An introduction to Auctions: Single Item Auctions (2) Maria Serna - PowerPoint PPT Presentation

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions An introduction to Auctions: Single Item Auctions (2) Maria Serna Fall 2016 AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions 1 English and Japanese auctions 2 FP auctions: Bayesian analysis 3 Revenue in auctions AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions English and Japanese auctions AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions English and Japanese auctions A much more complicated strategy space than sealed bid auctions AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions English and Japanese auctions A much more complicated strategy space than sealed bid auctions extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions English and Japanese auctions A much more complicated strategy space than sealed bid auctions extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids Nevertheless, the revealed information doesn’t make any difference in the independent private values setting. AGT-MIRI Single item auctions

English and Japanese auctions Prices FP auctions: Bayesian analysis Revenue in auctions English and Japanese auctions A much more complicated strategy space than sealed bid auctions extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids Nevertheless, the revealed information doesn’t make any difference in the independent private values setting. Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions 1 English and Japanese auctions 2 FP auctions: Bayesian analysis 3 Revenue in auctions AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. How do people behave? AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions. Bidders do not know their opponent’s values, i.e., there is incomplete information. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP auctions: Bayesian analysis There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions. Bidders do not know their opponent’s values, i.e., there is incomplete information. Each bidder’s strategy must maximize her expected payoff accounting for the uncertainty about opponent values. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Auctions with uniform distributions AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Auctions with uniform distributions A simple Bayesian auction model: AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Auctions with uniform distributions A simple Bayesian auction model: 2 buyers Values are between 0 and 1. Values are distributed uniformly on [0 , 1] AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Auctions with uniform distributions A simple Bayesian auction model: 2 buyers Values are between 0 and 1. Values are distributed uniformly on [0 , 1] What is the equilibrium in this game of incomplete information? AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. Let us show that b 1 ( v ) = v 1 / 2 is a best response to Bidder 2. (clearly, no need to bid above v 1 ). AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. Let us show that b 1 ( v ) = v 1 / 2 is a best response to Bidder 2. (clearly, no need to bid above v 1 ). Bidder 1’s utility is: AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. Let us show that b 1 ( v ) = v 1 / 2 is a best response to Bidder 2. (clearly, no need to bid above v 1 ). Bidder 1’s utility is: Prob [ b 1 > b 2 ] × ( v 1 − b 1 ) = = Prob [ b 1 > v 2 / 2] × ( v 1 − b 1 ) =2 b 1 ∗ ( v 1 − b 1 ) AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. Let us show that b 1 ( v ) = v 1 / 2 is a best response to Bidder 2. (clearly, no need to bid above v 1 ). Bidder 1’s utility is: Prob [ b 1 > b 2 ] × ( v 1 − b 1 ) = = Prob [ b 1 > v 2 / 2] × ( v 1 − b 1 ) =2 b 1 ∗ ( v 1 − b 1 ) maximizing for b 1 we have: AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Simple FP: Equilibrium 2 bidders uniform distribution Bidding b ( v ) = v / 2 is an equilibrium Assume that Bidder 2’s strategy is b 2 ( v ) = v 2 / 2. Let us show that b 1 ( v ) = v 1 / 2 is a best response to Bidder 2. (clearly, no need to bid above v 1 ). Bidder 1’s utility is: Prob [ b 1 > b 2 ] × ( v 1 − b 1 ) = = Prob [ b 1 > v 2 / 2] × ( v 1 − b 1 ) =2 b 1 ∗ ( v 1 − b 1 ) maximizing for b 1 we have: [2 b 1 ∗ ( v 1 − b 1 )] ′ = 2 v 1 − 4 b 1 = 0 which gives b 1 = v 1 / 2 AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP: uniform values AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP: uniform values We consider the simple Bayesian model n bidders Values drawn uniformly form [0 , 1] AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP: uniform values We consider the simple Bayesian model n bidders Values drawn uniformly form [0 , 1] Theorem In a FP auction, under the uniform values model, it is a Bayesian Nash equilibrium when, for any i, bidder i bids n − 1 n v i . AGT-MIRI Single item auctions

English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions FP uniform values: Efficiency AGT-MIRI Single item auctions