Transcript Lect

Topic 2: Designing the “optimal auction”
• Reminder of previous classes: Discussed 1st price and 2nd
price auctions. Found equilibrium strategies. Saw that the
revenue in both auctions is the same.
• Questions:
– Is this by accident?
– Can we design an auction with a higher revenue?
• This topic: Find an auction with a better revenue.
– We will understand the difference between direct and
indirect auctions.
– We will see what is the influence of prices and of the
winner determination rule. Expect many surprises.
The Revelation Principle
• Problem: there are infinitely many possible auction formats, so
it is hard to go over all of them…
• Reminder: In a direct-revelation auction, the strategy space of a
player is simply to report her type (value).
THM: Given any auction format with equilibrium strategies s(),
there exists a direct-revelation auction for which truthfulness is
an equilibrium, with the same outcome, and the same prices.
– Remark: This holds for any type of equilibrium (dominant
strategies, Bayesian-Nash,...)
• The implication: we need only search for a direct-revelation
auction.
Proof (sketch)
New Auction
Input v
“Proxy”
Output s(v)
Original
Auction
Original
Output
• A proxy mimics the equilibrium strategy: if others are truthful,
player i would like to play si(vi), so she needs to declare vi.
• Examples:
– In the 1st price auction, the proxy will convert vi to [(n-1)/n] vi
– In the English auction, the proxy will take vi and will keep
“raising the hand” until the price reaches vi.
How to set prices?
• As we saw, there can be many ways to set prices. Which is best?
• Surprisingly, prices also don’t matter!!
THM (The Revenue Equivalence Theorem):
Suppose vi is drawn independently from some Fi(x), which is
continuous and strictly increasing on some interval [a,b].
Take any two auctions such that, in both auctions:
- The expected utility of a player with value a is the same.
- The probability of a player that declares v to win is the
same (assuming players play the equilibrium strategies).
Then these two auctions raise the same expected revenue (in
equilibrium) .
• The 1st and 2nd price auctions satisfy all conditions (why??), so
our analysis is an example to this general theorem.
How to set prices?
• For simplicity we will assume that a=0 and that the expected
utility of a player with value 0 is 0. In this case the theorem is:
THM (The Revenue Equivalence Theorem):
Suppose vi is drawn independently from some Fi(x), which is
continuous and strictly increasing on some interval [0,b].
Take any two auctions such that, in both auctions, the probability
of a player that declares v to win is the same (assuming players
play the equilibrium strategies).
Then these two auctions raise the same expected revenue (in
equilibrium) .
Proof (1)
• By the revelation principle, we need only check truthful auctions
(that get values, and output a winner and a price). Denote by:
– Oi(v) the probability (over v-i) that i wins when declaring v.
– Pi(v) the expected payment of I when she declares v.
– ui(v) = v·Oi(v) – Pi(v) = i’s expected profit when declaring v.
Proof (1)
• By the revelation principle, we need only check truthful auctions
(that get values, and output a winner and a price). Denote by:
– Oi(v) the probability (over v-i) that i wins when declaring v.
– Pi(v) the expected payment of I when she declares v.
– ui(v) = v·Oi(v) – Pi(v) = i’s expected profit when declaring v.
• Truthfulness in equilibrium is equivalent to the condition that,
for every v,v’ :
ui(v) = v·Oi(v) – E[i’s price when declaring v]
> v·Oi(v’) – E[i’s price when declaring v’]
= ui(v’)+(v-v’) Oi(v’)
Proof (1)
• By the revelation principle, we need only check truthful auctions
(that get values, and output a winner and a price). Denote by:
– Oi(v) the probability (over v-i) that i wins when declaring v.
– Pi(v) the expected payment of I when she declares v.
– ui(v) = v·Oi(v) – Pi(v) = i’s expected profit when declaring v.
• Truthfulness in equilibrium is equivalent to the condition that,
for every v,v’ :
ui(v) = v·Oi(v) – E[i’s price when declaring v]
> v·Oi(v’) – E[i’s price when declaring v’]
= ui(v’)+(v-v’) Oi(v’)
• By exactly the same argument:
ui(v’) > ui(v) + (v’-v) Oi(v)
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
dv Oi(v) > ui(v) - ui(v’) > dv Oi(v’)
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
dv Oi(v) > ui(v) - ui(v’) > dv Oi(v’)
Oi(v’+dv) > [ui(v’+dv) - ui(v’)]/dv > Oi(v’)
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
dv Oi(v) > ui(v) - ui(v’) > dv Oi(v’)
Oi(v’+dv) > [ui(v’+dv) - ui(v’)]/dv > Oi(v’)
Letting dv approach zero => dui/dv = Oi(v)
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
dv Oi(v) > ui(v) - ui(v’) > dv Oi(v’)
Oi(v’+dv) > [ui(v’+dv) - ui(v’)]/dv > Oi(v’)
Letting dv approach zero => dui/dv = Oi(v)
=> ui(v) = 0 Oi(x)dx
v
Proof (2)
• Denote dv = v-v’. Rewriting what we have:
ui(v) > ui(v’)+dv Oi(v’) ; ui(v’) > ui(v) - dv Oi(v)
dv Oi(v) > ui(v) - ui(v’) > dv Oi(v’)
Oi(v’+dv) > [ui(v’+dv) - ui(v’)]/dv > Oi(v’)
Letting dv approach zero => dui/dv = Oi(v)
=> ui(v) = 0 Oi(x)dx
v
• We get Pi(v) = v·Oi(v) - ui(v) = v·Oi(v) - 0 Oi(x)dx
Thus Pi(v) depends only on the outcome function Oi(·).
Since these are identical for both mechanisms, so is the revenue.
v
Picture
Oi(vi)
Pi(v) = v·Oi(v) - 0 Oi(x)dx
v
1
Oi(vi)
0
Pi(vi)
vi
b
vi
Proof (3)
• Since E[i’s price] = a f(v) Pi(v) dv , it follows that this too is
identical to both mechanisms.
• Since E[Total revenue] = i E[i’s price], the total expected
revenue is also identical.
b
Conclusion: To design a revenue-maximizing auction, the
only question is who will be the winner (in equilibrium).
(And since the player with highest value wins both in the 1st and 2nd
price auctions, their revenues are equal)
A characterization of truthfulness
• We saw that, to maximize revenue, we only need to search for
allocation rules Oi(·). But what allocation rules can be
implemented? I.e. for which allocation rules we can add prices
that will make the auction truthful?
THM: An auction is truthful if and only if
(i) Oi(·) is non-decreasing.
v
(ii) Pi(v) = v·Oi(v) – a Oi(x)dx
• Note that (i) and (ii) were obtained as part of the proof of the
revenue equivalence theorem.
• The necessity part (truthfulness implies (i)+(ii)) was shown as
part of the proof of the Revenue Equivalence theorem.
Proof of sufficiency
• Since ui(v) = v·Oi(v) – Pi(v) we have ui(v) = a Oi(x)dx
• Recall that truthfulness is equivalent to the condition
ui(v’) > ui(v) + (v’-v) Oi(v)
v
Proof of sufficiency
• Since ui(v) = v·Oi(v) – Pi(v) we have ui(v) = a Oi(x)dx
• Recall that truthfulness is equivalent to the condition
ui(v’) > ui(v) + (v’-v) Oi(v)
v
• So fix any v’,v.
•
If v’ > v then:
ui(v’) - ui(v) =
v’
> v Oi(v)dx
= (v’ – v) Oi(v)
v’
v Oi(x)dx
( by (ii) )
( by (i) )
Proof of sufficiency
• Since ui(v) = v·Oi(v) – Pi(v) we have ui(v) = a Oi(x)dx
• Recall that truthfulness is equivalent to the condition
ui(v’) > ui(v) + (v’-v) Oi(v)
v
• So fix any v’,v.
•
If v’ > v then:
ui(v’) - ui(v) =
v’
> v Oi(v)dx
= (v’ – v) Oi(v)
•
If v’ < v then:
v’
v Oi(x)dx
v
-v’ Oi(x)dx
ui(v’) - ui(v) =
v
> -v’ Oi(v)dx
= -(v – v’) Oi(v)
( by (ii) )
( by (i) )
( by (ii) )
( by (i) )
Prices and virtual valuation
• Let (v) = v - (1-F(v))/f(v). Call it the “virtual value” function.
Claim: Evi[Pi(vi)] = Evi[(vi)·Oi(vi)]
• Interpretation: The best revenue that the auctioneer can hope to
extract from a player is vi ·Oi(vi). The claim tells us that (in
expectation) it can extract less, at most (vi)·Oi(vi).
Prices and virtual valuation
• Let (v) = v - (1-F(v))/f(v). Call it the “virtual value” function.
Claim: Evi[Pi(vi)] = Evi[(vi)·Oi(vi)]
Proof: Recall that Pi(v) = v·Oi(v) – a Oi(x)dx. Therefore:
v
Evi[Pi(vi)] = a Pi(v)f(v)dv =
b
= a v·Oi(v)·f(v)dv - a f(v) a Oi(x)dx dv
b
b
v
= a v·Oi(v)·f(v)dv - a Oi(v)(1-F(v)) dv
b
b
= a f(v)·Oi(v)·[v - (1-F(v))/f(v)]dv
b
= a f(v)·Oi(v)· (v) dv = Evi[(vi)·Oi(vi)]
b
Explanation to
this in the next
slide
Explanation delayed from previous slide
Let G(v) = a Oi(x)dx. Define F(v)·G(v) = W(v). Then:
v
F(b)·G(b) = W(b) =a W’(v)dv =
b
= a F’(v)·G(v)dv + a F(v)·G’(v)dv =
b
b
= a f(v)·G(v)dv + a F(v)·Oi(v) dv =
b
b
= a f(v)· a Oi(x)dx dv + a F(v)·Oi(v) dv
b
=>
v
b
a f(v)· a Oi(x)dx dv = F(b)·G(b) - a F(v)·Oi(v) dv =
b
v
b
= 1· a Oi(v)dv - a F(v)·Oi(v) dv = a Oi(v)(1-F(v)) dv
b
b
b
What is the revenue of the auction?
• Define:
– Oi(v1…vn) the probability that player i wins when all the
declarations are v1…vn. Note that Oi(vi) = Ev-i[Oi(vi,v-i) ].
– Pi(v1…vn) the probability that player i wins when all the
declarations are v1…vn. Note that Pi(vi) = Ev-i[Pi(vi,v-i) ].
What is the revenue of the auction?
• Define:
– Oi(v1…vn) the probability that player i wins when all the
declarations are v1…vn. Note that Oi(vi) = Ev-i[Oi(vi,v-i) ].
– Pi(v1…vn) the probability that player i wins when all the
declarations are v1…vn. Note that Pi(vi) = Ev-i[Pi(vi,v-i) ].
• Conclusion: the expected revenue of the auction is:
Ev1…vn [i Pi(v1…vn)] =i Ev1…vn [Pi(v1…vn)] =i Evi[Ev-i[Pi(vi,v-i) ]]
= i Evi[Pi(vi)] = i Evi[(vi)·Oi(vi)] = i Evi[(vi)·Ev-i[Oi(vi,v-i) ]]
= i Ev1…vn [(vi)·Oi(v1…vn)] = Ev1…vn [i (vi)·Oi(v1…vn)]
What is the optimal auction?
Conclusion: the expected revenue of the any auction must be
Ev1…vn [i (vi)·Oi(v1…vn)]
• How to maximize this? Choose the winner to be the player with
the maximal virtual valuation, unless all virtual valuations are
negative, and then no one wins.
• Is this truthful? If and only if the virtual valuation function is
monotone.
A more convenient formulation
•
Since virtual valuations are non-decreasing, this means that
the player with the highest value has the highest marginal
valuation.
•
One exception: if (v) < 0 then no one wins. This happens
for every v < v* where v* solves v = (1-F(v))/f(v).
•
By the revenue equivalence theorem, this means that a 2nd
price auction with a reservation price v* is a revenue
maximizing auction.
Numerical example
VAL 1
VAL 2
96
52
36
94
58
12
29
47
88
2
87
96
44
11
38
5
52
50
51
64
18
78
39
23
38
24
33
19
78
64
84
79
27
72
82
46
16
14
1st price revenue
48
26
39
47
29
19
14.5
23.5
44
39
43.5
48
39.5
13.5
36
41
26
25
25.5
33
2nd price revenue
64
18
36
39
23
12
24
33
19
2
64
84
44
11
38
5
46
16
14
31.15789
optimal revenue
64
50
50
50
50
0
0
0
50
50
64
84
50
0
50
50
50
50
50
42.73684
Remark for asymmetric bidders
•
•
•
If player values are drawn from different distributions, all the
analysis we’ve done still holds.
However, the marginal valuation functions will be different
between players, and so we cannot use the convenient format
just described.
This also means that it is not necessarily the case that the
player with the highest value will win. This depends on the
shape of the virtual valuation functions.
Some problems with assumptions of the
revenue equivalence theorem
• Budget constraints: bidders may not be able to pay their value.
• Asymmetries in probability distributions - this by itself does
not violate the statement but it may violate the outcome
equivalence (1st price may not give the item to the bidder
with the highest value).
• Bidders may not be risk-neutral.
• Interdependent valuations: values of players are related in
some way.
Risk Neutrality in the Analysis
DFN: The strategies s1,…, sn are in Bayesian-Nash equilibrium if
for any i, vi, ai : Ev-i[ui(si(vi),s-i(v-i)] > Ev-i[ui(ai,s-i(v-i)]
But for the analysis of the 1st price auction, and for the proof of the
revenue equivalence theorem, we actually assume that the player
chooses a bid b to maximize his expected profit:
Ev-i[value - price | i bids b]
• Thus, a crucial assumption is:
utility from expectation = expected profit
– This is termed risk-neutral players.
– Thus, it is an assumption for the revenue equivalence theorem.
– But sometimes this is not true. For example, we might care
about the variance (smaller variance might be better).
Risk aversion in 1st and 2nd price auctions
• In a 2nd price auction, the dominant strategy is to bid
truthfully, so risk-aversion does not change anything.
– There is no expectation in the considerations of a player
-- dominant strategy maximizes the player’s utility, no
matter what the others are doing.
• In a 1st price auction, the equilibrium we calculated is a
Bayesian-Nash equilibrium, so risk-aversion might change
the picture.
Sensitivity to risk
• This is usually modeled by a von-Neumann – Morgenstern utility:
– We have a utility function ui: R -> R. The utility of a given
profit x ( = value minus price) is ui(x).
– The utility of a lottery p is the expected utility:
ui(p)= x p(x) ui(x)
– Thus players maximize the expectation of utility of profit. This
might be different than the expectation of profit.
Example 1 – risk neutral players
u(x)
u(x)=x
u(4)=4
u(1)=1
x
1
4
The lottery: with probability 0.5 a profit of 4, with probability
0.5 a profit of 1
u(0.5 · 4 + 0.5 · 1) = u(2.5)=2.5
0.5 u(4) + 0.5 u (1) = 0.5 · 4 + 0.5 · 1 = 2.5
Conclusion: the lottery and receiving the mean for sure are
equivalent for the player.
Example 2 – risk averse players
u(x)
u(x)=x
u(4)=2
u(1)=1
x
1
4
The lottery: with probability 0.5 a profit of 4, with probability
0.5 a profit of 1
u(0.5 · 4 + 0.5 · 1) = u(2.5)1.58
0.5 u(4) + 0.5 u (1) = 0.5 · 2 + 0.5 · 1 = 1.5
Conclusion: the lottery is worse than receiving the mean for sure.
Risk-aversion in first-price auctions
THM: Suppose that players are risk-averse, with the same utility
function u, and that values are drawn independently from the
same distribution F (with bounded support). Then the expected
revenue of the symmetric equilibrium of a first-price auction is
not smaller than the expected revenue of the second-price
auction.
Remarks:
(1) There is only one unique symmetric equilibrium.
(2) No asymmetric equilibria exist (unless the support is not
bounded).
[We will not prove anything]
The picture
Revenue of 1st price with risk-aversion
>
Revenue of 1st price with risk-neutrality
=
Revenue of 2nd price with risk-neutrality
=
Revenue of 2nd price with risk-aversion
There are examples where the revenue is strictly higher in a 1st
price auction, and examples where the revenue is equivalent.