Question

Solve the following game \(\begin{array}{lllcl} & & \mathrm{B}: & \mathrm{B}_{1} & \mathrm{~B}_{2} & \mathrm{~B}_{3} \\ \mathrm{~A} & \mathrm{~A}_{1} & 1 & 2 & 3 \\ & \mathrm{~A}_{2} & & 3 & -1 \\ & \mathrm{~A}_{3} & -1 & -2 & 4\end{array}\)

Short answer

The optimal strategy for player A is \(A_1\) and for player B is \(B_2\). The expected value of the game for A is 2, while the expected value for B is -1/3.

Step-by-step solution

Identify the saddle points

A saddle point is a cell in the payoff matrix where the value is both the largest in its row and the smallest in its column. If a saddle point exists, it represents the optimal strategy for both players. In order to identify the saddle points, check each element in the matrix one by one: \( A_1B_1: 1 \) – Not the largest in row 1 and not the smallest in column 1 \( A_1B_2: 2 \) – Not the largest in row 1 and not the smallest in column 2 \( A_1B_3: 3 \) – Not the largest in row 1 but the smallest in column 3 \( A_2B_1: 3 \) – Not the largest in row 2 and not the smallest in column 1 \( A_2B_2: -1 \) – Largest in row 2 and not the smallest in column 2 \( A_3B_1: -1 \) – Not the largest in row 3 and smallest in column 1 \( A_3B_2: -2 \) – Not the largest in row 3 and not the smallest in column 2 \( A_3B_3: 4 \) – Largest in row 3 and not the smallest in column 3 No saddle points found.

Calculate the payoff for each player

Since there are no saddle points, we need to determine the optimal strategy for each player. First, calculate the value of each cell and its column and row averages: \(\mathrm{A}'s \) row averages: \( A_1 = (1+2+3) / 3 = 2 \) \( A_2 = (3+(-1)) / 2 = 1 \) \( A_3 = ((-1)+(-2)+4) / 3 = 1/3 \) \(\mathrm{B}'s \) column averages: \( B_1 = (1+3+(-1)) / 3 = 1 \) \( B_2 = (2+(-1)+(-2)) / 3 = -1/3 \) \( B_3 = (3+4) / 2 = 3.5 \)

Find the optimal strategy for each player

Find the optimal strategy for player A by selecting the row with the highest average for A. Since highest average value for A is 2 (row A1), A should always choose to play strategy \(A_1\). Find the optimal strategy for player B by selecting the column with the smallest average for B. Since the smallest average value for B is -1/3 (column B2), B should always choose to play strategy \(B_2\). The optimal strategy for player A is \(A_1\) and for player B is \(B_2\). The game is solved and the expected value of the game for A is 2, while the expected value for B is -1/3.

Key concepts

Saddle Point

In the context of game theory, a saddle point is a crucial concept when analyzing a payoff matrix. It represents a scenario where a particular value is both the largest in its row and the smallest in its column. This essentially means that this point is stable and optimal for both players; neither player can unilaterally improve their payoff by switching strategies. In practical terms, when a saddle point exists in a given matrix, it offers a straightforward solution to the game as it indicates the optimal strategies for both players involved. Here, the players can confidently choose the corresponding strategies without the risk of the opponent improving their outcome by changing their move. Unfortunately, not every game will have a saddle point, as seen in the exercise, where no value in the matrix meets both conditions of being the largest in its row and the smallest in its column. This leads players to figure out optimal strategies by alternative methods.

Payoff Matrix

A payoff matrix, in game theory, is a simple way to represent the potential outcomes of a strategic game between two players. This matrix lays out all possible actions available to each player and the resultant payoffs for every combination of choices. The table or matrix provides a clear visualization of strategies and outcomes, guiding players in decision-making.
In our example, the payoff matrix is presented with two players, player A and player B, and their corresponding strategies (\(A_1, A_2, A_3\) for player A and \(B_1, B_2, B_3\) for player B). Each cell in the matrix represents the outcome when a strategy combination is chosen. Understanding and correctly interpreting this matrix is vital for players to determine the best course of action under the given circumstances. In the absence of a saddle point, insights from the payoff matrix, such as row and column averages, can help players deduce the optimal strategies.

Optimal Strategy

Determining an optimal strategy is vital in strategic games to ensure the best possible outcome for each player. An optimal strategy involves selecting a course of action that maximizes a player's payoff given the strategies of the opponent.
In simpler terms, an optimal strategy is the best defense or offense a player can choose to minimize losses or maximize gains, irrespective of what the opponent does. In this exercise, after calculating the row and column averages of the payoff matrix, player A identifies their optimal strategy by choosing the row with the highest average payoffs (\(A_1\)), and player B chooses the column with the smallest average (\(B_2\)). Even without a saddle point, deriving these strategies involves analyzing the matrix to maximize each player's expected outcome. Thus, understanding and applying optimal strategies allow players to make the best choice under uncertain competition.

Expected Value

The expected value in the context of game theory is an essential concept for decision-making under uncertainty. It provides a numerical measure of the likely outcomes in a game when players follow their optimal strategies. The expected value is calculated by averaging payoffs for different strategies, considering the probability of each strategy being played.
For player A in the given exercise, the expected value of their choice (\(A_1\)) represents the mean payoff they can anticipate achieving through their best strategy, given player B's optimal response. Similarly, player B, by playing strategy \(B_2\), can calculate their own expected value. This helps in assessing their overall gain or loss over repeated plays. Essentially, expected value helps players understand the average benefit from a strategic choice, indicating how favorable a particular strategy combination is in the long run. By computing these values for optimal strategies, players can make informed decisions to achieve better outcomes.