Finding Nash equilibria through simulation
The article describes Python programs that find and visualize Nash Equilibria in simultaneous games, tailored for different matrix sizes, emphasizing their educational value in understanding game theory concepts.
Read original article
The article discusses a series of Python programs designed to find and visualize Nash Equilibria (NE) in simultaneous games, such as the Prisoner's Dilemma and Rock, Paper, Scissors. Developed as part of a course based on William Spaniel's "Game Theory 101," the programs include sim22.py, sim33.py, sim44.py, and simNN.py, each tailored to different sizes of payoff matrices. The sim22.py program analyzes a 2x2 matrix, demonstrating how players' strategies converge to a pure NE, while sim33.py handles a 3x3 matrix, revealing a mixed NE in Rock, Paper, Scissors. The sim44.py program extends this to a 4x4 matrix, visualizing outcomes in a battle scenario, and simNN.py accommodates larger matrices using generalized barycentric coordinates. Each program employs repeated gameplay to adjust strategy weights based on performance, ultimately visualizing the results through various graphical representations. The article emphasizes the educational value of these simulations in understanding game theory concepts and provides insights into the implementation details and potential improvements for efficiency.
- The Python programs simulate simultaneous games to find Nash Equilibria.
- Each program is designed for specific matrix sizes: 2x2, 3x3, 4x4, and NxN.
- Visualization techniques include barycentric coordinates and simplex graphs.
- The simulations adjust strategy weights based on player performance over multiple rounds.
- The code requires Matplotlib and python-ternary for graphical output.
Related
Using Stockfish to identify ideal squares
This blog post explores using Stockfish to determine optimal squares for chess pieces. The program developed evaluates positions, showing potential for positional analysis despite some limitations and areas for improvement.
HAMURABI.BAS and Its Dystopian Lessons
The article explores the game HAMURABI.BAS from 1973, highlighting financial and political lessons. Strategies involve resource management, disaster response, and controlled starvation for wealth. Automated testing led to a 99% win rate, emphasizing wise investments and strategic decisions in the game.
Simulating the evolution of rock, paper, scissors [video]
The YouTube video explores a simulation with blobs playing rock, paper, scissors for food, inspired by natural strategies. It discusses population dynamics, game theory, lack of Nash equilibrium, and evolving strategies. The creator suggests adjusting reward matrix values and mixed strategies for strategy coexistence.
Professional Poker Players Know the Optimal Strategy but Don't Always Use It
Professional poker players balance game theory optimal strategies with exploitative play to outsmart opponents. AI advancements challenge traditional approaches, pushing players to blend defensive and aggressive tactics for competitive success in evolving poker dynamics.
Fair Chess and Simultaneous Games
The article proposes a simultaneous chess variant to address first-move advantage, outlining rules for conflict resolution and legal moves, aiming to create fair simultaneous games across various types.
- Several commenters reference John Nash and his contributions to game theory, including mentions of the film "A Beautiful Mind."
- There is a discussion on the complexity of finding Nash equilibria, with references to computational challenges and the PPAD-completeness of the problem.
- Some users express interest in applying the concepts to specific games, such as poker and market dynamics.
- Questions arise about the choice of simulation methods versus analytical approaches for finding equilibria.
- Commenters share their own projects and experiences related to game theory, indicating a collaborative interest in the topic.
11 commentsAs Nash proved, under very general conditions (e.g., payoffs are finite), in every game there's always at least one equilibrium, i.e., at least one fixed point.
Alas, as Papadimitriou proved in the 90's, finding Nash equilibria is PPAD-complete.[a][b]
So, as games get larger and more complex -- say, with rules and payoffs that evolve over time -- finding equilibria can become... intractable: There will always exist at least one Nash equilibrium, but you'll never be able to reach it. Simulation may well be the only way to model such games.
---
[a] https://en.wikipedia.org/wiki/PPAD_(complexity)
[b] There's a great intro lecture on this by Papadimitriou himself at https://www.youtube.com/watch?v=TUbfCY_8Dzs
Find the Nash equilibrium for poker with an exact set of cards and a deck. There's a fun arena-based tree structure that should allow finding the optimal strategy for different bet sizes, etc. One of the most challenging parts of finding the equilibrium is ensuring the simulation has no edge cases where value is lost.
There's a bug somewhere, and the game state isn't matching the second time through a tree node. (I'd pay a bounty to whoever can get it finished)
What are good references for this?
I've used optimization of https://en.wikipedia.org/wiki/Lyapunov_function in my Bachelor thesis https://github.com/Artimi/neng to do that.
Related
Using Stockfish to identify ideal squares
This blog post explores using Stockfish to determine optimal squares for chess pieces. The program developed evaluates positions, showing potential for positional analysis despite some limitations and areas for improvement.
HAMURABI.BAS and Its Dystopian Lessons
The article explores the game HAMURABI.BAS from 1973, highlighting financial and political lessons. Strategies involve resource management, disaster response, and controlled starvation for wealth. Automated testing led to a 99% win rate, emphasizing wise investments and strategic decisions in the game.
Simulating the evolution of rock, paper, scissors [video]
The YouTube video explores a simulation with blobs playing rock, paper, scissors for food, inspired by natural strategies. It discusses population dynamics, game theory, lack of Nash equilibrium, and evolving strategies. The creator suggests adjusting reward matrix values and mixed strategies for strategy coexistence.
Professional Poker Players Know the Optimal Strategy but Don't Always Use It
Professional poker players balance game theory optimal strategies with exploitative play to outsmart opponents. AI advancements challenge traditional approaches, pushing players to blend defensive and aggressive tactics for competitive success in evolving poker dynamics.
Fair Chess and Simultaneous Games
The article proposes a simultaneous chess variant to address first-move advantage, outlining rules for conflict resolution and legal moves, aiming to create fair simultaneous games across various types.