Linear Algebra Game Strategy

[dawsob3]

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[soroban]

Hello, dawsob3!

Player R has two playing cards: a black Ace and a red Four.
Player C has two cards: a black Two and a red Three.
Each player secretly selects one of his or her cards.
If both selected cards are the same color, player C pays player R the sum of the face values in dollars.
If the cards are different colors, player R pays player C the sum of the face values.
What are the optimal strategies for both players and what is the value of the game?

The game matrix is: .\(\displaystyle \begin{array}{c||c|c|} & 2\clubsuit & 3 \diamondsuit \\ \hline \hline A\spadesuit & 3 & \text{-}4 \\ \hline 4\heartsuit & \text{-}6 & 7 \\ \hline \end{array}\)


There are formulas for this type of problem.

Given the game matrix: .\(\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

The optimal strategy for R is: .\(\displaystyle (p,\:1-p)\)

. . where: .\(\displaystyle p \:=\:\dfrac{|c-d|}{(a+d)-(b+c)} \)


The optimal strategy for C is: .\(\displaystyle (q,\:1-q)\)

. . where: .\(\displaystyle q \:=\:\dfrac{|b-d|}{(a+d)-(b+c)}\)


The value of the game is: .\(\displaystyle V \:=\:\dfrac{ad-bc}{(a+d)-(b+c)}\)

Go for it!