Games with saddle point

In game theory, a saddle point is a point in a game matrix where the minimum value in a row is equal to the maximum value in the corresponding column. If a game matrix has a saddle point, it can help simplify the analysis of the game and identify the equilibrium strategies of the players.

Here is an example of a game with a saddle point:

Consider a two-player game in which Player A can choose between two pure strategies, A1 and A2, and Player B can choose between two pure strategies, B1 and B2. The game matrix is as follows:

In this matrix, the minimum value in the first row is 2, which corresponds to the maximum value in the first column. This is a saddle point, and it indicates that Player A should choose A1 and Player B should choose B1 to achieve the equilibrium outcome.

To see why, consider the following:

• If Player A chooses A1, Player B’s best response is to choose B1, which gives them a payoff of 2.
• If Player A chooses A2, Player B’s best response is to choose B2, which gives them a payoff of 3.
• If Player B chooses B1, Player A’s best response is to choose A1, which gives them a payoff of 2.
• If Player B chooses B2, Player A’s best response is to choose A1, which gives them a payoff of 4.

Thus, the equilibrium outcome of this game is (A1, B1), with payoffs of (2, 2). The presence of the saddle point in the game matrix simplifies the analysis of the game and allows us to identify the equilibrium outcome without having to consider more complex strategies.