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Chapter 3: Problem 19

Decide whether the game is strictly determined. If it is, give the players'optimal pure strategies and the value of the game. $$ \begin{gathered} a & b & \multicolumn{1}{c} {c} \\ P & {\left[\begin{array}{llr} 1 & -1 & -5 \\ Q \\ 4 & -4 & 2 \\ R \\ S \end{array}\right.} & \left.\begin{array}{rrr} 3 & -3 & -10 \\ 5 & -5 & -4 \end{array}\right] \end{gathered} $$

Short Answer

Expert verified
The game is not strictly determined, as neither player has a strictly dominated strategy. Therefore, it is not possible to provide optimal pure strategies and the value of the game for this particular case.

Step by step solution

01

Understanding the game

First, let's understand the given payoff matrix. There are two players, Row Player (player 1) and Column Player (player 2). Row Player has strategies P, Q, R, and S, while Column Player has strategies a, b, and c. The matrix shows the payoff to the Row Player when both players choose respective strategies. For instance, if Row Player chooses strategy P and Column Player chooses strategy a, the payoff to Row Player is 1.
02

Identify strictly dominated strategies

Now, let's look for strictly dominated strategies for each player. A strategy is strictly dominated if there exists another strategy that has a higher payoff for every possible opponent's strategy. For a strategy of Row Player to be strictly dominated, it must have a smaller payoff for every choice of Column Player's strategy. For Row Player: - Comparing P and Q: - If Column Player chooses a: \( 1 > -5 \) - If Column Player chooses b: \( -1 > -4 \) - If Column Player chooses c: \( -5 > 2 \) (we found a situation where P is not better than Q, so we can stop comparing and look for another strictly dominated strategy) - Comparing P and R: - If Column Player chooses a: \( 1 > 3 \) (we found a situation where P is not better than R, so we can stop comparing and look for another strictly dominated strategy) - Comparing P and S: - If Column Player chooses a: \( 1 > 5 \) (we found a situation where P is not better than S, so we can stop comparing and look for another strictly dominated strategy) There are no strictly dominated strategies for Row Player. For a strategy of Column Player to be strictly dominated, it must result in a higher payoff for Row Player for every choice of Row Player's strategy. For Column Player: - Comparing a and b: - If Row Player chooses P: \( 1 < -1 \) - If Row Player chooses Q: \( 4 < -4 \) - If Row Player chooses R: \( 3 < -3 \) (we found a situation where a is not worse than b, so we can stop comparing and look for another strictly dominated strategy) - Comparating a and c: - If Row Player chooses P: \( 1 < -5 \) - If Row Player chooses Q: \( 4 < 2 \) (we found a situation where a is not worse than c, so we can stop comparing and look for another strictly dominated strategy) There are no strictly dominated strategies for Column Player.
03

Determine if the game is strictly determined

Since neither player has a strictly dominated strategy, the game is not strictly determined. Thus, we don't need to calculate the optimal pure strategies and the value of the game as the game is not strictly determined. We would have to use other methods, such as finding the mixed strategy Nash equilibrium, to analyze the optimal strategies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Strictly Determined Games
In game theory, a strictly determined game is one where each player's optimal strategy leads to a specific outcome that is predictable and consistent every time the game is played. This type of game possesses a saddle point, which means there is a particular strategy for both players that guarantees a payoff to one player while taking into account the opponent鈥檚 best countermove.
This concept is significant when assessing games because it implies both players can employ a pure strategy to achieve their best outcome without needing to consider complex mixed strategies.
If a game lacks strictly dominated strategies, as in the exercise, it generally indicates that the game does not have a straightforward optimal outcome for both players using pure strategies.
Consequently, the exercise highlights that when neither player can eliminate any strategies through dominance, the game is not strictly determined.
  • An understanding of strictly determined games helps in identifying scenarios where a predictable outcome can be achieved by both players.
  • Saddle points are crucial in determining if a game is strictly determined.
Payoff Matrix
A payoff matrix is a crucial tool in game theory that represents the possible outcomes of a strategic game, showing the payoff for each player given their choices of strategies. In the provided exercise, the payoff matrix lists the payoffs for the Row Player based on their strategy and the strategy selected by the Column Player.
The matrix gives a visual representation of all possible strategy combinations and their respective payoffs, simplifying the process of identifying optimal strategies. Each cell in the matrix corresponds to a specific strategy pair (one from each player) and displays the outcome for the Row Player.
This type of matrix is especially beneficial when analyzing simple strategic games, as it provides a clear, comprehensive view of the potential gains or losses dependent on combined strategies.
  • Payoff matrices assist in visualizing all possible outcomes of a game and their associated payoffs.
  • The ease of comparison they offer makes them valuable for determining dominated and dominant strategies.
Dominated Strategies
Dominated strategies occur when one strategy consistently results in a worse payoff than another, regardless of the opponent鈥檚 choice. In strategic decision-making, players can identify and eliminate dominated strategies to narrow down the possible best strategies to use.
A strictly dominated strategy is a strategy that is always worse than another option across all possible scenarios. Discovering a strictly dominated strategy indicates that rational players should never opt for it since there is always a more favorable choice.
In the given exercise, it was found that neither the Row Player nor the Column Player had any strictly dominated strategies. This elimination process is vital because it helps determine if a game is strictly determined or if players need to explore more complex solution methods such as mixed strategies.
  • Dominated strategies simplify decision-making by eliminating suboptimal choices.
  • Identifying and removing them can clarify the potential optimal strategies remaining in a game.

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Most popular questions from this chapter

In 1990 the U.S. population, broken down by regions, was \(50.8\) million in the Northeast, \(59.7\) million in the Midwest, \(85.4\) million in the South, and \(52.8\) million in the West. \({ }^{4}\) Between 1990 and 2000, the population in the Northeast grew by \(2.8\) million, the population in the Midwest grew by \(4.7\) million, the population in the South grew by \(14.8\) million, and the population in the West grew by \(10.4\) million. Set up the population figures for 1990 and the growth figures for the decade as row vectors. Assuming that the population will grow by the same numbers from 2000 to 2010 as they did from 1990 to 2000 , show how to use matrix operations to find the population in each region in 2010 .

Translate the given matrix equations into svstems of linear equations. $$ \left[\begin{array}{lrll} 1 & -1 & 0 & 1 \\ 1 & 1 & 2 & 4 \end{array}\right]\left[\begin{array}{r} x \\ y \\ z \\ w \end{array}\right]=\left[\begin{array}{r} -1 \\ 2 \end{array}\right] $$

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\\1 & 0 \\\\-1 & 2 \end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\\0 & 0.5 \\\\-1 & 3\end{array}\right], \text { and } \\\&C=\left[\begin{array}{rr}1 & -1 \\\1 & 1 \\\\-1 & -1\end{array}\right].\end{aligned}$$ $$ A^{T}+3 C^{T} $$

Reduce the payoff matrices by dominance. $$ \begin{array}{rrrr} & & \mathbf{B} & \\ & a & b & c \\ p & {\left[\begin{array}{rrr} 1 & -1 & -5 \\ q & 4 & 0 & 2 \\ \mathbf{A} & r & 3 & -3 & 10 \\ 3 & -5 & -4 \end{array}\right]} \end{array} $$

The Left Coast Bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of book: hardcover, softcover, and plastic (for infants). At the beginning of January, the central computer showed the following books in stock: $$ \begin{array}{|r|c|c|c|} \hline & \text { Hard } & \text { Soft } & \text { Plastic } \\ \hline \text { San Francisco } & 1,000 & 2,000 & 5,000 \\ \hline \text { Los Angeles } & 1,000 & 5,000 & 2,000 \\ \hline \end{array} $$ Suppose its sales in January were as follows: 700 hardcover books, 1,300 softcover books, and 2,000 plastic books sold in San Francisco, and 400 hardcover, 300 softcover, and 500 plastic books sold in Los Angeles. Write these sales figures in the form of a matrix, and then show how matrix algebra can be used to compute the inventory remaining in each store at the end of January.
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