Revenue equivalence
Revenue equivalence is a cornerstone theorem in auction theory stating that, under certain assumptions including risk-neutral bidders with independent private values drawn from continuous distributions, any auction mechanism that allocates the good to the highest-value bidder and provides zero expected utility to the lowest-type bidder will yield the same expected revenue to the seller, regardless of the specific format such as first-price sealed-bid, second-price sealed-bid, English, or Dutch auctions.[1] This principle, formalized as a corollary in Roger Myerson's seminal 1981 paper on optimal auction design, demonstrates that the seller's expected utility depends solely on the allocation probabilities and the baseline utilities for the lowest bidder types, rather than on the payment rules themselves.[1] Building on William Vickrey's earlier 1961 insights into bidding behavior, the theorem was independently generalized by John Riley and William Samuelson in the same year, highlighting its robustness across symmetric environments.[2] The theorem's key assumptions include stochastic independence of bidders' value estimates, additively separable and risk-neutral utilities for both bidders and the seller, and incentive-compatible mechanisms that satisfy individual rationality.[1] These conditions ensure that equilibrium bidding strategies adjust endogenously to equate revenues, as seen in standard auctions where the object is always awarded to the bidder with the highest signal.[2] However, revenue equivalence does not hold in asymmetric settings, with correlated values, risk-averse bidders, or multi-object auctions without adjustments, leading to divergences in expected payments and allocations.[2] Despite these limitations, the principle has profound implications for mechanism design, informing that revenue maximization often prioritizes efficient allocation over auction format, and it underpins Myerson's virtual valuation framework for identifying optimal mechanisms.[1] Empirically, field experiments on platforms like eBay have tested and largely supported the theorem's predictions in symmetric private-value contexts, though real-world frictions can introduce deviations.[3]Fundamentals
Notation
In auction theory, particularly for the revenue equivalence theorem, the notation for the independent private values (IPV) model defines v_i as the private valuation of bidder i for the single indivisible item, where i \in \{1, 2, \dots, n\} and n denotes the fixed number of risk-neutral bidders participating in the auction. These valuations v_i are drawn independently from a common continuous cumulative distribution function F(v) on a support [ \underline{v}, \overline{v} ] , with corresponding density f(v) = F'(v), allowing for general distributions such as uniform or others that are strictly increasing and atomless to facilitate equilibrium analysis.[4] The bidding strategy in equilibrium is represented by b_i(v_i), the bid submitted by bidder i as a monotonically increasing function of their valuation v_i, reflecting symmetric strategies under the assumption of identical information structures across bidders. The seller's expected revenue from the auction is denoted by R, which, under revenue equivalence, equals the expected value of the second-highest valuation among the bidders, \mathbb{E}[v_{(2)}], where v_{(2)} is the order statistic for the second-largest v_i.[4] For bidder utilities, the interim expected payoff for bidder i is given by u_i(v_i), the ex-ante expected utility conditional on v_i, often expressed in equilibrium as u_i(v_i) = \int_{\underline{v}}^{v_i} [F(t)]^{n-1} dt for the lowest type receiving zero payoff, capturing the bidder's surplus from participation. The symmetry assumption central to the model requires that all bidders face identical valuation distributions, drawn i.i.d. from F(v), ensuring that equilibrium bidding is symmetric and that the theorem's conditions on efficiency and participation hold uniformly.[4]Assumptions
The revenue equivalence theorem in auction theory applies specifically within the independent private values (IPV) model, where each bidder i has a private valuation v_i for the good, drawn independently from a common cumulative distribution function F that is continuous and strictly increasing on [ \underline{v}, \overline{v} ] with F(\underline{v}) = 0 (often normalized so \underline{v} = 0).[5] In this setup, each bidder knows only their own v_i, while the distribution F is common knowledge among all participants, ensuring symmetric information about the valuation process. This IPV framework assumes that valuations are purely private and unaffected by others' signals, distinguishing it from common value or affiliated value models.[5] Bidders are assumed to be risk-neutral, maximizing their expected utility u_i = (v_i - p_i) \Pr[\text{win}_i], where p_i is the expected payment conditional on winning and \Pr[\text{win}_i] is the probability of winning given their strategy. Symmetry among bidder types is required, meaning all bidders face identical valuation distributions F and have the same strategic incentives, which facilitates equilibrium analysis without heterogeneity in beliefs or preferences.[5] Additionally, there must be at least two bidders to ensure competition, as single-bidder scenarios collapse to posted-price mechanisms without equivalence implications. The theorem further presupposes an efficient allocation rule, under which the good is awarded to the bidder with the highest valuation with probability one (whenever a highest valuation exists), thereby maximizing social surplus.[5] A reservation price of zero is standard, implying the seller values the good at zero and imposes no minimum bid threshold, allowing the efficient rule to allocate without distortion from entry barriers. These conditions collectively ensure that expected revenues across qualifying auction formats are identical, as deviations—such as risk aversion or correlation in values—would violate the theorem's predictions.[5]Theorem Statement
The revenue equivalence theorem states that, under the standard assumptions of auction theory—such as independent private values drawn from symmetric continuous distributions, risk-neutral bidders, and symmetric equilibria—any two auction mechanisms that allocate the good to the bidder with the highest valuation in equilibrium and ensure that a bidder with the lowest possible valuation receives zero expected payoff will generate the same expected revenue for the seller. This result holds for any mechanisms satisfying these conditions, encompassing formats like first-price, second-price, English, and Dutch auctions.[1] The expected revenue in such mechanisms equals the expected value of the second-highest bidder's valuation, formally expressed as \mathbb{E}[v_{(2)}], where v_{(k)} denotes the k-th highest order statistic among n bidders' valuations v_1, \dots, v_n.[1] This equivalence arises because the mechanisms share identical allocation rules—awarding the item to the highest-valuation bidder—and impose the same boundary condition on bidder payoffs, such as zero utility for a bidder whose valuation equals the distribution's lower bound, ensuring that payment differences across formats do not affect overall expected revenue.[6]Basic Examples
Single-Item Auction Setup
In a canonical illustration of the revenue equivalence theorem, consider a seller auctioning a single indivisible item to two risk-neutral bidders, each with a privately known value v_i drawn independently from a uniform distribution on [0, 1].[7] The item is allocated to the bidder with the highest value, and the bidders' values are independent and identically distributed.[8] In the second-price sealed-bid auction format, each bidder's dominant strategy is to bid their true value, b(v_i) = v_i.[7] The winner pays the second-highest bid, ensuring truthful bidding regardless of the opponent's strategy.[8] In the first-price sealed-bid auction, the symmetric Bayesian Nash equilibrium bidding strategy is b(v_i) = \frac{1}{2} v_i, where each bidder shades their bid below their value to balance the trade-off between winning probability and payment conditional on winning.[7] The winner pays their own bid, and this linear strategy arises from the uniform value distribution and risk neutrality.[9] Under these assumptions, the expected revenue to the seller is \frac{1}{3} in both the second-price and first-price formats, as the theorem predicts identical outcomes for standard auctions satisfying the conditions of independent private values and allocation to the highest bidder.[7] This equivalence stems from the fact that both mechanisms yield the same expected payment from the second-highest value in the uniform case.[10]Revenue Comparison
In the single-item auction with n risk-neutral bidders having independent private values drawn from a uniform distribution on [0,1], the revenue equivalence theorem implies that the expected seller revenue is identical across standard auction formats satisfying the theorem's conditions.[11] To illustrate, consider the second-price auction, where the winner pays the second-highest bid (equal to the second-highest value, as bidders bid truthfully). The expected revenue is thus the expected second-highest value among n uniform [0,1] draws, which equals \frac{n-1}{n+1}. This can be derived by integrating the second-highest order statistic's density f(y) = n(n-1) y^{n-2} (1-y) over [0,1]: \int_0^1 y \cdot n(n-1) y^{n-2} (1-y) \, dy = n(n-1) \int_0^1 y^{n-1} (1-y) \, dy = n(n-1) \, B(n, 2) = \frac{n-1}{n+1}, where B(a,b) is the beta function.[11][12] In the first-price auction, the symmetric Nash equilibrium bidding strategy is b(v) = \frac{n-1}{n} v. The revenue is the expected highest bid, or \mathbb{E}[b(V_{(1)})] = \frac{n-1}{n} \mathbb{E}[V_{(1)}], where V_{(1)} is the highest order statistic with \mathbb{E}[V_{(1)}] = \frac{n}{n+1}. Thus, the expected revenue simplifies to \frac{n-1}{n} \cdot \frac{n}{n+1} = \frac{n-1}{n+1}, matching the second-price outcome exactly.[11] For n=2 bidders, both auctions yield an expected revenue of \frac{1}{3}, confirming the equivalence in this basic case:| Auction Type | Expected Revenue (n=2) |
|---|---|
| Second-Price | 1/3 |
| First-Price | 1/3 |