Auction Theory I

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price

Auction Theory I

Lecture 18

Auction Theory I Lecture 18, Slide 1 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Lecture Overview

1 Auctions

2 Canonical Single-Good Auctions

3 Comparing Auctions

4 Second-price auctions

Auction Theory I Lecture 18, Slide 2 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Motivation

Auctions are any mechanisms for allocating resources among self-interested agents Very widely used government sale of resources privatization stock market request for quote FCC spectrum real estate sales eBay

Auction Theory I Lecture 18, Slide 3 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price CS Motivation

resource allocation is a fundamental problem in CS increasing importance of studying distributed systems with heterogeneous agents markets for: computational resources (JINI, etc.) P2P systems network bandwidth currency needn’t be real money, just something scarce that said, real money trading agents are also an important motivation

Auction Theory I Lecture 18, Slide 4 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Lecture Overview

1 Auctions

2 Canonical Single-Good Auctions

3 Comparing Auctions

4 Second-price auctions

Auction Theory I Lecture 18, Slide 5 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Some Canonical Auctions

English Japanese Dutch First-Price Second-Price All-Pay

Auction Theory I Lecture 18, Slide 6 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price English Auction

English Auction auctioneer starts the bidding at some “reservation price” bidders then shout out ascending prices once bidders stop shouting, the high bidder gets the good at that price

Auction Theory I Lecture 18, Slide 7 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Japanese Auction

Japanese Auction Same as an English auction except that the auctioneer calls out the prices all bidders start out standing when the price reaches a level that a bidder is not willing to pay, that bidder sits down once a bidder sits down, they can’t get back up the last person standing gets the good

analytically more tractable than English because jump bidding can’t occur consider the branching factor of the extensive form game...

Auction Theory I Lecture 18, Slide 8 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Dutch Auction

Dutch Auction the auctioneer starts a clock at some high value; it descends at some point, a bidder shouts “mine!” and gets the good at the price shown on the clock

Auction Theory I Lecture 18, Slide 9 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price First-, Second-Price Auctions

First-Price Auction bidders write down bids on pieces of paper auctioneer awards the good to the bidder with the highest bid that bidder pays the amount of his bid

Second-Price Auction bidders write down bids on pieces of paper auctioneer awards the good to the bidder with the highest bid that bidder pays the amount bid by the second-highest bidder

Auction Theory I Lecture 18, Slide 10 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price All-Pay auction

All-Pay Auction bidders write down bids on pieces of paper auctioneer awards the good to the bidder with the highest bid everyone pays the amount of their bid regardless of whether or not they win

Auction Theory I Lecture 18, Slide 11 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Auctions as Structured Negotiations

Any negotiation mechanism that is: market-based (determines an exchange in terms of currency) mediated (auctioneer) well-speciﬁed (follows rules)

Deﬁned by three kinds of rules: rules for bidding rules for what information is revealed rules for clearing

Auction Theory I Lecture 18, Slide 12 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Auctions as Structured Negotiations

Deﬁned by three kinds of rules: rules for bidding who can bid, when what is the form of a bid restrictions on oﬀers, as a function of: bidder’s own previous bid auction state (others’ bids) eligibility (e.g., budget constraints) expiration, withdrawal, replacement rules for what information is revealed rules for clearing

Auction Theory I Lecture 18, Slide 12 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Auctions as Structured Negotiations

Deﬁned by three kinds of rules: rules for bidding rules for what information is revealed when to reveal what information to whom rules for clearing

Auction Theory I Lecture 18, Slide 12 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Auctions as Structured Negotiations

Deﬁned by three kinds of rules: rules for bidding rules for what information is revealed rules for clearing when to clear at intervals on each bid after a period of inactivity allocation (who gets what) payment (who pays what)

Auction Theory I Lecture 18, Slide 12 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Lecture Overview

1 Auctions

2 Canonical Single-Good Auctions

3 Comparing Auctions

4 Second-price auctions

Auction Theory I Lecture 18, Slide 13 How should agents bid in these auctions?

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Intuitive comparison of 5 auctions Intuitive Comparison of 5 auctions

English Dutch Japanese 1st-Price 2nd-Price

Duration #bidders, starting #bidders, fixed fixed increment price, clock increment speed Info 2nd-highest winner’s all val’s but none none Revealed val; bounds bid winner’s on others Jump bids yes n/a no n/a n/a

Price yes no yes no no Discovery Fill in “regret” after Regret no yes no yes no the fun game

• How should agents bid in these auctions? Auction Theory I Lecture 18, Slide 14 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Intuitive comparison of 5 auctions Intuitive Comparison of 5 auctions

English Dutch Japanese 1st-Price 2nd-Price

Price yes no yes no no Discovery Fill in “regret” after Regret no yes no yes no the fun game

How should agents bid in these auctions?

• How should agents bid in these auctions? Auction Theory I Lecture 18, Slide 14 The paper you are given contains four valuations independent valuations, normally distributed with mean 100, stdev 20 Bid in four auctions: English ﬁrst-price second-price Dutch

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Fun Game

Valuation models: the most important one: IPV valuations are iid draws from some commonly-known distribution do you see how we can write this as a Bayesian game?

Auction Theory I Lecture 18, Slide 15 ﬁrst-price second-price Dutch

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Fun Game

Valuation models: the most important one: IPV valuations are iid draws from some commonly-known distribution do you see how we can write this as a Bayesian game? The paper you are given contains four valuations independent valuations, normally distributed with mean 100, stdev 20 Bid in four auctions: English

Auction Theory I Lecture 18, Slide 15 second-price Dutch

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Fun Game

Auction Theory I Lecture 18, Slide 15 Dutch

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Fun Game

Auction Theory I Lecture 18, Slide 15 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Fun Game

Auction Theory I Lecture 18, Slide 15 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Intuitive comparison of 5 auctions

Intuitive Comparison of 5 auctions

English Dutch Japanese 1st-Price 2nd-Price

Price yes no yes no no Discovery Fill in “regret” after Regret no yes no yes no the fun game

Auction• Theory How I should agents bid in these auctions? Lecture 18, Slide 16 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Lecture Overview

1 Auctions

2 Canonical Single-Good Auctions

3 Comparing Auctions

4 Second-price auctions

Auction Theory I Lecture 18, Slide 17 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price

Theorem Truth-telling is a dominant strategy in a second-price auction.

In fact, we know this already (do you see why?) However, we’ll look at a simpler, direct proof.

Auction Theory I Lecture 18, Slide 18 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof

Theorem Truth-telling is a dominant strategy in a second-price auction.

Proof. Assume that the other bidders bid in some arbitrary way. We must show that i’s best response is always to bid truthfully. We’ll break the proof into two cases: 1 Bidding honestly, i would win the auction 2 Bidding honestly, i would lose the auction

Auction Theory I Lecture 18, Slide 19 or lose and get utility of zero.

i’s true i’s true i’s true value value value

i pays i pays winner pays

next-highest next-highest highest i’s bid i’s bid i’s bid bid bid bid

If i bids higher, he will still win and still pay the same amount If i bids lower, he will either still win and still pay the same amount. . .

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (2)

i’s true value

i pays

next-highest i’s bid bid

Bidding honestly, i is the winner

Auction Theory I Lecture 18, Slide 20 or lose and get utility of zero.

i’s true i’s true value value

i pays winner pays

next-highest highest i’s bid i’s bid bid bid

If i bids lower, he will either still win and still pay the same amount. . .

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (2)

i’s true i’s true value value

i pays i pays

next-highest next-highest i’s bid i’s bid bid bid

Bidding honestly, i is the winner If i bids higher, he will still win and still pay the same amount

Auction Theory I Lecture 18, Slide 20 i’s true value

winner pays

highest i’s bid bid

or lose and get utility of zero.

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (2)

i’s true i’s true i’s true value value value

i pays i pays i pays

next-highest next-highest next-highest i’s bid i’s bid i’s bid bid bid bid

Bidding honestly, i is the winner If i bids higher, he will still win and still pay the same amount If i bids lower, he will either still win and still pay the same amount. . .

Auction Theory I Lecture 18, Slide 20 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (2)

i’s true i’s true i’s true i’s true value value value value

i pays i pays i pays winner pays

next-highest next-highest next-highest highest i’s bid i’s bid i’s bid i’s bid bid bid bid bid

Bidding honestly, i is the winner If i bids higher, he will still win and still pay the same amount If i bids lower, he will either still win and still pay the same amount. . . or lose and get utility of zero.

Auction Theory I Lecture 18, Slide 20 or win and pay more than his valuation.

i pays i’s true i’s true i’s true value value value

highest highest next-highest i’s bid i’s bid i’s bid bid bid bid

If i bids lower, he will still lose and still pay nothing If i bids higher, he will either still lose and still pay nothing. . .

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (3)

i’s true value

highest i’s bid bid

Bidding honestly, i is not the winner

Auction Theory I Lecture 18, Slide 21 or win and pay more than his valuation.

i pays i’s true i’s true value value

highest next-highest i’s bid i’s bid bid bid

If i bids higher, he will either still lose and still pay nothing. . .

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (3)

i’s true i’s true value value

highest highest i’s bid i’s bid bid bid

Bidding honestly, i is not the winner If i bids lower, he will still lose and still pay nothing

Auction Theory I Lecture 18, Slide 21 i pays i’s true value

next-highest i’s bid bid

or win and pay more than his valuation.

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (3)

i’s true i’s true i’s true value value value

highest highest highest i’s bid i’s bid i’s bid bid bid bid

Bidding honestly, i is not the winner If i bids lower, he will still lose and still pay nothing If i bids higher, he will either still lose and still pay nothing. . .

Auction Theory I Lecture 18, Slide 21 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price Second-Price proof (3)

i pays i’s true i’s true i’s true i’s true value value value value

highest highest highest next-highest i’s bid i’s bid i’s bid i’s bid bid bid bid bid

Bidding honestly, i is not the winner If i bids lower, he will still lose and still pay nothing If i bids higher, he will either still lose and still pay nothing. . . or win and pay more than his valuation.

Auction Theory I Lecture 18, Slide 21 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.

Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price English and Japanese auctions

A much more complicated strategy space 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 intuitively, though, the revealed information doesn’t make any diﬀerence in the IPV setting.

Auction Theory I Lecture 18, Slide 22 Auctions Canonical Single-Good Auctions Comparing Auctions Second-Price English and Japanese auctions

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.

Auction Theory I Lecture 18, Slide 22