Games without saddle point

A saddle point in game theory refers to a strategy combination in a game where each player’s best response is the same strategy regardless of what the other player does. In other words, it’s a point in the payoff matrix where both players have a unique optimal strategy. However, not all games have a saddle point equilibrium.

Here are some examples of games that do not have a saddle point:

1. Matching Pennies: In this game, each player has a coin and can choose to show either heads or tails. The payoff matrix is as follows:

2. There is no saddle point in this game because there is no single strategy combination where both players have a unique best response.
3. Prisoner’s Dilemma: In this game, two players are arrested for a crime and are given the choice of whether to cooperate or defect. The payoff matrix is as follows:

4. There is no saddle point in this game because each player’s best response depends on what the other player does, so there is no single strategy combination where both players have a unique best response.
5. Chicken: In this game, two drivers are racing towards each other and must decide whether to swerve or stay on course. The payoff matrix is as follows:

6. There is no saddle point in this game because each player’s best response depends on what the other player does, so there is no single strategy combination where both players have a unique best response.

In each of these games, finding a Nash equilibrium requires the use of mixed strategies, as no pure strategy Nash equilibrium exists.