Mr. and Mrs. Ward typically vote oppositely in elections and so their votes "cancel each other out." They each gain 24 units of utility from a vote for their positions (and lose 24 units of utility from a vote against their positions). However, the bother of actually voting costs each 12 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward. Mr. Ward Vote Vote Mrs. Ward Mr. Ward: -12, Mrs. Ward: -12 Don't Vote Mr. Ward: -24, Mrs. Ward: 12 The Nash equilibrium for this game is for Mr. Ward to Don't Vote Mr. Ward: 12, Mrs. Ward: -24 Mr. Ward: 0, Mrs. Ward: 0 and for Mrs. Ward to . Under this outcome, Mr. Ward receives a

When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, the auctioneer actively woos representatives of other museums that have no chance of winning to attend anyway. Suppose a piece of art has recently become available for sale and will be auctioned off to the highest bidder, with the winner paying an amount equal to the second highest bid. Assume that most collectors know that Valerie places a value of $15,000 on the art piece and that she values this art piece more than any other collector. Suppose that if no one else shows up, Valerie simply bids $15,000/2=$7,500 and wins the piece of art. The expected price paid by Valerie, with no other bidders present, is $________.. Suppose the owner of the artwork manages to recruit another bidder, Antonio, to the auction. Antonio is known to value the art piece at $12,000. The expected price paid by Valerie, given the presence of the second bidder Antonio, is $_______. .

Suppose that Lionel Messi is negotiating a contract with FC Barcelona. Messi has an offer from Real Madrid for $20 million a year. If he signs with FC Barcelona, they will earn $90 million in revenue from the signing. FC Barcelona's next best option is to sign Cristiano Ronaldo. They would earn $70 million from signing Ronaldo and would pay him a contract of $10 million. Messi's bargaining power is w = 1/2. a) What is the negotiated salary between Messi and FC Barcelona under Nash Bargaining? What is Messi's surplus and what is FC Barcelona's surplus? b) Due to an injury, FC Barcelona would only earn $50 million from signing Ronaldo but everything else remains the same. What is the negotiated salary between Messi and FC Barcelona under Nash Bargaining? What is Messi's surplus and what is FC Barcelona's surplus?

The count is three balls and two strikes, and the bases are empty. The batter wants to maximize the probability of getting a hit or a walk, while the pitcher wants to minimize this probability. The pitcher has to decide whether to throw a fast ball or a curve ball, while the batter has to decide whether to prepare for a fast ball or a curve ball. The strategic form of this game is shown here. Find all Nash equilibria in mixed strategies.