Ex-Post Efficiency in Random Assignments

• Ex-post efficiency is a concept in randomized assignments that requires every deterministic outcome (from a lottery) to be Pareto-optimal based on agent preferences.
• It is achieved by representing a random assignment as a convex combination of Pareto-optimal deterministic outcomes, with its verification being NP-complete.
• Distinctions between ex-post, robust ex-post, and SD-efficiency guide mechanism design, as exemplified by methods like Random Serial Dictatorship.

Ex-post efficiency is a foundational concept in the analysis of randomized assignment, social choice, and mechanism design, referring to the requirement that a realized (deterministic) outcome drawn from a (possibly randomized) mechanism must be Pareto-optimal with respect to agent preferences. In the context of random assignments, ex-post efficiency is tied to the structure of lotteries over deterministic assignments, their convex decompositions, and the computational complexity of verifying and implementing such decompositions. Although ex-post efficiency offers a minimal and natural notion of welfare rationality for sampled assignments, it is strictly weaker than ex-ante efficiency notions such as stochastic dominance efficiency, and its verification raises intricate combinatorial and algorithmic challenges (Aziz et al., 2014).

1. Formal Foundations and Key Definitions

In the standard random assignment model with $n$ agents $N = \{1,\ldots, n\}$ and $n$ objects $O = \{o_1,\ldots, o_n\}$, each agent $i$ expresses a strict preference $\succ_i$ over objects. A random assignment $p$ is given as an $n \times n$ matrix $p(i, o) \in [0, 1]$, where each agent $i$ receives each object $o$ with probability $p(i, o)$, subject to each row and column summing to $1$.

Ex-post efficiency (EPE):

A random assignment $p$ is ex-post efficient if it can be represented as a convex combination of deterministic (i.e., 0–1 permutation) matrices $P_k$ such that each $P_k$ is Pareto-optimal: $$p = \sum_{k=1}^K \lambda_k P_k,\quad \lambda_k > 0,\; \sum_k \lambda_k = 1,$$ where each $P_k$ is a deterministic Pareto-optimal assignment. The Pareto-optimality of a deterministic assignment requires that no re-assignment of objects can strictly increase the satisfaction of at least one agent without strictly decreasing that of another (Aziz et al., 2014).

Robust ex-post efficiency (REPE):

A stricter variant, robust ex-post efficiency, requires that every Birkhoff-type decomposition of $p$ uses only Pareto-optimal deterministic assignments. Thus, $p$ is robust ex-post efficient if and only if no non–Pareto-optimal deterministic assignment $P$ is consistent with its support (Aziz et al., 2014).

Comparison with related efficiency concepts:

Concept	Demand	Relationship
Ex-post efficiency	All realized deterministic assignments (in the lottery support) are Pareto-optimal	Weakest among main efficiency notions
Robust ex-post	Every Birkhoff decomposition uses only Pareto-optimal assignments	Strictly implies ex-post, weaker than SD-eff
SD-efficiency	No assignment stochastically dominates the lottery	Strongest

SD-efficiency $\implies$ robust ex-post efficiency $\implies$ ex-post efficiency, none of which are equivalent in general (Aziz et al., 2014).

2. Structural Properties and Characterizations

A central insight is that ex-post efficiency is an inherently arithmetic property of the assignment matrix $p$: its validity depends on the exact convex decomposition into Pareto-optimal deterministic assignments. Tiny perturbations in the values of $p(i, o)$—even without changing its support—can induce or destroy ex-post efficiency. Thus:

• Ex-post efficiency is not combinatorial: Two assignment matrices $p$ and $q$ with identical zero/nonzero support may differ; one could be ex-post efficient, the other could not.
• Robust ex-post efficiency is combinatorial: It depends only on the support pattern (entries of $p$ which are zero vs. positive), since it can be characterized as the absence of any non–Pareto-optimal permutation matrix consistent with the support (Aziz et al., 2014).

Pareto-optimal deterministic assignments have a structural characterization: they are precisely those that can be generated by some serial dictatorship order (i.e., serially letting agents pick their favorite available object) or, equivalently, whose associated preference graph is acyclic—where agents point to objects they prefer over their assignment, and objects point to their assigned agents (Aziz et al., 2014).

3. Computational Complexity

The complexity landscape for ex-post efficiency is sharply differentiated by the problem version:

• Verification: Checking whether a given random assignment $p$ is ex-post efficient (i.e., can be written as a convex combination of Pareto-optimal deterministic assignments) is NP-complete [Theorem 5.1; (Aziz et al., 2014)]. The proof proceeds via a reduction from 3-SAT, by encoding boolean variable assignments into the decomposition, with "trading cycles" corresponding to unsatisfied clauses.
• Implementation: Generating a decomposition that implements a random assignment $p$ using only Pareto-optimal deterministic assignments, or checking membership in the convex hull of such assignments ("the ex post polytope"), is NP-hard/NP-complete.
• Optimization: Optimizing linear functions over the ex-post polytope is NP-complete.

However, ex-post efficiency is straightforward to guarantee by construction: any single run of serial dictatorship (or, more generally, any Pareto-optimal deterministic assignment) provides a degenerate (singleton) ex-post efficient random assignment. Sampling uniformly from the set of such assignments yields a tractable family of ex-post efficient mechanisms (e.g., Random Serial Dictatorship performs exactly this). The optimal support size for certificates (by Carathéodory’s theorem) is at most $n^2 + 1$ (Aziz et al., 2014).

Testing robust ex-post efficiency admits more efficient algorithms: when the number of agent types is constant, one can enumerate all small potential violation cycles and check consistency and Pareto-domination in polynomial time [Theorem 6.2; (Aziz et al., 2014)].

4. Examples and Distinctions

The subtlety of ex-post efficiency is illustrated by constructed examples:

• Two random assignments $p$ and $q$ may share identical support patterns but differ in ex-post efficiency. For instance, in a 4-agent/4-object instance where agents 1,2 prefer $o_1 \succ o_2 \succ o_3 \succ o_4$ and agents 3,4 prefer $o_2 \succ o_1 \succ o_4 \succ o_3$, the RSD outcome $p$ is ex-post efficient, while $q$ (obtained by rearranging the support) is not [Example 5.1; (Aziz et al., 2014)].
• RSD itself is not robust ex-post efficient; the support of its outcome could admit a non–Pareto-optimal permutation matrix, thus underlining the necessity of distinct analysis for these criteria [Example 5.2; (Aziz et al., 2014)].

The distinction between ex-post and ex-ante efficiency is also crucial beyond random assignment, such as general collective choice: ex-post efficient lotteries merely avoid dominated outcomes in their support, but may be ex-ante dominated in stochastic dominance by other feasible lotteries (Echenique et al., 2022). The two notions coincide only in restricted domains—e.g., single-peaked or dichotomous preferences with bounded numbers of agents.

5. Relationships with Other Notions and Mechanisms

In the deterministic setting, Pareto-optimality, ex-post efficiency, and SD-efficiency collapse. However, in the randomized context:

$$\text{SD-efficiency} \implies \text{robust ex-post efficiency} \implies \text{ex-post efficiency}$$

with neither implication reversible (Aziz et al., 2014). Ex-post efficiency underlies the main output of Random Serial Dictatorship (RSD) and other lottery-based rules, but these are not generally SD-efficient or robust ex-post efficient for arbitrary preference profiles.

Mechanisms such as Trading Cycles, Hierarchical Exchange, and various characterizations of object allocation rules exploit these relationships. Notably, in settings where ex-post non-wastefulness and probabilistic monotonicity are imposed, ex-post pairwise efficiency and ex-post Pareto efficiency become equivalent, further streamlining the analysis of standard mechanisms (Demeulemeester et al., 7 Aug 2025).

6. Robustness, Generalizations, and Hardness

Robust ex-post efficiency presents a combinatorial, easily checkable relaxation: it’s enough to verify absence of a non–Pareto-optimal deterministic assignment consistent with the nonzero entries of the assignment matrix. For settings with a bounded number of agent types, testers can enumerate all relevant cycles efficiently, demonstrating algorithmic tractability in structured domains (Aziz et al., 2014).

Nevertheless, in the general case, ex-post efficiency remains arithmetically delicate and computationally intractable, emphasizing the practical need for mechanisms that either guarantee ex-post efficiency by construction or restrict attention to settings where efficient verification is possible.

7. Summary Table: Properties and Complexity

Notion	Can Be Computed?	Can Be Verified?	Depends Only on Support?	Hardness
Ex-post efficiency (EPE)	Yes (via serial dictatorship)	No (NP-complete)	No (depends on values)	NP-complete (Aziz et al., 2014)
Robust ex-post efficiency (REPE)	Yes in restricted cases	Yes coNP / poly (if $|$types$|$ fixed)	Yes (support pattern)	coNP/poly(n) if $|$types$|=O(1)$ (Aziz et al., 2014)
SD-efficiency	Yes (comb LP/graph criterion)	Yes (polytime)	Yes (improving cycles)	Polytime

In conclusion, ex-post efficiency is a minimal but highly nontrivial efficiency requirement in randomized assignment, collective choice, and related domains. Its precise mathematical and computational characterization delineates the achievable boundary for realizable welfare guarantees and informs the design and analysis of allocation mechanisms in both theory and application (Aziz et al., 2014).