# The simplified method Analysis

In this problem, we will explore the “simplified method” from lectures on a game with
three strategies. Game with three strategies present new challenges.
Consider the following game:
l c r
L 3,3 -10,1 4,2
C 1,-10 10,10 1,11
R 2,4 11,1 0,0
(a) Find all pure strategy Nash equilibria or show that there are none.
(b) Suppose that player 1 plays her strategies L, C and R with probabilities α > 0, β >
0, γ > 0 and player 2 plays l, c and r with probabilities a > 0, b > 0 and c > 0. Find a
mixed-strategy Nash equilibrium.
(c) What are the expected payoffs of the players when they use this mixed strategy?
(d) Suppose β = γ = b = c = 0.5. Does player 1 have incentives to deviate (to a pure or
mixed strategy)? (Note that player 2 is symmetric, so your answer about player 1 applies
to player 2.)
(e) Suppose you use “simplified method” only for strategies C and R, ignoring L; that
is, you find a, b and c such that u1(C, σ2) = u1(R, σ2). What values of a, b and c you
obtain?
(f) Note that you would obtain the same equations for player 2. Given (d), what values
of a, b, c, α, β and γ correspond to mixed strategy Nash equilibrium where players do not
play strategies L and l? Explain.
(g) Does the mixed strategy Nash equilibrium you found in (f) continues to be a Nash
equilibrium if I change the payoff from (L, l) to (100,100)?
(h) Does the mixed strategy Nash equilibrium you found in (f) continues to be a Nash
equilibrium if I change the payoff from (L, c) to (10,1

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