Market Design and Analysis Lecture 4

Market Design and Analysis

Lecture 4

Lecturer: Ning Chen (陈宁 )

Email: [email protected]

2

Last Lecture

Dominant strategy Nash equilibrium

First price auction No ε-Nash

Second price auction Yes Yes

Vickrey auction Yes Yes

3

Revenue Consideration

Question: For first price and second price auction, which generates more revenue?

Second price auction may look bad - the winner’s price may be much lower than his bid.

(An extreme once happened in a New-Zealand government-auction: One firm bid NZ$100,000 for a license, and paid the second-highest price of only NZ$6)

4


In Economics, the (revenue) analysis of auctions is usually based on the assumption that the valuations of the buyers are drawn from a known distribution (e.g. [0,1]).

5


Let’s consider a simple case – there are 2 buyers with values v1,v2 drawn independently and uniformly from [0,1].

For second price auction, they bid bi = vi, hence the expected revenue is E[min(v1,v2)].

For first price auction, what do they bid? How can we compute the expected revenue without knowing bi?

6

Bayesian Game

A Bayesian game has, in addition to a normal game, for each player i, a type space Ti. There is a probability distribution P generating the type for each player. Player i knows his (fixed) type ti∈Ti

only knows the probability distribution of types of other players

has a payoff functionui(s1,…,sn; ti): S1 x …x Sn x Ti R

7

Bayesian Game

In a Bayesian game, a strategy for player i is a function si(ti), i.e. si: Ti Si.

For a given type ti, si(ti) gives the strategy selected by player i.

a strategy profile s* = (s*1,…,s*n) is called a Bayesian Nash equilibrium if for each player i and his type ti, s*i(ti) is the solution of

where

8

Describe Auctions as Bayesian Games For each buyer i,

there is a privately know true value (i.e. type) vi (assume drawn uniformly at random from [0,1])

decide bi, a function of vi, as his submitted bid

For first price auction, his utility is ui(b1,…,bn;vi) = vi – bi if i has largest bid and is a winner,

ui(b1,…,bn;vi) = 0 otherwise.

For second price auction, his utility is ui(b1,…,bn;vi) = vi – maxj≠i if i has largest bid and is a winner,

ui(b1,…,bn;vi) = 0 otherwise.

9

Bayesian Nash of First Price Auction Claim. Bidding bi = vi (n-1)/n for each buyer i is a Bayesian

Nash equilibrium for first price auction.

Proof. Consider any buyer i, assume all other buyers use this strategy. For a bid b ≤ (n-1) / n, the probability of winning is (b∙n/(n-

1))n-1. So the expected utility is (vi-b) (b∙n/(n-1))n-1

Derivative w.r.t b gives – (b∙n/(n-1))n-1 + (vi-b)(n-1)bn-2(n/(n-1))n-1

which should equal zero Hence, –b + (vi-b)(n-1) = 0, which gives b = vi(n-1)/n.

10


Let’s consider a simple case – there are 2 buyers with values v1,v2 drawn independently and uniformly from [0,1].

For second price auction, they bid bi = vi, hence the expected revenue is E[min(v1,v2)].

For first price auction, assume they bid according to Bayesian Nash equilibrium, i.e. bi = vi / 2. Then the expected revenue is E[max(v1,v2)] / 2.

11

A Probability Fact

Fact. A non-negative random variable X satisfies

Proof. Let fX(z) be the density function of X. The we have

Using the fact that

we have

12

Revenue Consideration – Second Price By uniform distribution, Pr[vi < x] = x Pr[vi ≥ x] = 1-x.

Since v1,v2 are independent,

Pr[min(v1,v2) ≥ x] = Pr[v1 ≥ x & v2 ≥ x] = (1-x)2

Hence,

13

Revenue Consideration – First Price E[max(v1,v2)] = 2/3.

Hence, E[max(v1,v2)] / 2 = 1/3.

14

Vickrey Auction with Reservation Price r The Vickrey auction with reservation price r, VAr,

sells the item if any buyer bids above r. The price the winning buyer pays is the maximum of the second highest bid and r.

Consider the revenue obtained when VA0.5, i.e. r = 0.5

there are 2 buyers, and their values v1,v2 are drawn independently at random from the uniform distribution in [0,1].

15

Vickrey Auction with Reservation Price r Analysis:

Case 1. both v1,v2 < 0.5. No one wins and no revenue

Case 2. one bid < 0.5, one bid ≥ 0.5. Get revenue 0.5 Case 3. both v1,v2 ≥ 0.5. This is the same as the above

analysis except that the two variables are drawn uniformly from [0.5,1]. Thus, the expected revenue is 0.5+0.5/3 = 2/3

Therefore, the expected revenue is 0 * 1/4 + 0.5 * 1/2 + 2/3 * 1/4 = 5/12

16

Single-Parameter Agents

There are n agents, each desires some item or service.

Agent i’s value for receiving service is vi, and 0 otherwise.

The mechanism solicits sealed bids (b1,…,bn) from agents and computes an outcome, consisting of allocation x = (x1,…,xn)

price / payment p = (p1,…,pn) ≥ 0

The utility of agent i is ui(bi) = vi∙xi(bi) – pi(bi).

17

Surplus and Profit

Assume there is a cost c(x) of producing the outcome x.

The surplus (or social welfare) of an outcome is defined by

∑i vixi - c(x)

The profit of an outcome is defined by ∑i pi - c(x)

18

Examples

Single item auction:

Combinatorial auction: There are n various items and each agent i desires a bundle of items Si. Then

19

Truthful Mechanisms

We say a mechanism is truthful if truth-telling is a dominant strategy for every agent. That is, for any agent i, vi, bi and b-i,

ui(vi,b-i) ≥ ui(bi,b-i)

We assume all mechanisms are voluntary participation, i.e. no agent has negative expected utility for participation pi(bi) ≤ bi∙xi(bi).

20

Characterization of Truthful Mechanisms Theorem. A mechanism is truthful if and only if, for

any agent i and bids of other agents b-i fixed,

xi(bi) is a monotone non-decreasing

pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz

For a fixed monotone allocation rule x(∙), the payment rule p(∙) is also fixed. Hence, when specifying a truthful mechanism, we need only specify a monotone allocation rule.

21

Characterization of Truthful Mechanisms Proof. Assume the mechanism is truthful, i.e. for any value vi and

possible lies bi, vi ∙xi(vi) – pi(vi) ≥ vi ∙xi(bi) – pi(bi)

Consider two possible values z1 and z2 with z1 < z2, and two scenarios: vi = z1, bi = z2 z1 ∙xi(z1) – pi(z1) ≥ z1 ∙xi(z2) – pi(z2)

vi = z2, bi = z1 z2 ∙xi(z2) – pi(z2) ≥ z2 ∙xi(z1) – pi(z1)

Adding them gives z1 ∙ xi(z1) + z2 ∙ xi(z2) ≥ z1 ∙ xi(z2) + z2 ∙ xi(z1)

Hence, (z2 – z1)∙(xi(z2) – xi(z1)) ≥ 0 xi(z2) ≥ xi(z1)

This is true for any z2 > z1, thus xi(∙) is monotone.

22

Characterization of Truthful Mechanisms Next we will show that the payment must be


Given ui(bi) = vi ∙xi(bi) – pi(bi), consider its partial derivative w.r.t bi:

u’i(bi) = vi ∙x’i(bi) – p’i(bi)

Since truthfulness implies that ui(bi) is maximized at vi = bi, we have

0 = u’i(vi) = vi ∙x’i(vi) – p’i(vi)which holds for any value of vi. In particular, when vi = z,

p’i(z) = z ∙ x’i(z)

23

Characterization of Truthful Mechanisms Hence,

By voluntary participation assumption, pi(0) = 0, and we are done (for the first direction).

24

Characterization of Truthful Mechanisms Proof. The other direction. Assume that xi(bi) is

monotone and payment is pi(bi) = bi ∙xi(bi) – ∫0

bi xi(z) dz

Consider two possible values z1 and z2 with z1 < z2. Suppose vi = z2, we will show that agent i does not benefit by shading his bid down to z1.

(The other case when vi = z1 and agent i does not benefit by shading his bid up to z2 is left as an assignment.)

25

Characterization of Truthful Mechanisms The following figure shows xi(∙) is monotone by

assumption.

26

Characterization of Truthful Mechanisms

valuation vi ∙ xi(vi)

payment pi(vi) = vi ∙xi(vi) – ∫0vi xi(z) dz

utility ui(vi) = vi ∙ xi(vi) – pi(vi)

bidding bi = vi = z2

27

Characterization of Truthful Mechanisms

valuation vi ∙ xi(bi)

payment pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz

utility ui(bi) = vi ∙ xi(bi) – pi(bi)

bidding bi = z1

28

Characterization of Truthful Mechanisms The difference of utility when bidding bi = vi = z2 and

bidding bi = z1 < z2. (The monotonicity of the allocation implies this difference is always non-negative.)

Hence, the mechanism is truthful.

29

Characterization of Truthful Mechanisms Theorem. A mechanism is truthful if and only if, for

any agent i and bids of other agents b-i fixed,

xi(bi) is a monotone non-decreasing


30

Deterministic Truthful Mechanism For any deterministic mechanism xi(∙)∈{0,1},

monotonicity implies that xi(∙) is a step function.

If agent i wins, his payment is pi(vi) = vi ∙xi(vi) – ∫0

vi xi(z) dz = infz {z: xi(z) = 1}

31

Maximizing Surplus

Recall that the surplus is defined by ∑i vixi – c(x).

Consider an allocation rule that maximizes the surplus, i.e. given b, output allocation x maximizing ∑i bixi – c(x).

Theorem. The surplus maximizing allocation is monotone. Proof. Let S(v,x) = ∑i vixi – c(x). Let v’ be the vector of

values where v’i = vi + δ, and v’j = vj for j ≠ i. ThenS(v’,x) = ∑i v’ixi – c(x) = δ∙xi + ∑i vixi – c(x) = δ∙xi+S(v,x) Consider any x maximizing S(v,x) with the largest xi, then x

maximizes S(v’,x) as well. Thus, the value of xi is monotone.

32

Virtual Valuation and Surplus

Assume that the value vi of agent i is drawn independently at random from Fi. Let F = F1 x F2 x … x Fn

The cumulative distribution function for vi is Fi(z) = Pr[vi < z]

The probability density function for vi is fi(z) = F’i(z)

33

Virtual Valuation and Surplus

The virtual valuation of agent i with value vi is

Given value vi, virtual valuation Φi(vi), and allocation x, the virtual surplus is

34

Myerson’s Optimal Mechanism

Theorem. (Myerson’81) The expected profit of a mechanism is equal to its expected virtual surplus.

Myerson’s optimal mechanism outputs x to maximize the virtual surplus. Given bids b, compute the “virtual bids” b’i = Φi(bi).

Compute the surplus maximizing allocation x’ and corresponding payment p’.

Output x = x’ and p, where pi= bi ∙xi(bi) – ∫0bi xi(z) dz

35


Since maximizing-surplus allocation rule is monotone, Myerson’s optimal mechanism is truthful if and only if Φi(vi) is monotone in vi for any i.

The condition that Φi(vi) is monotone is consistent with the monotone hazard rate assumption, which is a common assumption in economics. (The hazard rate is define as f(z) / (1-F(z)).)

36


Lemma. The expected payment of an agent, as a function of their bid bi and with all other bids fixed b-i, satisfies

Proof. (drop the subscript i) The bid b is selected at random from distribution F with density function f. Hence,

37


Focusing on the 2nd term and switching the order of integration, we have

Rename z to b and factor x(b)f(b) out

38


Proof of Myerson’s Theorem. Since the values of all agents are independently distributed and

by linearity of expectation, we have

39

Interpretation of Myerson’s Mechanism Single item auction:

In single-item auctions, the surplus maximizing allocation gives the item to the agent with the highest value, unless the highest value is less than 0. (Usually the values of all agents are assumed to be at least 0.)

Myerson’s mechanism computes virtual surplus maximizing allocation, given virtual valuations, which can be negative. The mechanism, thus, allocates the item to the agent with the largest positive virtual valuation.

40

Interpretation of Myerson’s Mechanism Consider there are only two agents.

Agent 1 wins when Φ1(b1) ≥ max{Φ2(b2), 0}.

This is a deterministic allocation rule. Thus the payment is p1 = inf {b: Φ1(b) > Φ2(b2) and Φ1(b) > 0}

Suppose that F1 = F2 = F, then Φ1(z) = Φ2(z) = Φ(z) If agent 1 wins, his payment is

p1 = inf {b: Φ(b) > Φ(b2) and Φ(b) > 0}which is

p1 = inf {b: b > b2 and b > Φ-1(0)}

If agent 2 winsp2 = inf {b: b > b1 and b > Φ-1(0)}

41

Interpretation of Myerson’s Mechanism Theorem. The optimal single-item auction for agents

with values drawn independently from F is precisely the Vickrey auction with reservation price Φ-1(0).

For example, when F = [0,1], then the cumulative distribution function F(z) = z the density function f(z) = F’(z) = 1 Thus Φ(z) = 2z – 1 and the virtual valuations are

drawn uniformly from [-1,1] Hence, Φ-1(0) = 1/2 and the optimal mechanism for

two agents with uniform value on [0,1] is the Vickrey auction with reservation price 1/2.

42

Summary

Characterization of truthful mechanisms. Steps to design an optimal mechanism to maximize

the expected profit by Myerson’s optimal mechanism: valuation distribution of each agent virtual valuation and surplus allocation rule maximizing virtual surplus payment rule (according to the characterization

theorem) simplify allocation and payment rule.

43

Revenue Equivalence Theorem

Assume that there are n agents, and the true value vi of each agent i is drawn independently from the same distribution F(v) on [vmin,vmax] (i.e. F(vmin)=0 and F(vmax)=1) with density function f(v). Assume that f(v) > 0 for any v∈[vmin,vmax].

In a Bayesian Nash equilibrium, agent i submits his bid (a function of vi) according to the equilibrium assume that the probability that i wins is qi(vi)

assume that the expected payment of agent i is pi(vi)

The utility function of agent i is ui(vi) = vi∙qi(vi) – pi(vi)

44


Lemma. For any value v and v’, ui(v) ≥ ui(v’) + (v – v’)∙qi(v’)

Proof. Let v be the true value of agent i, thenui(v’) + (v – v’)∙qi(v’) = v’qi(v’) – pi(v’) + (v – v’)∙qi(v’)= vqi(v’) – pi(v’)≤ ui(v)

45


Given the lemma, ui(v) ≥ ui(v’) + (v – v’)∙qi(v’) when v is the true value, we have

ui(v) ≥ ui(v+dv) + (– dv)∙qi(v+dv)

when v+dv is the true value, we have ui(v+dv) ≥ ui(v) + (dv)∙qi(v)

Hence,

Taking the limit dv 0, we have

46


ui(vmin

)

expected

utility

the slope of the utility function is qi(v)

47


Now consider any two mechanisms that have the same ui(vmin) and the same qi(v) functions for all v and agent i.

As

they have the same utility function ui(v).

Hence, for any given value vi, agent i makes the same expected payment in these two mechanisms.

This implies the expected payment of agent i, across all possible values vi, is also the same.

Therefore, the two mechanisms have the same expected revenue.

48


Consider any mechanisms which always give the item to the highest-bid agent. give an agent of lowest possible value no chance of

any positive utility, i.e. ui(vmin) = 0.

Then qi(v) = (F(v))n-1, and all these mechanisms have the same expected revenue.

49


Revenue Equivalence Theorem. Assume that there are k items and n agents, where each agent desires

one item. the values are independently drawn from a common

distribution [vmin,vmax], which is strictly increasing

Then any mechanism in which the items always go to the k agents with the highest values, any agent with value vmin expects 0 utility,

yields the same expected revenue, and each agent i has the same expected payment.

50


The condition that f(v) > 0 for any v∈[vmin,vmax] is crucial for the theorem.

For example, there are two agents and their values are independently and equally either 0 or 1.

Consider the following mechanism: offer a price p to two agents simultaneously if only one accepts, then sell him at p if both accept, then choose a winner at random at price p if both reject, then choose a winner at random at price 0

Then ui(0) = 0 and the higher-bid always wins. But the revenue strictly increases in p, so Revenue Equivalence fails.

Market Design and Analysis Lecture 4

Documents

Transcript of Market Design and Analysis Lecture 4