1. Charitable donations, public goods, cost sharing 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. Matching offers Matching offers are such a contract –One-sided –Involve only a single charity Promise to match donations Big institution Private individual(s) Two problems:

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

7. A different approach [Conitzer & Sandholm EC04] Donors can submit bids indicating their preferences over charities A center accepts all the bids and decides who pays what to whom

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

9. 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

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

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

12. 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

13. Tsunami event (Dagstuhl 05)

14. 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

15. 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

16. 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) ($)

17. 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) Definition. 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

18. 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

19. Avoiding indirect payments

20. No payments to disliked charities In general, this is a MAX-FLOW problem

21. Reduction from MAX2SAT (+v 1 OR –v 2 ) AND (-v 1 OR +v 2 ) (target # of clauses: 2) X +v u +v (X +v ) X -v u -v (X -v ) w v (u +v (X +v ) + u -v (X -v )) u v +v (X +v ) + u v -v (X -v ) One bidder corresponding to each literal l XlXl u l (X l ) w v (u l (X l )) u v l (X l ) One bidder corresponding to each clause c = (l c1 OR l c2 ) X lc1 u lc1 (X lc1 ) X lc2 u lc2 (X lc2 ) w c (u lc1 (X lc1 ) + u lc2 (X lc2 )) u c lc1 (X lc1 ) + u c lc2 (X lc2 ) One more bidder who likes all charities and will give a huge amount if that makes everyone else give enough Four charities, corresponding to literals (+v 1, -v 1, +v 2, –v 2 ) One bidder corresponding to each variable v

22. Hardness of clearing We showed how to model an NP-complete problem (MAX2SAT) as a clearing instance –Nonzero surplus/total donation possible iff MAX2SAT instance has solution So, 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. Good news… 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)

26. Can we do something faster than LP? For general concave bids, the answer is no We show how to model an arbitrary linear feasibility problem as a clearing instance

27. Clearing with quasilinear bids Quasilinear bids = bids where w(u) = u For surplus maximization, can solve the problem separately for each charity Not so for donation maximization –Weakly NP-hard to clear –But, if in addition, utility functions are concave, simple greedy approach is optimal

28. Mechanism design (quasilinear bids) Theorem. There does not exist a mechanism that is ex-post budget balanced, ex-post efficient, ex-interim incentive compatible (BNE), and ex-interim IR … …even in a setting when there is only one charity, two quasilinear bidders with identical type distributions (both for step functions and concave piecewise linear functions)

29. Proof (for step functions) Let there be a (nonexcludable) public good that costs 1 Two agents; each agent’s distribution –With probability.5, utility 1.25 for public good (H) –With probability.5, utility 0 for public good (L) Assume efficient, BB, ex-interim IR, BNE IC mechanism 1 should not misreport L instead of H, i.e. 5/4-π 1 (L,L)- π 1 (L,H)≤5/4+5/4-π 1 (H,L)-π 1 (H,H) By IR, -π 1 (L,L)-π 1 (L,H)≥0, hence π 1 (H,L)+π 1 (H,H)≤5/4, by symmetry π 2 (L,H)+π 2 (H,H)≤5/4 By BB, π 1 (H,H)+π 2 (H,H)+π 1 (L,H)+π 2 (L,H)+ π 1 (H,L)+π 2 (H,L)=3; hence π 1 (L,H)+π 2 (H,L)≥3-10/4=1/2 By BB, π 1 (L,L)+π 2 (L,L)=0, hence π 1 (L,L)+π 1 (L,H)+π 2 (L,L)+π 2 (H,L)≥1/2 But by IR, π 1 (L,L)+π 1 (L,H)≤0 and π 2 (L,L)+π 2 (H,L)≤0 Contradiction!

30. Mechanism design (quasilinear bids) – combining charities may help Proposition. There are settings with two charities, where for either charity alone there does not exist a mechanism that is ex-post budget balanced, ex-post efficient, ex-interim incentive compatible (BNE), and ex-interim IR … …but for both charities together, there is a mechanism that is ex-post budget balanced, ex-post efficient, ex-post incentive compatible (DS), and ex-post IR.

31. A multicast cost sharing problem [Feigenbaum et al. 00] [these slides inspired by a talk by Tim Roughgarden] Vertices (agents) are organized in a tree Need to select a subset that will receive service from root Need to construct the minimal subtree that connects all nodes that receive service Edges have costs Agents have utilities for service 3 2 1 4 5 6 1 10 5 1 6 2 Total utility = 10 + 5 + 1 + 6 = 22 Total cost = 3 + 5 + 6 + 1 + 2 = 17 Surplus = 22 - 17 = 5

32. Mechanism design considerations Agents’ utilities are private knowledge We need a mechanism that –takes utilities as input –produces a subset of served agents + payments as output Desiderata: –Efficiency: choose the surplus-maximizing subset –No deficit: payments collected should cover the costs incurred Even nicer would be (strong) budget balance: payments collected = costs incurred

33. “Marginal cost” mechanism (Clarke mechanism) Agent i pays: –surplus that would have resulted if i had reported 0, minus –surplus that did result, not counting i’s utility Efficient. Deficit? 3 2 1 4 5 6 1 10 5 1 6 2

34. Serious deficit for the Clarke mechanism 1 0 2 2

35. The Shapley mechanism Iterative mechanism Start with everyone included Divide the cost of an edge among agents that make use of it Note that this sets too high a price for some users… 3 2 1 4 5 6 1 10 5 1 6 2 3/2 = 1.5 3/2 + 5 = 6.5 6/4 = 1.5 6/4 + 4 = 5.5 6/4 + 1/2 = 2 6/4 + 1/2 + 2 = 4

36. Shapley mechanism continued… … so we drop those users Recalculate prices for remaining users Repeat until convergence ~ ascending auction (prices keep increasing) Might as well stay in until price too high Generalization: use any cost sharing function so that prices only increase (Moulin mechanisms) 3 2 1 4 5 6 1 10 5 1 6 2 3 + 5 = 8 6/2 = 3 6/2 + 1 + 2 = 6

37. Serious inefficiency for the Shapley mechanism 1.1 0 0 1/311/2 1/k … 000

38. Results Clarke requires less communication than Shapley [Feigenbaum et al. 00] More recent results approximate both budget balance and efficiency –Most recently Mehta et al. [EC 07] study a generalization of Moulin mechanisms (“acyclic” mechanisms) that correspond naturally to various approximation algorithms