Suppose I had a probability table which included the probability of scoring a penalty kick for 3 positions: Left, riGHT and center (9 cell table) and I had variables which would be in the mixed strategy for the player and keeper:
Player
x= kick left 
y= kick right
1-x-y= kick center

Keeper 
p=dive left
q=dive right
1-p-q= stay center

How would I compute these as percentage values in the mixed strategy for both player and keeper?Do i need partial derivatives?

To compute the mixed strategy percentages for both the player and the keeper in this penalty kick scenario, you would typically use **game theory** concepts, particularly the **Nash equilibrium** for a mixed strategy in a two-player zero-sum game. 

### Step 1: Setting Up the Payoff Matrix
Assume that you have a payoff matrix for the player. This matrix represents the probability of success for the player based on the combination of strategies chosen by both the player and the keeper. The general form of the payoff matrix for the player would look like this:

\[
\begin{array}{c|ccc}
\text{Player/ Keeper} & \text{Dive Left (p)} & \text{Stay Center (1-p-q)} & \text{Dive Right (q)} \\
\hline
\text{Kick Left (x)} & a_{11} & a_{12} & a_{13} \\
\text{Kick Center (1-x-y)} & a_{21} & a_{22} & a_{23} \\
\text{Kick Right (y)} & a_{31} & a_{32} & a_{33} \\
\end{array}
\]
Where \(a_{ij}\) represents the probability of scoring based on the player’s choice \(i\) and the keeper’s choice \(j\).

### Step 2: Express the Player's Expected Payoff
The expected payoff for the player (i.e., the likelihood of scoring) when they adopt a mixed strategy can be written as:
\[
\text{E(Player)} = x \cdot (p \cdot a_{11} + (1-p-q) \cdot a_{12} + q \cdot a_{13}) + (1-x-y) \cdot (p \cdot a_{21} + (1-p-q) \cdot a_{22} + q \cdot a_{23})

Learn how to compute mixed strategy percentages for both the player and the keeper in a penalty kick scenario using game theory concepts. Understand the Nash equilibrium and probability theory applied to determine optimal strategies. Suitable for advanced undergraduate students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation