Question

player 2 H T player 1 H 1,-1 -1,1 T -1,1 1,-1

Answer 1

Answer: Note that this is a zero sum game. The sum of payoffs of the two players in any given cell is exactly zero. Lets proceed to the answers.

(a) The strategy set for each player consists of only 2 possibilities - Head or Tail.

(b) To find a dominant strategy for player 1, let's consider the strategy available to player 2. If player 2 plays H, player one earns a positive payoff by playing H. If player 2 plays T, player 1 earns a positive payoff by playing T. Thus, there is no dominant strategy for player 1, which always gives him the largest payoff irrespective of the strategy played by player 2.

Likewise, there is no dominant strategy for player 2. It changes with the strategy played by player 1.

(c) No, there does not exist any Nash equilibrium in pure strategy in this game. A Nash equilibrium is one which is a mutual best response for all the players in the game, and does not provide an incentive to deviate to any player.

For example, consider the strategy set (H, H). Can it be a Nash equilibrium ? Clearly not, as player 2 has an incentive to deviate to "T" instead of playing "H".

Likewise, we can rule out the other strategy pairs - (H, T), (T, H), (T, T).

(d) To find the Nash equilibrium in mixed strategy, refer to the images below:

(6) H Player to) T 1-1,1) (1-1) (P) H playeft I (0-1) 11, -1) (-1) In order af the mixed find consider equilibrium, strategy following: the. Player I's when player payoff from playing H, probability playe Н. with q is, T, (H) = T. (Hl player 2 randomises) (1) v q: (1) + (1-9)(-1) q-1+q 29,- -1 9,1-1) + (1-2)(1) - 9+1-9 =1-aq indifferent between for be player H strategy 4 I A(H) 2q2 4q 1 te 214 - 12. for - q to prefer TilH) 7 T, (T) over player I

for of - Over H player I prefer en I Kill) ( ,(T) q ( 112 2 Likewise, for T2 (H) = player 2 (-1)(p) + 1(1-P) + I-R : 1 - 2p (1210) + (-1)(1-P) T2 (T) = 2p-1 Player a allade 2 H Lover I 2 is indifferent if T2 (H) - I 2(T) Player 2 prefers. er I if T2 (H) > T2 (7) P< 112 over t if Tz (H) < T2(T) P7112 Player 2 -prefers I Thus (7 bd) is the nash 2 paxnu strategy for this equilibrium game, response it is 6 the only mutual best 2 between the 2 players.

(e) To see how the mixed strategy equilibrium is obtained by the intersection of the best response curves, refer to the image below:

classmate Dole Page (5) Best response (player 1 1/2 Best Response (player 2) 0 12 clearly -9 1 G 1112 1/2 112 Р O 112 Player is Player a's best response best response clearly, the best Curves intersect one at only Thus, response point, chool es where (p=112, q = 112). play each. to TH? with player 12. probability