1 / 33

Charitable donations, public goods, cost sharing

Charitable donations, public goods, cost sharing. Vincent Conitzer conitzer@cs.duke.edu. One donor (bidder). u( ) = 1 u( ) = .8. U = 1. Two independent donors. u( ) = 1 u( ) = .8. u( ) = 1

teva
Download Presentation

Charitable donations, public goods, cost sharing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 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. Matching offers Promise to match donations • Matching offers are such a contract • Two problems: • One-sided Big institution Private individual(s) • Involve only a single charity

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

  7. A different approach [Conitzer & Sandholm EC04] • Donors can submit bidsindicating 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)” Bidder’s maximum payment Budget w(x) x = total payment to charity

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

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

  12. 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

  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 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

  15. 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

  16. 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)

  17. What to do with the bids? • Decide x1,x2,…,xm (total payment to each charity) • Decide y1,y2,…,yn (total payment by each bidder) • Definition. x1,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

  18. 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

  19. 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)

  20. 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

  21. 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)

  22. Good news… • 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) uj (utils)

  23. 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

  24. 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)

  25. 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-π1(L,L)-π1(L,H)≤5/4+5/4-π1(H,L)-π1(H,H) • By IR, -π1(L,L)-π1(L,H)≥0, hence π1(H,L)+π1(H,H)≤5/4, by symmetry π2(L,H)+π2(H,H)≤5/4 • By BB, π1(H,H)+π2(H,H)+π1(L,H)+π2(L,H)+ π1(H,L)+π2(H,L)=3; hence π1(L,H)+π2(H,L)≥3-10/4=1/2 • By BB, π1(L,L)+π2(L,L)=0, hence π1(L,L)+π1(L,H)+π2(L,L)+π2(H,L)≥1/2 • But by IR, π1(L,L)+π1(L,H)≤0 and π2(L,L)+π2(H,L)≤0 • Contradiction!

  26. 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 6 3 1 5 5 4 1 10 1 2 2 6 Total utility = 10 + 5 + 1 + 6 = 22 Total cost = 3 + 5 + 6 + 1 + 2 = 17 Surplus = 22 - 17 = 5

  27. 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

  28. “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? 6 3 1 5 5 4 1 10 1 2 2 6

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

  30. 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… 6 3 6/4 = 1.5 3/2 = 1.5 1 5 5 6/4 + 1/2 = 2 4 1 10 1 2 3/2 + 5 = 6.5 6/4 + 4 = 5.5 2 6 6/4 + 1/2 + 2 = 4

  31. 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) 6 3 6/2 = 3 1 5 5 4 1 10 1 2 3 + 5 = 8 2 6 6/2 + 1 + 2 = 6

  32. Serious inefficiency for the Shapley mechanism 1.1 0 0 0 0 0 … 1/k 1 1/2 1/3

  33. 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

More Related