Introduction to Game Theory

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Introduction to Game Theory
Lecture 9: Auctions

Preview Bayesian Games The last lecture
Imperfect information about the state of the world, just believes The last lecture At least one player knows only his own type Avery player consider expected payoffs given his believes Equilibrium – every type of every player cannot be better off by unilateral deviation given his believes

Today’s Plan Nobody knows neither his own payoff neither opponents’ payoffs Oil 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. Auctions Second-price sealed bid auction First-price sealed bid auction

Too Much Information Hurts
Classic 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 Hurts
Two states of the world: S1 and S2 Players 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 Probability = ½ Probability = ½ 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

Nobody Knows If P2 believes that P1 will choose T:
EP(L) = ½*4 + ½*4 = EP(M) = ½*0 + ½*6 = 3 EP(R) = ½*6 + ½*0 = 3 If P2 believes that P1 will choose B: EP(L) = ½*16 + ½*16 = EP(M) = ½*0 + ½*24 = 12 EP(R) = ½*24 + ½*0 = 12 Probability = ½ Probability = ½ 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

Everybody Expects Expected payoff of Player 1 (row)?
All problem collapses to 2x2 game. Equilibrium: (L,B) Expected payoff P2 P1 L M R T 1,4 1,3 B 2,16 0,12

Player Two Knows Assume 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 P2 P1 L M R T 1,4 1,0 1,6 B 2,16 0,0 0,24 P2 P1 L M R T 1,4 1,6 1,0 B 2,16 0,24 0,0

Too Much Information Hurts
Comparison of outcomes No information about state: NE = (B,L) with payoffs: Player1gets 2 and Player2 gets 16 Information 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 P1 L M R T 1,4 1,0 1,6 B 2,16 0,0 0,24 P2 P1 L M R T 1,4 1,6 1,0 B 2,16 0,24 0,0

Auctions Allocation of goods Forms Subject Information Oil tracts
Art works Treasury bills Forms Sequential vs. Simultaneous Subject Single unit vs. multiunit Information Private value vs. common value

Auctions First-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

Auctions English auction: Dutch 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 bid
n bidders with publicly known valuation of the object v1>v2 > v3 > … > vn Actions: Whatever nonnegative bid Payoffs: difference between the value and the second highest bid in the case of winning and zero otherwise Ties are resolved by such that player with higher valuation wins. Technical assumption.

Nash Equilibrium Many 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 cv2 Player i submits v1 and all other bids are not higher than her valuation.

Nash Equilibrium NE (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 Information
n players P1,P2,P3,…Pn Actions: whatever nonnegative bid Every player knows only her valuation of the object. All players have believes about the valuation of opponents - v is distributed such that P(vx)=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 domination Submitting own valuation weakly dominates any other bid. Consider player i. B is the highest bid by other players Bids lower than valuation vi: Player i cannot be better off by bidding less than her valuation. Value of B B<bi B=bi bi<B<vi Bvi bi<vi vi-B (vi-B)/m vi (vi-B) i’s bid

Weak Domination Bids 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]. Value of B Bvi vi<B<bi bi=B B=bi vi vi-B vi<bi (vi-B)/m i’s bid

Second-Price Sealed Bid Auction
Distinguished Nash Equilibrium Every player submits his valuation Seller’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 Auction
n bidders with publicly known valuation of the object v1>v2 > v3 > … > vn Actions: Whatever nonnegative bid Payoffs: difference between the value and bid in the case of winning and zero otherwise Ties are resolved such that player with higher valuation wins. Technical assumption.

First-Price Sealed Bid Auction
Nash 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 Auction
Equilibrium (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 v2 Seller’s revenue is same as in the second-price sealed bid auction.

Imperfect Information
Two players P1 and P2 Actions: whatever nonnegative bid Every 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 Equilibrium Any 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 valuation bi(vi)=vi/2, where bi is bid of player i and vi is her valuation

Nash Equilibrium Expected 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)*v1 By symmetry, the same holds for Player2 Sellers 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 case Generally:
For n players: bi(vi)=vi*(n-1)/n For 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 Equivalence Under very broad assumptions the second-price sealed bid auction results in the same seller’s expected revenue as the first-price sealed bid auction Does 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.

Homework Pareto efficient Pareto improvement 1 2 Confess Silent 1,1
Prisoner’s dilemma Confess Silent 1,1 3,0 0,3 2,2

Summary Bayesian game where no player knows the payoffs. Auctions
First vs. second-price sealed bid auctions Revenue equivalence Pareto efficiency