Player I holds a black Ace and a red 8. Player II holds a red 2 and a black 7. T
ID: 1169458 • Letter: P
Question
Player I holds a black Ace and a red 8. Player II holds a red 2 and a black
7. The players simultaneously choose a card to play. If the chosen cards are of the same
color, Player I wins. Player II wins if the cards are of dierent colors. The amount won
is a number of dollars equal to the number on the winner's card (Ace counts as 1.) Thus,
if Player I plays black Ace while Player II plays red 2, Player I gets a payo of -2 (loses
two dollars), and Player II gets a payo of 2 (wins two dollars); and likewise for the other
strategy proles.
Set up the payo matrix, nd the Nash equilibrium, and calculate each player's expected
payo in the equilibrium.
Explanation / Answer
The above game is a zero sum game.A two player zero sum game is one in which the sum of the payoffs of both the players is zero for any outcome. The payoff matrix for the above is given below.
player2
There is no pure nash quilibrium .Therfore. we can find nash equilibrium by using mixed strategies.
Let the probabability of player 2 playing Red 2 be p and Black 7 be (1-p).
Similarly, take probabibility of player 1 playing black Ace be q and Red 8 be (1-q).
Player 2
-2p + (1-p)
8p -7(1-p)
-2q + 8(1-q)
q -7(1-q)
Player 1 chooses the value of q such that it equates the value of player 2 choosing a Red 8 or Black Ace. Expected value for q in the equilibrium
-2q+8(1-q)=q-7(1-q)
q= 15/18
q=5/6
Player 2 chooses the value of p such that it equates the value of player 1 choosing a Red2 or Black 7Expected value for p in the equilibrium
-2p+(1-p)=8p-7(1-p)
p=4/9
A strictly mixed strategy Nash equilibrium in a 2 player, 2 choice (2x2) game is a p > 0 and a q > 0 such that q is a best response by the row player to column player’s choices, and p is a best response by the column player to the row player s player’s choices choices.
In this case the mix equilibrium strategy (p,q) is (4/9,5/6).
Expected payoff of player 1= -2*(5)/6 + 8(1-(5/6)) {substituting the value of q in above equation}
=1/3
Since, this is a zero sum game, expected pay off of player 2 is negative of player 1's payoff.
So, expected payoff of player 2 = -(1/3)
player1 red 2 Black 7 Black Ace -2,2 1,-1 Red 8 8,-8 -7,7