Zero Sum Games 3

Victor · #1 $Old$ May 12th, 2009, 03:34 AM

Find the expected payoff to each player, if Player I uses mixed strategy 0.2, 0.3, 0.5) and Player II uses mixed strategy (0.1, 0.5, 0.2, 0.2), in the two-person zero-sum game with the following payoff matrix.

(It's a 3x4 matrix, I couldn't type it any other way)

-1 0 1 1
2 1 -2 -1
1 3 -1 0

Soroban · #2 $Old$ May 12th, 2009, 05:48 AM

Hello, Victor!

Quote:

Find the expected payoff to each player,
if Player 1 uses mixed strategy $(0.2, 0.3, 0.5)$
and Player 2 uses mixed strategy $(0.1, 0.5, 0.2, 0.2)$
in the two-person zero-sum game
with the following payoff matrix: . $\begin{pmatrix}\text{-}1& 0& 1& 1 \\ 2 &1 &\text{-}2& \text{-}1 \\ 1& 3& \text{-}1& 0 \end{pmatrix}$

We have: . $\begin{array}{c||c|c|c|c|} & 0.1 & 0.5 & 0.2 & 0.2 \\ \hline \hline 0.2 & \text{-}1 & 0 & 1 & 1 \\ \hline 0.3 & 2 & 1 & \text{-}2 & \text{-}1 \\ \hline 0.5 & 1 & 3 & \text{-}1 & 0 \\ \hline \end{array}$

Player 1's expectation: . $\begin{array}{c} +(0.2)(0.1)(\text{-}1) + (0.2)(0.5)(0) + (0.2)(0.2)(1) + (0.2)(0.2)(1) \\ \\[-4mm] +(0.3)(0.1)(2) + (0.3)(0.5)(1) + (0.3)(0.2)(\text{-}2) + (0.3)(0.2)(\text{-}1) \\ \\[-4mm] + (0.5)(0.1)(1) + (0.5)(0.5)(3) + (0.5)(0.2)(\text{-}1) + (0.5)(0.2)(0)\end{array}$

. . . . . . . $= \;-0.02 + 0 + 0.04 + 0.04 + 0.06 + 0.15 - 0.12 - 0.06 + 0.05 + 0.75 - 0.10 + 0$

. . . . . . . $= \;0.79$

If the given units are measured in dollars,

. . Player 1 can expect to win an average of 79 cents per game,

. . Player 2 can expect to lose an average of 79 cents per game.