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

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

Jul 30, 2023 119 likes •468 views

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

slide-1
SLIDE 1

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

bi ≥ 0 on a piece of paper, and submits to the auctioneer in a sealed envelop.

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SLIDE 2

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.
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SLIDE 3

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.
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SLIDE 4

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.
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SLIDE 5

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.

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SLIDE 6

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 (b2, ..., bN). !

  • Bidder 1 needs to decide b1 to maximize his
  • payoff. !
  • We will show that choosing b1 =W will lead to

a payoff no smaller than any other choices, no matter what b* is.

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SLIDE 7

Case I

  • Assume b* = max (b2, ..., bN) = W. !
  • If b1 = 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 b1 > W, then bidder 1 is the winner with a payoff of

W- b* = 0.!

  • If b1 < W, then bidder 1 will not win and gets a 0

payoff.!

  • Hence bidding b1 = W leads to the maximum

payoff of 0.

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SLIDE 8

Case II

  • Assume b* = max (b2, ..., bN) > W. !
  • If b1 > b*, then bidder 1 is the winner with a payoff of

W - b* < 0.!

  • If b1 = 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 b1 < b*, then bidder 1 will not win and gets a 0

payoff.!

  • Hence choosing b1 = W < b* leads to the

maximum payoff of 0.

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SLIDE 9

Case III

  • Assume b* = max (b2, ..., bN) < W. !
  • If b1 > b*, then bidder 1 is the winner with a payoff of

W - b* > 0.!

  • If b1 = 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 b1 < b*, then bidder 1 will not win and gets a 0

payoff.!

  • Hence choosing b1 = W > b* leads to the

maximum payoff of W - b* > 0.

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SLIDE 10

Nash Equilibrium

  • Since bidding bi = 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.
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SLIDE 11

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

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SLIDE 12

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.
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SLIDE 13

Centipede Game

Player 1

Stop Continue

1, 1 Player 2

Stop Continue

0, 3 Player 1

Stop Continue

2, 2

...

Player 1

Stop Continue

99, 99

...

Player 2

Stop Continue

97, 100 Player 2

Stop Continue

98, 101 100, 100

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SLIDE 14

Centipede Game

Player 1

Stop Continue

1, 1 Player 2

Stop Continue

0, 3 Player 1

Stop Continue

2, 2

... ...

Player 1

Stop Continue

99, 99 Player 2

Stop Continue

97, 100 Player 2

Stop Continue

98, 101 100, 100

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SLIDE 15

Centipede Game

Player 1

Stop Continue

1, 1 Player 2

Stop Continue

0, 3 Player 1

Stop Continue

2, 2

... ...

Player 2

Stop Continue

97, 100 Player 2

Stop Continue

98, 101 100, 100 Player 1

Stop Continue

99, 99

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SLIDE 16

Centipede Game

Player 1

Stop Continue

1, 1 Player 2

Stop Continue

0, 3 Player 1

Stop Continue

2, 2

...

Player 2

Stop Continue

98, 101 100, 100 Player 1

Stop Continue

99, 99

...

Player 2

Stop Continue

97, 100

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SLIDE 17

Centipede Game

Player 1

Stop Continue

1, 1 0, 3 Player 1

Stop Continue

2, 2

...

Player 2

Stop Continue

98, 101 100, 100 Player 1

Stop Continue

99, 99 97, 100

...

Player 2

Stop Continue

Player 2

Stop Continue

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SLIDE 18

Centipede Game

Player 1

Stop Continue

1, 1 0, 3 Player 1

Stop Continue

2, 2

...

Player 2

Stop Continue

98, 101 100, 100 Player 1

Stop Continue

99, 99 97, 100

...

Player 2

Stop Continue

Player 2

Stop Continue

Player 1

Stop Continue

1, 1

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SLIDE 19

Centipede Game

Player 1

Stop Continue

1, 1 0, 3

Continue

2, 2

...

Continue

98, 101 100, 100

Continue

99, 99 97, 100

...

Continue Continue

Player 1

Stop Continue

1, 1 Player 2

Stop

Player 1

Stop

Player 1

Stop

Player 2

Stop

Player 2

Stop

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SLIDE 20

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.

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SLIDE 21

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

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SLIDE 22

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
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SLIDE 23

Fairness in Games

  • Let us look at another game where fairness

can play an important role.

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SLIDE 24

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
  • ffer.!
  • If player 2 accepts, then players’ payoffs are (10-x, x). !
  • If player 2 rejects, then players’ payoffs are (0,0).
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SLIDE 25

Ultimatum Game

10-x, x

Player 2

0, 0

Accept Reject

Player 1

x=0.01 x=10

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SLIDE 26

Backward Induction

Player 1

10-x, x

Player 2

0, 0

Accept Reject x=0.01 x=10

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SLIDE 27

Ultimatum Game

Player 1

10-x, x

Player 2

0, 0

Accept Reject x=0.01 x=10

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SLIDE 28

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.

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SLIDE 29

Experimental Results

  • Instead of offering the minimum value, majority
  • f 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?
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SLIDE 30

Possible Explanations

  • Many explanations!
  • One popular one: fairness is an important concern
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SLIDE 31
  • Each player i’s payoff is parameterized by two

coefficients: Ai and Bi.!

  • Ai is the fairness coefficient: Ai max(yj - yi, 0)!
  • Bi is the guilt coefficient: Bi max(yi - yj, 0)!
  • Player i’s payoff is !
  • Ui(yi, yj) = yi - Ai max(yj - yi, 0) - Bi max(yi - yj, 0)

Payoff Modification

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SLIDE 32

Payoff Modification

  • It is clear that the values of Ai and Bi will affect

the equilibrium behavior. !

  • Intuitively, player 2’s threshold of rejection

will increase with his fairness coefficient A2.

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SLIDE 33

Example

  • Let player 2’s fairness coefficient A2 = 2.!
  • Other parameters (A1, B1, and B2) are zero.
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SLIDE 34

Backward Induction

  • Stage 2: player 2’s payoff is !
  • U2(x, 10-x)= x-2 max(10-x-x,0) = x-2 max(10-2x,0)!
  • Player 2 will only accept the offer if U2(x, 10-x) ≥ 0,

which leads to x ≥ 4.!

  • Stage 1: player 1 will offer x=4. !
  • The SPNE payoff: (6, 4).