CPS 196 2 Expressive negotiation over donations Vincent

CPS 196. 2 Expressive negotiation over donations Vincent Conitzer conitzer@cs. duke. edu

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

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

Two donors with a contract u( u( )=1 ) =. 8 u( u( U =. 5 +. 8 = 1. 3 > 1 )=1 ) =. 8

Contracting using matching offers I’ll match any donation to u( )=1 u( ) = 1. 3

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

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

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)

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

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

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

Current solution 45 degree line w(x) total $125 payment bidders are $75 willing to make max donated $25 max surplus x = total payment $100 $43. 75 $500

Tsunami event (Dagstuhl 05)

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 wu(xu) $5000 xu wa(xa) $50 $2500 xa • Could get stuck paying $100! • Most general solution: w(x 1, x 2, …, xm) – Requires specifying exponentially many values

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: ua$50 u (x ) u (xu) a $50 1 util $100 xu $50 xa w(uu(xu)+ua(xa)) uu ( a 50 utils xu)+u ( xa)

The general form of a bid (utils) u 1(x 1) u 2(x 2) (utils) um(xm) … x 1 ($) x 2 ($) w(u 1(x 1) + u 2(x 2)+ … + um(xm)) u 1(x 1) + u 2(x 2)+ … + um(xm) (utils) xm ($)

What to do with the bids? • • Decide x 1, x 2, …, xm (total payment to each charity) Decide y 1, y 2, …, yn (total payment by each bidder) y 1 x 1 y 2 x 2 • Say x 1, x 2, …, xm ; y 1, y 2, …, yn is valid if – x 1+ x 2 + … + xm ≤ y 1 + y 2 + …+ yn (no more money given away than collected) – For any bidder j, yj ≤ wj(uj 1(x 1) + uj 2(x 2) + … + ujm(xm)) (nobody pays more than they wanted to)

Objective • Among valid outcomes, find one that maximizes • Total donated = x 1+ x 2 + … + xm y 1 x 1 y 2 x 2 • Surplus = y 1 + y 2 + …+ yn - x 1 - x 2 - … - xm y 1 x 1 y 2 x 2

Avoiding indirect payments

No payments to disliked charities

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)

General program formulation • Maximize – x 1+ x 2 + … + xm , OR – y 1 + y 2 + …+ yn - x 1 - x 2 - … - xm • Subject to – y 1 + y 2 + …+ yn - x 1 - x 2 - … - xm ≥ 0 – For all j: yj ≤ wj(uj 1 + uj 2 + … + ujm) – For all i, j: uji ≤ uji(xi) nonlinear

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

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