 # Mixed Strategy Equilibrium

Subject Mathematics Game Theory

## Question

1. Given the following payoff matrix which represents a game with no pure strategy equilibrium (best-response payoffs are shown in red), find the mixed strategy equilibrium.
Left       Right
Up    10, 2       4, 9
Down 1,7         8, 5
Solving this means finding probabilities that are best responses to each other.
3. Make up a 2x2 game in strategic form with no pure strategy equilibrium by using the lowest digit in your student number as the payoff Row gets at (Up, Left) and the highest digit the payoff that column gets at (Up, Left). You must also use three other digits from your student number.
4. Find the mixed strategy equilibrium for your game algebraically.
5. Provide a graphical demonstration of your solution.

## Solution Preview

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1. Let p be the probability that the Row player will play “Up” while q be the probability that the column player will play “Left”.

Then the mixed strategy values for p and q happens when

2p+7(1-p)=9p+5(1-p)
and
10q+4(1-q)=1q+8(1-q)...

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