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## PowerPoint Slideshow about ' Charitable donations, public goods, cost sharing' - cole-lynn

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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

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

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

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)”

Bidder’s maximum payment

Budget

w(x)

x = total payment to charity

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

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

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

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)

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

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

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
- 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

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)

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

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)

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!

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

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

“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

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

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

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

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