Lecture 9:Auctions

Introduction to Game Theory

PreviewBayesian Games

Imperfect information about the state of the world, just believes

The last lectureAt least one player knows only his own typeAvery player consider expected payoffs given

his believesEquilibrium – every type of every player

cannot be better off by unilateral deviation given his believes

Today’s PlanNobody knows neither his own payoff

neither opponents’ payoffsOil tracts

Two oil drilling companies consider whether to drill oil on the tract or not. Both has just believes bout how rich the tract is.

AuctionsSecond-price sealed bid auctionFirst-price sealed bid auction

Too Much Information HurtsClassic economic theory of single person

decision problem: Player cannot be worse off if she has additional information.She can ignore the information.

Strategic game: If player has additional information and other players are aware of it, she might be worse off.

Example:Two states of the world and no player knows

which is the true one. Players just have believes over the states.

Too Much Information HurtsTwo states of the world: S1 and S2Players preferences over action profiles

Both players have believes over the probability of S1 and S2

Player considers expected payoffs for each action profile given her believes

S1 L M R

T 1,4 1,0 1,6

B 2,16 0,0 0,24

S2 L M R

T 1,4 1,6 1,0

B 2,16 0,24 0,0

Probability = ½ Probability = ½

Nobody KnowsIf P2 believes that P1 will choose T:

EP(L) = ½*4 + ½*4 = 4 EP(M) = ½*0 + ½*6 = 3

EP(R) = ½*6 + ½*0 = 3If P2 believes that P1 will choose B:

EP(L) = ½*16 + ½*16 = 16 EP(M) = ½*0 + ½*24 = 12

EP(R) = ½*24 + ½*0 = 12S1 L M R

T 1,4 1,0 1,6

B 2,16 0,0 0,24

S2 L M R

T 1,4 1,6 1,0

B 2,16 0,24 0,0

Probability = ½ Probability = ½

Everybody ExpectsExpected payoff of Player 1 (row)?All problem collapses to 2x2 game.

Equilibrium:(L,B)

P2P1

L M R

T 1,4 1,3 1,3

B 2,16 0,12 0,12

Expected payoff

Player Two KnowsAssume Player 2 knows the state of the world.Player 1 faces two types of Player 2 – “Left” and

“Right” type with same probabilities ½ and ½.

Player 2 of “Left” type plays R – dominates L and M

Player 2 of “Right” type plays M – dominates L and R

Player 1 has higher expected value from playing T

P2P1

L M R

T 1,4 1,0 1,6

B 2,16 0,0 0,24

P2P1

L M R

T 1,4 1,6 1,0

B 2,16 0,24 0,0

Too Much Information HurtsComparison of outcomes

No information about state:NE = (B,L) with payoffs: Player1gets 2 and Player2

gets 16Information about state:

NE = (T,(R,M))with payoffs: Player 1gets 1and Player 2gets 6 (both types)

How would be proceed if P1’s payoff is different across states? P2

P1L M R

T 1,4 1,0 1,6

B 2,16 0,0 0,24

P2P1

L M R

T 1,4 1,6 1,0

B 2,16 0,24 0,0

AuctionsAllocation of goods

Oil tractsArt worksTreasury bills

FormsSequential vs. Simultaneous

SubjectSingle unit vs. multiunit

InformationPrivate value vs. common value

AuctionsFirst-price sealed bid auction:

Bidders simultaneously hand their bids to the auctioneer. The individual with the highest bid wins, paying a price equal to the exact amount that he or she bid.

Second-price sealed bid auction:Bidders simultaneously hand their bids to the

auctioneer. The individual with the highest bid wins, paying a price equal to the amount of second highest bid submitted

AuctionsEnglish auction:

The price is steadily raised by the auctioneer with bidders dropping out once the price becomes too high. This continues until there remains only one bidder who wins the auction at the current price.

Dutch auction:The price starts at a level sufficiently high to

deter all bidders and is progressively lowered until a bidder indicates that he is willing to buy at the current price. He or she wins the auction and pays the current.

Second-Price sealed bidn bidders with publicly known valuation of

the objectv1>v2 > v3 > … > vn

Actions: Whatever nonnegative bidPayoffs: difference between the value and

the second highest bid in the case of winning and zero otherwiseTies are resolved by such that player with

higher valuation wins.Technical assumption.

Nash EquilibriumMany Nash Equilibria

(b1,b2,b3,…,bn)=(v1,v2,v3,…,vn)Everybody submits her valuation.

(b1,b2,b3,…,bn)=(vn,0,0,…,0,v1)Player with lowest valuation gets the object

(b1,b2,b3,…,bn)=(c, v1,c,…,c,c,c,…), where cv2Player i submits v1 and all other bids are not higher than her valuation.

Nash EquilibriumNE (b1,b2,b3,…,bn)=(v1,v2,v3,…,vn) is special.Submitting own valuation weakly

dominates any other bid.Assume Player i

bi<vi : payoff is same if bi is still higher than second highest bid and zero otherwise.bi>vi : if vi is highest bid then payoff does not change and if it is lower than highest bid then payoff is negative.

Seller’s revenue is v2 given that second highest valuation is v2.

Imperfect Informationn players P1,P2,P3,…PnActions: whatever nonnegative bidEvery player knows only her valuation of the

object.All players have believes about the valuation of

opponents - v is distributed such thatP(vx)=F(x), where x positive.

Ties are resolved by chance – All players who submit the highest bid have same chance to become winner.

Expected payoff – Probability of winning when submitting bi times (vi-b), where b is highest bid done by bidder different from winner.

Weak dominationSubmitting own valuation weakly dominates

any other bid.Consider player i. B is the highest bid by other playersBids lower than valuation vi:

Player i cannot be better off by bidding less than her valuation.

B<bi

B=bi bi<B<vi Bvi

bi<vi

vi-B (vi-B)/m 0 0

vi vi-B (vi-B) (vi-B) 0

i’s bid

Value of B

Weak DominationBids higher than valuation:

Player i cannot be better off by bidding over her valuation.

Whatever type Player i is he cannot do better by bidding vi.

Seller’s revenue is E[X|X<v] given that v is highest valuation among players.

Expected revenue of the seller is expected value of random variable E[X|X<v].

Bvi vi<B<bi bi=B B=bi

vi vi-B 0 0 0

vi<bi

vi-B vi-B (vi-B)/m 0

i’s bid

Value of B

Second-Price Sealed Bid AuctionDistinguished Nash Equilibrium

Every player submits his valuationSeller’s revenue is expected second highest

valuation given that that winner has valuation is V.

Why second-price sealed bid auctions are not generally used?

First-Price Sealed Bid Auctionn bidders with publicly known valuation of

the objectv1>v2 > v3 > … > vn

Actions: Whatever nonnegative bidPayoffs: difference between the value and

bid in the case of winning and zero otherwiseTies are resolved such that player with

higher valuation wins.Technical assumption.

First-Price Sealed Bid AuctionNash equilibrium

(b1,b2,b3,…,bn)=(v2,v2,v3,…,vn) Any other Nash Equilibria?

In any Nash equilibrium, bidder with the highest valuation gets the object and the two highest bids are same from interval [v2,v1], where one of these bids is submitted by P1.For example: (v1,0,0,…,0,v1)

First-Price Sealed Bid AuctionEquilibrium (b1,b2,b3,…,bn)=(v2,v2,v3,…,vn) is

special:Does not require any bidder to bid above his valuation.Why bidder should place bid that brings him negative

payoff?Any bid over valuation is weakly dominated by bid

equal to valuation itself.Bidding more can result only in non-positive payoff.

To bid valuation does not dominated to bid less.Seller’s revenue is v2 given that second highest

valuation is v2Seller’s revenue is same as in the second-price

sealed bid auction.

Imperfect InformationTwo players P1 and P2Actions: whatever nonnegative bidEvery player knows only her valuation of the

object.All players have same believes about the valuation of

opponents - v is uniformly distributed between 0 and 1.P(v<x)=x, where x is from [0,1]

Ties are resolved by chance – All players who submit the highest bid have same chance to become winner.

Expected payoff – Probability of winning when submitting b times (v-b).

Nash EquilibriumAny bid higher than valuation itself is

weakly dominated by bid equal to the valuation.

Bidding valuation itself does not dominate submitting bid lower than valuation.

Suggestion: There might exist such NE that every player submits bid lower than her valuation.

Bid is in linear relationship to valuationbi(vi)=vi/2, where bi is bid of player i and vi is her valuation

Nash EquilibriumExpected payoff of Player 1given that Player2:

if b1 ½ : 2*b1*(v1-b1), because 2*b1 is probability that Player2 submits bid lower than b1.if b1> ½ : (v1-b1), because if b1>c, than Player2 submits bid lower than b1 for sure.

Maximize expected payoff: b1=(1/2)*v1By symmetry, the same holds for Player2Sellers revenue is half of the highest valuation.Second-price sealed bid with two bidders: E[X|

X<v]=v/2.Seller’s revenue is the same.

General caseGenerally:

For n players: bi(vi)=vi*(n-1)/nFor any distribution of valuations: bi(vi)=E[X|vi>X],

where E[X|vi>X] is expected value of second highest valuation given that vi is the highest valuation.

Player bids the value, she expects to be the second highest value given that her value is the highest.

Revenue to the seller is E[X|v>X] given that highest valuation is v.

Expected revenue of the seller is expected value of random variable E[X|v>X].

Revenue EquivalenceUnder very broad assumptions the second-

price sealed bid auction results in the same seller’s expected revenue as the first-price sealed bid auctionDoes the English auction as well?

Does not hold with risk aversion.Symmetric equilibrium in the first-price

sealed bid auction results in higher seller’s expected revenue than second-price sealed bid auction.

HomeworkPareto efficientPareto improvement

1 2

Confess Silent

Confess 1,1 3,0

Silent 0,3 2,2

Prisoner’s dilemma

SummaryBayesian game where no player knows the

payoffs.Auctions

First vs. second-price sealed bid auctionsRevenue equivalence

Pareto efficiency