A Revealed-Preference Activity Rule for Quasi-Linear Utilities with Budget Constraints Robert Day, University of Connecticut with special thanks to: Pavithra Harsha, Cynthia Barnhart, MIT and David Parkes, Harvard

Multi-Unit Auctions In auctions for spectrum licenses (for example), many items may be auctioned simultaneously through an iterative procedureIn auctions for spectrum licenses (for example), many items may be auctioned simultaneously through an iterative procedure We consider an environment in which bidders report demand amounts at the current price-vectorWe consider an environment in which bidders report demand amounts at the current price-vector Examples include the Simultaneous Ascending Auction (used by the FCC), and Ausubel, Cramton, and Milgrom’s Clock-Proxy AuctionExamples include the Simultaneous Ascending Auction (used by the FCC), and Ausubel, Cramton, and Milgrom’s Clock-Proxy Auction

Problem: Bidders in an iterative multi-unit auction can often benefit by waiting to reveal their intentionsBidders in an iterative multi-unit auction can often benefit by waiting to reveal their intentions This can slow auctions and undermine the purpose of the iterative auction to reveal accurate price information (price discovery)This can slow auctions and undermine the purpose of the iterative auction to reveal accurate price information (price discovery) Solution: Activity Rules

Summary 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

Notation For a specific bidder, let p t =Price vector announced at time t (non-decreasing in t) x t =Bid vector reported at time t v(x) =Value of the bundle x (to this bidder) u(p,x) =Utility of bundle x at price p

FCC Activity Rule Aggregate demand (expressed in MHz-pop) may not increase as prices increaseAggregate demand (expressed in MHz-pop) may not increase as prices increase Problem: bidders “park” their bids on licenses with the cheapest MHz-pops to maintain eligibility later, distorting price discoveryProblem: bidders “park” their bids on licenses with the cheapest MHz-pops to maintain eligibility later, distorting price discovery Ausubel, Cramton, & Milgrom argue that their Revealed Preference activity rule provides an improvementAusubel, Cramton, & Milgrom argue that their Revealed Preference activity rule provides an improvement

Revealed Preference Activity Rule (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 · x The rule enforces consistency of preferences for any pair of bid vectors x s and x t with s < tThe rule enforces consistency of preferences for any pair of bid vectors x s and x t with s < t that is...

Revealed Preference Activity Rule (Ausubel, Cramton, and Milgrom) v(x s ) – p s · x s ≥ v(x t ) – p s · x t and v(x t ) – p t · x t ≥ v(x s ) – p t · x s But since v(·) is unknown, we cancel and get rule RP (p t – p s ) · (x t – x s ) ≤ 0

Revealed Preference Activity Rule (Ausubel, Cramton, and Milgrom) (p t – p s ) · (x t – x s ) ≤ 0 For a single item: demand must decrease as price increasesFor a single item: demand must decrease as price increases Further ACM argue that the rule performs as desired for cases of perfect substitutes and perfect complements or a mix of bothFurther ACM argue that the rule performs as desired for cases of perfect substitutes and perfect complements or a mix of both

A Weakened Revealed Preference Activity Rule (p t – p s ) · (x t – x s ) ≤ α Recent presentations of the clock-proxy indicate that a weakened form may be desirableRecent presentations of the clock-proxy indicate that a weakened form may be desirable

Definition:Definition: Budget-constrained quasi-linear utility u B (p,x) = v(x) – p · xif p · x ≤ B 0otherwise Definition:Definition: An activity rule is consistent if an honest bidder never causes a violation of the rule

A Problem with the RP rule (due to Harsha et al.) RP is not consistent when bidders haveRP is not consistent when bidders have budget-constrained quasi-linear utility Counter example: A bidder for multiple units of two items has values: v(5,1) = 590v(4,3) = 505B = 515 Prices announced: p 1 = (100,10)p 2 = (110, 19)

Counter example (continued) At p 1 the bidder prefers (5,1) to (4,3): 590 – (100,10) · (5,1) > 505 – (100,10) · (4,3) But at p 2 the bidder cannot afford (5,1) so (4,3) is preferred. But according to RP we must have: (p t – p s ) · (x t – x s )= (10,9) · (-1,2)= 8 ≤ 0 Which is violated, so the bid of (4,3) would be rejected, despite honest bidding

Lemma 1: If an honest, budget-constrained quasi-linear bidder submits a bid x t that violates an RP constraint for some s < t, then it must be the case that: B < p t · x sLemma 1: If an honest, budget-constrained quasi-linear bidder submits a bid x t that violates an RP constraint for some s < t, then it must be the case that: B < p t · x s Proof: if p t · x t, p s · x s, p s · x t, and p t · x s ≤ B then RP must be satisfied by an honest bidder. p t · x t and p s · x s must be ≤ B by IR. If p s · x t this yields p s > p t, contradicting a monotonically increasing price rule. Therefore the only other possibility is B p t, contradicting a monotonically increasing price rule. Therefore the only other possibility is B < p t · x s.

Implication of Lemma 1 A violation of RP can be met by a budget constraint enforced by the auctioneerA violation of RP can be met by a budget constraint enforced by the auctioneer In practice a bidder will be warned that a bid will constrain future bidding activity, that all bids must be less than the implied or revealed budgetIn practice a bidder will be warned that a bid will constrain future bidding activity, that all bids must be less than the implied or revealed budget Should an arbitrarily large violation of the RP rule be accepted?Should an arbitrarily large violation of the RP rule be accepted?

No! Find the maximum violation for which every pair of bids is consistent Max (p t – p s ) · (x t – x s )s.t. v(x s ) – p s · x s ≥ v(x t ) – p s · x t v(x t ) – p t · x t ≥ 0 B ≥ p t · x t (LP) B ≥ p s · x t B ≥ p s · x s B < p t · x s We can soften this inequality to be ≤

Lemma 2: Closed form solution to LP Let B* = p t · x sLet B* = p t · x s Find item index j = argmax i (p i t – p i s )/p i tFind item index j = argmax i (p i t – p i s )/p i t Set x j *= p t · x s /p j tSet x j *= p t · x s /p j t Set x i *= 0for all i ≠ jSet x i *= 0for all i ≠ j Claim: B* and x* form a solution to the LP from the previous slide Proof: See paper. ( me.)

Refined Activity Rule RPB PSEUDO-CODE For demand vector x t submitted at time t Compute (p t – p s ) · (x t – x s ) for each s < t 1. If for all s < t, (p t – p s ) · (x t – x s ) ≤ 0 Then accept the bid with no stipulation (continued…)

Refined Activity Rule RPB (cont.) 2. If for some s < t, (p t – p s ) · (x* – x s ) ≥ (p t – p s ) · (x t – x s ) > 0 Accept bid with implied budget B < p t · x s 3. If for some s < t, (p t – p s ) · (x t – x s ) > (p t – p s ) · (x* – x s ) Reject bid as dishonest

In Summary: RPB is a strict relaxation of the RP activity ruleRPB is a strict relaxation of the RP activity rule Violations of the RP rule are limited and result in budget restrictions on future biddingViolations of the RP rule are limited and result in budget restrictions on future bidding This overcomes the inconsistency of the RP rule when bidders have budget-constrained quasi-linear utilitiesThis overcomes the inconsistency of the RP rule when bidders have budget-constrained quasi-linear utilities

Questions for future study Is RPB an adequate relaxation of RP, so that an arbitrary α-weakening is unnecessary?Is RPB an adequate relaxation of RP, so that an arbitrary α-weakening is unnecessary? Or will the need for Bayesian learning prove that even RPB is too restrictive?Or will the need for Bayesian learning prove that even RPB is too restrictive? How do we measure the effectiveness of any activity rule for encouraging price discovery/discouraging “parking”?How do we measure the effectiveness of any activity rule for encouraging price discovery/discouraging “parking”?