Collusion and the use of false names Vincent Conitzer

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

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 0 b = highest bid among other bidders Colluders report valuations to each other, Gains from colluding distributed evenly among colluders v 2 = second colluder’s true valuation v 1 = first colluder’s true valuation collusion gains to be distributed (evenly) among the two colluders i.e. first colluder pays second colluder half of this gain that first colluder would have had anyway 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 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? 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 p i, B (θ -i ’) is determined as a function of the other bids; –Every i is allocated a bundle B that maximizes v(θ i ’, B) - p i, B (θ -i ’) θ i ’, θ -i ’ are reported valuations Assume weakly anonymous pricing: p i, B (θ -i ’) = p B (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 B i is the bundle that i gets, –then it must be the case that Σ i in S p B i (θ -i ’) ≥ p U i in S B i (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 p i, B (θ -i ’) = max j≠i, B’ minimal for j, B∩B’ ≠ Ø v j (θ j ’, B’) Always possible to give each agent i a bundle B that maximizes v(θ i ’, B) - p i, 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: –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