<p>In an incentive-compatible mechanism,</p>
<p><span class="math-container">$$\max_{z_i} \{q_i(z_i) x_i - m_i(z_i)\}=q_i(x_i)x_i-m_i(x_i).$$</span></p>
<p>So once we know the mechanism is incentive compatible, we can forget about the global behavior.</p>
<p>Now, we have <span class="math-container">$U_i(x_i)=q_i(x_i)x_i-m_i(x_i)$</span> and <span class="math-container">$U_i(0)=q_i(0)0-m_i(0)=-m_i(0),$</span> so</p>
<p><span class="math-container">$$U_i(x_i) = U_i(0) + \int_{0}^{x_i} q_i(t_i) dt_i$$</span></p>
<p>translates to</p>
<p><span class="math-container">$$q_i(x_i)x_i-m_i(x_i) = -m_i(0) + \int_{0}^{x_i} q_i(t_i) dt_i,$$</span></p>
<p>from which</p>
<p><span class="math-container">$$m_i(x_i) = m_i(0) + q_i(x_i)x_i - \int_{0}^{x_i} q_i(t_i) dt_i$$</span></p>
<p>follows.</p>

CC BY-SA 4.0
2024-12-24T21:04:31.817Z
Thank you, that makes sense. Do you have any light to shed on the main question about why the result isn't vacuous?
Max
CC BY-SA 4.0
2024-12-24T21:19:43.903Z
Well, the formula depends only on $q_i$ and $m_i(0)$, so definitely not on all of $M$.
Michael Greinecker
CC BY-SA 4.0
2024-12-24T21:47:54.587Z
When I hear "revenue equivalence," I think: "Once you (the auctioneer) pick your allocation $Q$, you can choose any $M$ you want, but as long as the pair is incentive compatible, your revenue will be the same." But if we sum that $m_i(x_i)$ expression to get the auctioneer's revenue, we get a term that includes $\sum_i m_i(0)$, which the auctioneer could vary considerably by picking $M$. I think the theorem is saying, "Sure, but if IC holds, the optimal $x_i$ will cancel the profits out," but I don't see how that follows *from* the $m_i(x_i) = \dots$ expression. It feels like a new claim.
Max
CC BY-SA 4.0
2024-12-24T22:09:48.070Z
The point is exactly that you don't have much choice if you want the mechanism to be incentive compatible. Once you settled on the allocation rule, you can only change the entry fee.
Michael Greinecker
CC BY-SA 4.0
2024-12-30T15:10:06.730Z
So why not pick $M$ with $m_i(0) = 1000000$ to guarantee a high profit no matter how players bid?
Max
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2024-12-30T17:19:17.863Z
@Max One usually includes an individual rationality or participation constraint that rules that out.
Michael Greinecker
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2024-12-30T18:33:20.877Z
Yeah, that's kind of why I'm confused. If you add an assumption like "also, individual rationality holds and Q must award the item to the highest bidder," then the result becomes interesting, because now the theorem really says something about how much profit the auctioneer can make. But in the slides I linked, that discussion appears *after* the presentation of the theorem. I'm trying to make sense of this more general version of the theorem and whether it is useful other than the fact that it incorporates more familiar results as a special case.
Max
CC BY-SA 4.0
2024-12-30T18:41:49.123Z
What the result is useful for is different from.the question whether it is trivial.
Michael Greinecker
CC BY-SA 4.0
2024-12-30T19:00:43.813Z
The statement "all four-sided polygons are polygons" includes "all squares are polygons" as a special case, but the latter statement is much narrower than the former and could be proven directly using the definition of a polygon. And it would be a stretch to call the former the "generalized all-squares-are-polygons theorem" given that it includes shapes that are not squares.
Max
CC BY-SA 4.0
2024-12-30T19:00:46.827Z
Similarly, I know how to prove (directly) a revenue equivalence theorem for auctions with individual rationality and in which the item has to go to the highest bidder, but this is a much narrower result (in particular, $Q$ is deterministic). Why should I care about this more general result if its notion of "revenue" is so abstract that you have to add additional assumptions and take a special case to make sense of it?
Max
CC BY-SA 4.0
2024-12-30T19:13:02.817Z
What the theorem is useful for is a different question.
Michael Greinecker
CC BY-SA 4.0
2024-12-30T19:35:02.693Z
OK, then suppose that is the question I am asking.
Max
CC BY-SA 4.0
2024-12-24T20:56:46.683Z
2024-12-30T17:22:52.123Z
2024-12-30T17:22:52.123Z
Michael Greinecker
<p>I'm working through these <a href="https://ocw.mit.edu/courses/6-254-game-theory-with-engineering-applications-spring-2010/ce4daf120dcb728fc1fb128f6cd3b48f_MIT6_254S10_lec19.pdf" rel="nofollow noreferrer">slides from Asu Ozdaglar</a>, which give an exposition of the revenue equivalence theorem that is slightly more general than the examples I have studied before. However, the result she presents seems vacuous to me, and I am trying to obtain a deeper intuition.</p>
<p>In her notation, <span class="math-container">$x_i \geq 0$</span> is buyer <span class="math-container">$i$</span>'s private valuation, and a (direct) mechanism <span class="math-container">$(Q, M)$</span> maps stated valuations (bids) to probabilities of winning the object <span class="math-container">$Q_i(x)$</span> and payments <span class="math-container">$M_i(x)$</span> made by the buyers to the auctioneer.</p>
<p>Thus each player wants to maximize <span class="math-container">$Q_i(x) x_i - M_i (x)$</span>.</p>
<p>Ozdaglar also defines the helper functions <span class="math-container">$q_i(z_i)$</span> and <span class="math-container">$m_i(z_i)$</span>, which give the values of <span class="math-container">$Q_i(x)$</span> and <span class="math-container">$M_i(x)$</span> at <span class="math-container">$x_i = z_i$</span> in expectation over the other players' valuations (<span class="math-container">$x_{-i}$</span>), and
<span class="math-container">$$
U_i(x_i) = \max_{z_i} \{q_i(z_i) x_i - m_i(z_i)\}
$$</span>
which is player <span class="math-container">$i$</span>'s expected payoff when bidding optimally (i.e. in equilibrium).</p>
<p>On slides 11 and 12, Ozdaglar uses the second fundamental theorem of calculus to show that if the mechanism is incentive compatible, then
<span class="math-container">$$
U_i(x_i) = U_i(0) + \int_{0}^{x_i} q_i(t_i) dt_t
$$</span>
and writes:</p>
<blockquote>
<p>This shows that, up to an additive constant, the expected payoff to a buyer in an IC direct mechanism <span class="math-container">$(Q, M)$</span> depends only on the allocation rule <span class="math-container">$Q$</span>.</p>
</blockquote>
<p>I feel like the &quot;up to an additive constant&quot; bit renders this observation vacuous. The additive constant <span class="math-container">$U_i(0)$</span> is equal to <span class="math-container">$\max_{z_i} \{ - m_i(z_i)\}$</span> and thus plainly depends on <span class="math-container">$M$</span> (not just via <span class="math-container">$m_i(0)$</span>, but the entire curve of <span class="math-container">$m_i(z_i)$</span>). Thus, to me the above sentence reads as</p>
<blockquote>
<p>This shows that, up to an additive constant <strong>that depends on <span class="math-container">$M$</span>,</strong> the expected payoff to a buyer in an IC direct mechanism <span class="math-container">$(Q, M)$</span> depends only on the allocation rule <span class="math-container">$Q$</span>.</p>
</blockquote>
<p>which amounts to &quot;the expected payoff depends on <span class="math-container">$M$</span> and <span class="math-container">$Q$</span>,&quot; which is the entire mechanism.</p>
<p>What am I missing here?</p>
<p>I think that unpacking my misunderstanding above might help me make sense of Ozdaglar's statement of the revenue equivalence theorem on page 13, which is that</p>
<blockquote>
<p>If the direct mechanism is incentive compatible, then for all <span class="math-container">$x_i$</span>, the expected payment is <span class="math-container">$$m_i(x_i) = m_i(0) + q_i(x_i) x_i - \int_{0}^{x_i} q_i(t_i) dt_i.$$</span></p>
</blockquote>
<p>I don't see how this follows from the above. Where does <span class="math-container">$m_i(0)$</span> come from?</p>

<game-theory><auctions>
Filling in missing steps in a proof of the revenue equivalence theorem
Max
# Filling in missing steps in a proof of the revenue equivalence theorem

(Asked by: username_0 on 2024-12-24)

I'm working through these [slides from Asu Ozdaglar](https://ocw.mit.edu/courses/6-254-game-theory-with-engineering-applications-spring-2010/ce4daf120dcb728fc1fb128f6cd3b48f_MIT6_254S10_lec19.pdf), which give an exposition of the revenue equivalence theorem that is slightly more general than the examples I have studied before. However, the result she presents seems vacuous to me, and I am trying to obtain a deeper intuition.

In her notation, $x\_i \geq 0$ is buyer $i$'s private valuation, and a (direct) mechanism $(Q, M)$ maps stated valuations (bids) to probabilities of winning the object $Q\_i(x)$ and payments $M\_i(x)$ made by the buyers to the auctioneer.

Thus each player wants to maximize $Q\_i(x) x\_i - M\_i (x)$.

Ozdaglar also defines the helper functions $q\_i(z\_i)$ and $m\_i(z\_i)$, which give the values of $Q\_i(x)$ and $M\_i(x)$ at $x\_i = z\_i$ in expectation over the other players' valuations ($x\_{-i}$), and
$$
U\_i(x\_i) = \max\_{z\_i} \{q\_i(z\_i) x\_i - m\_i(z\_i)\}
$$
which is player $i$'s expected payoff when bidding optimally (i.e. in equilibrium).

On slides 11 and 12, Ozdaglar uses the second fundamental theorem of calculus to show that if the mechanism is incentive compatible, then
$$
U\_i(x\_i) = U\_i(0) + \int\_{0}^{x\_i} q\_i(t\_i) dt\_t
$$
and writes:

> This shows that, up to an additive constant, the expected payoff to a buyer in an IC direct mechanism $(Q, M)$ depends only on the allocation rule $Q$.

I feel like the "up to an additive constant" bit renders this observation vacuous. The additive constant $U\_i(0)$ is equal to $\max\_{z\_i} \{ - m\_i(z\_i)\}$ and thus plainly depends on $M$ (not just via $m\_i(0)$, but the entire curve of $m\_i(z\_i)$). Thus, to me the above sentence reads as

> This shows that, up to an additive constant **that depends on $M$,** the expected payoff to a buyer in an IC direct mechanism $(Q, M)$ depends only on the allocation rule $Q$.

which amounts to "the expected payoff depends on $M$ and $Q$," which is the entire mechanism.

What am I missing here?

I think that unpacking my misunderstanding above might help me make sense of Ozdaglar's statement of the revenue equivalence theorem on page 13, which is that

> If the direct mechanism is incentive compatible, then for all $x\_i$, the expected payment is $$m\_i(x\_i) = m\_i(0) + q\_i(x\_i) x\_i - \int\_{0}^{x\_i} q\_i(t\_i) dt\_i.$$

I don't see how this follows from the above. Where does $m\_i(0)$ come from?

**Tags:** game-theory, auctions

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## 1 answers

### Answer by username_1 on 2024-12-24. Score: 2

In an incentive-compatible mechanism,

$$\max\_{z\_i} \{q\_i(z\_i) x\_i - m\_i(z\_i)\}=q\_i(x\_i)x\_i-m\_i(x\_i).$$

So once we know the mechanism is incentive compatible, we can forget about the global behavior.

Now, we have $U\_i(x\_i)=q\_i(x\_i)x\_i-m\_i(x\_i)$ and $U\_i(0)=q\_i(0)0-m\_i(0)=-m\_i(0),$ so

$$U\_i(x\_i) = U\_i(0) + \int\_{0}^{x\_i} q\_i(t\_i) dt\_i$$

translates to

$$q\_i(x\_i)x\_i-m\_i(x\_i) = -m\_i(0) + \int\_{0}^{x\_i} q\_i(t\_i) dt\_i,$$

from which

$$m\_i(x\_i) = m\_i(0) + q\_i(x\_i)x\_i - \int\_{0}^{x\_i} q\_i(t\_i) dt\_i$$

follows.

> **Comments:**
> **username_0**: Thank you, that makes sense. Do you have any light to shed on the main question about why the result isn't vacuous?  
> **username_1**: Well, the formula depends only on $q_i$ and $m_i(0)$, so definitely not on all of $M$.  
> **username_0**: When I hear "revenue equivalence," I think: "Once you (the auctioneer) pick your allocation $Q$, you can choose any $M$ you want, but as long as the pair is incentive compatible, your revenue will be the same." But if we sum that $m_i(x_i)$ expression to get the auctioneer's revenue, we get a term that includes $\sum_i m_i(0)$, which the auctioneer could vary considerably by picking $M$. I think the theorem is saying, "Sure, but if IC holds, the optimal $x_i$ will cancel the profits out," but I don't see how that follows *from* the $m_i(x_i) = \dots$ expression. It feels like a new claim.  
> **username_1**: The point is exactly that you don't have much choice if you want the mechanism to be incentive compatible. Once you settled on the allocation rule, you can only change the entry fee.  
> **username_0**: So why not pick $M$ with $m_i(0) = 1000000$ to guarantee a high profit no matter how players bid?  
> **username_1**: @Max One usually includes an individual rationality or participation constraint that rules that out.  
> **username_0**: Yeah, that's kind of why I'm confused. If you add an assumption like "also, individual rationality holds and Q must award the item to the highest bidder," then the result becomes interesting, because now the theorem really says something about how much profit the auctioneer can make. But in the slides I linked, that discussion appears *after* the presentation of the theorem. I'm trying to make sense of this more general version of the theorem and whether it is useful other than the fact that it incorporates more familiar results as a special case.  
> **username_1**: What the result is useful for is different from.the question whether it is trivial.  
> **username_0**: The statement "all four-sided polygons are polygons" includes "all squares are polygons" as a special case, but the latter statement is much narrower than the former and could be proven directly using the definition of a polygon. And it would be a stretch to call the former the "generalized all-squares-are-polygons theorem" given that it includes shapes that are not squares.  
> **username_0**: Similarly, I know how to prove (directly) a revenue equivalence theorem for auctions with individual rationality and in which the item has to go to the highest bidder, but this is a much narrower result (in particular, $Q$ is deterministic). Why should I care about this more general result if its notion of "revenue" is so abstract that you have to add additional assumptions and take a special case to make sense of it?  
> **username_1**: What the theorem is useful for is a different question.  
> **username_0**: OK, then suppose that is the question I am asking.  

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