Collusion and the use of false names

1. Collusion and the use of false names Vincent Conitzer conitzer@cs.duke.edu

2. Collusion in the Vickrey auction • Example: two colluding bidders v1 = first colluder’s true valuation v2 = second colluder’s true valuation price colluder 1 would pay when colluders bid truthfully gains to be distributed among colluders b = highest bid among other bidders price colluder 1 would pay if colluder 2 does not bid 0

3. 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)?

4. Example • Colluders report valuations to each other, • Gains from colluding distributed evenly among colluders v1 = first colluder’s true valuation gain that first colluder would have had anyway v2 = second colluder’s true valuation collusion gains to be distributed (evenly) among the two colluders i.e. first colluder pays second colluder half of this b = highest bid among other bidders Which colluder has an incentive to lie? 0

5. 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

6. 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

7. Collusion under GVA in combinatorial auctions: example • Suppose there are two items for sale, A and B • Free disposal • 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

8. 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)

9. What if there is no free disposal? • 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!

10. 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)

11. 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]

12. 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

13. 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 strategy-proof, 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’)

14. 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

15. 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]

16. 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: • 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

17. 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

18. 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