Question

Consider the following payoff matrix: e to \(\begin{array}{lllll} & \mathrm{C}_{1} & \mathrm{C}_{2} & \mathrm{C}_{3} & \mathrm{C}_{4} \\ \mathrm{R}_{1} & 2 & 3 & -3 & 2 \\ \mathrm{R}_{2} & 1 & 3 & 5 & 2 \\ \mathrm{R}_{3} & 9 & 5 & 8 & 10 \\ \text { Find the value of this game. }\end{array}\)

Short answer

The value of the game is 8, which is the expected payoff when both players play their optimal strategies (Player A choosing Row 3 and Player B choosing Column 3).

Step-by-step solution

Identify Player A's Best Strategies

Player A has to choose a row to maximize their payoff. To identify the best strategies for Player A, we need to find the highest payoff in each row, also known as max-min strategy: - Row 1: Maximum payoff is 3 (in column 2) - Row 2: Maximum payoff is 5 (in column 3) - Row 3: Maximum payoff is 10 (in column 4)

Choose Player A's Optimized Strategy

Player A should choose the row with the highest minimum payoff. In this case, it's Row 3 (with a minimum payoff of 10).

Identify Player B's Best Strategies

Now that we know Player A's optimal strategy, we need to determine Player B's best strategies. Player B wants to minimize Player A's payoff, so they will choose the column with the lowest payoff in each row of the matrix as follows: - Column 1: Minimum payoff is 1 (in row 2) - Column 2: Minimum payoff is 3 (in row 1) - Column 3: Minimum payoff is -3 (in row 1) - Column 4: Minimum payoff is 2 (in row 1)

Choose Player B's Optimized Strategy

Player B should choose the column with the lowest maximum payoff. In this case, it's Column 3 (with a maximum payoff of -3).

Calculate the Value of the Game

The value of the game is the expected payoff when both players play optimally using the max-min strategy. In this scenario, Player A would choose Row 3, and Player B would choose Column 3. The value of the game is the payoff in Row 3 and Column 3, which is 8. So the value of this game is 8.

Key concepts

Payoff Matrix

In game theory, a payoff matrix is an extremely useful tool to visualize the possible outcomes of a game. Each cell in the matrix represents the outcome based on the actions chosen by the players in the game. For example, if Player A chooses a row and Player B chooses a column, the intersection of that row and column offers the payoff or the result of those choices.

In our case, the payoff matrix provides us with an overview of the potential gains or losses Player A might receive from Player B's choices. It involves organizing payoffs in a tabular format where rows represent Player A's strategies and columns represent Player B's strategies. Using the provided example, consider the matrix:

1. Row 1: [2, 3, -3, 2]
2. Row 2: [1, 3, 5, 2]
3. Row 3: [9, 5, 8, 10]

These values indicate what Player A receives if the corresponding row and column are chosen. Recognizing these payoffs helps both players determine their own optimal strategies.

Max-Min Strategy

The max-min strategy is a principle used to secure the best worst-case payoff. For Player A, it means choosing a strategy that maximizes their minimum gain. This involves looking at each row of their payoff matrix and identifying the maximum number across the minimums available in those options.

To perform this calculation in our example, we examine the potential payoffs:

• From Row 1, the maximum payoff is 3.
• From Row 2, it is 5.
• From Row 3, the maximum is 10.

The max-min strategy suggests that Player A select the strategy (or row) with the greatest minimum payoff, directing Player A towards choosing Row 3. This selection ensures that Player A achieves the highest potential gain out of the minimum outcomes available.

Optimal Strategy

An optimal strategy in game theory ensures that a player achieves the best possible outcome given that the other player is also playing optimally. To find Player A's optimal strategy, we apply the max-min strategy to identify the row that yields the best minimum payoff outcome. In our case, this led Player A to select Row 3.

Similarly, Player B's optimal strategy involves minimizing losses. Player B looks at each column and picks the strategy with the lowest maximum loss. In this context, it's Column 3 because it presents the least cost to Player B when considering the maximum payoffs Player A can secure.

Together, optimal strategies suggest a balance or equilibrium where both players are minimizing their potential losses while maximizing their potential gains.

Value of the Game

The 'value of the game' in game theory represents the payoff when both players choose their optimal strategies. It is the amount each player can expect to achieve if they act rationally and strategically.
In our example, Player A uses their max-min strategy, choosing Row 3, while Player B determines their optimal choice is Column 3. The intersection of Row 3 and Column 3 within the payoff matrix results in a value of 8.

This means that if both players choose their best possible strategies, the final outcome of the game will be a payoff of 8. This concept of the 'value of the game' helps in predicting outcomes in strategic situations, where players aim for decisions that result in the most favorable result.