In game theory, a "saddle point" refers to a specific point in the payoff matrix of a two-player, zero-sum game where the value of the game is optimized. A zero-sum game is one in which the total amount of utility or payoff available to all players is constant, meaning that one player's gain is exactly offset by another player's loss. The saddle point is a key concept in the analysis of such games.
Characteristics of a Saddle Point:
 |
| Characteristics of a Saddle Point by Study Terrain |
Optimal Value:
At the saddle point, the value of the game is optimized for the player whose turn it is to move. This optimal value is known as the "saddle point value."
Minimax Strategy:
The saddle point is associated with the concept of the minimax strategy. In a zero-sum game, the minimax strategy is a strategy that minimizes the maximum possible loss for a player. At the saddle point, each player is playing their minimax strategy, resulting in an equilibrium where neither player can improve their position by unilaterally changing their strategy.
Payoff Matrix:
The saddle point is located at the intersection of a row and a column in the payoff matrix. At this intersection, the value is the maximum payoff for the row player and the minimum payoff for the column player.
Value of the Game:
The value of the game at the saddle point is the same for both players. This common value represents the optimized outcome for the players in a zero-sum game.
To find a saddle point in a payoff matrix:
 |
| To find a saddle point in a payoff matrix by Study Terrain |
Compare Row Minima:
Look at each row and find the smallest number in each row.
Identify the Maximum of Row Minima:
Pick the largest number among the smallest numbers found in step 1. Let's call this number "A."
Compare Column Maxima:
Examine each column and find the biggest number in each column.
Identify the Minimum of Column Maxima:
Select the smallest number among the largest numbers found in step 3. Let's call this number "B."
Check for Equality:
If "A" is the same as "B," then there is a saddle point, and both "A" and "B" are the value of the game.
Example:
Consider the following payoff matrix:
Player 2
| A | B | C |
--------------------------------
Player 1| 3,2 | 0,1 | 4,3 |
|--------------------------------
Here, the minimum values in each row are 2, 0, and 3. The maximum of these row minima is 3. The maximum values in each column are 3, 2, and 3. The minimum of these column maxima is also 3. Since the maximum of row minima is equal to the minimum of column maxima, the matrix has a saddle point, and the value of the game is 3.
Applications of Saddle Points:
 |
| Applications of Saddle Points |
Saddle points are used to analyze and find equilibrium strategies in economic games such as pricing competition.
Game Theory:
In zero-sum games, saddle points provide insights into optimal strategies and the value of the game.
Optimization Problems:
Saddle points are relevant in optimization problems where the goal is to find the optimum in a multivariate function.
Resource Allocation:
In scenarios involving the allocation of limited resources, saddle points help identify optimal allocation strategies.
Saddle points play a fundamental role in zero-sum games, providing a solution concept that helps determine optimal strategies and the value associated with those strategies. The concept is valuable in various fields, contributing to the understanding of strategic interactions and decision-making.
