Auction Theory Laboratory
Monte Carlo simulation of the four canonical auction formats with private-value bidders. Values are drawn uniformly from $20–$100. Compare seller revenue, allocative efficiency, and how bidder risk preferences reshape the outcome — including why platforms like eBay use a second-price (Vickrey-style) proxy rule.
Uniform [L,H] with N bidders: 2nd-price revenue ≈ L + (H−L)(N−1)/(N+1). Risk-neutral bidders make all four formats revenue-equivalent. α<1 (risk aversion) raises first-price bids — and thus seller revenue — while Vickrey revenue is strategy-proof and unchanged.
Share of rounds won by the highest-value bidder. All standard auctions with symmetric bidders allocate efficiently.
Truth-telling is a dominant strategy. In a Vickrey auction the winner pays the second-highest bid, so shading your bid can only make you lose an item you value — it can never lower the price you pay. eBay's proxy bidding system implements exactly this: you submit your maximum, and the proxy bids one increment up on your behalf.
Revenue Equivalence holds. With risk-neutral bidders and independent private values, all four formats yield the same expected revenue to the seller (check the bars above — they should match within Monte Carlo noise). So the seller is free to choose the format that is most robust to bidder mistakes and easiest to participate in.
Naive bidders self-destruct in first-price. Switch the strategy to “naive”: bidders who bid their full value in a pay-your-bid auction win at zero profit every time — the winner's curse. Second-price protects them.
Risk aversion flips the ranking. Drag α toward 0.2: risk-averse bidders shade less in first-price auctions to avoid losing, pushing its revenue above Vickrey's. That is the one setting where a seller strictly prefers sealed first-price.