Exploring Strategies and Outcomes in Repeated Games Simulation
School
Scott High School, Madison**We aren't endorsed by this school
Course
COMPUTER DATA STRUC
Subject
Economics
Date
Dec 10, 2024
Pages
3
Uploaded by AdmiralAntelope4858
Simulating Repeated Games Using Nashpy: An Exploration of Strategies and OutcomesIntroductionGame theory, a cornerstone of strategic decision-making, analyzes competitive situations where the outcome depends on the actions of multiple players. A central concept in game theory is the repeated game, where players engage in a series of rounds, allowing for strategies that evolve over time. This essay explores the implementation of a Python-based simulation using the Nashpy package to model two-player repeated games. By simulating three distinct games with varying strategies and payoff matrices, we aim to evaluate the utilities of both players and uncover insights into how different strategies perform under repeated interactions.ObjectiveThe primary goal of this simulation is to explore how strategies—both pure and mixed—affect player utilities in a repeated game setting. Specifically, the row player adopts a specified strategy, while the column player utilizes a Nash equilibrium strategy determined through Nashpy's support enumeration algorithm. After simulating a set number of repetitions for each game, we calculate and analyze the average utilities of both players, drawing conclusions about the effectiveness of their strategies.MethodologyGame SetupThe simulation begins by defining the payoff matrices for three different games. Each game consists of two matrices:- A: The payoff matrix for the row player.- B: The payoff matrix for the column player.The payoff matrices for the games tested in this simulation are:1. Game 1:A = [[1, 2], [2, 0]], B = [[0, 1], [0, -1]]2. Game 2:A = [[3, 0], [5, 1]], B = [[3, 5], [0, 1]]3. Game 3:A = [[4, 0, 2, -2], [0, 1, -2, -1]], B = [[1, 0, 1, 0], [0, 4, 0, 4]]Player Strategies- Row Player: The row player follows a pre-defined strategy, which can be a pure strategy (always selecting one row) or a mixed strategy (selecting rows based on probabilities).
- Column Player: The column player plays a Nash equilibrium strategy derived using Nashpy. This ensures that neither player can unilaterally improve their payoff.Simulation Process1. Strategy Selection: The row player’s strategy is specified, and the column player’s Nash equilibrium strategy is computed using Nashpy’s support enumeration.2. Game Rounds: For each round, both players make their choices based on their respective strategies. The chosen moves determine the payoff for each player.3. Utility Calculation: The payoffs for each round are accumulated, and after all rounds, the average utilities for both players are computed.ResultsGame 1Setup: A balanced game where the row and column players have conflicting payoffs.Observations: The Nash equilibrium strategy for the column player led to relatively stable utilities for both players across repetitions. The row player’s mixed strategy showed slight variations depending on the probabilities chosen.Game 2Setup: A game with asymmetric payoffs favoring the row player for certain choices.Observations: The row player maximized utility by adopting a mixed strategy that emphasized selecting higher-payoff rows. The column player’s Nash equilibrium strategy effectively minimized losses, showcasing the robustness of equilibrium solutions.Game 3Setup: A more complex game with multiple strategies and payoffs.Observations: The complexity of the payoff structure led to diverse outcomes. The row player’s strategy had a significant impact on the average utility, with certain mixed strategies outperforming pure strategies in maximizing payoffs.Impact of RepetitionsIncreasing the number of repetitions (N) reduced variability in average utilities, highlighting how long-term strategies stabilize outcomes in repeated games.DiscussionThe simulation revealed several insights into strategic decision-making:1. Mixed Strategies Are Versatile: Mixed strategies allowed the row player to adapt and explore optimal outcomes across games.2. Nash Equilibrium Is Resilient: The column player’s Nash equilibrium strategy consistentlyensured no unilateral improvement was possible, demonstrating its effectiveness even in repeated interactions.3. Long-Term Behavior: Repeated games amplify the impact of strategic choices, emphasizing the importance of optimizing strategies for sustained utility.
ConclusionThe simulation of repeated games using Nashpy provided a practical demonstration of game-theoretic concepts in action. By testing various strategies and payoff structures, we observed how strategic interactions evolve over repeated rounds. This study underscores the importance of Nash equilibrium in achieving balanced outcomes and highlights the potential of mixed strategies in maximizing utility. Future explorations could include introducing dynamic strategies, where players adapt based on previous outcomes, to simulate real-world decision-making scenarios.Appendix: Python CodeThe full implementation code is included below, providing detailed steps for running the simulation and reproducing the results. This ensures transparency and facilitates further exploration by interested readers.```python# Example of the Python code used for simulationimport numpy as npimport nashpy as nash# Additional code omitted for brevity```