Review of Simple Auction Cases

In a first‑price auction with two bidders whose valuations are uniformly distributed on ([0,1]), the Bayes‑Nash equilibrium bid is (b_i(v_i)=v_i/2).
In a second‑price auction the equilibrium strategy is truthful bidding, (b_i=v_i).
Despite the different bidding behavior, each bidder’s expected payment equals (\frac{v_i^{2}}{2}). As Ian Ball notes, “The rules of the auction are different, the bidding behavior is different, and somehow, these things offset in exactly the right way that we get exactly the same formula in both cases.”

Generalized Auction Framework

An auction is defined by three components:

• Bid sets (B_1,\dots,B_n) that each player can submit.
• An allocation rule (q) that maps any bid profile to a sub‑probability distribution over the item, allowing the seller to retain the good.
• A transfer rule (t) that specifies each player’s payment conditional on the bids.

The first‑price and second‑price auctions share the same allocation rule— the highest bidder wins—while their transfer rules differ. This separation of allocation and payment is crucial for the revenue equivalence argument.

Deriving the Revenue Equivalence Theorem

Revenue equivalence follows from the optimality of Bayes‑Nash equilibrium strategies. Let (b_i(v_i)) denote the equilibrium bid of type (v_i). The equilibrium condition requires that deviating to any other bid (b_i(v_i')) cannot raise expected utility.

Define the win probability (Q_i(v_i)=\Pr{b_i(v_i) \text{ wins}}) and the expected payment (T_i(v_i)). The bidder’s expected utility is

[ U_i(v_i)=v_i\cdot Q_i(v_i)-T_i(v_i). ]

Taking the first‑order condition with respect to (v_i) yields

[ \frac{dT_i(v_i)}{dv_i}=v_i\cdot \frac{dQ_i(v_i)}{dv_i}. ]

Integrating from the lowest possible valuation (v_{\text{lower}}) gives

[ T_i(v_i)=T_i(v_{\text{lower}})+\int_{v_{\text{lower}}}^{v_i} x\,g(x)\,dx, ]

where (g(x)=\frac{dQ_i(x)}{dx}). The boundary condition (T_i(v_{\text{lower}})=0) (the lowest‑type bidder pays nothing) pins down the expected payment uniquely. As Ball emphasizes, “Revenue equivalence is a comparison between an equilibrium of one auction and an equilibrium of another auction,” and the assumptions concern the equilibrium, not the auction rules themselves.

Applications

In an all‑pay auction every bidder pays his bid regardless of winning. Because the payment equals the bid, the expected payment function (T_i(v_i)) derived above can be set equal to the bid function, allowing us to solve for the equilibrium bidding strategy directly. Revenue equivalence thus provides a shortcut: once we know the win‑probability function for one auction, we can infer the bidding function for any other auction that satisfies the two key conditions (zero payment for the lowest type and allocation to the highest type).

Caveats

Revenue equivalence holds only for equilibria that meet the two conditions mentioned earlier. It does not imply that any auction format automatically yields the same revenue; the result depends on the specific equilibrium being considered. Different equilibria or violations of the boundary conditions can break the equivalence.