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ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

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Page 1: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

ECO 426 (Market Design) - Lecture 8

Ettore Damiano

November 23, 2015

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 2: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 3: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N bidders

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 4: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 5: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Each vi is drawn independently from the same distribution F(e.g. U[0, 1])

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 6: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Each vi is drawn independently from the same distribution F(e.g. U[0, 1])

Theorem In any auction such that in equilibrium:

the winner with the highest valuation wins, andthe bidder with the lowest possible valuation pays nothing,

the average revenue are the same, and the average bidderprofits are the same.

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 7: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Each vi is drawn independently from the same distribution F(e.g. U[0, 1])

Theorem In any auction such that in equilibrium:

the winner with the highest valuation wins, andthe bidder with the lowest possible valuation pays nothing,

the average revenue are the same, and the average bidderprofits are the same.

DP, SP, FP and AP share the properties that

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 8: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Each vi is drawn independently from the same distribution F(e.g. U[0, 1])

Theorem In any auction such that in equilibrium:

the winner with the highest valuation wins, andthe bidder with the lowest possible valuation pays nothing,

the average revenue are the same, and the average bidderprofits are the same.

DP, SP, FP and AP share the properties that

equilibrium outcome is efficient (i.e. the highest value bidderwins the auction)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 9: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence

Model:

N biddersBidder i has valuation vi

Each vi is drawn independently from the same distribution F(e.g. U[0, 1])

Theorem In any auction such that in equilibrium:

the winner with the highest valuation wins, andthe bidder with the lowest possible valuation pays nothing,

the average revenue are the same, and the average bidderprofits are the same.

DP, SP, FP and AP share the properties that

equilibrium outcome is efficient (i.e. the highest value bidderwins the auction)a bidder with a 0 valuation pays nothing.

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 10: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 11: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 12: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

The solution b∗(v) is a function of v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 13: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

The solution b∗(v) is a function of v and satisfies the FOC

ub(b∗(v), v) = 0

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 14: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

The solution b∗(v) is a function of v and satisfies the FOC

ub(b∗(v), v) = 0

The value of the maximization problem is U(v) ≡ u(b∗(v), v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 15: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

The solution b∗(v) is a function of v and satisfies the FOC

ub(b∗(v), v) = 0

The value of the maximization problem is U(v) ≡ u(b∗(v), v)and by the chain rule of differentiation

U ′(v) = ub(b∗(v), v)b∗′(v) + uv (b∗(v), v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 16: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem

Consider a maximization problem

maxb

u(b, v)

where u() is differentiable in b and v

The solution b∗(v) is a function of v and satisfies the FOC

ub(b∗(v), v) = 0

The value of the maximization problem is U(v) ≡ u(b∗(v), v)and by the chain rule of differentiation

U ′(v) = ub(b∗(v), v)b∗′(v) + uv (b∗(v), v)

The envelope theorem says that

U ′(v) = uv (b∗(v), v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 17: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 18: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

maxb

vPr(win|b) − E[Payment|b]

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 19: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

maxb

vPr(win|b) − E[Payment|b]

In an auction the objective function is

u(b, v) = vPr(win|b) − E[Payment|b]

uv (b, v) = Pr(win|b)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 20: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

maxb

vPr(win|b) − E[Payment|b]

In an auction the objective function is

u(b, v) = vPr(win|b) − E[Payment|b]

uv (b, v) = Pr(win|b)

If b∗(v) is the equilibrium bidding strategy, the envelopetheorem says that the bidder expected profit U(v) satisfies

U ′(v) = Pr(win|b∗(v))

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 21: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

maxb

vPr(win|b) − E[Payment|b]

In an auction the objective function is

u(b, v) = vPr(win|b) − E[Payment|b]

uv (b, v) = Pr(win|b)

If b∗(v) is the equilibrium bidding strategy, the envelopetheorem says that the bidder expected profit U(v) satisfies

U ′(v) = Pr(win|b∗(v)) = Eq. Prob. value-v bidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 22: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Envelope theorem and auctions

A bidder with valuation v choosing to submit a bid b solves

maxb

vPr(win|b) − E[Payment|b]

In an auction the objective function is

u(b, v) = vPr(win|b) − E[Payment|b]

uv (b, v) = Pr(win|b)

If b∗(v) is the equilibrium bidding strategy, the envelopetheorem says that the bidder expected profit U(v) satisfies

U ′(v) = Pr(win|b∗(v)) = Eq. Prob. value-v bidder wins

Integrating

U(v)=U(0) +

∫ v

0Pr(win|v)dv

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 23: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 24: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 25: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 26: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 27: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))Both are identical across the four auction formats weconsidered

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 28: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))Both are identical across the four auction formats weconsidered

Critical assumptions

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 29: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))Both are identical across the four auction formats weconsidered

Critical assumptions

bidder know their own values (their values do not depend onothers private information)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 30: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))Both are identical across the four auction formats weconsidered

Critical assumptions

bidder know their own values (their values do not depend onothers private information)values are statistically independent

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 31: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Revenue equivalence theorem

U(v)=U(0) +

∫ v

0Pr(win|v)dv

A bidder expected profit only depends on

his probability of winning as function of his valuation (i.e.Pr(win|v))his expected profit when he has the lowest possible valuation(i.e. U(0))Both are identical across the four auction formats weconsidered

Critical assumptions

bidder know their own values (their values do not depend onothers private information)values are statistically independentbidders only care about their profit (i.e. payoff equals valuationminus price paid)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 32: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 33: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 34: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constant

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 35: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 36: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

In a second price auction:

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 37: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

In a second price auction:

the highest value bidder wins the object

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 38: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

In a second price auction:

the highest value bidder wins the objectequilibrium strategies are easily characterized (dominantstrategy)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 39: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

In a second price auction:

the highest value bidder wins the objectequilibrium strategies are easily characterized (dominantstrategy)bidders expected surplus and sellers revenue are easilycharacterized

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 40: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Using the Revenue equivalence theorem

The revenue equivalence theorem implies that: in any auctionwhere, in equilibrium, the highest valuation bidder wins theobject

the expected revenue to the seller is constantthe expected surplus to each bidder is constant

In a second price auction:

the highest value bidder wins the objectequilibrium strategies are easily characterized (dominantstrategy)bidders expected surplus and sellers revenue are easilycharacterized

Can use the bidder expected revenue characterization in asecond price auction to derive the (less obvious) equilibriumstrategies of other auctions

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 41: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 42: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 43: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuation

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 44: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 45: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 46: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 47: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 48: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 49: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auction

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 50: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 51: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 52: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 53: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)by the revenue equivalence theorem, his expected paymentupon winning must be the same as in a SP auction (i.e. v/2)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 54: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)by the revenue equivalence theorem, his expected paymentupon winning must be the same as in a SP auction (i.e. v/2)since the payment of the winner in a FP auction equals hisown bid, the equilibrium bid of a bidder with valuation v mustbe v/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 55: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction

Two bidders - valuations are independent draws from U[0, 1]Second price auction

Each bidder bids his valuationA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)the expected payment upon winning is v/2 (i.e. the expectedvaluation of his opponent, provided his opponent has avaluation smaller than v)

First price auctionSuppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)by the revenue equivalence theorem, his expected paymentupon winning must be the same as in a SP auction (i.e. v/2)since the payment of the winner in a FP auction equals hisown bid, the equilibrium bid of a bidder with valuation v mustbe v/2 (i.e. the equilibrium bidding strategy is b(v) = v/2.)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 56: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 57: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 58: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 59: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the object

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 60: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the objectEach bidder pays a price to the seller equal to his own bid

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 61: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the objectEach bidder pays a price to the seller equal to his own bid

What should bidders do?

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 62: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the objectEach bidder pays a price to the seller equal to his own bid

What should bidders do?

Suppose there is an equilibrium where the highest valuationbidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 63: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the objectEach bidder pays a price to the seller equal to his own bid

What should bidders do?

Suppose there is an equilibrium where the highest valuationbidder wins

use the revenue equivalence theorem to solve for the candidateequilibrium bidding strategies

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 64: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Each bidder submits a sealed bid

Bids are open

Bidder who submitted the highest bid wins the objectEach bidder pays a price to the seller equal to his own bid

What should bidders do?

Suppose there is an equilibrium where the highest valuationbidder wins

use the revenue equivalence theorem to solve for the candidateequilibrium bidding strategiesex-post verify that the strategies constitute an equilibrium (i.e.no bidder has any incentive to deviate)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 65: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 66: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 67: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 68: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 69: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)pays his own bid, b(v), regardless of whether he wins or not

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 70: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)pays his own bid, b(v), regardless of whether he wins or nothis expected profit is then

Prob(win|v) ∗ v − b(v) = v2 − b(v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 71: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)pays his own bid, b(v), regardless of whether he wins or nothis expected profit is then

Prob(win|v) ∗ v − b(v) = v2 − b(v)

in a second price auction has an expected profit of

v(v − v/2) = v2/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 72: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)pays his own bid, b(v), regardless of whether he wins or nothis expected profit is then

Prob(win|v) ∗ v − b(v) = v2 − b(v)

in a second price auction has an expected profit of

v(v − v/2) = v2/2

from the revenue equivalence theorem

v2 − b(v) = v2/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 73: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

All pay auction

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder winsA bidder with valuation v

wins with probability v (i.e. the probability his opponent valueis less than v)pays his own bid, b(v), regardless of whether he wins or nothis expected profit is then

Prob(win|v) ∗ v − b(v) = v2 − b(v)

in a second price auction has an expected profit of

v(v − v/2) = v2/2

from the revenue equivalence theorem

v2 − b(v) = v2/2

the equilibrium bidding strategy in an all pay auction must be

b(v) = v2/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 74: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 75: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 76: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 77: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

the highest bidder wins the object if bid > r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 78: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

the highest bidder wins the object if bid > rthe winner pays a price equal to the largest between thesecond highest bid and the reserve price r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 79: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

the highest bidder wins the object if bid > rthe winner pays a price equal to the largest between thesecond highest bid and the reserve price rExample 1: Two bids, 0.3 and 0.6, and reserve price r = 0.4.The high bidder wins and pays 0.4.

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 80: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

the highest bidder wins the object if bid > rthe winner pays a price equal to the largest between thesecond highest bid and the reserve price rExample 1: Two bids, 0.3 and 0.6, and reserve price r = 0.4.The high bidder wins and pays 0.4.Example 2: Two bids, 0.5 and 0.6, and reserve price r = 0.4.The high bidder wins and pays 0.5.

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 81: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Auction with a reserve price

A reserve price is a price below which the seller is not willingto give up the object

Second price auction with a reserve price r

the highest bidder wins the object if bid > rthe winner pays a price equal to the largest between thesecond highest bid and the reserve price rExample 1: Two bids, 0.3 and 0.6, and reserve price r = 0.4.The high bidder wins and pays 0.4.Example 2: Two bids, 0.5 and 0.6, and reserve price r = 0.4.The high bidder wins and pays 0.5.Example 1: Two bids, 0.3 and 0.36, and reserve price r = 0.4.Nobody wins, object remains with seller.

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 82: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 83: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 84: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 85: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 86: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 87: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 88: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 89: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

opponent value, v , if v > r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 90: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

opponent value, v , if v > r (happens with probability(v − r)/v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 91: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

opponent value, v , if v > r (happens with probability(v − r)/v)reserve price , r , if v ≤ r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 92: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

opponent value, v , if v > r (happens with probability(v − r)/v)reserve price , r , if v ≤ r (happens with probability r/v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 93: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Second price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

It is a dominant strategy to:

bid own valuation when v > rnot bid when v ≤ r (or bid own valuation)

A bidder with valuation v > r

wins with probability vwhen winning pays a price equal to:

opponent value, v , if v > r (happens with probability(v − r)/v)reserve price , r , if v ≤ r (happens with probability r/v)

expected payment when winning

(r/v) ∗ r + ((v − r)/v) ∗ (v + r)/2 = r + (v − r)2/(2v)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 94: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 95: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 96: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

A bidder with valuation v

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 97: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

A bidder with valuation v

does not bid if v ≤ r (dominant strategy)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 98: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

A bidder with valuation v

does not bid if v ≤ r (dominant strategy)bids b(v) = r + (v − r)2/(2v) if v > r (by the revenueequivalence theorem)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 99: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

First price auction with reserve price

Two bidders - valuations are independent draws from U[0, 1]

Suppose, in equilibrium, the highest valuation bidder wins

A bidder with valuation v

does not bid if v ≤ r (dominant strategy)bids b(v) = r + (v − r)2/(2v) if v > r (by the revenueequivalence theorem)

note that r + (v − r)2/(2v) is strictly increasing in v , so inequilibrium the highest value bidder wins

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 100: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 101: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 102: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted price

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 103: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted pricesell at price equal r if v > r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 104: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted pricesell at price equal r if v > rdo not sell otherwise

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 105: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted pricesell at price equal r if v > rdo not sell otherwise

Expected revenue is

Prob(v > r) ∗ r = (1 − r) ∗ r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 106: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted pricesell at price equal r if v > rdo not sell otherwise

Expected revenue is

Prob(v > r) ∗ r = (1 − r) ∗ r

monopolist’s revenue with demand function Q(p) = 1 − p

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 107: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

What reserve price maximizes the seller’s revenue?

Suppose there is just one bidder, with U[0, 1] valuation

reserve price is just a posted pricesell at price equal r if v > rdo not sell otherwise

Expected revenue is

Prob(v > r) ∗ r = (1 − r) ∗ r

monopolist’s revenue with demand function Q(p) = 1 − p

revenue maximizing reserve price r = 1/2

same as monopolist price

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 108: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 109: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 110: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 111: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 112: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r no impact on revenue

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 113: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

no impact on revenue

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 114: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

no impact on revenue

no impact on revenue

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 115: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

vl < r < vh

no impact on revenue

no impact on revenue

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 116: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

vl < r < vh

no impact on revenue

no impact on revenue

R increases by ε (probability 2r(1− r))

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 117: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

vl < r < vh

vl < r < vh < r + ε

no impact on revenue

no impact on revenue

R increases by ε (probability 2r(1− r))

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 118: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl vh

vl < vh < r

r + ε < vl < vh

vl < r < vh

vl < r < vh < r + ε

no impact on revenue

no impact on revenue

R increases by ε (probability 2r(1− r))

R decreases by r (probability 2εr)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 119: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl < vh < r

r + ε < vl < vh

vl < r < vh

vl < r < vh < r + ε

no impact on revenue

no impact on revenue

R increases by ε (probability 2r(1− r))

R decreases by r (probability 2εr)

Expected revenue change: ΔRevenue = ε2r(1 − r) − r2εr

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 120: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

Two bidders - independent U[0, 1] valuations

Compare the revenue from marginally increasing the reserveprice r to r + ε, across all possible pairs of valuations vl < vh

0 1r

r + ε

vl < vh < r

r + ε < vl < vh

vl < r < vh

vl < r < vh < r + ε

no impact on revenue

no impact on revenue

R increases by ε (probability 2r(1− r))

R decreases by r (probability 2εr)

Expected revenue change: ΔRevenue = ε2r(1 − r) − r2εr

must be zero at the optimal reserve price r∗ = 1/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 121: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 122: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bids

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 123: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenue

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 124: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 125: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 126: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 127: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theorem

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 128: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothing

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 129: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothingFor v ≥ r∗, solve for bidding strategy, b(v), using the revenueequivalence theorem

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 130: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothingFor v ≥ r∗, solve for bidding strategy, b(v), using the revenueequivalence theoremReserve price must be equal to b(r∗)

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 131: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothingFor v ≥ r∗, solve for bidding strategy, b(v), using the revenueequivalence theoremReserve price must be equal to b(r∗)

if it higher, the allocation rule is not the same as in thesecond price auction

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 132: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothingFor v ≥ r∗, solve for bidding strategy, b(v), using the revenueequivalence theoremReserve price must be equal to b(r∗)

if it higher, the allocation rule is not the same as in thesecond price auctionif it is lower a bidder with value r∗ would have an incentive tolower its bid

Ettore Damiano ECO 426 (Market Design) - Lecture 8

Page 133: ECO 426 (Market Design) - Lecture 8 · ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Ettore Damiano ECO 426 (Market Design) - Lecture 8

Optimal reserve price

With N > 2 bidders same argument applies

The revenue only depends on the highest two bidsSimilar calculation of impact on revenueOptimal reserve price remains r∗ = 1/2

First price, second price, ascending price and descending priceauctions all have the same optimal reserve price

Optimal reserve price in an all pay auction?

Use the revenue equivalence theoremBidders with valuation below r∗ bid nothingFor v ≥ r∗, solve for bidding strategy, b(v), using the revenueequivalence theoremReserve price must be equal to b(r∗)

if it higher, the allocation rule is not the same as in thesecond price auctionif it is lower a bidder with value r∗ would have an incentive tolower its bid

Homework: calculate the optimal reserve price with two bidders

Ettore Damiano ECO 426 (Market Design) - Lecture 8