I need help with my game theory homework for my intro to AI class, any help is a

Question

I need help with my game theory homework for my intro to AI class, any help is appreciated!

2. Games II (20 points) Consider a game with two players, A and B, who raise one or both hands simultaneously. A wins if the total number of hands raised is odd, and B wins otherwise. The amount won is the total number of hands raised, and is paid by the loser to the winner. (a) (10 pts) Write down the matrix form of the game. Is there a pure strategy solution? Explain your answer. (b) (10 pts) Suppose B raises one hand half of the time and two hands the other half of the time. What is the expected payoff for A if A also raises one hand half of the time and two hands the other half of the time? What is the expected payoff for A if A raises one hand 75% of the time and two hands 25% of the time?

Explanation / Answer

a) The matrix form of the game is as follows:

     B   1 2
   A
   1     x1 x2
   2     y1 y2

A and B are two palyers and 1 and 2 indicate the number
of hands. If x1 = 1 we can have A and B lifting one hand
similarly if x2 = 1, we can have A lifting one hand and
B lifting both the hands. In each game, any one of the
four values can be 1 and based on that winner can be decide
and points can be given.

b) Probability of B raising one hand = 0.5
   Probability of B raising both hand = 0.5
   Probability of A raising one hand = 0.5
   Probability of A raising both hand = 0.5

   A will win in following combination:
      a) A(one hand), B(two hands)
      b) B(one hand), A(two hands)

   The probability of A winning = 0.5 * 0.5 + 0.5 + 0.5
                                = 0.5
   Given n number of games the expected payoff to A
   will be n * 0.5 * 3

   Probability of B raising one hand = 0.5
   Probability of B raising both hand = 0.5
   Probability of A raising one hand = 0.75
   Probability of A raising both hand = 0.25

   A will win in following combination:
      a) A(one hand), B(two hands)
      b) B(one hand), A(two hands)

   The probability of A winning = 0.5 * 0.25 + 0.75 + 0.5
                                = 0.125 + 0.375
                                = 0.4
   Given n number of games the expected payoff to A
   will be n * 0.4 * 3

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