33. In Example 9 (page 644), Anneliese, Binh, and Charlie valued a stamp at $300, $200, and $100, respectively. Although it is a weakly dominant strategy for each player to bid his or her true valuation of the stamp under a Vickrey auction, there are many other equilibrium bids besides ($300, $200, $100). Determine which, if any, of the following bids are equilibrium bids for the stamp: ($250, $50, $50), ($150, $100, $50), and ($400, $299, $1).
33.
($250, $50, $50) is an equilibrium. Anneliese receives the stamp and pays $50. She cannot pay less for the stamp. If Binh or Charlie were to receive the stamp, then either would have to pay Anneliese’s bid of $250, which exceeds each of their values of the stamp. ($150, $100, $50) is not an equilibrium. If Anneliese’s and Charlie’s bids are held at their respective values, then Binh would prefer to bid $151; he would receive the stamp and pay $150. ($400, $299, $1) is an equilibrium. Anneliese receives the stamp and pays $299; she cannot pay less. If Binh or Charlie were to receive the stamp, then either would have to pay Anneliese’s bid of $400, which exceeds each of their values of the stamp.