Linkage Principle

The Linkage Principle is a cornerstone theorem in auction theory, mechanism design, and business and economics theory, established by American economists Paul R. Milgrom and Robert J. Weber in their landmark 1982 Econometrica paper "A Theory of Auctions and Competitive Bidding". Whereas William Vickrey (1961), Roger Myerson (1981), and Riley and Samuelson (1981) proved the Revenue Equivalence Theorem under Independent Private Values (IPV)—showing that English ascending, second-price sealed-bid, and first-price sealed-bid auctions generate identical expected revenue when each bidder's valuation is statistically independent and private—Milgrom and Weber introduced the general symmetric interdependent-values model with affiliated signals. Under affiliation (positive statistical dependence across bidders' noisy value estimates), bidders face the winner's curse and shade their bids defensively. The Linkage Principle proves that any auction rule or information policy that links the winning bidder's final payment more tightly to variables affiliated with the bidder's private signal dampens the strategic return to signal misrepresentation, compresses bidder informational rents, and raises the seller's expected revenue:

From Independent Private Values to Interdependent Values

In real-world competitive bidding—such as offshore oil and gas lease auctions, mineral rights sales, spectrum auctions, corporate takeovers, and timber or fine-art auctions—a bidder rarely knows the monetary value of the item in isolation. Instead, each of n symmetric risk-neutral bidders (i = 1, ..., n) observes a scalar private signal S_i ∈ [s_L, s_H] (such as a proprietary seismic survey of an oil tract), while the seller or nature may also possess a vector of unobserved state variables X = (X₁, ..., X_m).

Milgrom and Weber (1982) formulated a unified valuation function V_i = u(S_i, S_−i, X) that is nondecreasing in all arguments and symmetric across the other n − 1 bidders' signals S_−i = (S_j)_j≠i. This general specification encompasses both classic polar benchmarks as special cases:

1. Pure Private Values: u(S_i, S_−i, X) = S_i, where each bidder cares only about their own signal and learning a rival's signal does not alter their valuation.
2. Pure Common Value: u(S_i, S_−i, X) = V(X), where the asset has an identical ex-post monetary worth to all bidders (such as the barrel volume of an underground oil reservoir) and each signal S_i is a noisy statistical estimate of X.

Mathematical Definition of Affiliation

To model positive informational correlation across signals (S₁, ..., S_n, X) without restricting distributions to Gaussian or parametric families, Milgrom and Weber introduced affiliation (equivalent to the multivariate total positivity of order 2, or MTP₂, condition of Karlin and Rinott, 1980). Let Z = (S₁, ..., S_n, X) have a positive, smooth joint probability density function f(z). For any two realizations z and z′, denote the componentwise maximum (join) by z ∨ z′ = (max(z₁, z₁′), ..., max(z_k, z_k′)) and the componentwise minimum (meet) by z ∧ z′ = (min(z₁, z₁′), ..., min(z_k, z_k′)). The random variables Z are affiliated if and only if for all z and z′:

Taking logarithms shows that affiliation is equivalent to supermodularity of the log-density ln f(z): for all indices i ≠ j, the cross-partial derivative satisfies:

Affiliation implies the Monotone Likelihood Ratio Property (MLRP) across every pair of variables and ensures a fundamental comparative-statics property: for any nondecreasing function g(Z), the conditional expectation E[g(Z) | S_i = s] is a nondecreasing function of s. Observing a higher value of one's own signal S_i shifts the conditional probability distribution of all rivals' signals S_−i and the underlying state X upward in the sense of first-order stochastic dominance.

The Winner's Curse and Strategic Bid Shading

When signals are affiliated and values are interdependent (∂u/∂S_j > 0 for j ≠ i), a naive bidder who bids based on the unconditional expectation E[V₁ | S₁ = s] suffers the winner's curse: winning the auction in a symmetric equilibrium reveals the adverse event that every rival j ≠ 1 observed a signal S_j < s. Defining Y₁ = max_j≠1 S_j as the highest order statistic among the n − 1 rival signals, affiliation implies:

Rational equilibrium bidders anticipate this adverse selection on the event of winning (Y₁ ≤ s) and condition their bids on the pivotal boundary expectation v(s, y) ≡ E[V₁ | S₁ = s, Y₁ = y] evaluated at y = s.

Differential Revenue Representation in Symmetric Auctions

Consider any standard auction mechanism in which a symmetric, increasing equilibrium exists, so that Bidder 1 wins the object whenever their private signal S₁ = s exceeds the highest rival signal Y₁. Suppose the auction rule determines the winner's expected payment W(b | s) conditional on winning with report b and true signal S₁ = s (where dependence on s arises because the price paid in equilibrium—such as the runner-up bid b(Y₁) in a second-price auction—depends on variables Y₁ that are affiliated with S₁ = s).

Let F_Y1(y | s) and f_Y1(y | s) denote the conditional cumulative distribution function and conditional density of Y₁ given S₁ = s. Bidder 1 maximizes expected payoff:

At a symmetric truth-telling equilibrium (b*(s) = s), the first-order condition ∂Π(b | s) / ∂b |_b=s = 0 yields the differential equation governing the equilibrium expected payment:

where W₁(b | s) ≡ ∂W(b | s) / ∂b is the sensitivity of the winner's expected payment to their own report b, and W₂(b | s) ≡ ∂W(b | s) / ∂s is the statistical linkage of the winner's expected payment to their true private signal s through affiliated external information.

By the chain rule, the total derivative of the equilibrium payment schedule w(s) ≡ W(s | s) with respect to s is:

Comparing Two Auction Formats: Proof of the Linkage Principle

Suppose Auction Format A and Auction Format B both allocate the object to the bidder with the highest signal and satisfy w^A(s_L) = w^B(s_L) = 0 for the lowest signal type. Subtracting the differential equation for Format B from that for Format A yields a linear first-order differential equation for the payment difference Δw(s) ≡ w^A(s) − w^B(s):

Because f_Y1(s | s) / F_Y1(s | s) > 0 and Δw(s_L) = 0, a standard differential inequality comparison lemma implies that if Format A exhibits a stronger statistical linkage to the true signal s than Format B—that is, if:

then Δw(s) = w^A(s) − w^B(s) ≥ 0 for all s. Integrating over the distribution of the winning signal s establishes that Format A generates higher expected revenue for the seller than Format B.

The economic mechanism rests on information asymmetry and incentive compatibility: in a first-price auction, a bidder with signal s earns informational rent by shading their bid (reporting b < s to pay a lower price). When the auction links the winner's price to an affiliated variable (such as the runner-up signal Y₁ in a second-price auction), a bidder who considers shading their report b below s knows that their own high true signal s still statistically predicts a high realization of Y₁ (W₂ > 0). Because underreporting b cannot lower the affiliated component W₂(s | s) of the price, the marginal gain from bid shading is reduced, inducing bidders to bid closer to their pivotal valuation in equilibrium.

First-Price vs. Second-Price Sealed-Bid Auctions

To apply the Linkage Principle to the first-price sealed-bid auction (FPA) and the second-price sealed-bid (Vickrey) auction (SPA), compare their conditional payment functions W(b | s):

1. First-Price Sealed-Bid Auction (FPA): The winning bidder pays their own deterministic bid b^FPA(b), which depends on the bidder's chosen report b and not on the realized signals of rival bidders. Therefore, conditional on winning, the statistical linkage to the true signal s is zero:

1. Second-Price Sealed-Bid Auction (SPA): In the symmetric equilibrium of the Vickrey second-price auction, each bidder submits b^SPA(s) = v(s, s) = E[V₁ | S₁ = s, Y₁ = s], and the winner pays the highest rival bid b^SPA(Y₁). Conditional on winning with report b (so Y₁ < b) and holding true signal S₁ = s, the winner's expected payment is:

Because b^SPA(y) = v(y, y) is increasing in y and Y₁ is affiliated with S₁, increasing the true signal s shifts the conditional distribution of Y₁ on [s_L, b] upward, implying:

By the Linkage Principle, the second-price auction yields higher expected seller revenue than the first-price auction (R_SPA ≥ R_FPA).

English Ascending vs. Second-Price Sealed-Bid Auctions

When n ≥ 3 bidders have interdependent values and affiliated signals, Milgrom and Weber (1982) proved that the English ascending-clock auction dominates the second-price sealed-bid auction in expected revenue (R_English ≥ R_SPA).

In an irrevocable-exit English ascending clock auction, as the price clock rises, the lowest-signal bidders drop out one by one at observable dropout prices p_n ≤ p_n−1 ≤ ... ≤ p₃. Because the equilibrium dropout functions are invertible, the remaining two active bidders publicly infer the realized signals Y_n−1, Y_n−2, ..., Y₂ of all n − 2 exited rivals before the runner-up (holding Y₁) drops out. Consequently, the price paid by the winner in the English auction is linked to the entire vector of all n − 1 rival order statistics (Y₁, Y₂, ..., Y_n−1):

Because the full order-statistic vector (Y₁, Y₂, ..., Y_n−1) carries stronger affiliated information about S₁ than Y₁ alone (W₂^English ≥ W₂^SPA ≥ W₂^FPA = 0), the complete Milgrom–Weber revenue hierarchy follows:

Honesty Is the Best Policy for the Seller

A second major implication of the Linkage Principle is Milgrom and Weber's Public Information Revelation Theorem: suppose the seller possesses a private signal X₀ (such as an independent geological appraisal, an engineering inspection report, or an authenticity certificate) that is affiliated with the bidders' signals (S₁, ..., S_n) and enters the valuation function nondecreasingly. Then committing to publicly and fully disclose X₀ prior to bidding raises the seller's expected revenue in a first-price auction, a second-price auction, and an English auction compared to concealing X₀.

When the seller publicly announces X₀, equilibrium bids b(s, X₀) become explicit functions of X₀. From the perspective of ex-ante expected payments prior to the realization of X₀, conditioning bids on the affiliated variable X₀ introduces an additional positive statistical linkage term ∂E[W(b, X₀) | S₁ = s] / ∂s > 0, compressing the informational rent that a high-signal bidder can extract. By the same logic, linking the winner's payment to ex-post contingent royalties (such as a percentage share of extracted oil revenue) raises expected seller revenue because realized output is affiliated with the bidder's private signal S_i (Riley, 1988).

Risk Aversion, Asymmetries, and Multi-Unit Failures

Subsequent research identified settings where the Linkage Principle's revenue ranking can reverse:

1. Risk-Averse Bidders: In a first-price auction, a risk-averse bidder insures against losing the auction by submitting higher bids closer to true valuation, whereas in a second-price or English auction with private values, risk aversion does not alter the dominant strategy. When risk aversion is strong relative to affiliation, a first-price auction can yield higher revenue than an English auction.
2. Asymmetric Bidders and Entry Deterrence: Klemperer (1998) showed that if one bidder holds an ex-ante advantage, an English ascending auction worsens the winner's curse for weaker entrants, deterring entry and potentially lowering revenue below that of a sealed-bid first-price auction.
3. Multi-Unit Auctions (Perry and Reny, 1999): Motty Perry and Philip J. Reny (1999) demonstrated that the Linkage Principle can fail in multi-unit auctions when bidders demand multiple units and possess multi-dimensional or affiliated signals.
4. Full Surplus Extraction via Crémer–McLean Mechanisms: While the Linkage Principle ranks standard auction formats, Jacques Crémer and Richard P. McLean (1985, 1988) showed in general mechanism design that if the seller knows the joint probability distribution of correlated signals and bidders are risk-neutral without limited liability, side-bet lotteries conditioned on rivals' reports can extract 100% of the first-best social surplus.

For his pioneering work on the Linkage Principle, interdependent-values auction theory, and simultaneous multiple-round spectrum auctions, Paul Milgrom was awarded the 2020 Nobel Memorial Prize in Economic Sciences jointly with Robert B. Wilson.

Why does the Revenue Equivalence Theorem fail in the Milgrom–Weber model?

The classic Revenue Equivalence Theorem requires two assumptions: (1) each bidder's valuation depends only on their own signal (private values), and (2) signals are statistically independent across bidders. In the Milgrom–Weber model, signals are affiliated (positively correlated) and values are interdependent (common value components). Consequently, auction formats that reveal more affiliated information during bidding reduce the informational rent of high-signal bidders and yield higher expected seller revenue.

How does the English ascending auction alleviate the winner's curse compared to a sealed-bid auction?

In a first-price sealed-bid auction, a bidder must guess what rivals know without seeing any rival actions, inducing the bidder to shade their bid against the fear that winning means every rival saw a low signal. In an English ascending auction, each bidder observes the prices at which other bidders drop out. This real-time public aggregation of rivals' private signals narrows uncertainty about the common value and enables remaining bidders to bid higher without fear of overpaying in isolation.

Why should an auctioneer disclose even unfavorable appraisal information before an auction?

Under the Public Information Revelation Theorem, the seller commits ex ante (before seeing the appraisal report X₀) to a policy of full disclosure. Although an unfavorable appraisal realization lowers bids in that instance, a favorable appraisal raises bids by more on average, because removing the informational wedge permanently reduces the defensive winner's-curse shading applied by high-signal bidders across all states of nature.

1. Milgrom, P. R., & Weber, R. J. (1982). "A Theory of Auctions and Competitive Bidding." Econometrica, 50(5), 1089–1122. https://doi.org/10.2307/1911865
2. Perry, M., & Reny, P. J. (1999). "On the Failure of the Linkage Principle in Multi-Unit Auctions." Econometrica, 67(4), 895–900. https://doi.org/10.1111/1468-0262.00056
3. Klemperer, P. (1998). "Auctions with Almost Common Values: The 'Wallet Game' and Its Applications." European Economic Review, 42(3–5), 757–769. https://doi.org/10.1016/S0014-2921(97)00123-200123-2)
4. Crémer, J., & McLean, R. P. (1988). "Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions." Econometrica, 56(6), 1247–1257. https://doi.org/10.2307/1913096
5. Milgrom, P. (2004). Putting Auction Theory to Work. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511813825