Competitive Equilibrium from Random Incomes

• CERI is a framework that generalizes competitive equilibrium by incorporating random, generic, or price-dependent incomes to overcome nonexistence issues in indivisible goods markets.
• It employs methodologies like generic-budget regularization, random token budgets, and expected clearing to achieve fairness, ordinal efficiency, and envy-free allocations.
• Applications range from dynamic online assignments to combinatorial allocation, ensuring individual rationality and strategic fairness under varied market conditions.

Searching arXiv for relevant papers on Competitive Equilibrium from Random Incomes and closely related work. Competitive Equilibrium from Random Incomes (CERI) denotes a family of competitive-equilibrium constructions in which the income vector is not fixed as a deterministic equal-income profile, but is instead perturbed, drawn from a distribution, or endogenized as a function of prices and rights. Across the literature, this label covers two closely related perspectives. In one, competitive equilibrium is studied for almost all income vectors, so that if incomes are drawn from any continuous distribution, equilibrium exists with probability $1$. In the other, agents receive random token budgets and choose optimal bundles at common prices, with markets required to clear in expectation. Both perspectives arise as responses to the nonexistence of competitive equilibrium from equal incomes in discrete allocation problems and as devices for restoring fairness and efficiency in richer environments such as combinatorial assignment, dynamic online allocation, and probabilistic assignment with participation constraints (Segal-Halevi, 2017, Nguyen et al., 16 Sep 2025, Nguyen et al., 2023, Echenique et al., 2019).

1. Genealogy and basic idea

The immediate antecedent of CERI is competitive equilibrium from equal incomes (CEEI). In the classic divisible-goods setting, all agents receive equal artificial budgets and prices clear markets, yielding strong fairness properties such as proportionality and envy-freeness. With indivisible goods, however, exact CEEI can fail even in the simplest markets. A canonical failure is the two-agent, one-item market with equal budgets $b_1=b_2=1/2$: if the item price is at most $1/2$, both agents demand it; if the price exceeds $1/2$, neither can afford it, so no competitive equilibrium exists (Babaioff et al., 2017).

CERI generalizes this paradigm by relaxing the income side rather than the price system. One route is generic-budget regularization: add vanishingly small random perturbations to the budget vector, thereby destroying knife-edge equalities and recovering equilibrium existence generically. Another route is equilibrium from random token budgets: budgets themselves are random variables, and agents optimize at posted prices under realized budgets, while the market-clearing condition is imposed on expected demand. A third route, developed for participation constraints, replaces fixed or equal incomes by price-dependent incomes calibrated to reservation utilities and satiation levels (Babaioff et al., 2017, Nguyen et al., 16 Sep 2025, Echenique et al., 2019).

This broadening of the income dimension is not merely technical. In the cited literature, incomes encode entitlements, outside options, or randomized access to scarce resources. Small perturbations can regularize existence; carefully designed random incomes can recover all ordinally efficient lottery allocations; price-dependent incomes can internalize participation constraints and heterogeneous rights. The unifying theme is that competitive equilibrium is preserved as the allocation principle, while the income vector becomes an instrument for existence, fairness, or implementability.

2. Formal models of random incomes

A standard combinatorial-assignment formulation fixes a finite set of goods $M$, capacities $\mathbf{c}=(c_j)_{j\in M}$, agents $N$, and, for each agent $i$, an acceptable bundle set $\Psi_i \subseteq \mathbb{N}^m$ with outside option $0\in\Psi_i$. Preferences are strict ordinal rankings $b_1=b_2=1/2$0 over $b_1=b_2=1/2$1. A deterministic allocation $b_1=b_2=1/2$2 is feasible if

$b_1=b_2=1/2$3

componentwise. Because goods are indivisible, the primary equilibrium object is often a lottery allocation $b_1=b_2=1/2$4, feasible when $b_1=b_2=1/2$5 (Nguyen et al., 16 Sep 2025).

In the explicit CERI definition for combinatorial assignment, each agent $b_1=b_2=1/2$6 receives a nonnegative random income $b_1=b_2=1/2$7. Given prices $b_1=b_2=1/2$8, the random optimal bundle is

$b_1=b_2=1/2$9

A price vector $1/2$0 and a lottery allocation $1/2$1 form a CERI if, for every agent,

$1/2$2

and for every good $1/2$3,

$1/2$4

with equality whenever $1/2$5. Thus individual optimality is defined ex post with respect to realized budgets, while market clearing is imposed ex ante on expected aggregate demand (Nguyen et al., 16 Sep 2025).

A closely related construction appears in dynamic combinatorial assignment under the name $1/2$6-expected competitive equilibrium from equal incomes ($1/2$7-ECEEI). Budgets $1/2$8 are supported in $1/2$9, prices are deterministic, and expected demand exactly matches capacities. The technical novelty is that random budgets are not exogenous noise: they are constructed from a pseudoequilibrium over auxiliary utilities,

$1/2$0

so that the expected demand of each agent reproduces the pseudoequilibrium allocation. This produces exact clearing in expectation while keeping all realized budgets $1/2$1-close to $1/2$2 (Nguyen et al., 2023).

The measure-theoretic variant of CERI uses a different formalization. Instead of defining random demand directly, it studies the full income space $1/2$3 and asks whether competitive equilibrium exists for almost all income vectors, i.e., for all income vectors outside a Lebesgue-null subset. Under any continuous distribution over incomes, this is equivalent to probability-$1/2$4 existence. In this sense, “random incomes” means that bad income vectors are lower-dimensional exceptional sets, typically unions of hyperplanes such as $1/2$5 or $1/2$6 (Segal-Halevi, 2017).

3. Existence through genericity and almost-all incomes

The first systematic existence results along these lines show that competitive equilibrium with indivisible goods can often be restored by making budgets generic. In the two-agent additive setting, if equal budgets are perturbed by vanishingly small random shocks, equilibrium exists for all budgets except a measure-zero set. The key structural tool is combination pricing, where prices take the form

$1/2$7

so that, for additive valuations,

$1/2$8

A budget-exhausting combination pricing supporting a Pareto-optimal allocation is sufficient for competitive equilibrium. This yields existence for almost equal budgets and, under identical additive preferences, for arbitrary budgets outside explicit finite bad sets $1/2$9 (Babaioff et al., 2017).

The “almost all incomes” program extends this regularization beyond the two-agent additive case. For indivisible goods with general monotonically increasing preferences, competitive equilibrium exists for almost all incomes when there are up to $M$0 goods and any number of agents, and also for $M$1 goods and $M$2 agents. For $M$3 goods and $M$4 agents, nonexistence holds on a positive-measure subset for general monotone preferences, but there is a positive result for additive preferences: with $M$5 goods and $M$6 additive agents, a competitive equilibrium exists for almost all incomes and can be constructed via a sequential game with prices, or pixep, analyzed through subgame-perfect equilibrium. Conversely, with $M$7 goods and $M$8 agents, there exists an additive preference profile and a positive-measure subset of income space where no competitive equilibrium exists (Segal-Halevi, 2017).

The same paper shows that the chores domain is sharply less favorable. For two agents, chores allocation is dual to the two-agent goods problem and inherits its existence profile. For $M$9 agents and monotonically decreasing preferences, however, there exists an additive chore profile and a positive-measure set of income vectors with no competitive equilibrium. CERI, in the almost-all-incomes sense, therefore has a highly nontrivial domain of validity: it is not a generic existence theorem for all discrete economies, but a classification program whose conclusions depend sensitively on the number of goods, the number of agents, and the preference domain (Segal-Halevi, 2017).

A central methodological insight is that nonexistence is frequently a knife-edge phenomenon under equal incomes but a robust phenomenon once positive-measure bad regions appear. This sharply distinguishes cases where random perturbations are merely a regularizer from cases where no continuous income distribution can avoid equilibrium failure with probability $\mathbf{c}=(c_j)_{j\in M}$0.

4. Welfare, envy, and fairness guarantees

CERI inherits the standard welfare logic of competitive equilibrium, but its most distinctive contributions concern fairness under indivisibility and ordinal information. In the combinatorial-assignment model, a lottery allocation is ordinally efficient if no other feasible lottery allocation stochastically dominates it for all agents and strictly dominates it for at least one. The central characterization theorem states that a lottery allocation is ordinally efficient if and only if there exists a profile of random incomes for which it is a CERI allocation. This yields both a First Welfare Theorem and a Second Welfare Theorem for ordinal lotteries: every CERI allocation is ordinally efficient, and every ordinally efficient lottery can be supported by some random-income competitive equilibrium (Nguyen et al., 16 Sep 2025).

The supporting-price conditions have an explicit linear form. A feasible lottery allocation $\mathbf{c}=(c_j)_{j\in M}$1 is supportable if there exists $\mathbf{c}=(c_j)_{j\in M}$2 such that $\mathbf{c}=(c_j)_{j\in M}$3 whenever good $\mathbf{c}=(c_j)_{j\in M}$4 is not fully allocated in expectation, and such that whenever agent $\mathbf{c}=(c_j)_{j\in M}$5 receives bundle $\mathbf{c}=(c_j)_{j\in M}$6 with positive probability, every strictly preferred bundle $\mathbf{c}=(c_j)_{j\in M}$7 satisfies

$\mathbf{c}=(c_j)_{j\in M}$8

Through Farkas’ lemma, the nonexistence of such prices is equivalent to the existence of a feasible stochastic-dominance improvement, so price support and ordinal efficiency coincide (Nguyen et al., 16 Sep 2025).

Fairness properties depend directly on the design of the budget distributions. If all agents have identical budget distributions, then the resulting CERI allocation is ordinally envy-free: each agent’s lottery stochastically dominates every other agent’s lottery according to her own preferences. If all budget supports lie within $\mathbf{c}=(c_j)_{j\in M}$9, where $N$0 is the maximum bundle size, then any ex-post implementation is envy-free up to one good (EF1). The mechanism CERI-S combines these two conditions by using identical small-support budget distributions, thereby obtaining ordinal efficiency, $N$1-ex-post efficiency, ordinal envy-freeness, ex-post EF1, and strategyproofness in the large (Nguyen et al., 16 Sep 2025).

Earlier generic-budget work establishes related fairness guarantees for deterministic competitive equilibria with indivisible goods. Every competitive equilibrium guarantees each agent her $N$2-out-of-$N$3 maximin share for every rational $N$4, and competitive equilibrium is justified envy-free for coalitions: if an agent’s budget is at least the sum of the budgets of a coalition, she cannot prefer the union of that coalition’s bundles to her own. When exact budget-proportional shares are unattainable because of indivisibilities, truncated share identifies the best Pareto-optimal utility below the budget benchmark, and the existence results for generic budgets guarantee equilibria in which each agent receives at least her truncated share (Babaioff et al., 2017).

The almost-all-incomes literature pushes these fairness statements further by defining CE-fairness in terms of generalized local and multi-bundle maximin guarantees, denoted properties (P1) and (P2). Every competitive equilibrium is CE-fair, though not every CE-fair allocation is itself a competitive equilibrium. This distinction is useful in impossibility proofs: nonexistence is often shown by demonstrating that no CE-fair allocation exists on a positive-measure region of the income space (Segal-Halevi, 2017).

5. Price-dependent incomes, participation constraints, and income effects

One important extension replaces random or generic incomes by price-dependent incomes in order to accommodate reservation utilities, outside options, or property rights. In probabilistic assignment problems with participation constraints, strict envy-freeness can conflict with individual rationality because remedying envy by swapping bundles may violate the other agent’s reservation utility. The relevant fairness notion becomes justified envy: at allocation $N$5, agent $N$6 has justified envy toward $N$7 if

$N$8

so the swap would eliminate the envy without violating $N$9’s participation constraint. The target is therefore no justified envy, not no envy (Echenique et al., 2019).

The corresponding equilibrium construction uses expenditure functions

$i$0

and incomes

$i$1

If satiation is affordable for all, incomes are set to satiation expenditure plus an equal share of the residual budget. Otherwise, one chooses $i$2 such that $i$3 and sets $i$4. Under quasi-concavity and additional regularity conditions, there exist continuous income functions $i$5 and an equilibrium $i$6 such that $i$7 is individually rational, Pareto optimal, and has no justified envy. This is CERI in a broadened sense: incomes are neither equal nor exogenous, but endogenous functions of prices calibrated to rights and reservation utilities (Echenique et al., 2019).

A second extension concerns income effects. In models with indivisible goods and a numéraire, the Equilibrium Existence Duality shows that equilibrium existence depends on substitution effects, not income effects. For utility level $i$8, the Hicksian valuation

$i$9

represents the quasilinear valuation dual to the original non-quasilinear utility at fixed utility. Competitive equilibria exist in the original economy for all endowment allocations if and only if competitive equilibria exist in all associated Hicksian economies for all utility profiles. This permits existence results to be transported from transferable-utility settings to environments with income effects (Baldwin et al., 2020).

The same duality identifies net substitutability—defined via Hicksian rather than Marshallian demand—as the relevant existence condition. Gross substitutability implies net substitutability, but the latter is strictly weaker in the presence of income effects. The paper further extends the demand-type and unimodularity classification to this setting: competitive equilibria exist for all endowment allocations precisely for quasiconcave preferences whose Hicksian valuations are of a unimodular demand type. A plausible implication is that random-income equilibrium theory need not remain confined to quasilinear pseudomarkets; its existence envelope is governed by the geometry of compensated demand (Baldwin et al., 2020).

6. Dynamic assignment, implementation, applications, and limits

Dynamic combinatorial assignment provides a direct operational use of random budgets. The relevant solution concept is dynamic approximate competitive equilibrium from equal incomes (DACEEI). For parameters $\Psi_i \subseteq \mathbb{N}^m$0, an allocation $\Psi_i \subseteq \mathbb{N}^m$1, budgets $\Psi_i \subseteq \mathbb{N}^m$2, and prices $\Psi_i \subseteq \mathbb{N}^m$3 satisfy DACEEI if, for all late enough arrivals $\Psi_i \subseteq \mathbb{N}^m$4, agent $\Psi_i \subseteq \mathbb{N}^m$5 receives a most-preferred affordable bundle with budget $\Psi_i \subseteq \mathbb{N}^m$6, and cumulative demand for each good $\Psi_i \subseteq \mathbb{N}^m$7 satisfies

$\Psi_i \subseteq \mathbb{N}^m$8

whenever $\Psi_i \subseteq \mathbb{N}^m$9. A naive repeated ACEEI would compound market-clearing errors over time. The remedy is an $0\in\Psi_i$0-ECEEI with random budgets clearing exactly in expectation, used as the static backbone of the online combinatorial assignment mechanism (OCAM) (Nguyen et al., 2023).

OCAM samples an initial fraction of agents, computes prices and a type-based mapping from reported types to random budget distributions, and then assigns later agents their favorite affordable bundles under sampled prices and drawn budgets. Under random arrivals and a lower bound on capacities,

$0\in\Psi_i$1

the mechanism outputs an $0\in\Psi_i$2-DACEEI with probability at least $0\in\Psi_i$3. It is group-strategyproof up to one object and EF1 for a $0\in\Psi_i$4-fraction of agents. The same framework is proposed for refugee resettlement, daycare assignment, and airport slot allocation (Nguyen et al., 2023).

In the static combinatorial-assignment model, ex-post implementation of CERI is necessarily approximate in general. Any CERI has a $0\in\Psi_i$5-near-feasible ex-post implementation, meaning each good’s capacity may be exceeded by at most $0\in\Psi_i$6 units in any realized allocation. Every such realized allocation is $0\in\Psi_i$7-ex-post efficient. On the incentive side, the mechanism CERI-L uses identical small-support budgets together with a random grid over the type-count space. With grid step size $0\in\Psi_i$8, it is uniformly strategyproof, asymptotically ordinally efficient, asymptotically $0\in\Psi_i$9-ex-post efficient, ordinally envy-free, and ex-post EF1 (Nguyen et al., 16 Sep 2025).

The main limitations are equally sharp. Generic-budget existence with indivisible goods is proved in strong form only for restricted domains, especially two-agent additive markets. The almost-all-incomes program shows positive-measure nonexistence for broader domains, including $b_1=b_2=1/2$00 goods with $b_1=b_2=1/2$01 agents under general monotone preferences and chores with $b_1=b_2=1/2$02 agents. Exact ex-post feasibility is generally unavailable in combinatorial assignment; near-feasibility bounds such as $b_1=b_2=1/2$03 are therefore intrinsic to current implementations. Continuous or smoothed budget distributions are often required for clean existence theorems. These boundaries are not peripheral: they mark the precise extent to which randomizing or endogenizing incomes can regularize equilibrium without eliminating the combinatorial obstructions created by indivisibilities (Segal-Halevi, 2017, Nguyen et al., 16 Sep 2025).

CERI is therefore best understood not as a single mechanism but as a general equilibrium program for discrete allocation. Its core claim is that the budget vector—random, generic, almost equal, or price-dependent—is a powerful design lever. By moving from fixed equal incomes to carefully structured income distributions, the literature recovers competitive-equilibrium existence in measure-theoretic or expected-value senses, characterizes ordinally efficient lotteries, aligns fairness with entitlements and participation constraints, and supports dynamic online allocation with strong probabilistic guarantees.