# OK to post homework
# RDB, 2022-02-20, HW8

# 1. Done!

# 2.

(*

    Consider the game

            |[R, R] [S, T]|
        G = |[T, S] [P, P]|,

    where

        T(emptation) > R(eward) > P(unishment) > S(ucker).

    The (possibly mixed) Nash equilibria of a game is a selection of strategies
    (p1, p2) where p1 is a fixed point of the best-response function for player
    1, and p2 is a fixed point of the best-response function for player 2.

    The expected payoff for player 1 given the probabilities p1 and p2 is

        sum(p1[i] p2[j] G[i, j][1], i, j)
        =
        p1[1]p2[1] R + (1 - p1[1]) p2[1] T
        +
        p1[1](1 - p2[1]) S + (1 - p1[1]) (1 - p2[1]) P.

    This is a linear function in p1[1] with slope

        (P + R - S - T) p2[1] + (S - P)
        =
        (R - T) p2[1] + (1 - p2[1])(S - P),

    which is negative for all p2[1]. Therefore, the best response function
    gives p1[1] = 0. By symmetry considerations, the best response for player 2
    is always p2[1] = 0.

    So the only shared fixed point of the strategies is p1[1] = p2[1] = 0,
    i.e., both players fink.

*)

# 3.

(*
    Take the argument from before, but now say that

        T > R > S > P.

    That is, swap the place of S and P.

    The expected payoff for player 1 is still a linear function in p1 with
    slope

        (R - T) p2 + (1 - p2) (S - P),

    but now that S - P > 0 we are in some trouble. This is not always negative.

    If the slope *is* negative, then we take p1 = 0 again. This requires

        p2 > (S - P) / ((T - R) + (S - P)).

    which is possible. But p1 = 0 means that the slope for player 2 will be be
    positive (it's the mirror of player 1's situation), so the best response
    for player 2 is p2 = 1. So (0, 1) is one equilibrium.

    Reversing the signs and words above shows that (1, 0) is another
    equilibrium.

    What's left is the case where the slopes are both 0, which gives

        p1 = p2 = (S - P) / ((T - R) + (S - P)).

*)

# We can also do this in Maple as follows.

read "C7.txt":
G := [[ [R, R], [S, T] ],
      [ [T, S], [P, P] ]]:

# Prisoner's dilemma.
MNE22(G) assuming T > R and R > P and P > S;

# Snowdrift.
MNE22(G) assuming T > R and R > P and S > P;