Happy Pi Day! I have an economics midterm on Wednesday, so here is my attempt at studying.
1. Mechanisms
The idea is as follows.
- We have
people and a seller who wants to auction off a power drill.
- The
th person has a private value of at most
on the power drill. We denote it by
.
- However, everyone knows the
are distributed according to some measure
supported on
. (let’s say a Radon measure, but I don’t especially care). Tacitly we assume
.
Definition 1
Consider a game
played as follows:
We call the pair
a mechanism.
For experts: we require that each
and
is measurable. Right now this is not a very good definition, because there are no assumptions on what
and
look like. Nonetheless, we’ll give some examples.
2. Examples of mechanisms
In the auction that you’d see in real life, we usually set

which is the simple rule that the highest bidder gets the power drill with probability
; if there is a tie, we pick one of the highest bidders at random.
In all the examples that follow, for simplicity let’s take the case
, and call the two players Anna and Elsa. We assume that both Anna and Elsa the value they place on the drill is uniform between
. Finally, we give the auctioneer a name, say Hans.
2.1. First-price auction
The first-price auction is the one you’ve probably heard of: each player makes a bid and
-
, and
-
is defined by requiring the winner to pay their bid.
How do Anna and Elsa behave in this auction? Clearly no one will bid more than they think the drill is worth, because then they stand to lose utility if they happen to win the auction. But the truth is they actually will bid less than they think the drill is worth.
For concreteness, suppose Anna values the drill at
. It doesn’t make sense for Anna to bid more than
obviously. But perhaps she should bid
— save a dollar. After all, what’s the chance that Elsa would bid right between
and
? For that matter, Anna knows that Elsa has a
chance of valuing the drill at less than
. So if Anna bids
, she has at least a
of saving at least
; it makes no sense for her to bid her true
value.
It gets better. Anna knows that Elsa isn’t stupid, and isn’t going to bid
even if her true value is
. That is, Elsa is going to try and sell Hans short as well. Given this Anna can play the game of cheating Hans more aggressively, and so an ad infinitum.
Of course there’s a way to capture this idea of “I know that you know that I know that you know. . .”: we just compute the Nash equilibrium.
Proposition 2 (Nash eqilibirum of the first-price auction for
)
Consider a first-price auction where Anna and Elsa have values uniformly distributed in
. Suppose both Anna and Elsa bid
if they have value
. Then this is a Nash equilibrium.
Proof: Suppose Anna values the drill at
and wants to make a bid
. Elsa values the drill at
, and follows the equilibrium by bidding
. For a bid of
, Anna gets utility

The probability of winning with
is thus
(this the probability that
) so the expected utility is

Hence we see
maximizes Anna’s expected utility. 
The first-price auction is interesting because both players “lie” when bidding in the Nash equilibrium. For this reason we say that the first-price auction is not incentive compatible.
Just for interest, let’s compute how much money Hans is going to make off the drill in this equilibrium. The amount paid to him is equal to

To see this we had to use the fact that if two numbers in
are chosen at random, the expected value of the larger is
. Multiplying by
gives the answer: Hans expects to make
.
2.2. Second-price auction
The second-price auction is the other one you’ve probably heard of: each player makes a bid and
-
, and
-
is defined by requiring the winner to pay the smallest amount needed to win, i.e. the second highest bid.
The fundamental difference is that in a second-price auction, a player “doesn’t need to be shy”: if they place a large bid, they don’t have to worry about possibly paying it.
Another way to think about is as a first-price auction with the property that the winning player can retroactively change their bid, provided they still win the auction. So unlike before there is no advantage to being honest.
Indeed, the second-price auction is incentive compatible in a very strong sense: bidding your true value is the best thing to do regardless of whether your opponents are playing optimally.
Proposition 3 (Second-price auctions are incentive compatible)
In a second-price auction, bidding truthfully is a weakly dominant strategy.
Proof: Easy. Check it. 
Just for interest, let’s compute how much money Hans is going to make off the drill in this equilibrium. This time he amount paid to him is equal to

Here we had to use the fact that if two numbers in
are chosen at random, the expected value of the smaller is
. This might come as a surprise: the expected revenue is
in this auction too.
2.3. All-pay auction
The all-pay auction is like lobbying. Each player makes a bid, and
-
, and
-
is defined by requiring everyone to pay their bid, regardless of whether they win the power drill or not.
This is clearly not incentive compatible. In fact, the Nash equilibrium is as follows:
Proposition 4 (Nash equilibirum of the all-pay auction)
Consider a first-price auction where Anna and Elsa have values uniformly distributed in
. Suppose both Anna and Elsa bid
if they have value
. Then this is a Nash equilibrium.
Proof: Omitted, but a fun and not-hard exercise if you like integrals. It will follow from a later result. 
Just for interest, let’s compute how much money Hans is going to make off the drill in this equilibrium. This time he amount paid to him is equal to

Surprise — same value again! This is a very special case of the Revenue Equivalence Theorem later.
2.4. Extortion
We’ve seen three examples that all magically gave
as the expected gain. So here’s a silly counterexample to show that not every auction is going to give
as an expected gain. It blatantly abuses the fact that we’ve placed almost no assumptions on
and
:
-
, or any other
for that matter, and
-
is defined by requiring both Anna and Elsa to give
to Hans.
This isn’t very much an auction at all: more like Hans just extracting money from Anna and Elsa. Hans is very happy with this arrangement, Anna and Elsa not so much. So we want an assumption on our auctions to prevent this silly example:
Definition 5
A mechanism
is voluntary (or individually rational) if
for every
.
2.5. Second-price auction with reserve
Here is a less stupid example of how Hans can make more money. The second-price auction with reserve is the same as the second-price auction, except Hans himself also places a bid of
. Thus if no one bids more than
, the item is not sold.
For the same reason as in the usual second-price auction, bidding truthfully is optimal for each players. The cases are:
- If both Anna and Elsa bid less than
, no one gains anything.
- If both Anna and Elsa bids more than
, the higher bidder wins and pays the lower bid.
- If exactly one player bids more than
, that player wins and pays the bid for
.
So Hans suffers some loss in the first case, but earns some extra money in the last case (when compared to the traditional second-price auction.) It turns out that if you do the computation, then Hans gets an expected profit of

meaning he earns another
or so by setting a reserve price.
3. Direct mechanisms
As it stands,
might depend in complicated ways on the actual values
: for example in the first-price auction. We can capture this formalism as follows.
So in other words, I’m equipping the mechanism
with a particular Nash equilibrium
. This is not standard, but I think it is harder to state the theorems in a non-confusing form otherwise.
It’s important to realize the functions
,
,
carry much more information than just
,
,
. All three functions depend on the equilibrium strategy
, which in turn depend on both
and
. Moreover, all three functions also depend on
. Hence for example
actually depends indirectly on
as well, because the choice of
affects the resulting equilibrium
.
4. Equivalence of utility and payment
In what follows let
be any index.
We now prove that
Lemma 9 (Envelope theorem)
Assume
is a direct mechanism. Then
is convex, and

except for at most countably many points at which
is not differentiable.
Proof: Since
is an equilibrium, we have

i.e. there is no benefit in lying and bidding
rather than
.
First, let’s show that if
is differentiable, then
. We have that
![\displaystyle \lim_{h \rightarrow 0} \frac{u_i(x+h)-u_i(x)}{h} \ge \lim_{h \rightarrow 0} \frac{\left[ (x+h)q_i(x)-p_i(x) \right] - \left[ x \cdot q_i(x)-p_i(x) \right]}{h} = q(x).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clim_%7Bh+%5Crightarrow+0%7D+%5Cfrac%7Bu_i%28x%2Bh%29-u_i%28x%29%7D%7Bh%7D+%5Cge+%5Clim_%7Bh+%5Crightarrow+0%7D+%5Cfrac%7B%5Cleft%5B+%28x%2Bh%29q_i%28x%29-p_i%28x%29+%5Cright%5D+-+%5Cleft%5B+x+%5Ccdot+q_i%28x%29-p_i%28x%29+%5Cright%5D%7D%7Bh%7D+%3D+q%28x%29.+&bg=ffffff&fg=000000&s=0&c=20201002)
Similarly
![\displaystyle \lim_{h \rightarrow 0} \frac{u_i(x)-u_i(x-h)}{h} \le \lim_{h \rightarrow 0} \frac{\left[ x \cdot q_i(x)-p_i(x) \right] - \left[ (x-h)q_i(x)-p_i(x) \right]}{h} = q_i(x).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clim_%7Bh+%5Crightarrow+0%7D+%5Cfrac%7Bu_i%28x%29-u_i%28x-h%29%7D%7Bh%7D+%5Cle+%5Clim_%7Bh+%5Crightarrow+0%7D+%5Cfrac%7B%5Cleft%5B+x+%5Ccdot+q_i%28x%29-p_i%28x%29+%5Cright%5D+-+%5Cleft%5B+%28x-h%29q_i%28x%29-p_i%28x%29+%5Cright%5D%7D%7Bh%7D+%3D+q_i%28x%29.+&bg=ffffff&fg=000000&s=0&c=20201002)
This implies the limit, if it exists, must be
. We’ll omit the proof that
is differentiable almost everywhere, remarking that it follows from
being nondecreasing in
(proved later). 
Theorem 10 (Utility equivalence)
Let
be a direct mechanism. For any
,

Proof: Fundamental theorem of calculus. 
Theorem 11 (Payment equivalence)
Let
be a direct mechanism. For any
,

Thus
is determined by
up to a constant shift.
Proof: Use
and
. 
This means that both functions
and
are completely determined, up to a constant shift, by the expected allocation
in the equilibrium
.
5. Revenue equivalence
A corollary:
Corollary 12 (Revenue equivalence)
Let
be a mechanism. Then the expected revenue of the auctioneer, namely,

depends only on
and
.
Very often, textbooks will add the additional requirement that
or
, in which case the statements become slightly simpler, as the constant shifts go away.
Here are the two important corollaries, which as far as I can tell are never both stated.
Proof: In an incentive compatible situation where
we have

so
depends only on
and
up to a constant shift. Ditto for any other
. 
Proof: By the assumption on
we have

so it follows for example that

Once again
now depends only on
. 
As an application, we can actually use the revenue equivalence theorem to compute the equilibrium strategies of the first-price, second-price, and all-pay auctions with
players.
Proof: First, as we saw already the second-price auction has an equilibrium where everyone bids truthfully. In this case, the probability of winning is
and the expected payment when winning is
(this is the expected value of the largest of
numbers in
.) Now by revenue equivalence, we have

Now we examine the all-pay and first-price auction.
- We have
, i.e. in the equilibrium strategy for the all-pay auction, a player with type
pays on average
. But the payment in an all-pay auction is always the bid! Hence conclusion.
- We have
, and since in both cases the chance of paying at all is
, the payment if a player does win is
; hence the equilibrium strategy is to bid
.

6. Design feasibility
By now we have seen that all our mechanisms really depend mostly on the functions
, (from which we can then compute
and
) while we have almost completely ignored the parameters
,
,
which give rise to the mechanism
in the first place.
We would like to continue doing this, and therefore, we want a way to reverse the work earlier: given the functions
construct a mechanism
with those particular expected allocations. The construction need not even be explicit; we will be content just knowing such a mechanisms exists.
Thus, the critical question is: which functions
can actually arise? The answer is that the only real constraint is that
are nondecreasing.
Proof: First, we show that any
arising from a direct mechanism are nondecreasing. Indeed, if
then we have the inequalities

If we add these inequalities we recover
, which shows that
is nondecreasing. The second condition just reflects that
.
Conversely, suppose
are given nondecreasing function. We will construct an incentive compatible mechanism inducing them. First, define
as predicted by Revenue Equivalence. First, define
by
Then trivially
and
.
However, we haven’t specified a
yet! This is the hard part, but the crucial claim is that we can pick
: that is,
is an incentive compatible direct mechanism.
Thus we need to check that for all
and
that

We just do
since the other case is analogous; then the inequality we want to prove rearranges as

Since
is increasing, this is immediate. 
In particular, we can now state:
Corollary 17 (Individually Rational)
A direct mechanism
is voluntary if and only if

for every
.
Proof: Since
, the function
is nonnegative everywhere if and only if
. 
7. Optimal auction
Since we’re talking about optimal auctions, we now restrict our attention to auctions in which
for every
(hence the auction is voluntary).
Now, we want to find the amount of money we expect to extract for player
. Let’s compute the expected revenue given a function
and distribution
. Of course, we know that
![\displaystyle \mathbf E_{x_i}\left[ p_i(x_i) \right] = \int_{t=0}^{1000} p_i(t) \; d\mu_i](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+p_i%28x_i%29+%5Cright%5D+%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+p_i%28t%29+%5C%3B+d%5Cmu_i+&bg=ffffff&fg=000000&s=0&c=20201002)
but we also know by revenue equivalence that we want to instead use the function
, rather than the function
. So let’s instead use our theorem for the integral to compute it:
![\displaystyle \mathbf E_{x_i}\left[ p_i(x_i) \right] = p_i(0) + \mathbf E_{x_i}\left[ x_i \cdot q_i(x_i) \right] - \mathbf E_{x_i}\left[ \int_0^{x_i} q(t) \; dt \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+p_i%28x_i%29+%5Cright%5D+%3D+p_i%280%29+%2B+%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+x_i+%5Ccdot+q_i%28x_i%29+%5Cright%5D+-+%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+%5Cint_0%5E%7Bx_i%7D+q%28t%29+%5C%3B+dt+%5Cright%5D+&bg=ffffff&fg=000000&s=0&c=20201002)
Applying the definition of expected value and switching the order of summation,
![\displaystyle \begin{aligned} \mathbf E_{x_i}\left[ p_i(x_i) \right] &= p_i(0) + \int_{t=0}^{1000} t \cdot q_i(t) \; d\mu_i - \int_{x=0}^{1000} \int_0^{x} q(t) \; dt d\mu_i \\ &= \int_{t=0}^{1000} t \cdot q_i(t) \; d\mu_i - \int_{t=0}^{1000} q(t) \int_{x=t}^{1000} \int_0^{x} d\mu_i dt \\ &= \int_{t=0}^{1000} t \cdot q_i(t) \; d\mu_i - \int_{t=0}^{1000} q_i(t) \cdot \mu_i([t, 1000]) dt \\ &= \int_{t=0}^{1000} t \cdot q_i(t) \; d\mu_i - \int_{t=0}^{1000} q_i(t) \cdot \frac{1-\mu_i([0, t])} {\frac{d\mu_i}{dt}(t)} d\mu_i \\ &= \int_{t=0}^{1000} q_i(t) \left( t - \frac{1-\mu_i([0,t])} {\frac{d\mu_i}{dt}(t)} \right) d\mu_i \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Baligned%7D+%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+p_i%28x_i%29+%5Cright%5D+%26%3D+p_i%280%29+%2B+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+t+%5Ccdot+q_i%28t%29+%5C%3B+d%5Cmu_i+-+%5Cint_%7Bx%3D0%7D%5E%7B1000%7D+%5Cint_0%5E%7Bx%7D+q%28t%29+%5C%3B+dt+d%5Cmu_i+%5C%5C+%26%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+t+%5Ccdot+q_i%28t%29+%5C%3B+d%5Cmu_i+-+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q%28t%29+%5Cint_%7Bx%3Dt%7D%5E%7B1000%7D+%5Cint_0%5E%7Bx%7D+d%5Cmu_i+dt+%5C%5C+%26%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+t+%5Ccdot+q_i%28t%29+%5C%3B+d%5Cmu_i+-+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q_i%28t%29+%5Ccdot+%5Cmu_i%28%5Bt%2C+1000%5D%29+dt+%5C%5C+%26%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+t+%5Ccdot+q_i%28t%29+%5C%3B+d%5Cmu_i+-+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q_i%28t%29+%5Ccdot+%5Cfrac%7B1-%5Cmu_i%28%5B0%2C+t%5D%29%7D+%7B%5Cfrac%7Bd%5Cmu_i%7D%7Bdt%7D%28t%29%7D+d%5Cmu_i+%5C%5C+%26%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q_i%28t%29+%5Cleft%28+t+-+%5Cfrac%7B1-%5Cmu_i%28%5B0%2Ct%5D%29%7D+%7B%5Cfrac%7Bd%5Cmu_i%7D%7Bdt%7D%28t%29%7D+%5Cright%29+d%5Cmu_i+%5Cend%7Baligned%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
Thus, we define for the player
their virtual valuation to be the thing that we just obtained in this computation:
![\displaystyle \psi_i(t) = t - \frac{1-\mu_i([0,t])}{\frac{d\mu_i}{dt}(t)}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cpsi_i%28t%29+%3D+t+-+%5Cfrac%7B1-%5Cmu_i%28%5B0%2Ct%5D%29%7D%7B%5Cfrac%7Bd%5Cmu_i%7D%7Bdt%7D%28t%29%7D.+&bg=ffffff&fg=000000&s=0&c=20201002)
Thus
![\displaystyle \mathbf E_{x_i}\left[ p_i(x_i) \right] = \int_{t=0}^{1000} q_i(t) \cdot \psi_i(t) \; dt. \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathbf+E_%7Bx_i%7D%5Cleft%5B+p_i%28x_i%29+%5Cright%5D+%3D+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q_i%28t%29+%5Ccdot+%5Cpsi_i%28t%29+%5C%3B+dt.++%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002)
The virtual valuation
can be thought of as the expected value of the amount of money we extract if we give the object to player
when she has type
. Note it has the very important property of not depending on
.
Now, let’s observe that
![\displaystyle \begin{aligned} \sum_{i=1}^N \mathbf E_{x_i} \left[ p_i(x_i) \right] &= \sum_{i=1}^N \int_{t=0}^{1000} q_i(t) \psi_i(t) \; d\mu_i \\ &= \sum_{i=1}^N \int_{x_1=0}^{1000} \dots \int_{x_N=0}^{1000} Q_i(\vec x) \psi(x_i) \; d\mu_1 d\mu_2 \dots d\mu_N \\ &= \int_{x_1=0}^{1000} \dots \int_{x_N=0}^{1000} \sum_{i=1}^N \left( Q_i(\vec x) \psi(x_i) \right) \; d\mu_1 d\mu_2 \dots d\mu_N. \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Baligned%7D+%5Csum_%7Bi%3D1%7D%5EN+%5Cmathbf+E_%7Bx_i%7D+%5Cleft%5B+p_i%28x_i%29+%5Cright%5D+%26%3D+%5Csum_%7Bi%3D1%7D%5EN+%5Cint_%7Bt%3D0%7D%5E%7B1000%7D+q_i%28t%29+%5Cpsi_i%28t%29+%5C%3B+d%5Cmu_i+%5C%5C+%26%3D+%5Csum_%7Bi%3D1%7D%5EN+%5Cint_%7Bx_1%3D0%7D%5E%7B1000%7D+%5Cdots+%5Cint_%7Bx_N%3D0%7D%5E%7B1000%7D+Q_i%28%5Cvec+x%29+%5Cpsi%28x_i%29+%5C%3B+d%5Cmu_1+d%5Cmu_2+%5Cdots+d%5Cmu_N+%5C%5C+%26%3D+%5Cint_%7Bx_1%3D0%7D%5E%7B1000%7D+%5Cdots+%5Cint_%7Bx_N%3D0%7D%5E%7B1000%7D+%5Csum_%7Bi%3D1%7D%5EN+%5Cleft%28+Q_i%28%5Cvec+x%29+%5Cpsi%28x_i%29+%5Cright%29+%5C%3B+d%5Cmu_1+d%5Cmu_2+%5Cdots+d%5Cmu_N.+%5Cend%7Baligned%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
Now, our goal is to select the function
to maximize this. Consider a particular point
. Then we’re trying to pick
,
, \dots,
to maximize the value of

subject to
. The
‘s here depend only on the distribution
. This is thus a convex optimization problem, so the solution is obvious: put all the weight on the biggest positive
, breaking ties arbitrarily. In other words, if
and
, then set
and all the other coefficients to be zero. (Ties broken arbitrarily as usual.) To be explicit, we set

So we should think of this as a second-price auction with discriminatory reserve prices. Moreover, the “second-price” payment is done based on
, so rather than “pay second highest bid” we instead have “pay smallest amount needed to win”, i.e. the smallest
such that
for all
.
Unfortunately, since the
are arbitrary the resulting
might be strange enough that the game fails to have a reasonable strategy
; in other words we’re worried the maximum might not be achievable. The easiest thing to do is write down a condition to handle this:
Definition 18
We say
is regular if the virtual valuations
are strictly increasing for all
.
Theorem 19 (Regularity implies optimal auction is achievable)
Assume regularity of
. Consider a second-price auction with discriminatory reserve prices: the reserve price for player
is the smallest
such that
, and the winner
pays the smallest amount needed to win.
This is an incentive compatible mechanism which maximizes the expected revenue of the auctioneer.
Proof: The
described in the theorem is the one we mentioned earlier. The hypothesis defines
as follows:
- If
is maximal, then that player wins and pays the smallest amount
such that
still exceeds all other
.
- Otherwise, the item is not sold.
The fact that this mechanism
is incentive compatible is more or less the same as before (bidding truthfully is a weakly dominant strategy). Moreover we already argued above that this allocation
maximizes revenue. 
You can and should think of this as a “reserve price second price auction” except with virtual valuations instead. The winner is the player with the highest virtual valuation, who is then allowed to retroactively change their bid so long as they still win.
To see this in action:
Corollary 20 (Optimal symmetric uniform auction)
Consider an auction in which
players have uniform values in
. Then the optimal auction is a second-price auction with reserve price
. The expected earning of the auctioneer is

Proof: We compute each
as
![\displaystyle \psi_i(x) = x - \frac{1 - \mu_i([0,x])}{\frac{d\mu_i}{dt}(x)} = x - \frac{1-x/1000}{1/1000} = 2x - 1000.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cpsi_i%28x%29+%3D+x+-+%5Cfrac%7B1+-+%5Cmu_i%28%5B0%2Cx%5D%29%7D%7B%5Cfrac%7Bd%5Cmu_i%7D%7Bdt%7D%28x%29%7D+%3D+x+-+%5Cfrac%7B1-x%2F1000%7D%7B1%2F1000%7D+%3D+2x+-+1000.+&bg=ffffff&fg=000000&s=0&c=20201002)
Hence, we usually want to award the item to the player with the largest virtual valuation (i.e. the highest bidder), setting the reserve price at
for everyone since
. By (1) the expected payment from a player equals

Simplifying and multiplying by
gives the answer. 
More generally, in an asymmetric situation the optimal reserve prices are discriminatory and vary from player to player. As a nice exercise to get used to this:
Exercise 21
Find the optimal auction mechanism if two players are bidding, with one player having value distributed uniformly in
and the other player having value distributed uniformly in
.
8. Against revelation
Here’s a pedagogical point: almost all sources state early on the so-called “revelation principle”.
Proof: Consider
. Then
is played as follows:
- Each player
has an advisor to whom it tells their value
- The advisor and figures out the optimal bid
in
, and submits this bid on the player’s behalf.
- The game
is played with the advisor’s bids, and the payments / allocations are given to corresponding players.
Clearly the player should be truthful to their advisor. 
Most sources go on to say that this makes their life easier, because now they can instead say “it suffices to study incentive compatible mechanisms”. For example, one can use this to
- Replace “direct mechanism” with “incentive-compatible direct mechanism”.
- Replace “there exists a direct mechanism
” with “there exists a
which is incentive compatible”.
However, I personally think this is awful. Here is why.
The proof of Revelation is almost tautological. Philosophically, it says that you should define the functions
,
,
in terms of the equilibrium
. Authors which restrict to incentive compatible mechanisms are hiding this fact behind Revelation: now to the reader it’s no longer clear whether
should be interpreted as taking a bid or value as input.
Put another way, the concept of a bid/value should be kept segregated. That’s why I use
only for values,
only for bids, and build in the equilibrium
as into
,
,
; So
is the expected utility of bidding
in the equilibrium. Revelation does the exact opposite: it lets authors promiscuously mix the concepts of values and bids, and pushes
out of the picture altogether.