Game with three strategies. Consider the following game: (a) Find all pure strategy Nash equilibria or show that there

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Game with three strategies.
Consider the following game:
(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)?
Please
answer questions (e) to (h).
i only need e-h part answer