Game Theory Question: Find the pure nash equilibrium strategy(s) and payoffs. Fi
ID: 1129504 • Letter: G
Question
Game Theory Question: Find the pure nash equilibrium strategy(s) and payoffs. Find any mixed NE strategies and payoffs. What is the highest symmetric payoff with a public signal? What is the highest symmetric payoff with a private signal?
U 8,3 1,1 D 1,1 3,8 a. Pure NE & payoffs b. Mixed NE and payoffs c. Suppose a public signal can be observed by both players. What is the highest symmetric payoff that can be achieved in a correlated equilibrium d. What is the highest symmetric payoff that can be achieve in a correlated equilibrium w an arbitrary (possibly private) signal?Explanation / Answer
a. Player 1 will always get better off by choosing 8 (between 8&1) from left column and 3 (between 1&3) from right column.
Similarly Player 2, will always get better off by choosing 3 (between 3&1) from top row and 8 (between 1&8) from down row.
So the nash equilibrium in pure strategies are: (Up, Left)(Down, Right) and payoff : (8,3) & (3,8)
b) There exists a Nash equilibrium in mixed strategies.
Player 1 plays the mixed strategy of (Up, Down) = (0.7778,0.2222). Player has an expected payout = 2.5554.Player 2 plays the mixed strategy of (Left, Right) = (0.2222,0.7778). Player 2 has an expected payout = 2.5556.
c. There is a mixed strategy equilibrium where each player goes Up with probability 1/3. Now, if public signal observed. Now if a player assigned Up . he won't devaite for gaining maximum payoff . If a player is assigned down, other player will play down with probability 1/2 and Up with probability 1/2. the expected utility of UP = 8(1/2) + 1(1/2) = 4.5 and the expected utility of down = 1(1/2) + 3(1/2) = 2. Since neither player has an incentive to deviate, this is a correlated equilibrium. The expected payoff for this equilibrium is 8(1/3) + 3(1/3) + 1(1/3) = ,which is higher than the expected payoff of the mixed strategy Nash equilibrium.
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