1. Algorithmic Applications of Game Theory Lecture 8 1

2. Reading Chapter 9 and chapter 11 in Algorithmic Game Theory. 2

3. Reminder - model Each bidder i has a valuation function v i which takes a subset of the objects, and says how much the bidder wants this subset – In the simple case of a single item v i is just how much the bidder values the item Bidder i will pay a price p i. The utility (happiness) of bidder i is denoted u i, with u i (x i ) = v i (x i ) - p i where bidder i received x i (either the item or an empty set) Auctioneer does not know the bidder’s valuation function! 3

4. Second price auction Winner is highest bidder Price is the second highest bid This is what you really get in the ascending auction Dominant strategy truthful 4

5. Auctioning more than one good A player i has an additive valuation function if for every two sets of good S, T If all players have additive valuations, independent auctions make sense. 5

6. Non additive valuations I want to have a salad with tomatoes and cucumbers. I don’t want just tomatoes or just cucumbers. So: v({cucumber, tomato}) > v({cucumber}) + v({tomato}) I want to have a fruit – either a banana or an apple, and I don’t care which. So: v({A,B}) = v({A}) = v({B}) < v({A}) + v({B}) 6

7. Dominant strategies Suppose we have an auction, in which each player has a dominant strategy for every valuation function v i he has Revelation Principle: there exists a mechanism in which it is dominant for every player to report his valuation function v i 7

8. Efficiency Usually easier than revenue Defined as the sum of utilities of the players and the utility of the auctioneer But the utility of player i is v i (x i ) – p i And the utility of the auctioneer is  p i So the sum of utilities is  v i (x i ) which is independent of the prices 8

9. Why do we need prices then? Each player will report the value of each bundle to her The auctioneer will compute the best allocation, and this will be the outcome Problem: players will have incentive to report that the goods are very valuable to them We need prices to elicit the true values from the players 9

10. VCG auction 10

11. Making money 11

12. Clarke’s payment The rule satisfies: – No player looses from participating in the auction – The auctioneer never pays a player Meaning: player i is charged the damage that he caused to the world 12

13. Examples of VCG – Public Project VCG is not just for auctions… The government can build a bridge, which would cost C Each citizen has value v i for the bridge We want to build the bridge iff  v i ≥ C Use VCG to decide. Plugging in Clarke’s payment rule gives that i pays max(0, C-  k  i v k ) Each player pays at most v i The government will get its money back iff  v i = C 13

14. Example of VCG - trade A seller has an object, values it for v s Buyer values it for v b We want them to trade if v b > v s Doing this by VCG, means that the mechanism charges v s from the buyer, and pays v b to the seller, loosing money… 14

15. We are not making much money here… Bummer If you want citizens not to pay if the bridge is not built, or buyers not to pay when there is no trade, then these are the only possible prices Getting revenue truthfully is hard (and interesting!) 15

16. Disadvantages of VCG Complexity – the agents need to describe their valuation for every possible bundle (m items  2 m numbers) Even if valuations have succinct representations, the auctioneer has to find the optimal solution for VCG to be truthful – If you have an approximation algorithm, you loose truthfulness 16

17. Is it hard to find the optimal solution? Yes. A bidder i is single minded if there exists a set S i such that v i (S) = a i if S i  S, and v i (S)=0 otherwise Theorem: Even if all the bidders are single minded, it is NP hard to find the optimal solution. Moreover, even finding an approximate solution which would generate more than OPT/m 0.5 welfare is NP hard 17

18. Proof Reduction from independent set Given a graph G with m edges and n vertices: – Each edge becomes an item – For each vertex we add a player who wants exactly the set of items adjacent to each, and has value 1 for that set If two vertices are adjacent, the corresponding players can not be satisfied together – Getting a welfare of k gives an independent set of size k But approximating independent set better than n 1-  is NP hard – So approximating better than m 1/2 -  is also NP hard 18