Game Theory Arcade

Info: Nash equilibrium, best responses, and how to read the matrix.

What is a Nash equilibrium?

Plain English: a Nash equilibrium is a stable outcome. After you see the other player's move, you do not want to switch your move (and neither do they).
A simple test
  • Pick one square in the matrix (your action, bot action).
  • Hold the bot fixed: would you change your action to get a higher payoff? If yes, it is not Nash.
  • Hold you fixed: would the bot change its action to get a higher payoff? If yes, it is not Nash.
  • If both answers are no, that square is Nash.
In the arcade, Nash squares are labeled Nash and best responses are labeled BR-you and BR-bot.
Important notes
  • Nash is about incentives, not morality or fairness.
  • Stable does not mean good. Some games have a Nash outcome that is worse for both players than another outcome.
  • There can be multiple Nash outcomes (coordination problems).
  • Some games have no Nash in pure actions; stable play may require randomizing (a mixed strategy).

Examples (plain English)

Prisoner's Dilemma
The stable outcome is often (Defect, Defect). Given the other side defects, you do not gain by cooperating. It is stable, but it is not the best joint outcome.
Stag Hunt
There are typically two stable outcomes: a safe one (Hare, Hare) and a high-reward one (Stag, Stag). The tension is coordination: the best outcome is risky if you cannot trust the other side.
Matching Pennies
There is no stable pure outcome. Whatever you pick, the other side wants to switch. Stability comes from mixing: stay unpredictable.
Chicken (Hawk-Dove)
Two stable outcomes exist where one side backs down and the other does not. The unstable outcome is both refusing to back down (the crash).

How to use this arcade

Reading the matrix
  • Each cell shows payoffs as (row payoff, column payoff).
  • Green BR-you labels your best response against the bot's action in that column.
  • Blue BR-bot labels the bot's best response against your action in that row.
  • If a cell has both best responses, it is Nash.
Try this
  • Start with the suggested bot for each game and play 20-30 rounds.
  • Change only your first move and see how history-dependent bots react.
  • Use the Beliefs tab to see how expected payoffs change as you change a probability.
Privacy and persistence
Player names are saved in your browser for convenience and sessions are stored in the local SQLite database. Names are not authenticated.