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Econ-284 · economics · Economics

Inequality-aware allocation

LLM status Open unverified

Statement

Original subject group: Mechanism design and market design

Audit classification: Published research agenda. Dworczak’s verified §4 items 2–4 explicitly raises simple mechanisms, moments/robustness and multidimensional extensions. The displayed minimax problem is editorial.

Definitions and precise research target

Model and research target. A unit mass of agents demand at most one homogeneous indivisible good. Type $(r,w)$ comprises private willingness to pay $r\in[0,1]$ and social welfare weight $w\in[0,W]$, where finite $W>0$ is known. The designer has supply $q\in(0,1)$ and knows only specified moments of the joint type distribution; let $\mathcal F$ be all distributions consistent with them. A direct mechanism specifies allocation probability $x(r)\in[0,1]$ and payment $p(r)$, with utility $r x(r)-p(r)$, truthful reporting, voluntary participation, and feasibility under every $F\in\mathcal F$. With shadow value $\lambda\geq0$ for public revenue, evaluate \[ V_F(M)=E_F[w(r x(r)-p(r))+\lambda p(r)],\qquad \sup_M\inf_{F\in\mathcal F}V_F(M). \] Characterize robust optimal mechanisms and when price menus, subsidies, or rationing improve this redistributive objective. Derive sharp guarantees for posted-price mechanisms relative to the full robust optimum, under clearly stated budget constraints and welfare-weight bounds. Explain why privately varying $w$ affects the designer's objective without being directly reportable in utility; additional screening dimensions require an extended model.

Definitions and maintained assumptions (editorial unless a source definition is named). Use a unit-mass atomless population with type law $F$ on $[0,1]\times[0,W]$. The allocation depends only on reported willingness to pay $r$; $w$ is a designer welfare weight, not a private utility argument being truthfully screened. DSIC is $rx(r)-p(r)\ge rx(r')-p(r')$ for all $r,r'$, and IR is $rx(r)-p(r)\ge0$. Impose $-\bar s\le p(r)\le\bar p$ and a declared balanced-budget or public subsidy constraint; without a subsidy limit, high welfare weights can make unlimited payments to agents artificially optimal. Supply means $\mathbb E_F[x(r)]\le q$ for every $F\in\mathcal F$, interpreted as aggregate fractions in the atomless model. Specify which bounded moment functions and exact moment values define nonempty $\mathcal F$. The objective treats a unit of public revenue at shadow value $\lambda$ and agents’ net surplus at weight $w$; these are normative primitives. Approximation ratios need a nonnegative normalization and a strictly positive optimum; otherwise use additive losses.

Variants and assumption changes

  1. Moment ambiguity (source agenda, editorial model). Fix means and covariance of $(r,w)$, bounded transfers and a subsidy budget. Characterize the minimax DSIC menu and whether a posted price achieves a guaranteed fraction of its positive value.
  2. Multiple qualities (source agenda, editorial model). Replace the homogeneous good by finitely many qualities with explicit utility $v(q,\theta)$ and reportable multidimensional type $\theta$. Study robust mechanisms with redistributive weights; the one-dimensional monotonicity characterization no longer suffices.

References and source audit

  1. §4 items2–4, printed pp.90–91. Supports the stated research area; exact displayed specification is editorial. https://www.sigecom.hosting.acm.org/exchanges/volume_22/1/issue.pdf
  1. Piotr Dworczak. 2024. Inequality and Market Design. §4 items2–4, printed pp.90–91. https://www.sigecom.hosting.acm.org/exchanges/volume_22/1/issue.pdf.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-284-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-284

Inequality-aware allocation An exact robust allocation problem with unknown welfare-weight sorting

Question and scope

The canonical question asks for incentive-compatible allocation under uncertain correlation between willingness to pay and social welfare weights. Here values are uniform and their marginal distribution is known, while only the mean and bounds of welfare weights are known. This structured ambiguity set admits an exact reduction to at most two posted-price components. It also gives a sharp failure of guarantees for deterministic, unrationed posted prices. No claim is made for arbitrary moment ambiguity or multidimensional reports.

A specified ambiguity set and mechanism class

There is a unit mass of agents with $r\sim\mathrm{Unif}[0,1]$. A welfare weight $w\in[0,W]$ is unobserved and does not enter the agent's utility, where $W>0$. All joint laws with this value marginal and ${\mathbb{E}} w=\mu\in(0,W]$ are admissible. Write $a=\mu/W\in(0,1]$. This assumption fixes the full value marginal, which is stronger than finitely many moment restrictions; it is an explicit restricted variant.

A direct mechanism assigns probability $x(r)\in[0,1]$ and expected payment $p(r)\in[0,1]$. It is dominant-strategy incentive compatible and individually rational for utility $rx-p$. There are no subsidies. Supply requires $\int_0^1x(r)\,dr\le q$, where $0<q<1$. Feasibility is in expectation for this atomless population. Social welfare is \[ \inf_F{\mathbb{E}}_F[w\{rx(r)-p(r)\}+\lambda p(r)],\qquad\lambda\ge0. \] Payments accrue to the public sector and receive the fixed shadow value $\lambda$.

Lemma (Incentives and the utility envelope): A mechanism in this class is incentive compatible and individually rational if and only if $x$ is nondecreasing, $p(0)=0$ and \[ U(r):=rx(r)-p(r)=\int_0^r x(t)\,dt. \] Changes on null sets may be chosen consistently at jumps. These formulas imply $0\le p(r)\le r\le1$.
Proof: Incentive compatibility at two values $r>s$ gives $(r-s)x(s)\le U(r)-U(s)\le(r-s)x(r)$, and their sum gives $x(r)\ge x(s)$. Since $x$ is bounded, summing these inequalities over partitions sandwiches utility increments between lower and upper Riemann sums for a monotone function, yielding $U(r)-U(0)=\int_0^r x$. At zero, nonnegative payments and individual rationality force $p(0)=U(0)=0$. Conversely the integral of a nondecreasing $x$ lies between these two bounds, so truthful utility dominates the utility from every alternative report. The envelope is nonnegative and $\int_0^r x\le rx(r)$, giving the payment bounds. ∎
Theorem (The same adverse sorting works for every mechanism): For every feasible mechanism, \[ \inf_F{\mathbb{E}}_F[wU(r)+\lambda p(r)] =W\int_0^a U(r)\,dr+\lambda\int_0^1p(r)\,dr. \] The infimum is attained by $w=W{\mathbf 1}\{r<a\}$. In particular nature's worst sorting need not depend on which feasible monotone allocation is chosen.
Proof: Let $m(r)={\mathbb{E}}[w\mid r]/W$. It lies in $[0,1]$ and has integral $a$. Compare it with $m_0(r)={\mathbf 1}\{r<a\}$. Since $U$ is nondecreasing, \[ \int_0^1(m-m_0)U =\int_0^1(m-m_0)\{U-U(a)\}\ge0. \] For $r<a$, both factors are nonpositive; for $r>a$, both are nonnegative. The constant term canceled because the integrals of $m,m_0$ agree. Thus the proposed sorting minimizes weighted utility. It satisfies all ambiguity restrictions and attains the value; the payment term depends only on the fixed marginal. ∎

Reduction to mixtures of threshold allocations

A threshold $t\in[0,1]$ serves values $r\ge t$ and charges $t$ when served. It uses supply $1-t$ and has robust objective \[ R(t)=\frac W2(a-t)_+^2+\lambda t(1-t). \] For $t=1$ it is the null mechanism up to a zero-measure type. Any right-continuous nondecreasing allocation is the cumulative distribution of a subprobability measure of thresholds; adding unused mass at $t=1$ makes it a probability measure $\nu$. Its payment envelope is exactly the mixture of these threshold payments. Endpoint choices have no effect on the objective or supply.
Theorem (Exact two-threshold solution): The robust optimum equals \[ \max_{\nu}\int_0^1 R(t)\,d\nu(t) \quad\text{subject to}\quad \nu\text{ a probability law},\quad \int_0^1(1-t)\,d\nu(t)\le q. \] It is attained by a law supported on at most two thresholds. Equivalently, putting $z=1-t$ and $f(z)=R(1-z)$, its value is $\max_{0\le z\le q}\operatorname{cav}f(z)$, where $\operatorname{cav}f$ is the smallest concave majorant of $f$ on $[0,1]$.
Proof: The threshold representation and Fubini's theorem give the stated supply and objective; the preceding adverse-sorting theorem is what permits linearity of the robust objective in the mixture. Conversely any such mixture yields a feasible monotone allocation with the required envelope.

The attainable pairs of mean supply and mean value form the convex hull of the continuous compact curve $(z,f(z))$ in the plane. This hull is compact: in the plane every convex combination can be reduced to at most three points by removing a nonzero affine dependence among any four points, so it is the continuous image of a compact set of triples and their weights. At a fixed mean supply, the upper boundary is the smallest concave majorant of $f$.

A point on that upper boundary with supply strictly between zero and one admits a supporting line with finite slope. Every curve point used to represent it must lie on that line; otherwise its average would fall below the line. Along a line, any point between the leftmost and rightmost such curve points is a combination of those two. At supply zero or one only that endpoint is possible. The maximum over the compact interval $[0,q]$ therefore exists and has a two-threshold representation. ∎

The two thresholds refer to allocation probabilities for agents, not to secretly choosing a different welfare-weight law after observing a realized mechanism. A rationed free option can be one of the components. This is the economically relevant distinction in the next result.

Theorem (Exact zero-revenue optimum and a zero price guarantee): If $\lambda=0$, the optimal robust value is $qWa^2/2=q\mu^2/(2W)$, attained by the free lottery $x(r)=q$, $p(r)=0$. A feasible deterministic unrationed posted price has threshold $t\ge1-q$ and robust value at most \[ \frac W2(a-1+q)_+^2. \] Thus if $0<q\le1-a$ and $a<1$, every such deterministic price has value zero while the robust optimum is strictly positive. No positive multiplicative guarantee holds for that restricted price class, even with values uniformly distributed.
Proof: When $\lambda=0$, $f(z)=W(a-1+z)_+^2/2$ is convex, with $f(0)=0$ and $f(1)=Wa^2/2$. Convexity gives $f(z)\le zf(1)$. Any feasible mixture therefore has value at most $qf(1)$. The mixture placing probability $q$ at threshold zero and $1-q$ at the null threshold one attains this bound and has the stated allocation and payments. A deterministic price has supply $1-t$, so feasibility implies $t\ge1-q$; its value $W(a-t)_+^2/2$ is decreasing in $t$, proving the bound and the failure example. ∎

This impossibility expressly excludes rationed posted-price mechanisms: a zero price with uniform rationing is the optimal lottery just constructed. It also says nothing about positive revenue weights bounded away from zero or ambiguity sets imposing favorable sorting. Those are meaningful avenues for quantitative approximation guarantees.

Remaining gap and literature scope

The full canonical problem permits more general value distributions, finite moment ambiguity, transfer bounds, subsidy rules and revenue weights. In those settings the adverse sorting may vary with the mechanism, and a single linear concavification need not survive. We have not supplied a computational method for arbitrary moment descriptions or a sharp approximation ratio for an appropriately broad simple-mechanism class.

Dworczak's primary research letter, including its allocation-as-distribution transformation and final research directions, was checked. Concavification and randomized allocation are established tools in that literature. The claim here is the proved specialization to the displayed ambiguity set and the carefully qualified deterministic-price counterexample, not a claim to invent those tools.

Completion Estimate

43%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves robust allocation with a known uniform value marginal and unknown welfare-weight sorting through two-threshold concavification, including an exact lottery optimum and a deterministic-price counterexample. General finite moment ambiguity, subsidy and transfer constraints, and guarantees for a broader simple-mechanism class remain unresolved.

References

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