What are some common methods to solve for Nash equilibrium in mixed strategies?

1 Best response method

The best response method is based on the idea that each player will choose the strategy that maximizes their expected payoff, given the probabilities of the other players' strategies. To use this method, you need to calculate the expected payoff for each player for each possible strategy combination, and then compare them to find the best response. The Nash equilibrium is the strategy combination where each player's best response matches their actual strategy.

2 Payoff matrix method

The payoff matrix method is based on the idea that each player will be indifferent between their pure strategies at the Nash equilibrium, since they will have the same expected payoff. To use this method, you need to set up a payoff matrix that shows the payoffs for each player for each pure strategy combination, and then assign variables to the probabilities of each player's mixed strategy. The Nash equilibrium is the solution of the system of equations that equates the expected payoffs for each player.

3 Graphical method

The graphical method is based on the idea that each player will choose the strategy that minimizes their regret, which is the difference between their actual payoff and the best payoff they could have obtained. To use this method, you need to plot the regret curves for each player, which show how their regret varies with their probability of choosing a pure strategy. The Nash equilibrium is the point where the regret curves intersect, since both players have zero regret.

4 Examples and applications

To illustrate these methods, let's look at some examples and applications of mixed strategy Nash equilibrium. One classic example is the matching pennies game, where two players simultaneously choose heads or tails, and one player wins if they match and the other player wins if they differ. Another example is the rock-paper-scissors game, where two players simultaneously choose rock, paper, or scissors, and one player wins if they beat the other player according to the rules. A more realistic application is the entry deterrence game, where an incumbent firm and a potential entrant decide whether to enter or stay out of a market, and their payoffs depend on the competition and the costs.

5 Benefits and limitations

Finding Nash equilibrium in mixed strategies can help you understand and predict the behavior and outcomes of strategic interactions, such as games, auctions, bargaining, voting, and signaling. It can also help you design and evaluate strategies that maximize your payoff or minimize your risk. However, there are also some limitations and assumptions of this approach, such as the rationality and common knowledge of the players, the completeness and accuracy of the information, and the uniqueness and stability of the equilibrium.