Designing and analyzing auction mechanisms. Revenue equivalence.
Multi-session curriculum - substantial prior knowledge and complex material. Use mastery gates and deliberate practice.
Why do wildly different auction formats (first-price, second-price, all-pay, etc.) so often produce the same expected revenue—at least on paper? Auction theory explains when this “doesn’t matter” result is true, exactly what assumptions make it true, and what changes when those assumptions fail.
In the standard independent private-values, risk-neutral, single-item, single-parameter model, any two Bayesian incentive-compatible auctions with the same allocation rule (who wins for each bid/valuation profile) and the same payment normalization deliver the same expected payments for every bidder and thus the same expected seller revenue. This is the Revenue Equivalence Theorem. The key technical tool is the payment identity: once an allocation rule xᵢ(vᵢ) is fixed (monotone), incentive compatibility pins down payments up to a constant.
Auction theory is the study of mechanisms for allocating scarce resources—often a single item—to agents with private information, and of the strategic behavior those mechanisms induce.
At the core of modern auction theory is a mechanism-design viewpoint:
In a single-item auction with n bidders, each bidder i submits a bid bᵢ. The mechanism specifies:
Think of xᵢ(b) as the “who gets it” rule and pᵢ(b) as the “who pays what” rule. Together, they fully define the auction.
In the canonical model for revenue equivalence, each bidder i has a privately known valuation vᵢ for receiving the item. The bidder’s utility is quasi-linear:
uᵢ = vᵢ · xᵢ(b) − pᵢ(b).
This form matters because it lets us separate “value from allocation” from “cost of payment” and use calculus-like envelope arguments.
Revenue equivalence is powerful—but it is not magic. It relies on a carefully chosen environment:
1) Single item (or single-parameter allocation)
2) Independent private values (IPV)
3) Risk neutral
4) Bayesian setting
5) Incentive compatibility and equilibrium focus
Many real auctions look different on the surface:
Yet in the IPV, risk-neutral, symmetric setting with appropriate equilibrium behavior, expected revenue often coincides. Auction theory explains why: once you fix the probability with which each type wins, incentive compatibility forces a unique expected payment schedule up to an additive constant.
So auction theory is fundamentally about:
In this lesson, we’ll build toward the Revenue Equivalence Theorem (RET) as a central organizing principle—then connect it to concrete auction formats and to the limits of equivalence.
The key to understanding revenue equivalence is to stop thinking in terms of “first-price vs second-price” and start thinking in terms of (x, p).
A direct mechanism asks agents to report their type (here, valuation) directly. Formally:
An “ordinary” auction (like first-price) is an indirect mechanism: it asks for bids bᵢ, which are strategic actions rather than types. But the Revelation Principle says:
If an outcome can be achieved by some equilibrium of some mechanism, then there exists a direct mechanism that achieves the same outcome with truthful reporting.
So when proving general results like RET, it’s convenient to reason about direct, truthful mechanisms.
In a direct mechanism, bidder i with true value vᵢ chooses a report rᵢ to maximize expected utility:
Uᵢ(vᵢ; rᵢ) = E_{v_{−i}}[ vᵢ · xᵢ(rᵢ, v_{−i}) − pᵢ(rᵢ, v_{−i}) ].
The mechanism is Bayesian incentive compatible if truthful reporting is optimal:
Uᵢ(vᵢ; vᵢ) ≥ Uᵢ(vᵢ; rᵢ) for all vᵢ, rᵢ.
Define the interim allocation and interim payment (expectations over others’ values):
xᵢ(vᵢ) = E_{v_{−i}}[ xᵢ(vᵢ, v_{−i}) ]
pᵢ(vᵢ) = E_{v_{−i}}[ pᵢ(vᵢ, v_{−i}) ].
Then the interim expected utility under truthful reporting is:
Uᵢ(vᵢ) = vᵢ · xᵢ(vᵢ) − pᵢ(vᵢ).
This “interim” view is crucial: RET is a statement about expected payments, and these expected payments are pinned down by interim allocation probabilities.
In single-parameter settings, a foundational insight is:
Formally, xᵢ(vᵢ) must be non-decreasing in vᵢ.
Intuition:
This monotonicity is not just a “nice property.” In single-parameter domains, it is close to being the whole story.
Payments are only pinned down “up to a constant.” That constant is determined by a boundary condition such as:
Common choice:
Uᵢ(0) = 0.
With that normalization (and assuming vᵢ has a lower bound 0), we can solve for pᵢ(vᵢ) uniquely.
It helps to see how familiar auctions fit into xᵢ(b) and pᵢ(b) language:
| Auction format | Allocation xᵢ(b) | Payment pᵢ(b) | Notes |
|---|---|---|---|
| First-price | 1 if bᵢ is highest, else 0 | winner pays bᵢ | Indirect; equilibrium shading |
| Second-price (Vickrey) | 1 if bᵢ is highest, else 0 | winner pays 2nd-highest bid | Truthful in dominant strategies under IPV |
| All-pay | 1 if bᵢ is highest, else 0 | everyone pays bᵢ | Strong incentives; rent dissipation |
Notice: in all three, the allocation rule (deterministic) is the same—highest bid wins. What differs is the payment rule and therefore equilibrium behavior.
Revenue equivalence will say: if (in equilibrium) these mechanisms induce the same allocation probabilities for each type and satisfy the same normalization, then their expected revenues coincide.
At this point, we have the key objects:
Next we’ll derive the payment identity: the formula that makes revenue equivalence inevitable once you accept incentive compatibility and single-parameter structure.
Revenue equivalence is ultimately a consequence of one fact:
In single-parameter, quasi-linear environments, incentive compatibility forces expected utility to satisfy an envelope condition, which pins down payments from allocations.
We’ll go step by step and keep the “why” in view.
Suppose we fix an interim allocation rule xᵢ(vᵢ). Imagine you are the auction designer.
So the payment schedule pᵢ(vᵢ) is tightly constrained by incentive compatibility.
Let Uᵢ(v) be the maximum expected utility type v can obtain by choosing the best report:
Uᵢ(v) = max_{r} [ v · xᵢ(r) − pᵢ(r) ].
This is the maximum over a family of linear functions in v (each r defines a line with slope xᵢ(r)). The maximum of linear functions is convex and has a subgradient. Under standard regularity, we get an “envelope”:
Uᵢ′(v) = xᵢ(v) (almost everywhere).
Interpretation:
Start from the truthful interim utility:
Uᵢ(v) = v · xᵢ(v) − pᵢ(v).
Differentiate both sides with respect to v (where differentiable):
Uᵢ′(v) = xᵢ(v) + v · xᵢ′(v) − pᵢ′(v).
But envelope says Uᵢ′(v) = xᵢ(v). Therefore:
xᵢ(v) = xᵢ(v) + v · xᵢ′(v) − pᵢ′(v)
Cancel xᵢ(v) on both sides:
0 = v · xᵢ′(v) − pᵢ′(v)
So:
pᵢ′(v) = v · xᵢ′(v).
Now integrate from 0 to v:
pᵢ(v) − pᵢ(0) = ∫₀ᵛ t · xᵢ′(t) dt.
Integrate by parts to rewrite in the more standard “payment identity” form.
Let:
Then:
∫₀ᵛ t · xᵢ′(t) dt
= [ t · xᵢ(t) ]₀ᵛ − ∫₀ᵛ 1 · xᵢ(t) dt
= v · xᵢ(v) − ∫₀ᵛ xᵢ(t) dt.
So:
pᵢ(v) = v · xᵢ(v) − ∫₀ᵛ xᵢ(t) dt + pᵢ(0).
Equivalently, in utility form:
Uᵢ(v) = Uᵢ(0) + ∫₀ᵛ xᵢ(t) dt.
This is the central identity.
The term pᵢ(0) (or Uᵢ(0)) is a boundary condition:
In general, two mechanisms with the same xᵢ(v) can differ only by a constant shift in utilities/payments, as long as they remain feasible (e.g., no negative payments if not allowed).
Here is a standard version:
Revenue Equivalence Theorem (RET).
In an independent private-values, risk-neutral, single-parameter environment, consider two BIC mechanisms (x, p) and (x, p̃) that implement the same interim allocation rule xᵢ(vᵢ) for each bidder i and satisfy the same normalization (e.g., Uᵢ(0) = 0 for all i). Then for every i and every vᵢ:
pᵢ(vᵢ) = p̃ᵢ(vᵢ),
and therefore the expected seller revenue is the same:
E[ ∑ᵢ pᵢ(vᵢ) ] = E[ ∑ᵢ p̃ᵢ(vᵢ) ].
Once you accept:
1) Single-parameter + quasi-linear utility
2) Incentive compatibility
3) A fixed allocation rule xᵢ(vᵢ)
…the payment identity shows pᵢ(vᵢ) is determined up to a constant. With a shared boundary condition, the constant matches—so payments match.
RET is often paraphrased as “all standard auctions have the same revenue.” That’s too strong.
RET actually says:
So we still need to check:
When those match, the revenue comparison becomes trivial—because it is already fixed by x.
In the next section, we will connect this to concrete auction formats and show how to compute payments and revenues using the identity.
Revenue equivalence becomes practical when you can argue that different auction formats induce the same allocation probabilities and satisfy the same boundary condition.
Assume:
In many standard auctions, equilibrium behavior is such that the highest value bidder wins (monotone bidding strategies). Then the interim allocation probability for type v is:
x(v) = P(v is the highest) = P(v ≥ maxⱼ≠i vⱼ) = F(v)^{n−1}.
If we also normalize U(0) = 0, then payments are pinned down:
p(v) = v · F(v)^{n−1} − ∫₀ᵛ F(t)^{n−1} dt.
This formula is astonishingly general: it gives the expected payment of a type v bidder in any BIC mechanism (or equilibrium outcome) that allocates efficiently.
Expected seller revenue is:
Rev = E[∑ᵢ pᵢ(vᵢ)].
In symmetric settings, you can compute for one bidder and multiply by n:
Rev = n · E[p(V)] where V ∼ F.
You can also derive revenue via order statistics (expected second-highest value for second-price auctions), and RET guarantees agreement.
Thus both auctions share the same x(v) = F(v)^{n−1} and the same U(0) = 0, so their expected revenues match.
RET saves you from solving the first-price equilibrium bid function just to compute expected revenue (though solving it is still useful for other reasons).
All-pay auctions look very different: everyone pays.
But in the symmetric IPV model, the equilibrium allocation is still efficient (higher types bid more). If normalization aligns (lowest type gets zero expected utility), RET implies expected revenue matches again.
The distribution of payments differs (who pays and when), but expected seller revenue matches.
RET is fragile in the best possible way: it tells you exactly which modeling choices matter.
If bidders are risk-averse, first-price and second-price auctions no longer have the same revenue in general.
In common-value environments (oil drilling, spectrum with uncertain market size), bidders’ signals are correlated and values depend on others’ information.
If each bidder has a vector type (e.g., value for multiple items, or value and budget), there is no single monotone allocation curve x(v) that pins down payments.
RET is fundamentally a single-parameter phenomenon.
Even under IPV risk-neutrality, if auctions allocate differently, revenues can differ.
Example: a reserve price changes x(v): low types are less likely to win (possibly nobody wins). This can increase revenue by trading off efficiency for extracting surplus.
Two mechanisms with identical x(v) can still differ if:
RET requires a shared normalization like U(0)=0.
RET is not the end of auction theory; it’s the starting point.
Myerson’s optimal auction theory uses the same single-parameter machinery (monotonicity + payment identity) but optimizes expected revenue by choosing the allocation rule to maximize a virtual surplus objective.
RET teaches you this mindset:
This allocation-first view is one of the most important conceptual upgrades mechanism design offers.
Single bidder i in a direct BIC mechanism (interim view). Value v ∈ [0,1]. Suppose the interim allocation probability is x(v) = v (higher values win with higher probability). Normalize U(0)=0. Find p(v).
Start with the payment identity (with U(0)=0):
p(v) = v · x(v) − ∫₀ᵛ x(t) dt.
Plug in x(v)=v:
p(v) = v · v − ∫₀ᵛ t dt
= v² − [ t²/2 ]₀ᵛ
= v² − v²/2.
Simplify:
p(v) = v²/2.
Check utility:
U(v) = v · x(v) − p(v)
= v · v − v²/2
= v²/2.
Also U(v) = ∫₀ᵛ x(t) dt = ∫₀ᵛ t dt = v²/2, consistent.
Insight: Once x(v) is fixed and monotone, incentive compatibility determines payments (up to a constant). You can treat the mechanism as “choose x, then compute p.”
n bidders, i.i.d. values V ∼ Uniform[0,1], risk neutral. Consider any BIC auction that allocates the item to the highest valuation (efficient allocation) and satisfies U(0)=0. Compute the interim expected payment p(v) for a bidder of type v.
Efficient allocation implies a type v wins iff it is the highest among n values. So the interim allocation probability is:
x(v) = P(V_{−i} ≤ v) = F(v)^{n−1}.
For Uniform[0,1], F(v)=v. Thus:
x(v) = v^{n−1}.
Use the payment identity (U(0)=0):
p(v) = v · x(v) − ∫₀ᵛ x(t) dt
= v · v^{n−1} − ∫₀ᵛ t^{n−1} dt
= v^n − [ t^n / n ]₀ᵛ
= v^n − v^n/n.
Simplify:
p(v) = (1 − 1/n) v^n
= ((n−1)/n) v^n.
Insight: This p(v) must hold in expectation for every auction that (in equilibrium or truthfully) implements efficient allocation with the same normalization—first-price, second-price, all-pay, etc. The payment rule may differ ex post, but interim expected payments coincide.
Continue the previous setting: n bidders, i.i.d. Uniform[0,1], efficient allocation, U(0)=0. Compute expected revenue Rev = E[∑ᵢ p(Vᵢ)].
From the previous example, one bidder with value V has expected payment:
p(V) = ((n−1)/n) V^n.
Compute E[V^n] for V ∼ Uniform[0,1]:
E[V^n] = ∫₀¹ v^n dv
= [ v^{n+1}/(n+1) ]₀¹
= 1/(n+1).
Therefore:
E[p(V)] = ((n−1)/n) · 1/(n+1).
Total expected revenue is n times that (symmetry):
Rev = n · E[p(V)]
= n · ((n−1)/n) · 1/(n+1)
= (n−1)/(n+1).
Insight: You computed expected revenue without solving equilibrium bids in first-price (or handling second-highest order statistics in second-price). RET lets you compute revenue from the allocation rule alone.
An auction mechanism is fully described by allocation functions xᵢ(b) and payment functions pᵢ(b).
In the IPV, risk-neutral, single-parameter model, Bayesian incentive compatibility implies monotonicity: xᵢ(vᵢ) must be non-decreasing in vᵢ.
The payment identity: pᵢ(v) = v · xᵢ(v) − ∫₀ᵛ xᵢ(t) dt + pᵢ(0). A normalization like Uᵢ(0)=0 (often implying pᵢ(0)=0) removes the constant.
Revenue Equivalence Theorem: with the same interim allocation rule and the same normalization, all BIC auctions yield the same interim expected payments and the same expected revenue.
RET compares auctions through their induced allocation rule x(·), not through surface rules like “pay your bid” vs “pay second price.”
RET does not say all auctions always have equal revenue; it depends on assumptions (IPV, risk neutrality, single-parameter types) and on matching allocation rules and normalizations.
RET guides optimal design thinking: choose allocation to optimize the objective; payments follow from incentive compatibility.
Misstating RET as “first-price and second-price always have the same revenue” without checking assumptions (risk neutrality, IPV, efficient allocation in equilibrium, same participation normalization).
Forgetting the boundary condition: payments are determined only up to an additive constant unless you fix something like U(0)=0 or an explicit participation/outside option.
Mixing ex post payment rules with interim expected payments: RET is fundamentally about expected payments given types (interim), not necessarily pointwise equality of payments in each realized bid profile.
Assuming monotonicity is automatic: some allocation rules are not implementable; if x(v) is not non-decreasing, no incentive-compatible payment rule can fix it in a single-parameter setting.
Let values be i.i.d. with CDF F on [0, v̄], risk neutral, single item. Suppose an auction is BIC and allocates efficiently (highest value wins). (a) Write x(v). (b) Using U(0)=0, derive p(v) in terms of F.
Hint: Efficient allocation means you win iff all other n−1 values are ≤ v. Then use the payment identity p(v)=v x(v)−∫₀ᵛ x(t)dt.
(a) x(v) = P(V_{−i} ≤ v) = F(v)^{n−1}.
(b) With U(0)=0:
p(v) = v · F(v)^{n−1} − ∫₀ᵛ F(t)^{n−1} dt.
Assume n bidders with i.i.d. Uniform[0,1] values and efficient allocation. Using RET, compute the expected revenue for n=2 and n=3. Then compare to the expected second-highest value in each case.
Hint: Use Rev = (n−1)/(n+1) for Uniform[0,1]. For comparison, the expected second-highest order statistic for n bidders has a known formula: E[V_(n−1)] = (n−1)/(n+1).
Using Rev = (n−1)/(n+1):
In a second-price auction with truthful bidding, revenue equals the expected second-highest value. For Uniform[0,1], E[V_(n−1)] = (n−1)/(n+1), matching the above revenues.
Consider a BIC mechanism with interim allocation x(v)=v² on v∈[0,1]. (a) Compute p(v) under normalization U(0)=0. (b) What is U(v)? (c) Verify the envelope condition U′(v)=x(v).
Hint: Use p(v)=v x(v)−∫₀ᵛ x(t)dt and U(v)=∫₀ᵛ x(t)dt when U(0)=0.
(a) p(v) = v·v² − ∫₀ᵛ t² dt = v³ − [t³/3]₀ᵛ = v³ − v³/3 = (2/3) v³.
(b) U(v) = v x(v) − p(v) = v·v² − (2/3)v³ = v³/3. Also U(v)=∫₀ᵛ t² dt = v³/3.
(c) U′(v) = (d/dv)(v³/3) = v² = x(v).
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