Constrained Serial Dictatorships can be Fair

TL;DR

The paper studies strategyproof allocation of indivisible goods and shows that constrained serial dictatorships are essentially the viable class under mild conditions. It introduces a flexible framework with a scoring vector, a welfare functional (USW, ESW, NSW), and a prior Ψ over rankings to define and compute optimal CSDs. It develops aGreedyESW algorithm for egalitarian welfare, and a prefix-independence-based dynamic program for utilitarian and Nash welfare, with exact or approximated EU computations under FC, IC, and PL models; it also proves NP-hardness for optimally assigning agents to positions. Numerical experiments illustrate how the optimal CSDs shift with the number of goods and the correlation structure of preferences, confirming that later pickers can compensate for early-picking advantages under appropriate settings. The work provides practical guidance for designing fair, strategyproof allocations and highlights the tradeoffs between ex ante fairness and ex post guarantees.

Abstract

When allocating indivisible items to agents, it is known that the only strategyproof mechanisms that satisfy a set of rather mild conditions are constrained serial dictatorships: given a fixed order over agents, at each step the designated agent chooses a given number of items (depending on her position in the sequence). Agents who come earlier in the sequence have a larger choice of items; however, this advantage can be compensated by a higher number of items received by those who come later. How to balance priority in the sequence and number of items received is a nontrivial question. We use a previous model, parameterized by a mapping from ranks to scores, a social welfare functional, and a distribution over preference profiles. For several meaningful choices of parameters, we show that the optimal sequence can be computed exactly in polynomial time or approximated using sampling. Our results hold for several probabilistic models on preference profiles, with an emphasis on the Plackett-Luce model. We conclude with experimental results showing how the optimal sequence is impacted by various parameters.

Figures (9)

• Figure 1: Portion of total utility (plots on the left) and of goods (right) received by each of 5 agents with $m$ increasing from 5 to 300 in steps of 5. Maximizing USW (plots at the top), NSW (bottom), or ESW (middle), using Borda scoring vector and $\mathtt{IC}$.
• Figure 2: Number of goods received per agent (top); expected utility value per agent (bottom) as a function of $\phi$ for $\mathtt{Mll}_{\phi,\mu}$ and $x$ for $\mathtt{PL}_{\boldsymbol{\nu}^x}$. Maximizing ESW, Borda scoring vector, $n=5$, $m=70$.
• Figure 3: Portion of the total utility (plot on the left) and of goods (right) received by each of 5 agents with $m$ increasing from 5 to 300 in steps of 5. Maximizing ESW and using the lexicographic scoring vector and $\mathtt{FI}$.
• Figure 4: Comparison of the Borda scoring score (rescaled to [0,100]) with the scoring vector obtained through our experiment by taking the expectation of the participants' answers.
• Figure 5: Scoring vectors obtained through our experiment. The scoring vector "Lottery-based" (resp. "Simpler") was obtained by averaging the answers of the participants receiving the directives which mentioned (resp. did not mention) a probabilistic interpretation for values assigned to ice-cream flavours.

Theorems & Definitions (42)

• Example 1
• Proposition 1
• Lemma 1
• proof
• proof : Proof of Proposition \ref{['thm:Algo1']}
• Definition 1
• Proposition 2
• Proposition 3
• Proposition 4
• proof