Affiliated Values and Auction Revenue Comparisons

Why Dependence Among Signals Matters

In an auction, one bidder’s information can affect what another bidder should infer about the object’s value. If bidders’ signals tend to move together, winning against the others is informative: a bidder may have received a high signal, but the rivals’ lower bids can be evidence that their signals—and perhaps the object’s value—are lower. The resulting adjustment is central to the winner’s curse in common-value auctions and to comparisons of expected seller revenue.

The relevant dependence condition is affiliation. It is stronger than simply saying that two signals have positive correlation. Under affiliation, observing higher values of some signals shifts the distribution of the remaining signals upward in a precise sense. That structure can make information disclosed during an auction, or by the seller before it, valuable to bidders and to the seller.

For two binary signals, this condition has a simple form. Writing \(p_{ij}=P(S_1=i,S_2=j)\), affiliation requires \(p_{11}p_{00}\geq p_{10}p_{01}\). This determinant condition captures positive dependence for the binary case. With continuous signals, affiliation is a joint-distribution restriction; positive correlation alone does not establish it.

Values, Signals, and Auction Information

A bidder’s private value depends on that bidder’s own information, such as \(V_i=v_i(S_i)\). An interdependent value can depend on several bidders’ signals, \(V_i=v_i(S_1,\ldots,S_n)\). A common value is a special case in which all bidders value the object at the same underlying amount, though they may observe different signals about it. In the common-value case, a bidder’s estimate should account for the information conveyed by winning, not just the bidder’s own signal.

Affiliation matters because it gives a directional implication for inference. Under standard regularity conditions, if signals are affiliated, the conditional distribution of other signals shifts upward as a bidder’s own signal increases. If values are also increasing in those signals, the bidder’s expected value tends to increase with the bidder’s own signal. These properties support monotone bidding and the revenue comparisons discussed below; affiliation by itself does not determine an auction’s revenue.

Why Disclosure Can Raise Expected Revenue

The linkage principle describes a channel through which disclosure can benefit the seller. Suppose the seller has information about the object or a signal related to bidders’ information. If that information is affiliated with bidders’ signals and values increase with the relevant signals, disclosing it can make bidders’ estimates more informative. The bidders then condition their bids on information that would otherwise remain hidden. In a competitive auction, this can raise expected seller revenue.

One way to see the logic is to consider a bidder’s expected value conditional on the bidder’s information and on a signal the seller could disclose. If that additional signal is affiliated with the bidder’s information and value, learning it changes the conditional distribution used to form the bid. Information can also reduce uncertainty about the signals of competing bidders, which matters when bidders’ values are interdependent. The revenue effect is not simply “more information means higher bids”: the result depends on how the information is related to value, how bidders use it, and the auction rules.

The principle is a comparison of expected revenue under specified conditions, not a guarantee that disclosure increases revenue in every realization. Its usual benchmark assumes risk-neutral bidders, a well-defined symmetric equilibrium, affiliated signals, increasing values, and comparable rules with no features such as entry differences that alter participation. Changing disclosure may also change the information available to bidders in ways not covered by the benchmark.

Comparing Auction Formats

The familiar symmetric independent private-value benchmark provides a useful contrast. With risk-neutral bidders whose values are independent and identically distributed, standard regularity conditions give revenue equivalence across common formats when they allocate the object efficiently and give the same expected payoff to a bidder with the lowest possible value. Affiliation changes the informational environment: when signals are dependent, one bidder’s behavior or observed dropout can convey information about another bidder’s value.

In the canonical symmetric affiliated-values comparison, with monotone equilibrium strategies and otherwise comparable rules, expected revenue is weakly higher in an English auction than in a second-price sealed-bid auction, which is weakly higher than in a first-price sealed-bid auction; a Dutch auction is strategically equivalent to the first-price auction in this benchmark. The English auction can reveal information as bidders drop out. That additional information can affect the remaining bidders’ estimates and bids. A second-price sealed-bid auction does not provide the same ongoing public revelation. A first-price bidder’s payment depends on the bidder’s own bid, creating a different incentive to shade it.

These are conditional rankings, not universal laws about labels on auction formats. The precise comparison depends on the value model and assumptions. For example, private values that are independent satisfy affiliation, but the informational advantage of an English auction may disappear when there is no relevant information to reveal. With asymmetric bidders, risk aversion, reserve prices, entry costs, or different participation, the ranking may not apply. Expected revenue comparisons must hold those features fixed or model their differences explicitly.

A Short Justification for the Binary Dependence Result

For two binary signals with positive cell probabilities, the affiliation condition \(p_{11}p_{00}\geq p_{10}p_{01}\) also implies the conditional-probability comparison used above. To verify this, write the two conditional probabilities over their respective marginal probabilities: $$ P(S_2=1\mid S_1=1)=\frac{p_{11}}{p_{10}+p_{11}},\qquad P(S_2=1\mid S_1=0)=\frac{p_{01}}{p_{00}+p_{01}}. $$

Because the denominators are positive, comparing the fractions is equivalent to cross-multiplying: $$ p_{11}(p_{00}+p_{01})\geq p_{01}(p_{10}+p_{11}) \quad\Longleftrightarrow\quad p_{11}p_{00}\geq p_{01}p_{10}. $$

Thus the binary affiliation inequality implies that observing \(S_1=1\), rather than \(S_1=0\), raises the conditional probability that \(S_2=1\). This is a discrete illustration of the upward-shifting information property used in auction reasoning; it is not, by itself, a proof of a revenue ranking.

Common Mistakes and Practical Limits

• Equating positive correlation with affiliation. Positive correlation is only a summary of dependence. Affiliation imposes a stronger joint-distribution condition; verify the relevant condition rather than inferring it from a correlation coefficient.
• Claiming that any disclosure raises revenue. The linkage principle requires a suitable relationship among the disclosed information, bidders’ signals, and values, together with equilibrium and comparability conditions. Unrelated or negatively linked information need not help the seller.
• Ignoring the information in winning. In common-value settings, a bidder who wins has generally beaten the other bidders’ signals. A value estimate that ignores this selection can be too high—the winner’s curse.
• Treating format rankings as unconditional. Revenue rankings can depend on symmetry, risk preferences, entry, reserves, and value structure. State which assumptions make the comparison valid.