Auction Examples

Game Theory · Axiom Academy

EXAMPLE Auction Equilibrium Analysis Computing equilibrium strategies in first-price and second-price auctions Number of bidders: 2 (Bidder 1 and Bidder 2) Possible valuations: for each bidder i Probability distribution: P(v_i = 100) = P(v_i = 200) = 0.5 Information structure: Private values (each bidder knows only their own valuation) Auction formats: First-price sealed-bid and Second-price sealed-bid Beats any low-value opponent ( v_j = 100 bids b_L = 100 ) Ties with another high-value opponent (both bid 150 , win with probability ) Zero payoff if opponent has high value (loses the auction) Zero payoff if opponent has low value (ties, wins with probability , but pays full value) No profitable deviation: bidding higher risks negative payoff Bidding above value risks winning and paying more than the item is worth Bidding below value risks losing when you could have won profitably Truthful bidding maximizes expected payoff regardless of opponents' strategies Both high ( v_1 = v_2 = 200 ): Each wins with probability , pays 200 , expected payoff = 0 Both low ( v_1 = v_2 = 100 ): Each wins with probability , pays 100 , expected payoff = 0 One high, one low: High-value bidder wins, pays 100 , payoff = 100 Both low: Winner pays 100 , probability 0.25 , revenue = 100 Both high: Winner pays 150 , probability 0.25 , revenue = 150 One high, one low: High bidder wins, pays 150 , probability 0.50 , revenue = 150 Both low: Winner pays 100 (second bid), probability 0.25 , revenue = 100

This is the written version of the interactive lesson above. See the full Game Theory course.