Two risk neutral players, A and B, simultaneously choose bets on the outcome of a fair coin toss. Each player can place a $4 bet on either outcome (heads or tails), pay $8 to bet on both outcomes, or pay nothing to bet on neither outcome. If exactly one player bets on the correct outcome, that player wins $12 and the other player wins nothing. If both players bet on the correct outcome, they each win $6. If neither player bets on the correct outcome, they both win nothing. Thus, for example, if A bets on both heads and tails and B bets only on heads, then if heads comes up, A ends up with a net loss of $2 while B ends up with a net gain of $2, and if tails comes up, A ends up with a net gain of $4 while B ends up with a net loss of $4. (a) Find all pure strategy Nash equilibria of this game. (b) Find a mixed strategy Nash equilibrium of this game in both players choose to bet on both outcomes with positive probability, or show that no such equilibrium exists.
Two risk neutral players, A and B, simultaneously choose bets on the outcome of a fair coin toss. Each player can place a $4 bet on either outcome (heads or tails), pay $8 to bet on both outcomes, or pay nothing to bet on neither outcome. If exactly one player bets on the correct outcome, that player wins $12 and the other player wins nothing. If both players bet on the correct outcome, they each win $6. If neither player bets on the correct outcome, they both win nothing. Thus, for example, if A bets on both heads and tails and B bets only on heads, then if heads comes up, A ends up with a net loss of $2 while B ends up with a net gain of $2, and if tails comes up, A ends up with a net gain of $4 while B ends up with a net loss of $4. (a) Find all pure strategy Nash equilibria of this game. (b) Find a mixed strategy Nash equilibrium of this game in both players choose to bet on both outcomes with positive probability, or show that no such equilibrium exists.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Two risk neutral players, A and B, simultaneously choose bets on the outcome of a fair coin
toss. Each player can place a $4 bet on either outcome (heads or tails), pay $8 to bet on both
outcomes, or pay nothing to bet on neither outcome. If exactly one player bets on the correct
outcome, that player wins $12 and the other player wins nothing. If both players bet on the
correct outcome, they each win $6. If neither player bets on the correct outcome, they both
win nothing. Thus, for example, if A bets on both heads and tails and B bets only on heads,
then if heads comes up, A ends up with a net loss of $2 while B ends up with a net gain of
$2, and if tails comes up, A ends up with a net gain of $4 while B ends up with a net loss of
$4.
(a) Find all pure strategy Nash equilibria of this game.
(b) Find a mixed strategy Nash equilibrium of this game in both players choose to bet on
both outcomes with positive probability, or show that no such equilibrium exists.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2Fd10966a5-ac0a-4216-bd91-a07360785c18%2Fh9f9p1_processed.png&w=3840&q=75)
Transcribed Image Text:Two risk neutral players, A and B, simultaneously choose bets on the outcome of a fair coin
toss. Each player can place a $4 bet on either outcome (heads or tails), pay $8 to bet on both
outcomes, or pay nothing to bet on neither outcome. If exactly one player bets on the correct
outcome, that player wins $12 and the other player wins nothing. If both players bet on the
correct outcome, they each win $6. If neither player bets on the correct outcome, they both
win nothing. Thus, for example, if A bets on both heads and tails and B bets only on heads,
then if heads comes up, A ends up with a net loss of $2 while B ends up with a net gain of
$2, and if tails comes up, A ends up with a net gain of $4 while B ends up with a net loss of
$4.
(a) Find all pure strategy Nash equilibria of this game.
(b) Find a mixed strategy Nash equilibrium of this game in both players choose to bet on
both outcomes with positive probability, or show that no such equilibrium exists.
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