Second Price Auction A painting worth of W dollars is for sale - PowerPoint PPT Presentation

Second Price Auction • A painting worth of W dollars is for sale through an auction. ! • N bidders (buyers) participate in the auction. ! • Each bidder i writes down his name and a bid b i ≥ 0 on a piece of paper, and submits to the auctioneer in a sealed envelop.

Second Price Auction • The auctioneer opens up all envelops, gives the painting to the highest bidder, and charges the winner the second highest bid. ! • Break the tie randomly. ! • Payoff of the winner = W - second highest bid ! • Payoff of all other bidders = 0.

An Example • W =1M USD, N =3, Bids = (1.05M, 0.9M, 1.1M). ! • The 3rd bidder is the winner: ! • The highest bid is 1.1M. ! • The second highest bid is 1.05M. ! • His payoff is 1M - 1.05M = -0.05M ! • The 1st and 2nd bidders have zero payoffs.

Another Example • W =1M USD, N =3, Bids = (0.9M, 0.9M, 0.8M). ! • Break the tie (between the 1st and 2nd bidders) randomly. ! • Say the 1st bidder is the winner: ! • The second highest bid is 0.9M. ! • His payoff is 1M - 0.9M = 0.1M ! • The 2nd and 3rd bidders have zero payoffs.

Nash Equilibrium • What will be the bids at a Nash equilibrium? ! • We will show that bidding W (hence truthfully) is a weakly dominate strategy for all bidders.

Best Response • Without loss of generality, let us compute the best response of the 1st bidder. ! • Assume the highest bid from bidders 2 to N is b* = max ( b 2 , ..., b N ). ! • Bidder 1 needs to decide b 1 to maximize his payoff. ! • We will show that choosing b 1 = W will lead to a payoff no smaller than any other choices, no matter what b* is.

Case I • Assume b* = max ( b 2 , ..., b N ) = W. ! • If b 1 = W , then bidder 1 either is the winner with a payoff of W - b* = 0, or is not the winner with a 0 payoff. ! • If b 1 > W , then bidder 1 is the winner with a payoff of W - b* = 0. ! • If b 1 < W , then bidder 1 will not win and gets a 0 payoff. ! • Hence bidding b 1 = W leads to the maximum payoff of 0.

Case II • Assume b* = max ( b 2 , ..., b N ) > W. ! • If b 1 > b* , then bidder 1 is the winner with a payoff of W - b* < 0. ! • If b 1 = b* , then bidder 1 either is the winner with a payoff of W - b* < 0, or is not the winner with a 0 payoff. ! • If b 1 < b* , then bidder 1 will not win and gets a 0 payoff. ! • Hence choosing b 1 = W < b* leads to the maximum payoff of 0.

Case III • Assume b* = max ( b 2 , ..., b N ) < W. ! • If b 1 > b* , then bidder 1 is the winner with a payoff of W - b* > 0. ! • If b 1 = b* , then bidder 1 either is the winner with a payoff of W - b* > 0, or is not the winner with a 0 payoff. ! • If b 1 < b* , then bidder 1 will not win and gets a 0 payoff. ! • Hence choosing b 1 = W > b* leads to the maximum payoff of W - b* > 0.

Nash Equilibrium • Since bidding b i = W leads to the maximum payoff for bidder i independent of other bidders’ bids, it is a weakly dominant strategy (and hence a best response). ! • Everyone bidding W is a Nash equilibrium.

Bounded Rationality • So far we have assumed that players are fully rational ! • They are able to derive the equilibrium strategies even facing many choices or many stages ! • In reality, humans are often bounded rational ! • In chess, even a master seldom thinks beyond 5 moves

Centipede Game • Two players start with $1 in front of each. ! • Starting from player 1, they alternate saying “stop” or “continue”. ! • If a player chooses “continue”, then $1 is taken from his pile and $2 are put in his opponent’s pile. ! • If a player chooses “stop”, both players get what are currently in their piles. ! • The game also stops if both piles reach $100.

Centipede Game Player 1 Player 2 Player 1 Continue Continue Continue Stop Stop Stop ... 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 2 Player 1 Continue Continue Continue Stop Stop Stop ... 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 2 Player 1 Continue Continue Continue Stop Stop Stop ... 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 2 Player 1 Continue Continue Continue Stop Stop Stop ... 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 2 Player 1 Continue Continue Continue Stop Stop Stop ... 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 1 Player 2 Player 1 Continue Continue Continue Continue Stop Stop Stop Stop ... 1, 1 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game Player 1 Player 1 Player 2 Player 1 Continue Continue Continue Continue Stop Stop Stop Stop ... 1, 1 1, 1 0, 3 2, 2 Player 2 Player 1 Player 2 Continue Continue Continue 100, 100 Stop Stop Stop ... 97, 100 99, 99 98, 101

Centipede Game • Unique SPNE: ! • Each player chooses “stop” whenever possible. ! • The SPNE is very bad ! • Each player can get only $1. ! • If they cooperate, each of them can get $100. ! • Empirical results suggest that people rarely play according to SPNE.

Possible Explanations • Players might make mistakes ! • The inability of correctly performing backward induction - bounded rationality ! • When the opponent makes mistake (chooses “continue” instead of “stop”), it is beneficial to deviate from the SPNE (hence also chooses “continue”) ! • Evidence: chess masters usually stop at the first stage when playing the centipede game

Possible Explanations • Players might be altruistic (unselfish) ! • A player cares about the welfare of his opponent. ! • An extreme example: both players have the same payoff equal to the summation of the money ! • With this revision, the unique SPNE is to “continue” whenever possible ! • The theory is good enough ! • We need to choose the right payoff functions

Fairness in Games • Let us look at another game where fairness can play an important role.

Ultimatum Game • Two players try to divide 10 dollars. ! • Stage 1: Player 1 offers x dollars to player 2. ! • Assume that the min value of x is 0.01 dollar (1 cent). ! • Stage 2: Player 2 can accept or reject player 1’s offer. ! • If player 2 accepts, then players’ payoffs are (10 -x , x ). ! • If player 2 rejects, then players’ payoffs are (0 , 0).

Ultimatum Game Player 1 x=0.01 x=10 Player 2 Accept Reject 10-x, x 0, 0

Backward Induction Player 1 x=0.01 x=10 Player 2 Accept Reject 10-x, x 0, 0

Ultimatum Game Player 1 x=0.01 x=10 Player 2 Accept Reject 10-x, x 0, 0

SPNE • Unique subgame perfect Nash equilibrium: (0.01, Accept) ! • Player 2 will get the minimum amount, and player 1 will get almost all the money.

Experimental Results • Instead of offering the minimum value, majority of the offers are between 40% to 50% of the total amount. ! • Very few offers are below 20% of the total amount. ! • Very low offers are often rejected. ! • Why not humans follow SPNE?

Possible Explanations • Many explanations ! • One popular one: fairness is an important concern

Payoff Modification • Each player i ’s payoff is parameterized by two coefficients: A i and B i . ! • A i is the fairness coefficient: A i max( y j - y i , 0) ! • B i is the guilt coefficient: B i max( y i - y j , 0) ! • Player i ’s payoff is ! • U i ( y i , y j ) = y i - A i max( y j - y i , 0) - B i max( y i - y j , 0)

Payoff Modification • It is clear that the values of A i and B i will affect the equilibrium behavior. ! • Intuitively, player 2’s threshold of rejection will increase with his fairness coefficient A 2 .

Example • Let player 2’s fairness coefficient A 2 = 2. ! • Other parameters ( A 1 , B 1 , and B 2 ) are zero.

Backward Induction • Stage 2: player 2’s payoff is ! • U 2 ( x , 10-x )= x - 2 max( 10-x - x ,0) = x - 2 max( 10-2x ,0) ! • Player 2 will only accept the offer if U 2 ( x , 10-x ) ≥ 0, which leads to x ≥ 4. ! • Stage 1: player 1 will offer x=4. ! • The SPNE payoff: (6, 4).