Homogeneous Relative Externality in Economic Systems

• HRE is a structural restriction that requires external impact ratios to be scale invariant, supporting tractable equilibrium analysis across various models.
• In contest theory and data markets, HRE enables derivation of logit forms with additive luck and ensures predictable welfare and fairness outcomes.
• Applied to multiplex economies and sharing networks, HRE aligns exposure vectors to achieve efficient allocations and fair comparative statics.

Searching arXiv for the specified papers and topic. Homogeneous Relative Externality (HRE) denotes a family of restrictions on externality structure under which the impact of one agent’s action on others is homogeneous in a relative sense rather than arbitrary across identities or scales. The term is used explicitly as an axiom in contest success functions, where it requires scale invariance of the ratio of pairwise relative externalities (Yu, 18 Aug 2025). In adjacent literatures the label is not always used verbatim, but closely related structures appear as anonymous additive externalities in data markets, centrality-parallel exposure conditions in multiplex exchange economies, symmetric distance-based spillovers in homogeneous sharing networks, and common externality weights in chore division (Hossain et al., 2023, Li et al., 20 May 2026, Mane et al., 2018, Sanpui, 2023). Across these settings, HRE serves to constrain cross-effects sufficiently to recover tractable equilibrium, welfare, or fairness characterizations.

1. Conceptual scope across research areas

The term is not uniform across fields. In contest theory, HRE is an explicit axiom on the relative change in one contestant’s allocation caused by another contestant’s effort. In data markets and chore division, by contrast, HRE is a mapped or natural specialization of more general externality formalisms rather than a primitive definition. In multiplex network economies, the corresponding condition is topological: planner-relevant exposure vectors must be pairwise proportional across layers. In homogeneous ring networks, relative externality is operationalized through symmetric gain or loss shares induced by endogenous link formation (Yu, 18 Aug 2025, Hossain et al., 2023, Li et al., 20 May 2026, Mane et al., 2018, Sanpui, 2023).

Domain	HRE formulation	Main implication
Contest success functions	Scale invariance of $\frac{d_{ij}(x)}{d_{ji}(x)}$	Logit form with luck baselines
Data markets	Anonymous/symmetric specialization of externalities	Equilibrium and welfare guarantees under intervention
Multiplex economies	$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$	First and Second Welfare Theorems
Sharing networks	Symmetry among equidistant agents in a ring	Structured distribution of beneficiaries and losers
Chore division	Common externality share relative to baseline density	Fairness and algorithmic guarantees under specialization

A plausible unifying interpretation is that HRE rules out identity-specific idiosyncrasy in how externalities compare across agents or across layers, while still permitting heterogeneity in levels, baselines, or network positions. The precise form of “relative” therefore depends on the ambient model: ratios of deviations in contests, sums of anonymous spillovers in data markets, proportionality of exposure vectors in network economies, and piece-size- or valuation-relative spillovers in fair division.

2. Axiomatic HRE in contest success functions

In contest success functions with luck, contestants choose nonnegative efforts on the active domain $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$, and a CSF assigns probabilities $p_i^M(x^M)$ summing to one over every sub-contest $M \subseteq N$, $|M| \ge 2$. Under strict monotonicity (SM) and Luce’s choice axiom (LCA), the CSF has the logit representation

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$

where each $f_j(\cdot)$ is strictly increasing and nonnegative (Yu, 18 Aug 2025).

The paper defines relative externality through the percentage change in contestant $j$’s allocation when contestant $i$ turns active from inactive:

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$0

Under the logit form,

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$1

HRE is then the requirement

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$2

Equivalently,

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$3

This is weaker than homogeneity of absolute allocations or relative homogeneity of odds, because it permits positive baselines $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$4 and therefore accommodates luck (Yu, 18 Aug 2025).

The main characterization states that SM, LCA, and HRE hold iff

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$5

for parameters $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$6, $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$7, and common $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$8. Here $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$9 are bias weights on effort, $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$0 are luck baselines, and $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$1 is the discriminating parameter. The symmetric case under anonymity becomes

$X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$2

Setting $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$3 recovers the classic Tullock form, whereas HRE rather than HOM or RH is what allows additive luck (Yu, 18 Aug 2025).

The same structure appears in comparative statics. With $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$4, the cross-effect satisfies

$X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$5

and the cross-elasticity is

$X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$6

Hence the ratio of cross-elasticities inherits the HRE invariance. This makes HRE an externality restriction on comparative statics, not merely an equilibrium normalization (Yu, 18 Aug 2025).

3. HRE as a specialization of data-market externalities

In data markets with posted prices, sellers offer datasets, buyers choose arbitrary bundles $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$7, and the simultaneous-move game is driven by expected utility

$X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$8

where $X \equiv \mathbb{R}_+^n \setminus \{\mathbf{0}\}$9 is expected net gain. The paper does not use “HRE” explicitly. In this context, HRE arises as a symmetric specialization of the independent model $p_i^M(x^M)$0 and of the richer joint model $p_i^M(x^M)$1 (Hossain et al., 2023).

Under the independent model, HRE corresponds to anonymity:

$p_i^M(x^M)$2

so that buyer $p_i^M(x^M)$3 suffers

$p_i^M(x^M)$4

Without intervention, it is a dominant strategy for buyer $p_i^M(x^M)$5 to choose

$p_i^M(x^M)$6

and a pure Nash equilibrium exists, but welfare can be poor: there exist instances with welfare regret WRaE equal to $p_i^M(x^M)$7. WRaE is defined as

$p_i^M(x^M)$8

where $p_i^M(x^M)$9 maximizes social welfare and $M \subseteq N$0 denotes PNE profiles (Hossain et al., 2023).

The platform intervention is a revenue-neutral transaction cost

$M \subseteq N$1

with $M \subseteq N$2. The best-response-relevant effective utility becomes

$M \subseteq N$3

and buyer $M \subseteq N$4 has dominant strategy

$M \subseteq N$5

The resulting WRaE bound is

$M \subseteq N$6

where

$M \subseteq N$7

As $M \subseteq N$8 and prediction bias tends to zero, equilibrium welfare approaches optimum (Hossain et al., 2023).

The richer joint externality model goes beyond HRE. Without intervention, there exist instances with no $M \subseteq N$9-PNE for any $|M| \ge 2$0, and even when pure equilibria exist, WRaE can approach $|M| \ge 2$1. With intervention and symmetrization, however, there always exists an $|M| \ge 2$2-PNE with

$|M| \ge 2$3

When externalities are symmetric and prediction bias is small, $|M| \ge 2$4 as $|M| \ge 2$5, and WRaE is bounded by $|M| \ge 2$6. The paper identifies $|M| \ge 2$7 as a “sweet spot” in this setting (Hossain et al., 2023).

The learning extension shows that the same intervention remains feasible under repeated interaction with bandit feedback. Buyers learn over $|M| \ge 2$8 rounds, and the paper studies individual effective regret

$|M| \ge 2$9

and welfare regret

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$0

A zooming/UCB-type algorithm in Hamming metric yields worst-case

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$1

and under the metric/Hamming structure,

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$2

Moreover,

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$3

so increasing $$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$4 as learning improves aligns equilibrium and welfare (Hossain et al., 2023).

4. Centrality-parallel HRE in multiplex market economies

In an Arrow–Debreu economy with multiplex network externalities, consumers $$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$5 consume goods or layers $$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$6, and effective consumption is

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$7

Preferences are log-Cobb-Douglas across layers,

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$8

with the usual competitive budget constraint and layer-by-layer market clearing (Li et al., 20 May 2026).

The paper identifies HRE with a network-topological condition. Let

$$p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M} f_j(x_j)},$$9

and define the unweighted transposed Katz–Bonacich exposure vector

$f_j(\cdot)$0

HRE holds iff, for any pair of layers $f_j(\cdot)$1,

$f_j(\cdot)$2

that is, there exists $f_j(\cdot)$3 such that

$f_j(\cdot)$4

This means that every agent’s relative externality exposure is aligned across layers up to a scalar multiple (Li et al., 20 May 2026).

Competitive equilibrium consumption in layer $f_j(\cdot)$5 is

$f_j(\cdot)$6

and equilibrium prices are

$f_j(\cdot)$7

The effective endowment vector $f_j(\cdot)$8 is determined by

$f_j(\cdot)$9

The paper’s welfare result is sharp: if the interior competitive equilibrium exists and HRE holds, then both the First Welfare Theorem and the Second Welfare Theorem hold despite externalities. If HRE fails, both fail: no interior competitive equilibrium is efficient because private and social marginal rates of substitution cannot be equalized across agents (Li et al., 20 May 2026).

Two sufficient conditions generate HRE. First, if every $j$0 is regular, then $j$1 for all $j$2. Second, if $j$3 is identical across layers, then all $j$4 coincide and so do the exposure vectors. When neither condition holds, a centrality wedge appears between private and social first-order conditions (Li et al., 20 May 2026).

The efficiency-restoring instrument is a Lindahl equilibrium with personalized prices. Up to normalization,

$j$5

The resulting allocation is Pareto efficient, but the paper emphasizes that highly central agents may be worse off relative to the competitive equilibrium because they pay for externalities that market prices had left unpriced (Li et al., 20 May 2026).

The framework also quantifies inefficiency when HRE fails. For Pareto weights $j$6,

$j$7

and the coefficient of resource utilization satisfies

$j$8

Under HRE, the $j$9 coincide and $i$0, so approximate proportionality of exposure vectors implies approximate efficiency (Li et al., 20 May 2026).

5. Homogeneity and relative externality in sharing-economy networks

In endogenous sharing economy networks, a social cloud is an undirected graph $i$1 of $i$2 agents with links $i$3. The paper studies externalities generated when a pair of agents forms a link and thereby changes resource availability for third parties. Distances are shortest-path distances $i$4, and the relevant closeness notion is harmonic centrality

$i$5

Resource availability for agent $i$6 is

$i$7

so the externality from a new link is $i$8 (Mane et al., 2018).

The homogeneous case studied in detail is the populated ring network. Agents are homogeneous, the topology is vertex-transitive, each node has degree $i$9, and harmonic centrality is uniform. This symmetry generates a relative-externality interpretation: agents at the same ring distance from the newly formed link are equivalent, and their gain or loss shares can be compared through

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$00

and

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$01

In the ring, these relative externalities are identical for agents equidistant from the new chord and decay with ring distance from it (Mane et al., 2018).

A central empirical finding is that an increment in closeness is necessary but not sufficient for positive externalities. The paper conjectures, and experimentally supports, that if $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$02, then $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$03, and in observed cases of positive externality one has $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$04. Yet some agents satisfy $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$05 while still experiencing $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$06. The mechanism is the normalization in $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$07: when link endpoints become more central, their larger $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$08 dilutes supply probability to many others, offsetting shorter-path gains for some agents (Mane et al., 2018).

The ring simulations establish a clear size effect. For $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$09, no agent experiences positive externalities. For $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$10, one or more agents do experience positive externalities, and the number of beneficiaries generally increases with the initial ring distance between the link endpoints. Across all tested cases, however, the number of beneficiaries is strictly less than the number of non-beneficiaries, and beneficiaries constitute between $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$11 and $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$12 of agents. This suggests that homogeneity of topology does not imply uniformly positive spillovers; it merely makes the distribution of those spillovers structurally analyzable (Mane et al., 2018).

6. Homogeneous-relative externalities in chore division

In chore division with externalities, the heterogeneous bad is the interval $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$13, and for each ordered pair $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$14 agent $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$15 has an integrable nonnegative disutility density $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$16. If agent $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$17 receives piece $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$18, then the disutility to $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$19 is

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$20

and total disutility under allocation $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$21 is

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$22

The paper itself works with general $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$23 and does not introduce HRE explicitly. A natural specialization consistent with the framework is to fix $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$24 and baseline density $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$25 with $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$26, then define

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$27

This makes externalities homogeneous across targets and relative to agent $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$28’s baseline density (Sanpui, 2023).

Within the general model, proportionality requires

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$29

and swap envy-freeness requires

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$30

Swap stability is stronger:

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$31

The paper proves that for $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$32, proportionality and swap envy-freeness are equivalent; for $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$33, neither implies the other; and swap stability implies proportionality (Sanpui, 2023).

The structured classes give constructive consequences under homogeneous-relative specializations. For piecewise constant disutilities, the uniform allocation is swap stable and uses $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$34 cuts. For piecewise linear disutilities, the sandwich allocation is swap stable and uses $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$35 cuts. For piecewise constant inputs, a linear program computes an optimal allocation that is proportional and swap envy-free in polynomial time in $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$36 and $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$37. For general $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$38-Lipschitz disutilities lower bounded by $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$39, the paper gives polynomial-time approximation schemes for proportionality and swap envy-freeness (Sanpui, 2023).

The homogeneous-relative specialization makes these results transparent. With $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$40, $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$41, and $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$42, equal thirds satisfy

$\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$43

so proportionality holds with equality, and the swap envy-freeness inequalities bind at equality as well. With uneven shares, swap envy-freeness can fail, and for sufficiently large $\widetilde{b}^{(l)} \parallel \widetilde{b}^{(t)}$44 proportionality can fail too. Thus homogeneous-relative externality in chore division preserves the general tension between fairness and allocation shape while remaining compatible with the paper’s existence and algorithmic results (Sanpui, 2023).

Homogeneous Relative Externality is therefore best understood not as a single invariant formula shared across all models, but as a recurrent structural restriction that makes externality comparisons homogeneous in a relative sense. In contest theory it yields the canonical logit form with luck; in data markets it isolates anonymous relative spillovers that can be internalized by revenue-neutral transaction costs; in multiplex exchange economies it becomes the centrality-parallel condition under which competitive equilibrium is efficient; in homogeneous sharing networks it organizes the spatial distribution of gains and losses from link formation; and in chore division it identifies a tractable externality specialization consistent with proportionality, swap envy-freeness, and algorithmic optimization (Yu, 18 Aug 2025, Hossain et al., 2023, Li et al., 20 May 2026, Mane et al., 2018, Sanpui, 2023).