Discrete Choice with Endogenous Peer Selection

Nail Kashaev; Natalia Lazzati

REVIEW 3 major objections 5 minor 17 references

Discrete Choice with Endogenous Peer Selection

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read From a long panel of choices alone, an econometrician can recover the social network, the probability each agent attends to each peer, and the preferences that drive decisions — no covariates or network surveys required.

desk verdict Genuinely new identification idea undone by an unproven and false-as-stated full-support equilibrium claim; fixable, but currently the central results are conditional on unobservable CCPs. read the letter →

arxiv 2511.21446 v2 pith:5VGPHER2 submitted 2025-11-26 econ.EM econ.TH

Nail Kashaev , Natalia Lazzati This is my paper · ORCID

classification econ.EMecon.TH

keywords peereffectslimitedattentionidentificationcontinuous-timeMarkovprocessconditionalchoiceprobabilitiesnetworknonparametricpaneldata

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a social network and the way agents pay attention to their peers can be recovered from the record of their choices alone. The authors build a continuous-time model in which each agent, when her alarm rings, first selects a random subset of peers (with probabilities depending on recent choices) and then makes a choice influenced by that subset. They show that if the econometrician observes choices over a long period, the reference groups, the peer-selection mechanism, and the choice rules are all nonparametrically identified — no exogenous covariates or network surveys needed. The key is that variation in the number of peers within an agent type creates a monotone relation between observed choice probabilities and the unobserved attention probabilities. This matters because ignoring limited attention in peer-effect models can bias estimates and mislead counterfactuals.

What carries the argument

The engine of the argument is the mixture representation of conditional choice probabilities, P_a(v|y) = Σ_{N⊆N_a} R_a(v|y,N) S_a(N|y,N_a), combined with independent peer selection (Assumption 1), which makes S_a a product over individual peers. The identification lever is the strictly monotone function f(x) = (1 - x^{n3})/(1 - x^{n2}) linking observed CCP differences across agents with different numbers of peers to the selection probability q; because f is monotone, q is uniquely recoverable. The recursive step then identifies R for all peer subsets.

What would settle it

Construct two structurally distinct models satisfying Assumptions 1, 2, 4, and 5 that generate identical conditional choice probabilities for every possible history; such a pair would refute Proposition 4.4. Empirically, run a controlled experiment where the true network and attention probabilities are known, then apply the method and check whether it recovers those links and probabilities exactly.

Watch

Extended reading notes

Core claim

The central claim, formalized as Proposition 4.4, is that the network (the set of potential peers for each agent), the peer-selection mechanism Q, and the choice rules R are all identified from the conditional choice probabilities (CCPs) alone, under Assumptions 1–5. The proof first shows that the CCP difference P_a(v|0^v_{a'}) - P_a(v|0) is zero exactly when a' is not a peer of a (Proposition 4.1), so the network is recoverable. Then, comparing same-type agents with different numbers of peers, the ratio of CCP differences equals a strictly monotone function f(x) = (1 - x^{n3})/(1 - x^{n2}) of the selection probability q, allowing q to be solved from data. With q in hand, choice rules for em

Load-bearing premise

The load-bearing premise is Assumption 3, a rank/relevance condition asserting that changing a peer's choice always alters an agent's choice probability in a way that is not cancelled by simultaneous changes in attention; if this condition fails, the network cannot be distinguished from the CCPs and the whole identification chain collapses.

Editorial extensions

If this is right

If the identification result is correct, researchers can map who influences whom from standard panel data, without administering network surveys.
The model provides a direct test of the full-attention assumption: if the rank condition behind Proposition 4.1 fails in a systematic way, agents likely do not consider all peers.
For linear-in-means logit peer effects, the paper implies that only the choice rules under no peer and one peer are needed for estimation, simplifying practical implementation.
The recovered attention probabilities Q allow separate measurement of two channels of peer influence: who you notice and how you react once you notice them.
Counterfactual policy analysis (e.g., informing a firm of competitor moves) becomes feasible from observational data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The monotone-ratio identification strategy may transfer to other settings where consideration sets vary in size, such as consumer choice over products with endogenous consideration, suggesting a general template for nonparametric identification of consideration-set models.
The paper's reliance on variation in reference-group sizes within types means that empirically, identification is strongest in populations with heterogeneous degree distributions; in nearly regular networks, the model may be only partially identified unless parametric restrictions are imposed.
The full-support equilibrium claim (Proposition 3.1) appears to require an unstated strict positivity of the choice and selection probabilities; if some configurations are unreachable, the CCP-based argument would need adjustment.
A potentially testable extension: use the recursive identification to check the independent-selection assumption by comparing Q estimates derived from different active-set sizes; systematic disagreement would signal correlated peer selection.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real contribution here is the identification strategy: recovering the reference network, peer-attention probabilities, and choice rules from a single long panel of choices, using only variation in the number of potential peers within types. That is new relative to the cited literature, and the recursive argument in Proposition 4.2 is clever and algebraically sound. The two-agent example showing how attention based on similarity can flip comparative statics is also nice. This is a solid theoretical core. The problem is at the base. Proposition 3.1 asserts a unique, full-support invariant distribution under Assumptions 1 and 2. It is not proven, and it is false as stated. The model explicitly allows agents with empty reference groups (Agent 4 in Example 1). If such an agent has a degenerate choice rule, say always chooses 0, then configurations where she chooses 1 have zero probability in the long run. Even for non-isolated agents, nothing in Assumptions 1 and 2 rules out a degenerate choice rule conditional on an empty active set. So the CCPs used in Proposition 4.1's network test, P_a(v|0^v_{a'}) - P_a(v|0), are not recoverable from a long panel when they involve off-support configurations. This is not a minor technicality: it breaks the chain from data to the identified structural objects. The identification theorems themselves are fine as statements conditional on knowing P_a for all configurations, but the paper's claim that they are recoverable from data needs a strict-positivity amendment. That is a fixable gap, add an assumption that all choice rules have full support on Y conditional on any active set, but it is load-bearing. Two smaller issues. Assumption 3, the rank condition that makes Proposition 4.1 work, is opaque. It is an explicit algebraic inequality, so not circular, but it deserves more discussion or a primitive sufficient condition; as written, you have to trust that it is the right sort of condition. Second, the abstract promises an application to fast-food restaurant expansion and contraction, but there is no empirical section in the body. Either include the exercise or drop the claim from the abstract. Who is this for? People working on empirical social interactions, especially dynamic discrete choice with networks. The central idea is worth a serious referee, and the equilibrium gap is specific enough that a good referee could help the authors fix it. I would not cite it in its current form, but I would recommend sending it to review with a request to amend and prove Proposition 3.1, and to sort out the empirical content.

Referee Report

3 major / 5 minor

Summary. The paper develops a continuous-time discrete choice model with endogenous peer selection. Each agent, upon a Poisson alarm, first selects a subset of potential peers based on recent choices and then chooses an alternative conditional on the selected peers. Under type-homogeneity and independent peer selection, the paper claims that the social network, the peer-selection probabilities, and the choice rules are nonparametrically identified from a single long panel of choices, without exogenous covariate variation. The identification argument proceeds from conditional choice probabilities (CCPs): Proposition 4.1 recovers the network from CCP differences; Proposition 4.2 recovers selection probabilities and choice rules for empty and singleton peer sets using variation in reference-group size; Propositions 4.3 and 4.4 extend to parametric and fully nonparametric settings; Proposition 4.5 connects CCPs to panel data. The abstract also announces an empirical application to fast-food restaurant expansion and contraction, but no such application appears in the manuscript.

Significance. If the identification results hold, this is a meaningful contribution to the peer-effects and limited-attention literatures. The idea of using variation in the number of potential peers as a source of identification is elegant, and the monotonicity argument in the proof of Proposition 4.2 is a nice construction. The paper is generally clearly written and the examples are helpful. However, the full-support equilibrium result that underpins the empirical content is neither proved nor implied by the stated assumptions, and the abstract-promised empirical application is entirely missing. These issues must be addressed before the claims can be taken at face value.

major comments (3)

[Section 3, Proposition 3.1] The proposition asserts existence, uniqueness, and full support of the invariant distribution under Assumptions 1 and 2, but the stated assumptions are insufficient. Assumption 2(iii) is vacuous when N_a is empty, and empty reference groups are explicitly allowed (Example 1, Agent 4). If such an agent's choice rule R_a(·|y,∅) is degenerate at 0, then any configuration with y_a=1 has zero transition rate into it and zero invariant mass, so full support fails. No proof of Proposition 3.1 appears in Appendix A. This is load-bearing because Propositions 4.1–4.4 require CCPs at configurations y=0 and y=0^v_{a'}; if full support fails, those CCPs are not consistently estimable from a single long panel, and the network recovery test in Proposition 4.1 cannot be implemented for links involving isolated agents. The authors should add explicit strict positivity conditions on R_a(·|y,∅) (or rule ou
[Abstract / main text] The abstract announces an empirical application to fast-food restaurant expansion and contraction and claims 'evidence of limited attention to actions of competitors.' No such application, data description, estimation, or results appear anywhere in the manuscript; the body ends with Section 5, Concluding Remarks. As submitted, this is an unsupported empirical claim and cannot be evaluated. The authors should either include the application in a revised version or remove the claim from the abstract.
[Section 4.1, Assumption 3] Assumption 3 is the rank condition that makes the CCP difference in the proof of Proposition 4.1 nonzero, and therefore is load-bearing for network identification. It is stated in terms of Q, R, and the unknown network N_a, no primitive sufficient conditions are given, and it is not shown to be testable from the CCPs. Because the entire recursive identification collapses if Assumption 3 fails, the paper should provide at least one class of primitive models (e.g., logit or linear-in-means specifications) in which Assumption 3 holds, or an explicit discussion of when it is violated.

minor comments (5)

[General] The manuscript states 'All proofs can be find in Appendix A'; the appendix contains proofs of Propositions 4.1–4.3 only. Propositions 3.1, 4.4, and 4.5 are either unproved or deferred to related work. Please add proofs or explicit references.
[Section 2.1] In Example 1, 'NC3 = {2}' appears to be a typo for N_3 = {2}.
[Section 4] The paragraph following Assumption 5 says 'Assumption 4.4 means that...' and should read 'Assumption 5 means that...'. Also, Assumption 2(i) writes Q_a(a'|y,N_a), but Q_a is defined as a function of y only; please be consistent.
[Section 4.1] The notation 0^v_{a'} is used before it is defined; please define it in the main text.
[Section 4.2] The identification of P and λ from Dataset 2 is delegated to Kashaev et al. (2025). Please state the identification condition more explicitly, since the current manuscript only mentions a generic eigenvalue restriction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular identification: the derivation inverts CCPs under stated rank/menu conditions; the main red flags are an omitted full-support proof and external/self citations, which are correctness rather than circularity concerns.

full rationale

The identification chain is a nonparametric inversion of the structural map P_a(v|y)=∑_{N⊆N_a} R_a(v|y,N)S_a(N|y,N_a) under Assumptions 1–5. Proposition 4.1 tests a′∈N_a by P_a(v|0^v_{a'})−P_a(v|0); the proof shows the difference is zero for non-peers by Assumption 2 and nonzero for peers under Assumption 3. Assumption 3 is a stated rank/relevance inequality, not a normalization that assumes the conclusion; identification is explicitly conditional on it. Proposition 4.2 identifies Q_t(v*,v′) by inverting a strictly monotone function of observed CCP-difference ratios across agents with different |N_a|, then solves linearly for R_a(·|∅) and R_a(·|{a′}); Proposition 4.4 recurses using Assumption 5. No fitted parameter is relabeled as a prediction, and no primitive is defined in terms of the target object. The self-citations (Kashaev–Lazzati 2019; Kashaev–Lazzati–Xiao 2025) are background or delegate the matrix-exponential step to Blevins (2017, 2018), an external theorem source, so they do not make the argument circular. The serious issue in the manuscript is not circularity: Proposition 3.1 asserts a unique, full-support invariant distribution from Assumptions 1–2, but the appended Appendix A contains no proof of Proposition 3.1 (proofs are given only for Propositions 4.1–4.3), and the stated assumptions do not obviously deliver full support (e.g., an isolated agent with a degenerate R_a(·|y,∅)). Full support is load-bearing for estimating CCPs at configurations such as 0 and 0^v_{a'}; if it fails, the Proposition 4.1 network test is not implementable. This is a correctness/omitted-proof concern, not an equivalence-by-construction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The central claim rests on a set of structural assumptions about how attention and preferences work; none of these are fitted to data in this version. The main unstated premise is strict positivity of choice probabilities, needed for the asserted full-support equilibrium.

assumptions (8)

domain assumption Independent peer selection (Assumption 1: S_a(N|y,N_a)=∏_{a'}Q_a(a'|y)∏(1-Q_a(a'|y)))
Restricts the attention mechanism to independent Bernoulli draws, excluding correlated selection of peers; all identification formulas depend on this.
domain assumption Type-homogeneity and average-choice dependence (Assumption 2)
Q depends only on the pair (own choice, target choice) and R depends only on own choice and the average of selected peers' choices; this is what makes degree variation substitute for covariates.
ad hoc to paper Regularity inequality (Assumption 3)
Ensures the CCP difference detecting an edge is nonzero; it is a rank condition not derived from more primitive behavioral assumptions.
domain assumption At least three different reference-group sizes per type (Assumption 4)
Needed to identify q from the monotone ratio of CCP differences in Proposition 4.2.
domain assumption Full degree chain per type (Assumption 5)
Needed for recursive nonparametric identification of choice rules for active sets larger than singletons.
domain assumption Known type mapping h
The paper assumes the researcher knows which agents share preference/attention types; unknown heterogeneity would break the recursive identification.
domain assumption Full-support positivity of choice probabilities
Proposition 3.1 asserts full support, but Assumption 2 allows degenerate R_a; positivity is effectively an unstated assumption needed for CCP frequency estimates.
standard math Identification of transition-rate matrix from P(Δ) (Blevins 2017, 2018; Kashaev et al. 2025)
Used for Proposition 4.5; the eigenvalue condition is framed as generically satisfied.

invented entities (1)

Endogenous peer-attention probabilities Q_{h(a)}(y_a,y_{a'}) independent evidence
purpose: Governing which peers enter the active set at each revision, so attention can depend on recent choices and generate homophily or polarization.
The paper proves these probabilities are identified from observed CCPs (Proposition 4.2), giving a falsifiable handle through choice-panel data.

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Cite this review

Pith. "Pith review of Discrete Choice with Endogenous Peer Selection." pith.science (2026). https://pith.science/paper/5VGPHER2

@misc{pith2026251121446,
  author       = {Pith},
  title        = {Pith review of: Discrete Choice with Endogenous Peer Selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5VGPHER2}},
  note         = {Machine review of arXiv:2511.21446}
}

read the original abstract

We develop a continuous time discrete choice model of peer effects. The distinctive feature of the model is that agents might not consider all peers at the moment of making a decision. Instead, they select some of them on the basis of a mechanism that depends on recent choices. We characterize the equilibrium behavior and study the empirical content of the limited attention peer effect model. We allow changes in the choices of peers to affect both the set of peers to which the agent pays attention and her preferences over the alternatives. We exploit variation in choices together with variation in the size of the set of potential peers (or reference groups) to recover the preferences of the agents and the peer selection mechanisms. We apply our results to model expansion and contraction decisions by fast-food restaurants and find evidence of limited attention to actions of competitors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Random utility and limited consideration,

Aguiar, Victor H, Maria Jose Boccardi, Nail Kashaev, and Jeongbin Kim (2023) “Random utility and limited consideration,”Quantitative Economics, 14 (1), 71–116

2023

[2] [2]

Nash equilibria on (un)stable networks,

Badev, Anton I. (2021) “Nash equilibria on (un)stable networks,”Econometrica, 89 (3), 1263–1294

2021

[3] [3]

Identifying restrictions for finite parameter continuous time models with discrete time data,

Blevins, Jason R (2017) “Identifying restrictions for finite parameter continuous time models with discrete time data,”Econometric Theory, 33 (3), 739–754. (2018) “Identification and estimation of continuous time dynamic discrete choice games,”Technical report, Mimeo, Ohio State University

2017

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My friend far, far away: a random field approach to exponential random graph models,

Boucher, Vincent and Ismael Mourifie (2017) “My friend far, far away: a random field approach to exponential random graph models,”The econometrics journal, 20 (3), S14–S46

2017

[5] [5]

Econometrics of network formation

Chandrasekhar, Arun G (2016) “Econometrics of network formation.” De Paula, Aureo (2017) “Econometrics of network models,” inAdvances in Economics and Econo- 16 metrics: Theory and Applications, Eleventh World Congress, 268–323, Cambridge University Press Cambridge

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Durlauf, Steven N and H Peyton Young (2001)Social dynamics, 4: Mit Press

2001

[7] [7]

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Graham, Bryan S (2015) “Methods of identification in social networks,”Annu. Rev. Econ., 7 (1), 465–485

2015

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An econometric model of network formation with an application to board interlocks between firms,

Gualdani, Chiara (2021) “An econometric model of network formation with an application to board interlocks between firms,”Journal of Econometrics, 224 (2), 345–370

2021