A Revealed-Preference Activity A Revealed-Preference Activity Rule for Quasi-Linear Utilities Rule for Quasi-Linear Utilities

with Budget Constraintswith Budget Constraints

Robert Day, University of ConnecticutRobert Day, University of Connecticutwith special thanks to:with special thanks to:

Pavithra Harsha, Cynthia Barnhart, MITPavithra Harsha, Cynthia Barnhart, MITand David Parkes, Harvardand David Parkes, Harvard

Multi-Unit AuctionsMulti-Unit Auctions

• In auctions for spectrum licenses (for example), In auctions for spectrum licenses (for example), many items may be auctioned simultaneously many items may be auctioned simultaneously through an iterative procedurethrough an iterative procedure

• We consider an environment in which bidders We consider an environment in which bidders report demand amounts at the current price-vectorreport demand amounts at the current price-vector

• Examples include the Simultaneous Ascending Examples include the Simultaneous Ascending Auction (used by the FCC), and Ausubel, Auction (used by the FCC), and Ausubel, Cramton, and Milgrom’s Clock-Proxy AuctionCramton, and Milgrom’s Clock-Proxy Auction

Problem:Problem:

• Bidders in an iterative multi-unit auction Bidders in an iterative multi-unit auction can often benefit by waiting to reveal their can often benefit by waiting to reveal their intentionsintentions

• This can slow auctions and undermine the This can slow auctions and undermine the purpose of the iterative auction to reveal purpose of the iterative auction to reveal accurate price information (price discovery)accurate price information (price discovery)

Solution: Activity RulesSolution: Activity Rules

Summary of TalkSummary of Talk

• Ausubel, Cramton & Milgrom’s Rule: RPAusubel, Cramton & Milgrom’s Rule: RP

• A Problem with RP (from Harsha et al.)A Problem with RP (from Harsha et al.)

• A New Activity Rule: RPBA New Activity Rule: RPB

NotationNotation

For a specific bidder, letFor a specific bidder, let

pptt = = Price vector announced at time Price vector announced at time tt (non-decreasing in (non-decreasing in tt))

xxtt = = Bid vector reported at time Bid vector reported at time tt

v(x)v(x) = = Value of the bundle Value of the bundle xx (to this (to this bidder)bidder)

u(p,x)u(p,x) = = Utility of bundle Utility of bundle xx at price at price pp

FCC Activity RuleFCC Activity Rule

• Aggregate demand (expressed in MHz-pop) may Aggregate demand (expressed in MHz-pop) may not increase as prices increasenot increase as prices increase

• Problem: bidders “park” their bids on licenses Problem: bidders “park” their bids on licenses with the cheapest MHz-pops to maintain eligibility with the cheapest MHz-pops to maintain eligibility later, distorting price discoverylater, distorting price discovery

• Ausubel, Cramton, & Milgrom argue that their Ausubel, Cramton, & Milgrom argue that their Revealed Preference activity rule provides an Revealed Preference activity rule provides an improvementimprovement

Revealed Preference Activity RuleRevealed Preference Activity Rule(Ausubel, Cramton, and Milgrom)(Ausubel, Cramton, and Milgrom)

• Bidder Preferences are assumed to be quasi-linear:Bidder Preferences are assumed to be quasi-linear:

u(p,x) = v(x) – p u(p,x) = v(x) – p · · xx

• The rule enforces consistency of preferences for The rule enforces consistency of preferences for any pair of bid vectors any pair of bid vectors xxss and and xxtt with with s < ts < t

that is...that is...

Revealed Preference Activity RuleRevealed Preference Activity Rule(Ausubel, Cramton, and Milgrom)(Ausubel, Cramton, and Milgrom)

v(xv(xss) – p) – pss · · xxss ≥ ≥ v(xv(xtt) – p) – pss · · xxtt

andand

v(xv(xtt) – p) – ptt · · xxtt ≥ ≥ v(xv(xss) – p) – ptt · · xxss

But since But since v(v(·)·) is unknown, we cancel and get rule is unknown, we cancel and get rule RPRP

(p(pt t – p– pss) · (x) · (xtt – x – xss) ≤ 0) ≤ 0

Revealed Preference Activity RuleRevealed Preference Activity Rule(Ausubel, Cramton, and Milgrom)(Ausubel, Cramton, and Milgrom)

(p(pt t – p– pss) · (x) · (xtt – x – xss) ≤ 0) ≤ 0

• For a single item: demand must decrease as For a single item: demand must decrease as price increasesprice increases

• Further ACM argue that the rule performs Further ACM argue that the rule performs as desired for cases of perfect substitutes as desired for cases of perfect substitutes and perfect complements or a mix of bothand perfect complements or a mix of both

A Weakened Revealed Preference A Weakened Revealed Preference Activity RuleActivity Rule

(p(pt t – p– pss) · (x) · (xtt – x – xss) ≤ ) ≤ αα

• Recent presentations of the clock-proxy Recent presentations of the clock-proxy indicate that a weakened form may be indicate that a weakened form may be desirabledesirable

• Definition:Definition:Budget-constrained quasi-linear utilityBudget-constrained quasi-linear utility

uuBB(p,x) = (p,x) = v(x) – p v(x) – p · · xx if p if p · · x x ≤ B≤ B

00 otherwiseotherwise

• Definition:Definition:An activity rule is An activity rule is consistentconsistent if an if an

honest honest bidder never causes a violation of bidder never causes a violation of the rulethe rule

A Problem with the RP ruleA Problem with the RP rule(due to Harsha et al.)(due to Harsha et al.)

• RP is not consistent when bidders haveRP is not consistent when bidders havebudget-constrained quasi-linear utilitybudget-constrained quasi-linear utility

Counter example:Counter example:A bidder for multiple units of two items has values:A bidder for multiple units of two items has values:

v(5,1) = 590v(5,1) = 590 v(4,3) = 505v(4,3) = 505 B = 515B = 515

Prices announced:Prices announced:pp11= (100,10)= (100,10) pp22 = (110, 19) = (110, 19)

Counter example (continued)Counter example (continued)

At At pp11 the bidder prefers the bidder prefers (5,1)(5,1) to to (4,3)(4,3)::

590 – (100,10) 590 – (100,10) · · (5,1) > 505 – (100,10) (5,1) > 505 – (100,10) · · (4,3)(4,3)

But at But at pp22 the bidder cannot afford the bidder cannot afford (5,1)(5,1) so so (4,3)(4,3) is is preferred.preferred.

But according to RP we must have:But according to RP we must have:

(p(pt t – p– pss) · (x) · (xtt – x – xss)= (10,9) · (-1,2)= 8 ≤ 0 )= (10,9) · (-1,2)= 8 ≤ 0

Which is violated, so the bid of Which is violated, so the bid of (4,3)(4,3) would be would be rejected, despite honest biddingrejected, despite honest bidding

• Lemma 1: If an honest, budget-constrained Lemma 1: If an honest, budget-constrained quasi-linear bidder submits a bid quasi-linear bidder submits a bid xxtt that violates that violates an RP constraint for some an RP constraint for some s < ts < t, then it must be , then it must be the case that: the case that: B < pB < ptt ·· x xss

• Proof: if Proof: if pptt ·· x xtt, p, pss ·· x xss, p, pss ·· x xtt, , andand p ptt ·· x xss ≤ B≤ B then then RP must be satisfied by an honest bidder. RP must be satisfied by an honest bidder. pptt ·· x xtt and and ppss ·· x xss must be must be ≤ B≤ B by IR. If by IR. If ppss ·· x xtt this yields this yields ppss > p > ptt, contradicting a monotonically increasing , contradicting a monotonically increasing price rule. Therefore the only other possibility is price rule. Therefore the only other possibility is B < pB < ptt ·· x xss..

Implication of Lemma 1Implication of Lemma 1

• A violation of RP can be met by a budget A violation of RP can be met by a budget constraint enforced by the auctioneerconstraint enforced by the auctioneer

• In practice a bidder will be warned that a In practice a bidder will be warned that a bid will constrain future bidding activity, bid will constrain future bidding activity, that all bids must be less than the implied or that all bids must be less than the implied or revealed budgetrevealed budget

• Should an arbitrarily large violation of the Should an arbitrarily large violation of the RP rule be accepted?RP rule be accepted?

No! Find the maximum violation for No! Find the maximum violation for which every pair of bids is consistentwhich every pair of bids is consistent

Max Max (p(pt t – p– pss) · (x) · (xtt – x – xss)) s.t.s.t.

v(xv(xss) – p) – pss · · xxss ≥ ≥ v(xv(xtt) – p) – pss · · xxtt

v(xv(xtt) – p) – ptt · · xxtt ≥ 0≥ 0

B ≥ B ≥ pptt · · xxtt (LP)(LP)

B ≥ B ≥ ppss · · xxtt

B ≥ B ≥ ppss · · xxss

B < B < pptt · · xxssWe can soften this inequality to be ≤

Lemma 2: Closed form solution to LPLemma 2: Closed form solution to LP

• Let Let B* =B* = pptt · · xxss

• Find item index Find item index j j = argmax= argmaxii (p (piitt – p – pii

ss)/p)/piitt

• Set Set xxjj*= *= pptt · · xxss/p/pjjtt

• Set Set xxii*= 0*= 0 for all for all i ≠ ji ≠ j

Claim:Claim: B* B* and and x* x* form a solution to the LP form a solution to the LP from the previous slidefrom the previous slide

Proof: See paper. (Email me.)Proof: See paper. (Email me.)

Refined Activity Rule RPBRefined Activity Rule RPB

PSEUDO-CODEPSEUDO-CODE

For demand vector For demand vector xxtt submitted at time submitted at time tt

Compute Compute (p(pt t – p– pss) · (x) · (xtt – x – xss)) for each for each s < ts < t

1. If for all 1. If for all s < ts < t, , (p(pt t – p– pss) · (x) · (xtt – x – xss)) ≤ 0 ≤ 0

Then accept the bid with no stipulationThen accept the bid with no stipulation

(continued…)(continued…)

Refined Activity Rule RPB (cont.)Refined Activity Rule RPB (cont.)

2. If for some 2. If for some s < ts < t,,

(p(pt t – p– pss) · (x* – x) · (x* – xss)) ≥ ≥ (p(pt t – p– pss) · (x) · (xtt – x – xss)) > 0 > 0

Accept bid with implied budget Accept bid with implied budget B < pB < pt t · x· xss

3. 3. If for some If for some s < ts < t,,

(p(pt t – p– pss) · (x) · (xtt – x – xss)) > > (p(pt t – p– pss) · (x* – x) · (x* – xss))

Reject bid as dishonestReject bid as dishonest

In Summary:In Summary:

• RPB is a strict relaxation of the RP activity RPB is a strict relaxation of the RP activity rulerule

• Violations of the RP rule are limited and Violations of the RP rule are limited and result in budget restrictions on future result in budget restrictions on future biddingbidding

• This overcomes the inconsistency of the RP This overcomes the inconsistency of the RP rule when bidders have budget-constrained rule when bidders have budget-constrained quasi-linear utilitiesquasi-linear utilities

Questions for future studyQuestions for future study

• Is RPB an adequate relaxation of RP, so Is RPB an adequate relaxation of RP, so that an arbitrary that an arbitrary αα-weakening is -weakening is unnecessary?unnecessary?

• Or will the need for Bayesian learning Or will the need for Bayesian learning prove that even RPB is too restrictive?prove that even RPB is too restrictive?

• How do we measure the effectiveness of How do we measure the effectiveness of any activity rule for encouraging price any activity rule for encouraging price discovery/discouraging “parking”?discovery/discouraging “parking”?