[Paper Review] Fairness and efficiency for probabilistic allocations with participation constraints

Federico Echenique, Antonio Miralles+1 more|arXiv (Cornell University)|Aug 12, 2019

Game Theory and Voting Systems15 references4 citations

TL;DR

This paper proposes a fairness notion—absence of justified envy—where envy is only considered unfair if a swap between agents would not violate any agent’s participation constraint. It establishes the existence of fair, efficient, and individually rational allocations using a competitive equilibrium with price-dependent incomes, ensuring compatibility with reservation utilities and constraints.

ABSTRACT

We propose a notion of fairness for allocation problems in which different agents may have different reservation utilities, stemming from different outside options, or property rights. Fairness is usually understood as the absence of envy, but this can be incompatible with reservation utilities. It is possible that Alice's envy of Bob's assignment cannot be remedied without violating Bob's participation constraint. Instead, we seek to rule out {\em justified envy}, defined as envy for which a remedy would not violate any agent's participation constraint. We show that fairness, meaning the absence of justified envy, can be achieved together with efficiency and individual rationality. We introduce a competitive equilibrium approach with price-dependent incomes obtaining the desired properties.

Motivation & Objective

• To define fairness in allocation problems where agents have different reservation utilities due to outside options or property rights.
• To reconcile fairness with individual rationality and efficiency in settings where traditional envy-freeness may be incompatible with participation constraints.
• To show that fair, efficient, and individually rational allocations exist even when agents have unequal outside options.
• To extend the framework to handle quantitative constraints, such as minimum course requirements or diversity objectives in school choice.
• To demonstrate that fair outcomes can be implemented as market equilibria using carefully constructed price-dependent income functions.

Proposed method

• Introduces 'justified envy' as envy that could be remedied without violating any agent’s participation constraint, particularly through pairwise swaps.
• Constructs price-dependent income functions that ensure individual rationality and prevent overspending by unsatiated agents.
• Uses a continuous, utility-maximizing allocation rule $\phi(\lambda)$ that maps weight vectors $\lambda$ to allocations, optimizing a trade-off between utility and deviation from unit demand.
• Applies a variational approach to prove existence of equilibria via compactness and continuity arguments, showing that the set of fair allocations is closed.
• Employs a contradiction argument based on cycles of $\varepsilon$-justified envy to prove that no agent can have justified envy in equilibrium.
• Extends the model to accommodate constraints such as minimum course requirements or diversity goals in school choice via modified feasibility sets.

Experimental results

Research questions

• RQ1 Can fairness be meaningfully defined in allocation problems where agents have different reservation utilities?
• RQ2 Is it possible to achieve both efficiency and individual rationality while eliminating justified envy?
• RQ3 Can fair and efficient allocations be implemented as competitive equilibria with price-dependent incomes?
• RQ4 How can quantitative constraints—such as minimum course requirements or diversity objectives—be incorporated into fair allocation mechanisms?
• RQ5 Can more complex remedies beyond pairwise swaps be accommodated while preserving fairness and efficiency?

Key findings

• A fair, efficient, and individually rational allocation exists under the proposed notion of justified envy, even when agents have different reservation utilities.
• The existence of such allocations is proven via a continuous allocation rule $\phi(\lambda)$ that optimizes a trade-off between utility and deviation from unit demand.
• The set of allocations with no justified envy is closed, ensuring robustness under limits of sequences of allocations.
• Fair outcomes can be implemented as competitive equilibria when income functions are constructed to depend on prices in a way that preserves individual rationality.
• The model accommodates constraints such as minimum course requirements or diversity objectives in school choice, extending beyond standard random assignment.
• Cycles of $\varepsilon$-justified envy cannot exist in equilibrium, as they would contradict the optimality of the allocation under the objective function.