Robustness of Persuasion to Receiver Preferences

Fengming Hu; Rann Smorodinsky; Ronen Gradwohl

arxiv: 2605.28265 · v1 · pith:36SONXZZnew · submitted 2026-05-27 · 💰 econ.TH

Robustness of Persuasion to Receiver Preferences

Ronen Gradwohl , Fengming Hu , Rann Smorodinsky This is my paper

Pith reviewed 2026-06-29 09:26 UTC · model grok-4.3

classification 💰 econ.TH

keywords Bayesian persuasionrobustnesscontinuityKnightian uncertaintypreference uncertaintygenericityinformation design

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The pith

Bayesian persuasion is continuous and robust to small preference uncertainty exactly when the two properties coincide, and both hold generically.

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

The paper distinguishes two responses to uncertainty about the receiver's preferences in Bayesian persuasion. Continuity means the modeler's predictions remain accurate even without precise knowledge of preferences. Robustness means the sender's chosen outcome stays the same despite the sender's own ignorance. Modeling this uncertainty as infinitesimally small Knightian uncertainty and letting the sender either minimize regret or maximize minimum utility, the authors establish that the two properties are logically equivalent and that each holds for a generic set of persuasion problems. This matters because it separates atypical fragile cases from the typical ones that remain stable under minor ignorance.

Core claim

We show that continuity holds if and only if robustness holds, and that both notions are generic. Thus, while some instances of Bayesian persuasion are fragile, typical instances are both continuous and robust with respect to a small amount of ignorance.

What carries the argument

The logical equivalence between continuity and robustness under infinitesimal Knightian uncertainty about receiver preferences, together with the genericity of both properties in the space of persuasion problems.

Load-bearing premise

Preference uncertainty is modeled as infinitesimally small, non-probabilistic Knightian uncertainty, with the sender either minimizing regret or maximizing minimum utility.

What would settle it

A concrete Bayesian persuasion instance in which continuity holds but robustness fails, or a demonstration that the set of problems satisfying both properties fails to be dense.

Figures

Figures reproduced from arXiv: 2605.28265 by Fengming Hu, Rann Smorodinsky, Ronen Gradwohl.

**Figure 2.** Figure 2: Example 2, utilities of the receiver. at which the receiver is indifferent among three actions, a1, a2, and a3 (the red solid point in Figure 2b). Moreover, this posterior is the unique belief at which a2 is a best response. Finally, the viability of action a2 is fragile: for some of the utility realizations of the receiver it is never played. A naive modeler, working under complete information (δ = 0), pr… view at source ↗

**Figure 3.** Figure 3: Example 2, sender’s indirect utilities under two realizations [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Partition of the simplex at θ0 and its three perturbations. In the next step, we use a reusable technical device—the adjustment signal policy— which will also play a role in proving later results (detailed in Appendix A). The intuition is as follows. In general, an optimal policy for one type of receiver may be suboptimal for 6Throughout, we use the terms upper and lower semicontinuous for real-valued func… view at source ↗

**Figure 5.** Figure 5: Partition of the simplex at (a) Original type: [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Motivation for the adjustment signal policy [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗

**Figure 7.** Figure 7: lower-dimension set in R 2 (a): A point, the 0-dimensional case (b): A segment, the 1-dimensional case. Among the implicit equalities, those defining the simplex, are therefore linear independent. By Corollary 1, the implicit equalities are linear dependent. Thus, at least one of the equality defines the best-response constraints. For that equality, we find the other action b, satisfying B∗ θ (a) ⊂ B∗ θ (… view at source ↗

read the original abstract

We study the robustness of Bayesian persuasion to uncertainty about the receiver's preferences. We analyze two conceptually distinct notions: continuity, in which only the modeler lacks precise knowledge, but where the model's predictions are nonetheless accurate; and robustness, in which the sender also lacks precise knowledge, but where the outcome is insensitive to this ignorance. We model preference uncertainty as infinitesimally small, non-probabilistic (Knightian) uncertainty, and the sender's behavior as either minimizing the regret or maximizing the minimum utility. We show that continuity holds if and only if robustness holds, and that both notions are generic. Thus, while some instances of Bayesian persuasion are fragile, typical instances are both continuous and robust with respect to a small amount of ignorance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves that continuity and robustness to infinitesimal Knightian preference uncertainty are equivalent and generic when the sender uses min-regret or max-min, but the result is tightly tied to those modeling choices.

read the letter

The main result is that, in Bayesian persuasion, continuity of the outcome (when only the modeler is uncertain) is equivalent to robustness (when the sender is also uncertain), and both properties hold generically under their setup. They model preference uncertainty as infinitesimally small Knightian uncertainty and let the sender either minimize regret or maximize minimum utility. This equivalence plus the genericity claim is the new piece; it gives a clean way to say when typical persuasion mechanisms are stable to small ignorance about receiver tastes.

The paper does a decent job separating the two notions and showing they coincide in this framework. The abstract states the claims directly, and the modeling choice lets them get an if-and-only-if result plus a genericity statement in a suitable topology.

The limitation is that the equivalence and the conclusion about typical instances being robust rest entirely on the infinitesimal Knightian setup plus those two sender behaviors. Small probabilistic perturbations or other sender objectives are not covered, and the stress-test note is right that those alternatives could break the equivalence. Without the full proofs it is hard to check how the genericity is established or how sensitive it is to the exact topology, but the central claim is conditional on the stated assumptions.

This is for people working on information design and robustness questions in mechanism design. The math is formal and the question is well-posed, so it deserves a serious referee even if the modeling restrictions need to be discussed in revision.

Referee Report

1 major / 2 minor

Summary. The paper claims that in Bayesian persuasion, two notions of robustness to receiver preference uncertainty—continuity (accurate predictions despite modeler ignorance) and robustness (sender outcomes insensitive to sender ignorance)—are equivalent, and that both hold generically, when preference uncertainty is modeled as infinitesimally small Knightian uncertainty and the sender either minimizes regret or maximizes minimum utility. Thus typical instances are both continuous and robust.

Significance. If the equivalence and genericity results hold under the stated assumptions, the paper provides a positive characterization showing that fragility is nongeneric in Bayesian persuasion. This is a clean mathematical contribution; credit is due for establishing the if-and-only-if link and the genericity argument in an appropriate topology on preference spaces.

major comments (1)

[Introduction and §3 (main result)] The central equivalence and genericity claims are derived under the specific modeling of infinitesimal non-probabilistic uncertainty together with min-regret or max-min sender behavior. The manuscript should explicitly state whether the equivalence is an artifact of this behavioral assumption or holds more broadly; if the former, the scope of the 'typical instances are robust' conclusion needs to be qualified in the introduction and conclusion.

minor comments (2)

[§2 (model)] Clarify the precise topology used to define 'generic' in the space of receiver preferences; this is essential for interpreting the genericity claim.
[§2] Ensure that the definitions of continuity and robustness are stated formally before the equivalence theorem, with explicit reference to the sender's objective functions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on clarifying the scope of our results. We address the point below and will incorporate a minor revision to qualify the claims in the introduction and conclusion.

read point-by-point responses

Referee: [Introduction and §3 (main result)] The central equivalence and genericity claims are derived under the specific modeling of infinitesimal non-probabilistic uncertainty together with min-regret or max-min sender behavior. The manuscript should explicitly state whether the equivalence is an artifact of this behavioral assumption or holds more broadly; if the former, the scope of the 'typical instances are robust' conclusion needs to be qualified in the introduction and conclusion.

Authors: The equivalence between continuity and robustness, as well as the genericity result, are derived specifically under infinitesimal Knightian uncertainty together with the sender's min-regret or max-min behavior; these are the modeling choices adopted throughout the analysis. We do not claim the equivalence holds for arbitrary sender objective functions, and we will revise the introduction and conclusion to state this limitation explicitly, qualifying the conclusion that typical instances are both continuous and robust. revision: yes

Circularity Check

0 steps flagged

No circularity; equivalence is a direct proof under explicit modeling assumptions

full rationale

The paper proves that continuity holds iff robustness holds (and both are generic) by modeling preference uncertainty as infinitesimal Knightian uncertainty with the sender using min-regret or max-min utility. This is a standard mathematical derivation under stated assumptions; the equivalence does not reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The modeling choices are declared upfront and are not smuggled in via prior work. The result is self-contained as a theorem in the given framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical continuity concepts and domain-specific behavioral assumptions for the sender under Knightian uncertainty; no free parameters or new entities are introduced.

axioms (2)

standard math Standard mathematical definitions of continuity and robustness applied to Bayesian persuasion outcomes.
Invoked to define the two notions whose equivalence is proven.
domain assumption Sender behavior is either regret minimization or max-min utility under Knightian uncertainty.
This behavioral model is used to formalize robustness.

pith-pipeline@v0.9.1-grok · 5647 in / 1198 out tokens · 27780 ms · 2026-06-29T09:26:00.079736+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 11 canonical work pages

[1]

Theoretical Economics18(1), 15–36 (2023)

Arieli, I., Babichenko, Y., Smorodinsky, R., Yamashita, T.: Optimal persuasion via bi-pooling. Theoretical Economics18(1), 15–36 (2023)

2023
[2]

arXiv preprint arXiv:2302.00281 (2023) 18

Arieli, I., Gradwohl, R., Smorodinsky, R.: Informationally robust cheap-talk. arXiv preprint arXiv:2302.00281 (2023) 18

work page arXiv 2023
[3]

MIT press (1995)

Aumann, R.J., Maschler, M., Stearns, R.E.: Repeated Games with Incomplete Infor- mation. MIT press (1995)

1995
[4]

Games and Economic Behavior136(2022)

Babichenko, Y., Talgam-Cohen, I., Xu, H., Zabarnyi, K.: Regret- minimizing Bayesian persuasion. Games and Economic Behavior136(2022). https://doi.org/10.1016/j.geb.2022.09.001

work page doi:10.1016/j.geb.2022.09.001 2022
[5]

Artificial Intelligence314(2023)

Castiglioni, M., Celli, A., Marchesi, A., Gatti, N.: Regret minimiza- tion in online Bayesian persuasion: Handling adversarial receiver’s types under full and partial feedback models. Artificial Intelligence314(2023). https://doi.org/10.1016/j.artint.2022.103821

work page doi:10.1016/j.artint.2022.103821 2023
[6]

International Journal of Game Theory 50(4), 911–925 (2021)

Diehl, C., Kuzmics, C.: The (non-) robustness of influential cheap talk equilibria when the sender’s preferences are state independent. International Journal of Game Theory 50(4), 911–925 (2021)

2021
[7]

American Economic Review: Insights5(1), 111–24 (2023)

Dilm´ e, F.: Robust information transmission. American Economic Review: Insights5(1), 111–24 (2023)

2023
[8]

Econometrica90(5) (2022)

Dworczak, P., Pavan, A.: Preparing for the Worst but Hoping for the Best: Robust (Bayesian) Persuasion. Econometrica90(5) (2022). https://doi.org/10.3982/ecta19107

work page doi:10.3982/ecta19107 2022
[9]

In: Proceedings of the An- nual ACM-SIAM Symposium on Discrete Algorithms

Feng, Y., Ho, C.J., Tang, W.: Rationality-Robust Information Design: Bayesian Persuasion under Quantal Response. In: Proceedings of the An- nual ACM-SIAM Symposium on Discrete Algorithms. vol. 2024-January (2024). https://doi.org/10.1137/1.9781611977912.19

work page doi:10.1137/1.9781611977912.19 2024
[10]

Mathematical Pro- gramming pp

Gan, J., Han, M., Wu, J., Xu, H.: Robust stackelberg equilibria. Mathematical Pro- gramming pp. 1–41 (2025)

2025
[11]

Economic Theory 72(3) (2021)

Hu, J., Weng, X.: Robust persuasion of a privately informed receiver. Economic Theory 72(3) (2021). https://doi.org/10.1007/s00199-020-01299-5

work page doi:10.1007/s00199-020-01299-5 2021
[12]

Automatica43(10), 1808– 1816 (2007)

Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: Lexicographic perturbation for multi- parametric linear programming with applications to control. Automatica43(10), 1808– 1816 (2007)

2007
[13]

Econo- metrica: Journal of the Econometric Society pp

Kajii, A., Morris, S.: The robustness of equilibria to incomplete information. Econo- metrica: Journal of the Econometric Society pp. 1283–1309 (1997)

1997
[14]

Annual Review of Eco- nomics11(2019)

Kamenica, E.: Bayesian Persuasion and Information Design. Annual Review of Eco- nomics11(2019). https://doi.org/10.1146/annurev-economics-080218-025739 19

work page doi:10.1146/annurev-economics-080218-025739 2019
[15]

URLhttps://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590

Kamenica, E., Gentzkow, M.: Bayesian persuasion. American Economic Review101(6) (2011). https://doi.org/10.1257/aer.101.6.2590

work page doi:10.1257/aer.101.6.2590 2011
[16]

Theoretical Economics17(3) (2022)

Kosterina, S.: Persuasion with unknown beliefs. Theoretical Economics17(3) (2022). https://doi.org/10.3982/te4742

work page doi:10.3982/te4742 2022
[17]

In: In- ternational Conference on Learning Representations

Lin, T., Chen, Y.: Generalized principal-agent problem with a learning agent. In: In- ternational Conference on Learning Representations. vol. 2025, pp. 89757–89788 (2025)

2025
[18]

Journal of Political Economy130(10), 2705–2730 (2022)

Lipnowski, E., Ravid, D., Shishkin, D.: Persuasion via Weak Institutions. Journal of Political Economy130(10), 2705–2730 (2022)

2022
[19]

Journal of Political Economy Microeconomics (2025)

Lipnowski, E., Ravid, D., Shishkin, D.: Perfect bayesian persuasion. Journal of Political Economy Microeconomics (2025). https://doi.org/10.1086/740148,https://doi.org/ 10.1086/740148

work page doi:10.1086/740148 2025
[20]

Journal of Economic Theory211, 105678 (2023)

Matyskov´ a, L., Montes, A.: Bayesian persuasion with costly infor- mation acquisition. Journal of Economic Theory211, 105678 (2023). https://doi.org/https://doi.org/10.1016/j.jet.2023.105678,https://www. sciencedirect.com/science/article/pii/S0022053123000741

work page doi:10.1016/j.jet.2023.105678 2023
[21]

Journal of Economic Theory124(1), 45–78 (2005)

Morris, S., Ui, T.: Generalized potentials and robust sets of equilibria. Journal of Economic Theory124(1), 45–78 (2005)

2005
[22]

Journal of the American Statistical Association46(253), 55–67 (1951)

Savage, L.J.: The theory of statistical decision. Journal of the American Statistical Association46(253), 55–67 (1951)

1951
[23]

John Wiley & Sons (1998)

Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons (1998)

1998
[24]

Steg, J.H., Garashli, E., Greinecker, M., Kuzmics, C.: Robust equilibria in binary cheap-talk games (2023)

2023
[25]

Econometrica69(5), 1373–1380 (2001)

Ui, T.: Robust equilibria of potential games. Econometrica69(5), 1373–1380 (2001)

2001
[26]

Wilson, R.: Game theoretic approaches to trading processess (in truman bewley, ed., advances in economic theory: Fifth world congress) (1987)

1987
[27]

Advances in Neural Information Processing Systems37, 134430–134458 (2024) 20

Yang, K., Zhang, H.: Computational aspects of bayesian persuasion under approximate best response. Advances in Neural Information Processing Systems37, 134430–134458 (2024) 20

2024
[28]

In: Proceedings of the 22nd ACM Conference on Economics and Computation

Zu, Y., Iyer, K., Xu, H.: Learning to Persuade on the Fly: Robustness Against Igno- rance. In: Proceedings of the 22nd ACM Conference on Economics and Computation. pp. 927–928 (2021) 21 Appendix A Technical Device: The Adjustment Signal Policy The idea of the technical device dates back at least to Kamenica and Gentzkow [15] (for ex- ample, proof of Propo...

2021
[29]

Now he knows that the receiver is indeed ofθ 2, and he knows that there existµ ′ 1 = 0.49 andµ ′ 2 = 0.501, which are very close toµ 1 andµ 2, and satisfyv(µ i, θ) =v(µ i, θ2),∀i≤2

His expected utility here is 1 4 ×1 + 3 4 ×0.97 = 0.9775. Now he knows that the receiver is indeed ofθ 2, and he knows that there existµ ′ 1 = 0.49 andµ ′ 2 = 0.501, which are very close toµ 1 andµ 2, and satisfyv(µ i, θ) =v(µ i, θ2),∀i≤2. If he neither chooses to deduce the new optimal signal policy, nor applies the adjustment signal policy, but just upd...

2000
[30]

B Missing Proofs for Section 3.1 We use the following geometrical observation: Lemma 8.LetG={Ω, A, µ 0, uθ0, v}be an arbitrary BP model

However, the adjustment signal policy can bridge different optimal signal policies corresponding to different types without dependence on the optimal concavification. B Missing Proofs for Section 3.1 We use the following geometrical observation: Lemma 8.LetG={Ω, A, µ 0, uθ0, v}be an arbitrary BP model. Then for everyγ >0there existsδ 0 >0such that the fol...
[31]

it is purely supported on extreme points
[32]

We utilize the following observation

its associated weights are uniquely determined by its supporting extreme points. We utilize the following observation. Lemma 15.For any typeθthere exists an optimal signal policyπthat is a basic signal policy. The proof appears in Section D.5. D.2 Step 1: Pseudo Type and Its properties Compared toθ 0, the pseudo typeθ p has the following properties:
[33]

Every action with a lower-dimensional set atθ 0 corresponds to an empty set atθ p
[34]

Every action that is stable atθ 0 remains unchanged atθ p
[35]

Formally, Lemma 16.LetG θ0 ={Ω, A, µ 0, uθ0, v}be a BP model

Every action with non-empty and full-dimensional set but with duplicates for the receiver is divided by imposing the sender’s least-preferred choice. Formally, Lemma 16.LetG θ0 ={Ω, A, µ 0, uθ0, v}be a BP model. Consider a pseudo typeθ p whose best-reply regions satisfy, for everya∈A, Bθp(a) =    ∅,ifa∈A low θ0 Bθ0(a),ifa∈A stable θ0 µ∈B θ0(a) ...
[36]

X ω∈Ω v(a, ω)µk j (ω) # ≥ X j≤I k X ω∈Ω v(ak, ω)βk j µk j (ω) = X ω∈Ω v(ak, ω) X j≤I k βk j µk j (ω) = X ω∈Ω v(ak, ω)µk(ω) = max a∈optA(µk)

There exists aγ-adjustmentbπ θ′ ofπ θ′—the optimal signal policy atθ ′—with the fol- lowing properties: Evθ′(πθ′)≤Ev θp(bπθ′) +D(γ)≤Ev θp(optΠ(θp)) +D(γ). Proof.Note thatθ 0 is an interior type of theδ-BP model wrapping it. For anyδ >0 and anyδ-BP model, chooseδ ∗ >0 small enough so that the typeθ ′ defined below belongs to theδ-BP model. Sincev(a, ω)∈[0,...
[37]

14For any setBand anyx 0 ∈B, define the linear subspaceL:= span{x−x 0 :x∈B}

Letθ 1 andθ 2 be two types such thatB θ1(a), Bθ2(a)̸=∅, ifmax x∈B∗ θ1(a) miny∈B ∗ θ2(a) ||x− y||2 ≤γ, thenmax x∈Bθ1(a) miny∈Bθ2(a) ||x−y|| 2 ≤(1 + √ N−1)γ. 14For any setBand anyx 0 ∈B, define the linear subspaceL:= span{x−x 0 :x∈B}. The dimension of Bis defined by the linear dimension ofL, dim(B) := dim(L). This definition is independent of the choice of ...
[38]

Third,we establish the stability of contracted system

will complete the proof of (1) in Lemma 1. Third,we establish the stability of contracted system. If the original feasible set is nonempty and has propertiesU 1 andU 2, then for everyγ >0 there existsδ 0 >0 such that, whenever the perturbation size is smaller thanδ 0, every point in the original feasible set lies within distanceγof some point in the pertu...
[39]

there exists a pointx 1 ∈ {C =,−1x=D =,−1}such that⟨c, x 1⟩< d
[40]

If the first one is true, there existsδ 1 such that theδ 1 ball aroundx 1 satisfies that for anyx∈B δ1(x1),⟨c, x⟩< d

there exists a pointx 2 ∈ {C =,−1x=D =,−1}such that⟨c, x 2⟩=d. If the first one is true, there existsδ 1 such that theδ 1 ball aroundx 1 satisfies that for anyx∈B δ1(x1),⟨c, x⟩< d. Notice thatx 1 ∈B δ1(x1)∩ {x|C =,−1x=D =,−1}, and so we have B δ1(x1)∩ {x|C =,−1x=D =,−1}is theδ 1 ball aroundx 1 in the subspace defined by {x|C=,−1x=D =,−1}. Also, B δ1(x1)∩ ...

[1] [1]

Theoretical Economics18(1), 15–36 (2023)

Arieli, I., Babichenko, Y., Smorodinsky, R., Yamashita, T.: Optimal persuasion via bi-pooling. Theoretical Economics18(1), 15–36 (2023)

2023

[2] [2]

arXiv preprint arXiv:2302.00281 (2023) 18

Arieli, I., Gradwohl, R., Smorodinsky, R.: Informationally robust cheap-talk. arXiv preprint arXiv:2302.00281 (2023) 18

work page arXiv 2023

[3] [3]

MIT press (1995)

Aumann, R.J., Maschler, M., Stearns, R.E.: Repeated Games with Incomplete Infor- mation. MIT press (1995)

1995

[4] [4]

Games and Economic Behavior136(2022)

Babichenko, Y., Talgam-Cohen, I., Xu, H., Zabarnyi, K.: Regret- minimizing Bayesian persuasion. Games and Economic Behavior136(2022). https://doi.org/10.1016/j.geb.2022.09.001

work page doi:10.1016/j.geb.2022.09.001 2022

[5] [5]

Artificial Intelligence314(2023)

Castiglioni, M., Celli, A., Marchesi, A., Gatti, N.: Regret minimiza- tion in online Bayesian persuasion: Handling adversarial receiver’s types under full and partial feedback models. Artificial Intelligence314(2023). https://doi.org/10.1016/j.artint.2022.103821

work page doi:10.1016/j.artint.2022.103821 2023

[6] [6]

International Journal of Game Theory 50(4), 911–925 (2021)

Diehl, C., Kuzmics, C.: The (non-) robustness of influential cheap talk equilibria when the sender’s preferences are state independent. International Journal of Game Theory 50(4), 911–925 (2021)

2021

[7] [7]

American Economic Review: Insights5(1), 111–24 (2023)

Dilm´ e, F.: Robust information transmission. American Economic Review: Insights5(1), 111–24 (2023)

2023

[8] [8]

Econometrica90(5) (2022)

Dworczak, P., Pavan, A.: Preparing for the Worst but Hoping for the Best: Robust (Bayesian) Persuasion. Econometrica90(5) (2022). https://doi.org/10.3982/ecta19107

work page doi:10.3982/ecta19107 2022

[9] [9]

In: Proceedings of the An- nual ACM-SIAM Symposium on Discrete Algorithms

Feng, Y., Ho, C.J., Tang, W.: Rationality-Robust Information Design: Bayesian Persuasion under Quantal Response. In: Proceedings of the An- nual ACM-SIAM Symposium on Discrete Algorithms. vol. 2024-January (2024). https://doi.org/10.1137/1.9781611977912.19

work page doi:10.1137/1.9781611977912.19 2024

[10] [10]

Mathematical Pro- gramming pp

Gan, J., Han, M., Wu, J., Xu, H.: Robust stackelberg equilibria. Mathematical Pro- gramming pp. 1–41 (2025)

2025

[11] [11]

Economic Theory 72(3) (2021)

Hu, J., Weng, X.: Robust persuasion of a privately informed receiver. Economic Theory 72(3) (2021). https://doi.org/10.1007/s00199-020-01299-5

work page doi:10.1007/s00199-020-01299-5 2021

[12] [12]

Automatica43(10), 1808– 1816 (2007)

Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: Lexicographic perturbation for multi- parametric linear programming with applications to control. Automatica43(10), 1808– 1816 (2007)

2007

[13] [13]

Econo- metrica: Journal of the Econometric Society pp

Kajii, A., Morris, S.: The robustness of equilibria to incomplete information. Econo- metrica: Journal of the Econometric Society pp. 1283–1309 (1997)

1997

[14] [14]

Annual Review of Eco- nomics11(2019)

Kamenica, E.: Bayesian Persuasion and Information Design. Annual Review of Eco- nomics11(2019). https://doi.org/10.1146/annurev-economics-080218-025739 19

work page doi:10.1146/annurev-economics-080218-025739 2019

[15] [15]

URLhttps://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590

Kamenica, E., Gentzkow, M.: Bayesian persuasion. American Economic Review101(6) (2011). https://doi.org/10.1257/aer.101.6.2590

work page doi:10.1257/aer.101.6.2590 2011

[16] [16]

Theoretical Economics17(3) (2022)

Kosterina, S.: Persuasion with unknown beliefs. Theoretical Economics17(3) (2022). https://doi.org/10.3982/te4742

work page doi:10.3982/te4742 2022

[17] [17]

In: In- ternational Conference on Learning Representations

Lin, T., Chen, Y.: Generalized principal-agent problem with a learning agent. In: In- ternational Conference on Learning Representations. vol. 2025, pp. 89757–89788 (2025)

2025

[18] [18]

Journal of Political Economy130(10), 2705–2730 (2022)

Lipnowski, E., Ravid, D., Shishkin, D.: Persuasion via Weak Institutions. Journal of Political Economy130(10), 2705–2730 (2022)

2022

[19] [19]

Journal of Political Economy Microeconomics (2025)

Lipnowski, E., Ravid, D., Shishkin, D.: Perfect bayesian persuasion. Journal of Political Economy Microeconomics (2025). https://doi.org/10.1086/740148,https://doi.org/ 10.1086/740148

work page doi:10.1086/740148 2025

[20] [20]

Journal of Economic Theory211, 105678 (2023)

Matyskov´ a, L., Montes, A.: Bayesian persuasion with costly infor- mation acquisition. Journal of Economic Theory211, 105678 (2023). https://doi.org/https://doi.org/10.1016/j.jet.2023.105678,https://www. sciencedirect.com/science/article/pii/S0022053123000741

work page doi:10.1016/j.jet.2023.105678 2023

[21] [21]

Journal of Economic Theory124(1), 45–78 (2005)

Morris, S., Ui, T.: Generalized potentials and robust sets of equilibria. Journal of Economic Theory124(1), 45–78 (2005)

2005

[22] [22]

Journal of the American Statistical Association46(253), 55–67 (1951)

Savage, L.J.: The theory of statistical decision. Journal of the American Statistical Association46(253), 55–67 (1951)

1951

[23] [23]

John Wiley & Sons (1998)

Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons (1998)

1998

[24] [24]

Steg, J.H., Garashli, E., Greinecker, M., Kuzmics, C.: Robust equilibria in binary cheap-talk games (2023)

2023

[25] [25]

Econometrica69(5), 1373–1380 (2001)

Ui, T.: Robust equilibria of potential games. Econometrica69(5), 1373–1380 (2001)

2001

[26] [26]

Wilson, R.: Game theoretic approaches to trading processess (in truman bewley, ed., advances in economic theory: Fifth world congress) (1987)

1987

[27] [27]

Advances in Neural Information Processing Systems37, 134430–134458 (2024) 20

Yang, K., Zhang, H.: Computational aspects of bayesian persuasion under approximate best response. Advances in Neural Information Processing Systems37, 134430–134458 (2024) 20

2024

[28] [28]

In: Proceedings of the 22nd ACM Conference on Economics and Computation

Zu, Y., Iyer, K., Xu, H.: Learning to Persuade on the Fly: Robustness Against Igno- rance. In: Proceedings of the 22nd ACM Conference on Economics and Computation. pp. 927–928 (2021) 21 Appendix A Technical Device: The Adjustment Signal Policy The idea of the technical device dates back at least to Kamenica and Gentzkow [15] (for ex- ample, proof of Propo...

2021

[29] [29]

Now he knows that the receiver is indeed ofθ 2, and he knows that there existµ ′ 1 = 0.49 andµ ′ 2 = 0.501, which are very close toµ 1 andµ 2, and satisfyv(µ i, θ) =v(µ i, θ2),∀i≤2

His expected utility here is 1 4 ×1 + 3 4 ×0.97 = 0.9775. Now he knows that the receiver is indeed ofθ 2, and he knows that there existµ ′ 1 = 0.49 andµ ′ 2 = 0.501, which are very close toµ 1 andµ 2, and satisfyv(µ i, θ) =v(µ i, θ2),∀i≤2. If he neither chooses to deduce the new optimal signal policy, nor applies the adjustment signal policy, but just upd...

2000

[30] [30]

B Missing Proofs for Section 3.1 We use the following geometrical observation: Lemma 8.LetG={Ω, A, µ 0, uθ0, v}be an arbitrary BP model

However, the adjustment signal policy can bridge different optimal signal policies corresponding to different types without dependence on the optimal concavification. B Missing Proofs for Section 3.1 We use the following geometrical observation: Lemma 8.LetG={Ω, A, µ 0, uθ0, v}be an arbitrary BP model. Then for everyγ >0there existsδ 0 >0such that the fol...

[31] [31]

it is purely supported on extreme points

[32] [32]

We utilize the following observation

its associated weights are uniquely determined by its supporting extreme points. We utilize the following observation. Lemma 15.For any typeθthere exists an optimal signal policyπthat is a basic signal policy. The proof appears in Section D.5. D.2 Step 1: Pseudo Type and Its properties Compared toθ 0, the pseudo typeθ p has the following properties:

[33] [33]

Every action with a lower-dimensional set atθ 0 corresponds to an empty set atθ p

[34] [34]

Every action that is stable atθ 0 remains unchanged atθ p

[35] [35]

Formally, Lemma 16.LetG θ0 ={Ω, A, µ 0, uθ0, v}be a BP model

Every action with non-empty and full-dimensional set but with duplicates for the receiver is divided by imposing the sender’s least-preferred choice. Formally, Lemma 16.LetG θ0 ={Ω, A, µ 0, uθ0, v}be a BP model. Consider a pseudo typeθ p whose best-reply regions satisfy, for everya∈A, Bθp(a) =    ∅,ifa∈A low θ0 Bθ0(a),ifa∈A stable θ0 µ∈B θ0(a) ...

[36] [36]

X ω∈Ω v(a, ω)µk j (ω) # ≥ X j≤I k X ω∈Ω v(ak, ω)βk j µk j (ω) = X ω∈Ω v(ak, ω) X j≤I k βk j µk j (ω) = X ω∈Ω v(ak, ω)µk(ω) = max a∈optA(µk)

There exists aγ-adjustmentbπ θ′ ofπ θ′—the optimal signal policy atθ ′—with the fol- lowing properties: Evθ′(πθ′)≤Ev θp(bπθ′) +D(γ)≤Ev θp(optΠ(θp)) +D(γ). Proof.Note thatθ 0 is an interior type of theδ-BP model wrapping it. For anyδ >0 and anyδ-BP model, chooseδ ∗ >0 small enough so that the typeθ ′ defined below belongs to theδ-BP model. Sincev(a, ω)∈[0,...

[37] [37]

14For any setBand anyx 0 ∈B, define the linear subspaceL:= span{x−x 0 :x∈B}

Letθ 1 andθ 2 be two types such thatB θ1(a), Bθ2(a)̸=∅, ifmax x∈B∗ θ1(a) miny∈B ∗ θ2(a) ||x− y||2 ≤γ, thenmax x∈Bθ1(a) miny∈Bθ2(a) ||x−y|| 2 ≤(1 + √ N−1)γ. 14For any setBand anyx 0 ∈B, define the linear subspaceL:= span{x−x 0 :x∈B}. The dimension of Bis defined by the linear dimension ofL, dim(B) := dim(L). This definition is independent of the choice of ...

[38] [38]

Third,we establish the stability of contracted system

will complete the proof of (1) in Lemma 1. Third,we establish the stability of contracted system. If the original feasible set is nonempty and has propertiesU 1 andU 2, then for everyγ >0 there existsδ 0 >0 such that, whenever the perturbation size is smaller thanδ 0, every point in the original feasible set lies within distanceγof some point in the pertu...

[39] [39]

there exists a pointx 1 ∈ {C =,−1x=D =,−1}such that⟨c, x 1⟩< d

[40] [40]

If the first one is true, there existsδ 1 such that theδ 1 ball aroundx 1 satisfies that for anyx∈B δ1(x1),⟨c, x⟩< d

there exists a pointx 2 ∈ {C =,−1x=D =,−1}such that⟨c, x 2⟩=d. If the first one is true, there existsδ 1 such that theδ 1 ball aroundx 1 satisfies that for anyx∈B δ1(x1),⟨c, x⟩< d. Notice thatx 1 ∈B δ1(x1)∩ {x|C =,−1x=D =,−1}, and so we have B δ1(x1)∩ {x|C =,−1x=D =,−1}is theδ 1 ball aroundx 1 in the subspace defined by {x|C=,−1x=D =,−1}. Also, B δ1(x1)∩ ...