1. False-name Bids “The effect of false-name bids in combinatorial
auctions: new fraud in internet auctions”
Makoto Yokoo, Yuko Sakurai, and Shigeo Matsubara

2. Outline What are false-name bids? Why are false-name bids a problem?
Making an auction false-name-proof
Collusion in general
Conclusion

3. What are false-name bids?
A bidder submits additional bids under other identities
These are binding bids: the bidder pays the cost for the bid and gets the item if it wins.
Easy to do on the Internet
It's easy to get multiple addresses.
Paper doesn't mention other ways to prove identity on the Internet.

4. What are false-name bids?
Every agent has a set of identities they can use.
Every agent must use at least one identity (i.e. they must submit at least one bid).
Every identity belongs to exactly one agent.
No one can impersonate another agent.
Auctioneer does not know which agent each identity belongs to, or even how many real agents there are.
No agent knows who an identity belongs to, unless it's their own.

5. What are false-name bids?
The auction:
Feasibility: Cannot allocate an item to multiple identities, even if they belong to the same agent.
Budget: The payment to the auctioneer is equal to the total paid by all the identities.
Non-participation: If an identity makes no bids, then it gets no items and pays nothing.
Almost anonymous: The order of identities does not affect the outcome, except if there is a tie.
If two bids are the same, then
v(kidi(θ), θidi) + tidi(θ) = v(kidj(θ), θidj)+ tidj(θ) holds.

6. What are false-name bids?
The auction (continued):
The strategy for an agent is the set of the bids of all their identities.
An agent's utility is the sum of their values for all the items their identities win plus the total payments for their identities.
An agent's strategy is dominant if the agent has no reason to change the bids of any of its identities, regardless of the bids of all other identities.

7. Why are false-name bids a problem?
Example:
We run a combinatorial auction with a VCG mechanism. There are two items and two bidders. They specify their values as a triple: their values for the two items individually, and then their value for the pair. Their values are as follows:
Agent 1: (7, 7, 14)
Agent 2: (0, 0, 12)
Agent 1 wins both items.
Cost: 12 – 0 = 12

8. Why are false-name bids a problem?
But what if agent 1 could make a false-name bid?
Agent 1b is agent 1's super-secret identity:
Agent 1: (7, 0, 7)
Agent 2: (0, 0, 12)
Agent 1b: (0, 7, 7)
Cost to agent 1: 12 – 7 (utility of agent 1b) = 5
Cost to agent 1b: 12 – 7 (utility of agent 1) = 5
Total cost to agent 1 is 10.

9. Why are false-name bids a problem?
What we want is a combinatorial auction that's false-name-proof.
False-name-proof:
“For all bidder i, s∗(θi, φ(i)) = (θi, 0, . . ., 0) is a dominant strategy.”

10. Why are false-name bids a problem?
But, “[i]n combinatorial auctions, there exists no false-name-proof auction protocol that satisfies Pareto efficiency.” :(
Satisfies Pareto efficiency:
The mechanism distributes the items optimally, maximizing social welfare.

11. Making an auction false-name-proof
Want a sufficient condition condition which makes the VCG mechanism false-name-proof.
Surplus function U:
U(B, Y) = maxk∈KB,Y Σ(yi, θyi)∈Yv(kyi, θyi)
The value of the optimal allocation of the items in B to the agents in Y
UA(Y) = U(A, Y), where A contains all the items in the auction.

12. Making an auction false-name-proof
UA(·) is concave over bidders if
UA(Z ∪ W) − UA(Z) <= UA(Y ∪ W) − UA(Y)
for all sets of bidders Y, Z, and W, where Y is a subset of Z.

13. Making an auction false-name-proof
The VCG mechanism is false-name-proof if:
Θ satisfies that UA(·) is concave for every subset of bidders with types in Θ.
Each declared type is in Θ ∪ {0}
Paper does not claim that this is the only sufficient condition.
Paper does not claim that this condition is necessary to be false-name-proof.

14. Making an auction false-name-proof
Gross substitutes condition:
If an item j is demanded at price p, then it will still be demanded at price p if the prices of some other items increase.
No proof given that this condition implies concavity.

15. Making an auction false-name-proof
The gross substitutes condition holds in auctions where marginal utility of each item never increases.
This condition is convenient because auctions that satisfy it have Walrasian equilibria:
No agent has reason to change their bids because they cannot afford any better allocation.

16. Making an auction false-name-proof
U is submodular for all set of bidders X ⊆ N if
U(B, X) + U(C, X) >= U(B ∪ C, X) + U(B ∩ C, X)
for all B ⊆ A and C ⊆ A.
If U is submodular for all set of bidders X ⊆ N, then UA is concave.
In fact, submodularity is both sufficient necessary for concavity.

17. Making an auction false-name-proof
U(B, X) + U(C, X) >= U(B ∪ C, X) + U(B ∩ C, X)
Submodularity has nothing to do with the payments in the auction.
It cannot be achieved by changing the formula for determining payments.
To achieve submodularity, the allowed bids must be restricted.

18. Making an auction false-name-proof
Applying submodularity to our example:
U(B, X) + U(C, X) >= U(B ∪ C, X) + U(B ∩ C, X)
Let B = {item a}, C = {item b}
X = {Agent 1 (7, 0, 7)}:
7 + 0 >= Ok
X = {Agent 2 (0, 0, 12)}:
0 + 0 >= Contradiction

19. Making an auction false-name-proof
Submodularity requires that if any bundle of items is valued:
Each item in the bundle must be valued on its own.
The sum of the values of the items individually must equal or exceed the value for the bundle.
The value for every subset of the bundle must meet these criteria as well.

20. Making an auction false-name-proof
Submodularity eliminates one of the main reasons for using a combinatorial auction in the first place!
Since submodularity is necessary for concavity, any VCG mechanism which uses concavity to ensure it's false-name-proof will also have this problem.
This includes using gross substitution.

21. Making an auction false-name-proof
The submodular restrictions on individual bids are not enough to guarantee concavity.
Also need submodularity among bids from all subsets of agents.
The paper says nothing on how to make an auction submodular.

22. Making an auction false-name-proof
To sum up:
Concavity is sufficient but not necessary for making a VCG mechanism false-name-proof.
Gross substitution implies concavity, and so is also sufficient.
Submodularity is both necessary and sufficient for concavity.
Therefore gross substitution also implies submodularity.
But submodularity does not allow bundles to be valued higher than the sum of their items' values.
Concavity and gross substitution also have this problem.

23. Collusion in general
A mechanism is group-strategy proof if there can be no group of bidders such that:
Every member is untruthful.
Every member obtains at least the same utility as if they told the truth.
Group-strategy-proof as an auction property is independent of false-name-proof.

24. Collusion in general Possible types:
(10, 9, 18), (9, 10, 18), (10, 0, 10), (0, 10, 10)
All satisfy the gross substitutes condition.
This auction is false-name-proof.
Agent 1: (10, 9, 18)
Agent 2: (9, 10, 18)
Agent 1 gets item 1, agent 2 gets item 2, they both pay 8.

25. Collusion in general But if the agents coordinate their strategies:
Again agent 1 gets item 1 and agent 2 gets item 2. But the cost to each is 10 – 10 = 0.
Therefore this auction is not group-strategy-proof

26. Collusion in general
Group-strategy-proof but not false-name-proof example:
Auction with reserve price p
Auctioneer picks a winner at random. The winner pays p and gets the item.

27. Collusion in general
What if a group of agents can increase their total value by lying, but some agents get lower value?
Payments and exchange of goods afterwards can make every agent come out even or ahead.
If a legitimate bidder has no interest in a set of items, they can “sell” their bid.
In effect, they are selling a false-name bid to another agent.
The two agents can then split the amount the second saves by using the false-name bid.

28. Conclusion
The Internet makes false-name bids feasible because it's hard to prove identity.
No auction can be both Pareto efficient and false- name-proof.
If an auction is submodular, then it is false-name- proof.
False-name-proof is independent of group- strategy-proof.