText.text = "Player 1 Strategy: " + gameManager.player1.strategy;
        player2StrategyText.text = "Player 2 Strategy: " + gameManager.player2.strategy;
        player1PayoffText.text = "Player 1 Payoff: " + gameManager.player1.payoff;
        player2PayoffText.text = "Player 2 Payoff: " + gameManager.player2.payoff;
    }
}

Step 4: Implementing Strategy Selection

1. Button Handlers:

csharp

using UnityEngine;

public class ButtonHandler : MonoBehaviour
{
    public GameManager gameManager;

    public void SetPlayer1Strategy(int strategy)
    {
        gameManager.player1.ChooseStrategy(strategy);
        gameManager.CalculatePayoffs();
    }

    public void SetPlayer2Strategy(int strategy)
    {
        gameManager.player2.ChooseStrategy(strategy);
        gameManager.CalculatePayoffs();
    }
}

2. UI Setup: Add buttons to the canvas and link them to the SetPlayer1Strategy and SetPlayer2Strategy methods in the ButtonHandler script.

Step 5: Extending Functionality with Advanced Concepts

1. Nash Equilibrium Calculation: Implement a method to find Nash Equilibria in the payoff matrices.

csharp

public void FindNashEquilibria()
{
    bool isNashEquilibrium = true;

    for (int s1 = 0; s1 < payoffMatrixPlayer1.GetLength(0); s1++)
    {
        for (int s2 = 0; s2 < payoffMatrixPlayer1.GetLength(1); s2++)
        {
            int payoff1 = payoffMatrixPlayer1[s1, s2];
            int payoff2 = payoffMatrixPlayer2[s1, s2];

            for (int s1Alt = 0; s1Alt < payoffMatrixPlayer1.GetLength(0); s1Alt++)
            {
                if (payoffMatrixPlayer1[s1Alt, s2] > payoff1)
                {
                    isNashEquilibrium = false;
                    break;
                }
            }

            for (int s2Alt = 0; s2Alt < payoffMatrixPlayer1.GetLength(1); s2Alt++)
            {
                if (payoffMatrixPlayer2[s1, s2Alt] > payoff2)
                {
                    isNashEquilibrium = false;
                    break;
                }
            }

            if (isNashEquilibrium)
            {
                Debug.Log($"Nash Equilibrium found at strategies: Player1 - {s1}, Player2 - {s2}");
            }
            isNashEquilibrium = true;
        }
    }
}

2. Graphical Representation of Strategies and Payoffs: Use Unity’s UI elements or graphical tools to visually represent the strategy space and payoffs. This can include graphs or matrices showing the possible outcomes for different strategies.
3. Advanced Player Models: Implement more sophisticated player models that can adapt their strategies based on past outcomes or learn over time.

csharp

using System.Collections.Generic;

public class AdvancedPlayer : MonoBehaviour
{
    public string playerName;
    public int strategy;
    public int payoff;
    private List<int> pastStrategies = new List<int>();
    private List<int> pastPayoffs = new List<int>();

    public void ChooseStrategy(int chosenStrategy)
    {
        strategy = chosenStrategy;
        pastStrategies.Add(chosenStrategy);
    }

    public void SetPayoff(int payoffValue)
    {
        payoff = payoffValue;
        pastPayoffs.Add(payoffValue);
    }

    public void LearnFromPast()
    {
        // Implement a learning algorithm (e.g., Q-learning, strategy adaptation)
    }
}

4. Networking and Multiplayer: Implement networking to allow multiple players to participate in the game over a network. Use Unity’s networking solutions like Unity Multiplayer or third-party tools like Photon Unity Networking (PUN).

Conclusion

By extending our implementation to include advanced concepts and more sophisticated models, we can create a rich and interactive simulation of game theory scenarios in Unity. This deeper exploration allows for better understanding and analysis of strategic interactions in games, which can be valuable for educational purposes, research, or creating complex strategic games.