Expressive Negotiation over Donations to Charities Vincent Conitzer and Tuomas Sandholm

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

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

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

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

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

Our contribution We take an expressive negotiation approach to donations and matching offers Donors can submit bids indicating their preferences over charities A center accepts all the bids and decides who pays what to whom

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)

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

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

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

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) ($)

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

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

Avoiding indirect payments

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

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

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)

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

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) 

That’s 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)

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

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-complete to clear –But, if in addition, utility functions are concave, simple greedy approach is optimal

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)

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.

Conclusion We showed how to take an expressive negotiation approach to charitable donations –Introduced a bidding language –Defined the associated clearing problem We studied the complexity of the clearing problem –Completely inapproximable in general –Linear program for concave bids (best possible) –Easier with quasilinear bids We studied mechanism design aspects –Impossibility of efficiency in general –Combining charities may help

Future research Finish our web-based implementation Study computational scalability experimentally Identify other classes of bids for which clearing is tractable Is the bidding language ever not general enough? More on mechanism design for this setting

Thank you for your attention!