Question

Kim and Kanye are playing a single game of Rock-Paper-Scissors. They have complete information. The winner must pay the loser $1,000. If they make the same choice they each get nothing. What is the Nash equilibrium for the game?

Answer 1

Answer:--
1 2 r p s
R (0,0) (-1,1) (1,-1)
P (1,-1) (0,0) (-1,1)
S (-1,1) (1,-1) (0,0)
Rock Paper Scissors game pay offs
Rock-Paper-Scissors, don't have a pure strategy equilibrium. In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2 chooses p, Player 1 should choose S. This continues with Player 2 choosing r in response to the choice S by Player 1, and so forth.

In games like Rock-Paper-Scissors, a player will want to randomize over several actions. If a player randomly chooses a pure strategy, we say that the player is using a "mixed strategy." In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.

Probability that one of the players is choosing amongst rock, paper and scissors randomly = 1/3

Probability that both players pick the same item = Probability that both players pick the rock + Probability that both players pick the paper + Probability that both players pick the scissors

= (1/3) * (1/3) + (1/3) * (1/3) + (1/3) * (1/3)

= 1/3

Probability that both players pick the different item = 1 - (1/3) = 2/3

Given X be the number of times the game is played until someone wins with probability = 2/3.

So, X will follow geometric distribution where the probability of success on each trial is p = 2/3 and the probability function of X is given as,

Pr(X = k) = (1/3)k-1 (2/3)

for k = 1, 2, 3, .