博弈论课件 Lecture 15_Auction Theory

Game Theory

1

Spring2015

1Lecture 15Auction Theory

Readings ?Watson: Strategy_ An introduction to game

theory

–Ch27.

2

–Exercise:

Auction

common values private values

value of good for sale is same for all value of good is different for all and my value is irrelevant to you

oil well home pure consumption

goods

Type of Auctions

?A: First-price sealed bid

?B: Second-price sealed bid (winner pays the second bid, Vickery)

?C: Ascending open auction

?D: Descending open auction

拍卖的四种基本类型

?第一价格密封拍卖：众多买方以书面投标方式竞买拍卖品，出家最高者将以其出价水平获取拍卖品；

?第二价格密封拍卖：众多买方以书面投标方式竞买拍卖品，出家最高者将以次高价获取拍卖品；?英式拍卖：竞标者不断提高竞标价格，直到没有人再出更高价格为止，出价最高的竞标者获得该商品；

?荷式拍卖：出售者在确定的时间内从某个高价开始逐渐向下降价，直到有人喊出“购买”为止。

Type of Auctions

D = A B ≈ C

(Difference comes when

there’s a common value

component, then in C you

see this information, but

in B you don’t)

Private Value Auctions

?A: First-price sealed bid

?B: Second-price sealed bid (winner pays the second bid, Vickery)

?C: Ascending open auction

?D: Descending open auction

1211: Every type of player assigns probability ()()()()() to the event that the valuation of every other player is at most ;

Beliefs Payoff functions : Player ' Bernoull i i n j i F v F v F v F v F v j v i s -+ 11i payoff in state (,,)is 0 if her bid is not the highest bid, and (())/ if no bid is higher than and bid (including ) are equal to :

(())/ if for all (,(,,))n i i i i i i j i i n v v b v P b m b m b b v P b m b b u b v v --≤= and for players

0 if for some j i j i j i b b m b b j i ≠=????>≠?

Each value of gets the same weight when calculating the expected payoff to as it does when calculating the expected payoff to .i i b b v

Nash equilibrium in a second-price

sealed-bid auction

In a second-price sealed-bid auction with imperfect information about valuations, a player's bid equal to her valuation weakly dominates all her other bids.

We conclude, in particular, that a second-price sealed -bid auciton with imperfect information about valuations has a Nash equilibrium in which every type of every player bids her valuation.

not weakly dominates all her other bids in a first-price sealed-bid auction i i b v

Nash equilibrium in a first-price sealed-bid auction {}{} Find the equilibrium condition that the bid of each type of each player be a best response to the bids of each type of every other player. max(-)Pr (-)Pr v b Highest bid is b v b All other bids b =≤ If each bidder's valuation is drawn independently from the same continuous and increasing cumulative distribution, a first-price sealed -bid auction (with imperfect information about valuations) has a []

(symmetric) Nash equilibrium in which each type of each player bids |, the expected value of the highest of the other players'bids conditional on being higher than all the other valuations. v E X X v v < If the distribution is uniform from 0 to 1, each player with valuation bids -/ in a symmetric Nash equilibrium.F v v v n

Revenue equivalence of auctions

If each bidder is risk-neutral and each bidder's valuaton is drawn independently from the same continuous and increasing cumulative distribution, then the Nash equilibrium of a second-price sealed-bid auction (with imperfect information about valuations) in which each player bids her valuation yields the same revenue as the symmetric Nash equilibrium of the corresponding first-price sealed-bid auction.

If each bidder is risk-averse and each bidder's valuaton is drawn independently from the same continuous and increasing cumulative distribution, then the auctioneer's expected revenue is greater in a

a first-price auction than it is in a second-price auction.

1 Players St : The set of bidders, say 1,,;

: The set of all profiles (,,) of signals ates (and imperfect information about valuations)First- and second-price auctions with common valuations n n t t 1that the players may receive;

: Each player's set of actions is the set of possible bids (nonnegative numbers);

: The signal function of each player is given by Ac tions Signals (,,)i i n i i t t t ττ= (each player observes her own signal);