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Game Theory

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Game Theory

Lecture 11

problem set 11

from Binmore’s

Fun and Games.

p. 563 Exs. 35, (36)

p. 564. Ex. 39

first & second price auctions with independent private valuations

- Set of bidders 1,2….n
- The states of nature: Profiles of valuations (v1,v2,…..vn), Each is informed about his own valuation only.
- Given a profile (v1,v2,…..vn), the probability of having a profile (w1,w2,…..wn), s.t. vi ≥ wi is F(v1)F(v2)…...F(vn). Where F() is a cummulative distribution function.

first & second price auctions with independent private valuations

- Actions: Set of possible (non negative) bids
- Payoffs: In a state (v1,v2,…..vn), if player i’s bid is the highest and there are m such bids he gets [vi-P(b)]/m . If there are higher bids he gets 0.
- P(b) is what the winner pays when the profile of bids is b. It is the highest bid in a first price auction, and the second highest bid in a second price auction.

first & second price auctions with independent private valuations

- In a second price (sealed bid) auction, bidding the true value is a weakly dominating strategy.
- If the highest bid of the others is lower than my valuation I can only win by bidding my valuation.
- If the highest bid of the others is higher than my valuation I can possibly win by lowering my bid to my valuation.

Hence, truth telling is a Nash equilibrium

(there may be others)

first & second price auctions with independent private valuations

- In a first price (sealed bid) auction, bidding the true value is not a dominating strategy: It is better to bid lower when the highest bid of the others is lower than my valuation.

A simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

A simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

Assume all types of player 2 bid as above.

If player 1 bids more than ½ he certainly wins (v-b).

If player 1 bids b < ½, he wins if player 2’s bid is lower than b,

i.e. if player 2’s valuation is lower than 2b. This happens with probability 2b.

In this case his expected gain is 2b(v-b).

A simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

for v > ½the payoff function is:

The maximum is at b =½v

2b(v-b)

v-b

b

v

½v

½

A simple case:

for v < ½the payoff function is somewhat different but the

maximum is as before at b =½v

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

2b(v-b)

i.e. each type of player 1 wants to use the same strategy

½

v

b

½v

v-b

A simple case:

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

2b(v-b)

i.e. each type of player 1 wants to use the same strategy

½

v

b

½v

v-b

A simple case:

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

A player with valuation v, bids v/2 and will pay it if his valuation is the highest, this happens with probability v, i.e. he expects to payv2/2

Now consider this simple case for a second price auction

second price (sealed bid) auction.

(2 bidders)

- Each player’s valuation is independently drawn from the uniform distribution on [0,1]
- A player whose valuation is v, will bid v.
- He wins with probability v, and expects to pay v2/2.

same expected payoff as in the first price auction

v

b

β-1(b)

β(v)

v

b

First price (sealed bid) auction with n bidders.

Valuations are independently drawn from the cumulative distribution F( ).

We look for a symmetric equilibrium

The inverse function

Is this an increasing functionofv ???

An example:

- A seller sells an object whose value to him is zero, he faces two buyers.
- The seller does not know the value of the object to the buyers.
- Each of the buyers has the valuation 3 or 4 with probability p, 1-p (respc.)
- The seller wishes to design a mechanism that will yield the highest possible expected payoff.

?

Consider the ‘first best’ case:

the probability that both buyers value the object at 3

the probability that at least one buyer values the object at 4

Posted Prices

(take it or leave it offer)

only 3,4

the probability that at least one buyer values the object at 4

Posted Prices

(take it or leave it offer)

the first best

Second price auction

Posted price 4.

the probability that both buyers value the object at 4.

Modified Second price auction

Modified Second price auction

Modified Second price auction

Second price auction

Posting price 4