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 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
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
Summary of Talk
• Ausubel, Cramton & Milgrom’s Rule: RP
• A Problem with RP (from Harsha et al.)
• A New Activity Rule: RPB
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
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
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...
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
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
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
• Definition:
Budget-constrained quasi-linear utility
uB(p,x) =
v(x) – p · x
0
if p · x ≤ B
otherwise
• 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 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
Prices announced:
p1= (100,10)
B = 515
p2 = (110, 19)
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
• 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 · xt
and 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.
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?
No! Find the maximum violation for
which every pair of bids is consistent
Max (pt – ps) · (xt – xs)
v(xs) – ps · xs ≥ v(xt) – ps · xt
v(xt) – pt · xt ≥ 0
B ≥ pt · xt
B ≥ ps · xt
B ≥ ps · xs
B < pt · xs
s.t.
(LP)
We can soften this
inequality to be ≤
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.)
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…)
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
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
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”?