1. CPS 196.2 Expressive negotiation over donations Vincent Conitzer conitzer@cs.duke.edu

2. One donor (bidder) u( ) = 1 u( ) =.8 U = 1

3. Two independent donors u( ) = 1 u( ) =.8 u( ) = 1 u( ) =.8 U = 1

4. Two donors with a contract u( ) = 1 u( ) =.8 u( ) = 1 u( ) =.8 U =.5 +.8 = 1.3 > 1

5. Contracting using matching offers I’ll match any donation to u( ) = 1 u( ) = 1.3

6. Limitations of matching offers One-sided Involve only a single charity

7. Two charities u( ) = 1 u( ) =.8 u( ) =.3 u( ) = 1 u( ) =.3 u( ) =.8 U = 1.1

8. A different approach Donors can submit bids indicating their preferences over charities A center accepts all the bids and decides who pays what to whom

9. What do we need? A general bidding language for specifying “complex matching offers” (bids) Algorithms for the clearing problem (given the bids, who pays what to whom)

10. One charity A bid for one charity: “Given that the charity ends up receiving a total of x (including my contribution), I am willing to contribute at most w(x)” w(x) x = total payment to charity Bidder’s maximum payment Budget

11. Bid 1 w(x) x = total payment maximum payment Budget $10 $100 $30 $500 $50

12. Bid 2 w(x) x = total payment maximum payment Budget $15 $10 $100 $45 $500 $75

13. Current solution w(x) x = total payment total payment bidders are willing to make $25 $10 $100 $75 $500 $125 $43.75 45 degree line max surplus max donated

14. Tsunami event (Dagstuhl 05)

15. Problem with more than one charity Willing to give $1 for every $100 to UNICEF Willing to give $2 for every $100 to Amnesty Int’l BUDGET: $50 $5000 $50 $2500 $50 Could get stuck paying $100! w a (x a ) w u (x u ) xuxu xaxa Most general solution: w(x 1, x 2, …, x m ) –Requires specifying exponentially many values

16. Solution: separate utility and payment; assume utility decomposes $100 1 util $50 1 util $50 xaxa xuxu u u (x u ) u a (x a ) Willing to give $1 for every $100 to UNICEF Willing to give $2 for every $100 to Amnesty Int’l Budget constraint: $50 w(u u (x u )+u a (x a )) u u (x u )+ u a (x a ) 50 utils

17. The general form of a bid x1x1 u 1 (x 1 ) w(u 1 (x 1 ) + u 2 (x 2 )+ … + u m (x m )) u 1 (x 1 ) + u 2 (x 2 )+ … + u m (x m ) x2x2 u 2 (x 2 ) … xmxm u m (x m ) (utils) ($)

18. What to do with the bids? Decide x 1, x 2, …, x m (total payment to each charity) Decide y 1, y 2, …, y n (total payment by each bidder) Say x 1, x 2, …, x m ; y 1, y 2, …, y n is valid if –x 1 + x 2 + … + x m ≤ y 1 + y 2 + …+ y n (no more money given away than collected) –For any bidder j, y j ≤ w j (u j 1 (x 1 ) + u j 2 (x 2 ) + … + u j m (x m )) (nobody pays more than they wanted to) y1y1 y2y2 x1x1 x2x2

19. Objective Among valid outcomes, find one that maximizes Total donated = x 1 + x 2 + … + x m y1y1 y2y2 x1x1 x2x2 Surplus = y 1 + y 2 + …+ y n - x 1 - x 2 - … - x m y1y1 y2y2 x1x1 x2x2

20. Avoiding indirect payments

21. No payments to disliked charities

22. Hardness of clearing NP-complete to decide if there exists a solution with objective > 0 That means: the problem is inapproximable to any ratio (unless P=NP)

23. General program formulation Maximize –x 1 + x 2 + … + x m, OR –y 1 + y 2 + …+ y n - x 1 - x 2 - … - x m Subject to –y 1 + y 2 + …+ y n - x 1 - x 2 - … - x m ≥ 0 –For all j: y j ≤ w j (u j 1 + u j 2 + … + u j m ) –For all i, j: u j i ≤ u j i (x i ) nonlinear

24. Concave piecewise linear constraints b(x) x y ≤ b(x) l 1 (x) y ≤ l 1 (x) l 2 (x) y ≤ l 2 (x) l 3 (x) y ≤ l 3 (x)

25. Linear programming So, if all the bids are concave… –All the u j i are concave xixi u j i (x i ) (utils) ($) –All the w j are concave ujuj w j (u j ) (utils) ($) Then the program is a linear program (solvable to optimality in polynomial time) Even if they are not concave, can solve as MIP