Determine the best strategy for Becky.

Understand the Problem

The question involves a zero-sum game where two players, Alex and Becky, choose options to maximize their values. It outlines a mathematical problem involving linear programming and asks for the determination of expected outcomes based on the strategies chosen by each player.

Answer

The probabilities for Becky are $ p_1 = \frac{1}{3} $, $ p_2 = \frac{1}{3} $, $ p_3 = \frac{1}{3} $ with value $ V = 1 $.

Steps to Solve

Understanding Player Strategies Alex has two options, $Q$ and $R$, and Becky has three options, $X$, $Y$, and $Z$. The payoff for Alex when he plays $Q$ is represented as $P_Q$, and for playing $R$ as $P_R$.
Setting Up the Payoff Matrix Based on the provided payoffs:

For Option $Q$:
- Against option $X$: $3$
- Against option $Y$: $0$
- Against option $Z$: $2$
For Option $R$:
- Against option $X$: $-2$
- Against option $Y$: $1$
- Against option $Z$: $-1$

Finding Payoff Values Set up the equations for payoffs:

When Alex chooses $Q$, the expected payoff is given by: $$ P_Q = p_1 \cdot 3 + p_2 \cdot 0 + p_3 \cdot 2 $$
When Alex chooses $R$, the expected payoff is: $$ P_R = p_1 \cdot (-2) + p_2 \cdot 1 + p_3 \cdot (-1) $$

Establish Constraints The probabilities for Becky's options must sum to 1: $$ p_1 + p_2 + p_3 = 1 $$ and each probability must be non-negative: $$ p_1 \geq 0, \quad p_2 \geq 0, \quad p_3 \geq 0 $$
Setting Objective to Minimize Maximum Payoff To find the optimal strategy for Becky:

Minimize the worst-case payoff for Alex, which leads to the inequalities: $$ P_Q = p_1 \cdot 3 + p_3 \cdot 2 \geq V $$ $$ P_R = p_1 \cdot (-2) + p_2 \cdot 1 \geq V $$ Where $V$ is the value of the game for Alex.

Linear Programming Formulation Create the linear programming problem: Minimize ( V ): $$ V = \max (P_Q, P_R) $$ subject to the inequalities derived above.
Solving for Probability Values Use optimization methods such as the Simplex method to solve for:

Maximum Payoff Values of Alex
Corresponding probabilities ( p_1, p_2, p_3 )

The probabilities for Becky to minimize Alex's expected payoff are:

Option $X$ (probability ( p_1 )): ( \frac{1}{3} )
Option $Y$ (probability ( p_2 )): ( \frac{1}{3} )
Option $Z$ (probability ( p_3 )): ( \frac{1}{3} )

The value of the game for Alex is ( V = 1 ).

More Information

In a zero-sum game, one player's gain is equal to the other player's loss. By determining the optimal strategies for both Alex and Becky, we can analyze how they can maximize or minimize their expected payoffs.

Tips

Confusing the payoff structure can lead to incorrect setups of the equations.
Failing to recognize that probabilities must sum to 1 and be non-negative may result in an invalid solution.

AI-generated content may contain errors. Please verify critical information

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