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Coordinating Charitable Donations with Leontief Preferences

Felix Brandt, Matthias Greger, Erel Segal-Halevi, Warut Suksompong

TL;DR

This work addresses coordinating divisible charitable donations across multiple complementary public goods under Leontief (minimum-based) preferences. It introduces the equilibrium distribution rule (EDR), showing that every profile yields a unique equilibrium distribution that maximizes Nash welfare and corresponds to a Lindahl equilibrium; EDR is group-strategyproof and monotone in both preferences and contributions, and it can be implemented by natural best-response spending dynamics. The analysis covers both binary and general Leontief weights, with special results for the binary-weight case linking EDR to lexicographic notions of fairness and enabling linear-programming solutions. Beyond charities, the framework applies to donation programs and public budgets, offering robust coordination without heavy central control and providing convergence guarantees for decentralized processes.

Abstract

We consider the problem of funding public goods that are complementary in nature. Examples include charities handling different needs (e.g., protecting animals vs. providing healthcare), charitable donations to different individuals, or municipal units handling different issues (e.g., security vs. transportation). We model these complementarities by assuming Leontief preferences; that is, each donor seeks to maximize an individually weighted minimum of all contributions across the charities. Decentralized funding may be inefficient due to a lack of coordination among the donors; centralized funding may be undesirable as it ignores the preferences of individual donors. We present a mechanism that combines the advantages of both methods. The mechanism efficiently distributes each donor's contribution so that no subset of donors has an incentive to redistribute their donations. Moreover, it is group-strategyproof, satisfies desirable monotonicity properties, maximizes Nash welfare, returns a unique Lindahl equilibrium, and can be implemented via natural best-response spending dynamics.

Coordinating Charitable Donations with Leontief Preferences

TL;DR

This work addresses coordinating divisible charitable donations across multiple complementary public goods under Leontief (minimum-based) preferences. It introduces the equilibrium distribution rule (EDR), showing that every profile yields a unique equilibrium distribution that maximizes Nash welfare and corresponds to a Lindahl equilibrium; EDR is group-strategyproof and monotone in both preferences and contributions, and it can be implemented by natural best-response spending dynamics. The analysis covers both binary and general Leontief weights, with special results for the binary-weight case linking EDR to lexicographic notions of fairness and enabling linear-programming solutions. Beyond charities, the framework applies to donation programs and public budgets, offering robust coordination without heavy central control and providing convergence guarantees for decentralized processes.

Abstract

We consider the problem of funding public goods that are complementary in nature. Examples include charities handling different needs (e.g., protecting animals vs. providing healthcare), charitable donations to different individuals, or municipal units handling different issues (e.g., security vs. transportation). We model these complementarities by assuming Leontief preferences; that is, each donor seeks to maximize an individually weighted minimum of all contributions across the charities. Decentralized funding may be inefficient due to a lack of coordination among the donors; centralized funding may be undesirable as it ignores the preferences of individual donors. We present a mechanism that combines the advantages of both methods. The mechanism efficiently distributes each donor's contribution so that no subset of donors has an incentive to redistribute their donations. Moreover, it is group-strategyproof, satisfies desirable monotonicity properties, maximizes Nash welfare, returns a unique Lindahl equilibrium, and can be implemented via natural best-response spending dynamics.
Paper Structure (42 sections, 54 theorems, 78 equations, 2 figures)

This paper contains 42 sections, 54 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Public goods
  4. Participatory budgeting
  5. Nash product rule
  6. Many-to-many matchings
  7. The model
  8. Equilibrium distributions
  9. Critical charities
  10. Efficient distributions
  11. Existence, uniqueness, and computation
  12. Public good markets
  13. The equilibrium distribution rule
  14. Strategyproofness and participation
  15. Monotonicity conditions
  16. ...and 27 more sections

Key Result

Lemma 1

A vector of individual distributions $(\delta_i)_{i \in N}$ is in equilibrium if and only if $\delta_i(x)=0$ for every charity $x\not\in T_{\delta,i}$. Equivalently, it has to satisfy the following equality instead of eq:dec-ci-A in def:decomposition:

Figures (2)

  • Figure 1: An instance with four charities (named $w,x,y,z$), $\delta^{t}=(3,2, 6, 9)$, and an agent $i_t$ with $\delta^{t}_{i_t}=(0,2,2,2)$ and Leontief utilities with binary weights $v_{i_t} = (0,1,1,1)$. Then, $\delta^\mathit{best}_{i_t}=(0,5,1,0)$, $\delta^{t+1} = (3, 5, 5,7)$, $c_t=3$, $A^-_{i_t} = \{y,z\}$, $A^+_{i_t} = \{x\}$.
  • Figure 2: Charity sets in the proof of \ref{['thm: cd-is-fmaximal']}, for the case $r>s$. The horizontal position of a charity denotes its allocation in $\delta^*$; the vertical position denotes its allocation in $\delta$.

Theorems & Definitions (108)

  • Example 1
  • Example 2
  • Definition 1: Decomposition
  • Definition 2: Equilibrium distribution
  • Lemma 1
  • Corollary 1
  • Definition 3: Efficiency
  • Lemma 2
  • Lemma 3
  • Example 3
  • ...and 98 more