A Revealed-Preference Activity Rule for Quasi-Linear Utilities with Budget Constraints

1. 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

2. Multi-Unit Auctions • In 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-vector • Examples include the Simultaneous Ascending Auction (used by the FCC), and Ausubel, Cramton, and Milgrom’s Clock-Proxy Auction

3. Problem: • Bidders 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) Solution: Activity Rules

4. Summary of Talk • Ausubel, Cramton & Milgrom’s Rule: RP • A Problem with RP (from Harsha et al.) • A New Activity Rule: RPB

5. Notation For a specific bidder, let pt = Price vector announced at time t (non-decreasing in t) xt = 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

6. FCC Activity Rule • Aggregate 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 discovery • Ausubel, Cramton, & Milgrom argue that their Revealed Preference activity rule provides an improvement

7. Revealed Preference Activity Rule(Ausubel, Cramton, and Milgrom) • 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 xs and xt with s < t that is...

8. Revealed Preference Activity Rule(Ausubel, Cramton, and Milgrom) v(xs) – ps· xs≥ v(xt) – ps· xt and v(xt) – pt· xt≥ v(xs) – pt· xs But since v(·) is unknown, we cancel and get rule RP (pt – ps) · (xt – xs) ≤ 0

9. Revealed Preference Activity Rule(Ausubel, Cramton, and Milgrom) (pt – ps) · (xt – xs) ≤ 0 • For 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 both

10. A Weakened Revealed Preference Activity Rule (pt – ps) · (xt – xs) ≤ α • Recent presentations of the clock-proxy indicate that a weakened form may be desirable

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

12. A Problem with the RP rule(due to Harsha et al.) • RP 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) = 590 v(4,3) = 505 B = 515 Prices announced: p1= (100,10) p2 = (110, 19)

13. Counter example (continued) At p1 the bidder prefers (5,1) to (4,3): 590 – (100,10) · (5,1) > 505 – (100,10) · (4,3) But at p2 the bidder cannot afford (5,1) so (4,3) is preferred. But according to RP we must have: (pt – ps) · (xt – xs)= (10,9) · (-1,2)= 8 ≤ 0 Which is violated, so the bid of (4,3) would be rejected, despite honest bidding

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

15. Implication of Lemma 1 • A 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 budget • Should an arbitrarily large violation of the RP rule be accepted?

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

17. Lemma 2: Closed form solution to LP • Let B* =pt· xs • Find item index j = argmaxi (pit – pis)/pit • Set xj*= pt· xs/pjt • Set xi*= 0 for all i ≠ j Claim:B* and x* form a solution to the LP from the previous slide Proof: See paper. (Email me.)

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

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

20. In Summary: • RPB is a strict relaxation of the RP activity rule • Violations 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 utilities

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