Complex Matching Offers in Donation Negotiation

1. CPS 196.2Expressive 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 U = 1

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

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

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

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

8. A different approach • Donors can submit bidsindicating 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)” Bidder’s maximum payment Budget w(x) x = total payment to charity

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

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

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

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 wa(xa) wu(xu) $50 $50 xa xu $2500 $5000 • Could get stuck paying $100! • Most general solution: w(x1, x2, …, xm) • Requires specifying exponentially many values

16. Solution: separate utility and payment; assume utility decomposes • Willing to give $1 for every $100 to UNICEF • Willing to give $2 for every $100 to Amnesty Int’l • Budget constraint: $50 ua(xa) uu(xu) w(uu(xu)+ua(xa)) $50 1 util 1 util uu(xu)+ua(xa) 50 utils $100 xu $50 xa

17. The general form of a bid (utils) (utils) u2(x2) u1(x1) um(xm) (utils) … x2 ($) x1 xm ($) ($) ($) w(u1(x1) + u2(x2)+ … + um(xm)) u1(x1) + u2(x2)+ … + um(xm) (utils)

18. What to do with the bids? • Decide x1,x2,…,xm (total payment to each charity) • Decide y1,y2,…,yn (total payment by each bidder) • Sayx1,x2,…,xm ;y1,y2,…,yn is valid if • x1+x2 +… +xm ≤y1 +y2 +…+yn (no more money given away than collected) • For any bidder j, yj ≤wj(uj1(x1)+uj2(x2) + … +ujm(xm)) (nobody pays more than they wanted to) x1 y1 x2 y2

19. Objective • Among valid outcomes, find one that maximizes • Total donated = x1+x2 +… +xm x1 y1 x2 y2 • Surplus = y1 +y2 +…+yn -x1 -x2 -… -xm x1 y1 x2 y2

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 • x1+x2+… +xm , OR • y1+y2 +…+yn-x1-x2-… -xm • Subject to • y1+y2 +…+yn-x1-x2-… -xm≥ 0 • For all j: yj≤ wj(uj1+ uj2+ … + ujm) • For all i, j: uji≤ uji(xi) nonlinear nonlinear

24. Concave piecewise linear constraints l2(x) l1(x) l3(x) b(x) x y ≤ l1(x) y ≤ b(x) y ≤ l2(x) y ≤ l3(x)

25. Linear programming • So, if all the bids are concave… • All the uji are concave uji(xi) (utils) • All the wj are concave ($) xi wj(uj) ($) • Then the program is a linear program (solvable to optimality in polynomial time) • Even if they are not concave, can solve as MIP uj (utils)