Collusion and the use of
false names
Vincent Conitzer
conitzer@cs.duke.edu
Collusion in the Vickrey auction
• Example: two colluding bidders
v1 = first colluder’s
true valuation
v2 = second
colluder’s true
valuation
b = highest bid
among other bidders
0
price colluder 1 would pay when
colluders bid truthfully
gains to be distributed among colluders
price colluder 1 would pay if
colluder 2 does not bid
Rules for colluding
• How do the colluders split the gains?
• If the colluders do not initially know each other’s
valuations, how do the colluders communicate their
valuations to each other?
• Do colluders have incentives to lie to each other?
• Do colluders have incentives to deviate from their
agreed behavior (submit a different bid than they said
they would)?
Example
• Colluders report valuations to each other,
• Gains from colluding distributed evenly among
colluders
v1 = first colluder’s
true valuation
v2 = second
colluder’s true
valuation
b = highest bid
among other bidders
0
gain that first colluder would have
had anyway
collusion gains to be distributed
(evenly) among the two colluders
i.e. first colluder pays
second colluder half of this
Which colluder has an
incentive to lie?
Bidding rings
• Bidding ring = organized collusion protocol for subset of agents
• Suppose there is an agent with no interest in the item for sale,
but who is willing to organize the collusion (potentially at a profit)
– The ring center
• Collusion protocol for the Vickrey auction:
– Every colluder submits a bid to the ring center in a pre-auction,
– Ring center submits (only) the highest of these bids in the auction,
– If ring center wins, then
• she must pay the second-highest bid in the auction (p),
• she awards the item to the colluder with the highest bid,
• this colluder pays the ring center:
– the maximum of p, and the second-highest bid in the pre-auction
• From perspective of colluders, same as standard Vickrey
auction
• Ring center can make a profit
– Center can pay agents some constant amount k to participate in ring
– Then strictly better for agents to join ring
Other reasons colluders may
respect arrangements
• Repeated interaction with other colluders
– Breaking the collusion agreement may imply never being
able to collude again
– Other colluders may even try to “punish” the deviants
– ~ repeated games, folk theorems
• “Colluders” act on behalf of one agent
– False-name bidding, coming up shortly
Collusion under GVA in
combinatorial auctions: example
• Suppose there are two items for sale, A and B
– Free disposal
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•
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•
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Bidder 1 bids: ({A, B}, b)
Bidder 2 bids: ({A, B}, b-ε)
If these are the only bids, bidder 1 wins and pays b-ε
Now suppose two more bids arrive:
Bidder 3 bids: ({A}, b’) (where b’ > b)
Bidder 4 bids: ({B}, b’)
Now bidders 3 and 4 win, pay nothing
Bidders 3 and 4 may well be colluding
– E.g. maybe they really each value their item at < b, or even < b/2
• Also, if b’ is sufficiently large, neither colluder has an incentive
to deviate from this collusive agreement
Under what conditions can the colluders get
everything for free? [Conitzer & Sandholm AAMAS06]
• Theorem: can do so if and only if there is some way of
assigning the items to the colluders so that:
– each item is assigned to exactly one colluder,
– for each (positive) bid by a noncolluder, at least two colluders have
items in that bid assigned to them
• Proof:
– “If” direction:
• Let each colluder bid a huge amount on the bundle of items assigned to him
• Why does this work?
– “Only if” direction:
• Suppose such an assignment is not possible
• Suppose the colluders win everything
• There must be a (positive) noncolluder bid, all of whose items are contained
in one colluder’s bundle
• Then that colluder must pay at least that bid’s value
• But: NP-complete to decide whether such an assignment is
possible (even with two colluders)
What if there is no free disposal?
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•
•
•
Suppose there are two items for sale, A and B
Bidder 1 bids: ({A, B}, b)
Bidder 2 bids: ({A, B}, b-ε)
If these are the only bids, bidder 1 wins and pays b-ε
Now suppose two more bids arrive (colluders):
Bidder 3 bids: ({A}, b’) (where b’ > b)
Bidder 4 bids: ({B}, b’)
Now bidders 3 and 4 win, and each is paid b’ - b
Note: b’ can be arbitrarily large!
Characterization without free disposal
• Theorem: the colluders can receive all items and each be paid
an arbitrary large amount, if and only if
– there is some way of assigning the items to the colluders so that:
– for each colluder, the bundle of items assigned to him cannot be
covered exactly with (i.e. partitioned into) noncolluder bids
• Proof:
– “If” direction:
• Let each colluder bid a huge amount on the bundle of items assigned to him
• Why does this work?
– “Only if” direction:
• Suppose such an assignment is not possible
• Suppose the colluders win everything
• There must be a colluder whose bundle can be covered exactly with
noncolluder bids
• Then that colluder cannot be paid an arbitrarily large amount
• Again, NP-complete to decide whether such an assignment is
possible (even with two colluders)
What if colluders only care about the
total (sum) payment to them?
• Theorem: without free disposal, two (or more) colluders can
receive all items and be paid an arbitrary large amount in total,
if and only if:
– there is at least one item s that does not receive a singleton bid (i.e. a
bid on {s}) from a noncolluder
• Proof:
– “If” direction:
• Have one colluder bid on {s}
• Have another colluder bid on the complement I-{s} with a huge value
– “Only if” direction:
• If every item has a noncolluder singleton bid on it, then every colluder
bundle can be covered exactly with noncolluder bids
• Computationally easy to decide
• More characterizations (including combinatorial reverse
auctions and exchanges) in [Conitzer & Sandholm AAMAS06]
False-name bidding [Yokoo et al. AIJ2001]
• Suppose a combinatorial auction for items A and B is being run
over the Internet, using GVA
• You know that the other bids are
– Bidder 1 bids: ({A, B}, b)
– Bidder 2 bids: ({A, B}, b-ε)
• You would like to own both items
• You can sign up for as many accounts as you like, and bid from
each of them
• Auctioneer cannot detect whether two accounts belong to the
same person, so must treat each account as a different bidder
• What will you do?
– Hint: you can “collude with yourself” using multiple accounts
• We say that a mechanism is false-name proof if it is (weakly)
dominant to use only one account and report your true value
• GVA is not false-name proof: you (sometimes) have an
incentive to open multiple accounts
• Theorem: no efficient false-name proof CA mechanism exists
Characterization of false-name proof
combinatorial auctions [Yokoo IJCAI03]
• Strategy-proof (not false-name proof) combinatorial auctions
can always be characterized as follows:
– For every bidder i, for every bundle B, a price pi, B(θ-i’) is determined as a
function of the other bids;
– Every i is allocated a bundle B that maximizes v(θi’, B) - pi, B(θ-i’)
• θi’, θ-i’ are reported valuations
• Assume weakly anonymous pricing: pi, B(θ-i’) = pB(θ-i’)
– … makes sense in settings where bidders are anonymous…
• A mechanism is false-name proof if and only if it is strategyproof, and it satisfies No SuperAdditive price increase (NSA),
which means that the following must always hold:
– For a subset S of bidders,
– if Bi is the bundle that i gets,
– then it must be the case that Σi in SpBi(θ-i’) ≥ pUi in SBi(θ-S’)
When is GVA false-name proof?
[Yokoo et al. Games and Economic Behavior 2003]
• For a subset of bidders X, let V(X) be the maximum
allocation value that can be obtained using only
bidders in X
• Say V is concave if for all subsets of bidders X, Y, Z
where Y is a subset of Z, V(XUY) - V(Y) ≥ V(XUZ) V(Z)
• GVA is false-name proof if bidders report types from a
set such that V is always concave
Max-Minimal Bundle (M-MB)
mechanism [Yokoo IJCAI03]
• Bundle B is minimal for bidder j if any smaller bundle
will give j a lower utility (according to the reported
type)
• Set price pi, B(θ-i’) = maxj≠i, B’ minimal for j, B∩B’ ≠ Øvj(θj’, B’)
• Always possible to give each agent i a bundle B that
maximizes v(θi’, B) - pi, B(θ-i’) (why?)
• Satisfies NSA/false-name proofness (why?)
• Other false-name proof combinatorial auction
mechanisms:
– Leveled Division Set [Yokoo et al. AIJ01]
– Groves Mechanism-Submodular Approximation [Yokoo et
al. AAMAS06]
Collusion and false names in
coalitional game theory [Yokoo et al. AAAI05]
• Suppose there is a set of skills T that agents can contribute
– E.g. agents are working on a computer science project
– Skills: Theory (T), Coding (C), Writing (W)
• There is a characteristic function v(S) (for S subset of T)
– Value that agents can achieve when union of agents’ skills is S
– Increasing in skills
– E.g. v({T}) = 0, v({C}) = 2, v({W}) = 0, v({T, C}) = 5, v({T, W}) = 5, …
• Assume each skill is held by at most one agent
• Agents report which skills they have
– Agents cannot report skills that they do not have
• When the time comes to use the skill, their lie would be discovered
• Agents can:
–
–
–
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hide skills,
use false names (and split up their skills across multiple names),
collude (join their skills under a single name),
combinations of all of these
How should we distribute the value?
• Consider the following example:
– v({T, C, W}) = 1
– v = 0 everywhere else
• Suppose agent 1 can do Theory, 2 can Code, 3 can Write
• Characteristic function over agents:
– w({1, 2, 3}) = 1,
– w = 0 everywhere else
• Reasonable solution concepts that only use w (Shapley value,
nucleolus) will give each agent 1/3
• Now suppose 1 can do Theory and Coding, 2 can Write
• Characteristic function over agents:
– w({1, 2}) = 1,
– w = 0 everywhere else
• Reasonable solution concepts that only use w (Shapley value,
nucleolus) will give each agent 1/2
• But then, agent 1 is better off pretending to be two agents (one
who can do Theory and one who can Code) to get 1/3 + 1/3
Why not use v?
• What if we just use v, and award payoffs to the skills rather
than the agents?
– … using Shapley value, nucleolus…
• Now there is no incentive to use false names/collusion
– A skill will get the same payoff no matter who it is submitted by
• What about hiding skills?
• Consider
– v({T, W}) = v({C, W}) = v({T, C, W}) = 1,
– v = 0 everywhere else
• Suppose all three skills are present
– Shapley value will give 2/3 to Writing, nucleolus 1 to Writing
• Suppose agent 1 can do Theory and Code, 2 can Write
• 1 is better off just reporting Theory:
– Characteristic function will be v({T, W}) = 1, v = 0 everywhere else
– 1 gets ½
• To make hiding suboptimal, a greater set of skills must be
rewarded more