The Mixed Strategy Nash Equilibrium Calculator is designed to solve games in strategic form, identifying the probabilities with which players should randomly select their strategies to ensure that no player can increase their expected payoff by choosing a different strategy, given the strategies of the other players. This concept, integral to game theory, allows for the analysis of competitive situations where the outcome depends on the strategies of all participants.
Formula
To understand the calculator’s workings, we delve into the mathematical formulae it employs:
- Expected Payoff for Player 1:
E_1(i) = p_1 * u_1(i, j_1) + (1 - p_1) * u_1(i, j_2)
- Expected Payoff for Player 2:
E_2(j) = p_2 * u_2(j, i_1) + (1 - p_2) * u_2(j, i_2)
- Probability of Player 1 Choosing Strategy i:
p_1 = E_1(i') / (E_1(i') + E_1(i))
- Probability of Player 2 Choosing Strategy j:
p_2 = E_2(j') / (E_2(j') + E_2(j))
These formulae encapsulate the essence of strategic equilibrium in mixed strategies, where u_1 and u_2 represent the payoffs, E_1 and E_2 the expected payoffs, and p_1 and p_2 the strategy selection probabilities. The indices i, j, i’, and j’ symbolize the chosen strategies and their alternatives.
Practical Applications
| Player \ Strategy | High Price (H) | Low Price (L) |
|---|---|---|
| Probability of Choosing H | p | 1-p |
| Probability of Choosing L | q | 1-q |
| Expected Payoff for Choosing H | E_1(H) | E_2(H) |
| Expected Payoff for Choosing L | E_1(L) | E_2(L) |
Where:
- p and 1-p are the probabilities of Player 1 choosing High Price (H) and Low Price (L), respectively.
- q and 1-q are the probabilities of Player 2 choosing High Price (H) and Low Price (L), respectively.
- E_1(H) and E_1(L) are the expected payoffs for Player 1 when choosing H and L, respectively, calculated based on the mixed strategy equilibrium.
- E_2(H) and E_2(L) are the expected payoffs for Player 2 when choosing H and L, respectively, calculated based on the mixed strategy equilibrium.
Example
Consider a simple game where two firms compete on pricing strategies: High Price (H) or Low Price (L). The calculator helps determine the probability mix that maximizes each firm’s expected payoff, guiding them to a strategy that mitigates risks and maximizes returns in a competitive market.
Most Common FAQs
It’s a situation in a game where each player chooses their strategy based on a probability distribution. No player can improve their payoff by changing their strategy mix, given the strategies of the others.
By applying the formulae for expected payoffs and calculating the probabilities that equate these payoffs across different strategies. Ensuring that no player has an incentive to deviate.
Yes, it’s designed for games of two or more players with finite strategies. Making it a versatile tool for analyzing a wide range of strategic interactions