A $100 gift card is being auctioned off to Ann and Brandon. Ann has $99 while...

 

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A $100 gift card is being auctioned off to Ann and Brandon. Ann has $99 while Brandon has $200. Each bidder will separately write down how much they are willing to pay for the gift card. They cannot bid more money than what they have, and only whole number bids are permitted, i.e., bids can be $0, $1, $2, etc. The highest bidder wins the gift card and pays his/her bid. The lowest bidder gets nothing and pays nothing. If the two bidders tie, they each get the gift card with probability half, in which case the one who gets the card pays his/her bid. For example, if both bid $5, then Ann's average payoff would be 1 (99+1005) + Ann gets gift card with probability 0.5 1 -99 2 Brandon gets gift card with probability 0.5 while Brandon's average payoff would be (200 + 100 5) + Brandon gets gift card with probability 0.5 1 200 Ann gets gift card with probability 0.5 (8 points total) (a) Given a profile of pure strategies (8₁, 82), what is Ann's payoff U₁(S₁, S₂)? What is Brandon's payoff UB (S1, S2)? (2 points) (b) Show Ann does not have a strictly dominated strategy in this game. (1 point) (c) For Brandon, show that any bid SB ≥ 100 is a strictly dominated strategy. (1 point) (d) Is there a dominant strategy equilibrium? (1 point) (e) Derive Ann's (pure strategy) best response function BRA(SB) as a function of Bran- don's pure strategy SB. (1 point) (f) Derive Brandon's (pure strategy) best response function BRÂ(SA) as a function of Ann's pure strategy SA. (1 point) (g) Find all pure strategy Nash equilibria of this game. (1 points) A $100 gift card is being auctioned off to Ann and Brandon. Ann has $99 while Brandon has $200. Each bidder will separately write down how much they are willing to pay for the gift card. They cannot bid more money than what they have, and only whole number bids are permitted, i.e., bids can be $0, $1, $2, etc. The highest bidder wins the gift card and pays his/her bid. The lowest bidder gets nothing and pays nothing. If the two bidders tie, they each get the gift card with probability half, in which case the one who gets the card pays his/her bid. For example, if both bid $5, then Ann's average payoff would be 1 (99+1005) + Ann gets gift card with probability 0.5 1 -99 2 Brandon gets gift card with probability 0.5 while Brandon's average payoff would be (200 + 100 5) + Brandon gets gift card with probability 0.5 1 200 Ann gets gift card with probability 0.5 (8 points total) (a) Given a profile of pure strategies (8₁, 82), what is Ann's payoff U₁(S₁, S₂)? What is Brandon's payoff UB (S1, S2)? (2 points) (b) Show Ann does not have a strictly dominated strategy in this game. (1 point) (c) For Brandon, show that any bid SB ≥ 100 is a strictly dominated strategy. (1 point) (d) Is there a dominant strategy equilibrium? (1 point) (e) Derive Ann's (pure strategy) best response function BRA(SB) as a function of Bran- don's pure strategy SB. (1 point) (f) Derive Brandon's (pure strategy) best response function BRÂ(SA) as a function of Ann's pure strategy SA. (1 point) (g) Find all pure strategy Nash equilibria of this game. (1 points)