Auction Mechanisms

In a first‑price auction each bidder shades the bid below the true valuation to avoid paying the full amount and ending with zero surplus. The optimal bid balances a higher winning probability against a lower profit margin. In a second‑price auction the bidder submits the true valuation; this bid influences only the chance of winning, while the price paid equals the second‑highest bid, making truthful bidding a weakly dominant strategy.

Graphical Analysis of Second‑Price Auctions

When a bidder bids above the true valuation, the probability of winning rises but the payment remains the opponent’s bid, so the expected utility does not improve. Bidding below the true valuation reduces the chance of winning without lowering the price paid when winning. Consequently, bidding truthfully maximizes expected payoff regardless of the opponent’s behavior.

Revenue Comparison with Uniform Distribution

Consider two bidders whose valuations are independently drawn from a uniform ([0,1]) distribution. The expected revenue in a first‑price auction equals the expected value of the highest bid, while the expected revenue in a second‑price auction equals the expected value of the second‑highest bid. Both expectations evaluate to (1/3), a result that can be visualized by placing valuation pairs on a unit square and noting the geometric symmetry of the regions that determine the winning and payment outcomes.

General SIPV Model

The symmetric independent private values (SIPV) framework assumes (n) bidders, each drawing a valuation (v) from the same distribution (F) with density (f). Valuations are private and independent. In a first‑price auction the equilibrium bidding function (\beta(v)) satisfies the differential equation
[ \beta'(v)G(v)+\beta(v)g(v)=v\cdot g(v), ]
where (G(v)=F(v)^{\,n-1}) is the CDF of the maximum opponent valuation and (g(v)=(n-1)F(v)^{\,n-2}f(v)) its density. Solving this equation yields (\beta(v)) as the expected highest opponent valuation conditional on being below the bidder’s own valuation.

Revenue Equivalence Theorem

The revenue equivalence theorem states that, under the SIPV assumptions and when bidders have the same risk preferences, the expected payment of a bidder with valuation (v) is identical across auction formats that award the item to the highest bidder and charge the winner the expected payment of the second‑price rule. Because each bidder’s expected payment matches, the auctioneer’s expected revenue is the same in first‑price, second‑price, and many other standard auction designs. This deep result explains why the uniform two‑bidder case yields the same (1/3) revenue for both formats.