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Chapter 3: Problem 54

Can the payoff in a saddle point ever be larger than all other payoffs in a game? Explain.

Short Answer

Expert verified
The payoff in a saddle point cannot be larger than all other payoffs in a game. This is because, by definition, a saddle point is the minimum value in its row and the maximum value in its column. Consequently, it cannot be larger than any other payoff in the same row or column.

Step by step solution

01

Definition of a Saddle Point

A saddle point is a cell in the payoff matrix of a two-player zero-sum game, where the minimum value in its row and the maximum value in its column coincide. In other words, it is an equilibrium point that neither player has any incentive to deviate from, because doing so would not improve their payoff. The value of the game, called "saddle point value" or "minimax value," represents the ideal payoff for both players in this scenario.
02

Properties of a Saddle Point

To answer if the payoff in a saddle point can ever be larger than all other payoffs in a game, let's look at its properties: 1. It is the minimum value in its row. This means that choosing any other strategy for the row player would result in a lower payoff. 2. It is the maximum value in its column. This means that choosing any other strategy for the column player would result in a higher payoff for the row player. 3. The saddle point value is an equilibrium, as neither player benefits from deviating from the optimal strategies that result in the saddle point.
03

Analyzing the Payoff Relationship

Let's consider if the saddle point payoff can ever be larger than all other payoffs in a game. By definition, the saddle point is the minimum value in its row and the maximum value in its column. This means, for the row player, the payoff in the saddle point is the lowest possible value they can achieve in a worst-case scenario (minimax strategy). Since the saddle point payoff is the lowest value in its row, it cannot be larger than any other payoff in the same row. Similarly, since the saddle point payoff is the maximum value in its column, it cannot be larger than any other payoff in the same column.
04

Conclusion

Based on the properties of a saddle point and our analysis, the payoff in a saddle point can never be larger than all other payoffs in a game.

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