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arxiv: 2406.07186 · v4 · submitted 2024-06-11 · 💰 econ.TH

Information Aggregation with Costly Information Acquisition

Spyros Galanis , Sergei Mikhalishchev This is my paper

Pith reviewed 2026-05-24 00:07 UTC · model grok-4.3

classification 💰 econ.TH
keywords information aggregationcostly signalsseparable securitiesdynamic tradingpayoff structureprivate informationequilibrium
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The pith

Lowering information acquisition costs causes almost all securities to suddenly aggregate private trader information.

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

The paper establishes that κ separable securities characterize information aggregation in a dynamic trading model where traders can buy signals at a cost κ in each period. As this cost decreases, almost all securities become κ separable no matter the traders' starting private information. The transition is discontinuous, so even small cost reductions can enable full aggregation. The classification of how well securities aggregate depends solely on their payoff structure.

Core claim

κ separable securities characterize information aggregation and, as the cost decreases, almost all securities become κ separable, irrespective of the traders' initial private information. The switch to κ separability happens not gradually but discontinuously, and the complete classification of securities depends only on their payoff structure.

What carries the argument

κ separable securities, defined as those that aggregate information when traders optimally acquire signals at cost κ each period.

Load-bearing premise

Traders choose to acquire signals optimally in every period.

What would settle it

Observe in a controlled trading setting whether a security classified as not κ separable fails to aggregate information at a given cost level while doing so at a slightly lower cost.

Figures

Figures reproduced from arXiv: 2406.07186 by Sergei Mikhalishchev, Spyros Galanis.

Figure 1
Figure 1. Figure 1: Prediction accuracy (A(µ) = EµA(ω, µ)) for markets and polls. market would benefit from learning the intrinsic value of X, and the corresponding event, faster. Unfortunately, there are no good news. Even when traders are myopic, we show that information acquisition can make the process of information aggrega￾tion both faster and slower, depending on the parameters. This is true for all securities except fo… view at source ↗
read the original abstract

We study information aggregation in a dynamic trading model with partially informed traders. Ostrovsky [2012] showed that `separable' securities aggregate information in all equilibria, however, determining whether a security is separable requires knowing the exact information structure of agents. To remedy this problem, we allow traders to acquire signals with cost $\kappa$, in every period. We show that `$\kappa$ separable securities' characterize information aggregation and, as the cost decreases, almost all securities become $\kappa$ separable, irrespective of the traders' initial private information. Moreover, the switch to $\kappa$ separability happens not gradually but discontinuously, hence even a small decrease in costs can result in a security aggregating information. We provide a complete classification of securities in terms of how well they aggregate information, which surprisingly depends only on their payoff structure.

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.

Referee Report

2 major / 1 minor

Summary. The paper studies information aggregation in a dynamic trading model with partially informed traders who can acquire signals at cost κ in every period. Building on Ostrovsky (2012), it defines 'κ separable securities' that characterize information aggregation in all equilibria. The central claims are that as κ decreases, almost all securities become κ separable irrespective of initial private information, the transition occurs discontinuously rather than gradually, and a complete classification of securities by aggregation quality depends only on payoff structure.

Significance. If the characterization holds, the work offers a practically useful extension of Ostrovsky (2012) by rendering separability robust to unknown initial information structures via endogenous costly acquisition. The discontinuous switch result, if established, would be a notable contribution with potential implications for understanding threshold effects in market information efficiency. No machine-checked proofs or reproducible code are mentioned, but the parameter-free classification claim (depending only on payoffs) would be a strength if verified.

major comments (2)
  1. [Abstract / central result] Abstract / central result: the claim that κ-separable securities characterize aggregation 'irrespective of the traders' initial private information' requires that optimal acquisition each period produces endogenous information structures that continue to satisfy the fixed-structure conditions of Ostrovsky (2012) for separability to be well-defined. Because acquisition is strategic and can depend on the security, it is not immediate that this preservation holds; an explicit argument or counter-example check is needed.
  2. [Abstract] Abstract: the statement that 'almost all securities become κ separable' as cost decreases, and that the switch is discontinuous, needs a precise definition of the measure on the space of securities and a demonstration that the discontinuity survives once information acquisition is chosen in equilibrium rather than fixed exogenously.
minor comments (1)
  1. [Abstract] Abstract: the term 'partially informed traders' is used without specifying the initial information partition or how it evolves with acquisition; a brief clarification would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed report. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract / central result] Abstract / central result: the claim that κ-separable securities characterize aggregation 'irrespective of the traders' initial private information' requires that optimal acquisition each period produces endogenous information structures that continue to satisfy the fixed-structure conditions of Ostrovsky (2012) for separability to be well-defined. Because acquisition is strategic and can depend on the security, it is not immediate that this preservation holds; an explicit argument or counter-example check is needed.

    Authors: We thank the referee for highlighting this important point. In our framework, κ-separability is defined to incorporate the equilibrium acquisition behavior, which ensures that the resulting information structures allow the application of Ostrovsky's result. Specifically, the definition of κ-separability requires that for the security, there exist acquisition strategies such that the induced beliefs satisfy the separability condition in every period. We will add an explicit lemma showing that optimal acquisition preserves the necessary conditions for the characterization to hold irrespective of initial information. This will be included in the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'almost all securities become κ separable' as cost decreases, and that the switch is discontinuous, needs a precise definition of the measure on the space of securities and a demonstration that the discontinuity survives once information acquisition is chosen in equilibrium rather than fixed exogenously.

    Authors: We agree that a precise definition of the measure is necessary. In the paper, the space of securities is identified with the space of payoff functions over the finite state space, and we use the standard Lebesgue measure on this finite-dimensional space. The discontinuity result is established by showing that the set of κ-separable securities jumps in measure as κ decreases, and the proof explicitly considers equilibrium acquisition choices rather than exogenous signals. To address the concern, we will include a more detailed explanation of the measure and verify that the equilibrium acquisition does not smooth out the discontinuity in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: extends external Ostrovsky result with endogenous costly signals; κ-separability derived from model, not tautological

full rationale

The derivation introduces costly signal acquisition (cost κ) into a dynamic trading model and defines κ-separable securities to characterize aggregation irrespective of initial private information. This builds directly on the independent, externally cited Ostrovsky [2012] conditions for separability, without self-citations, fitted parameters renamed as predictions, or self-definitional reductions. The discontinuity result and payoff-structure classification follow from the equilibrium analysis under optimal acquisition; they do not reduce by construction to the inputs or prior assumptions. The model remains falsifiable against the cited benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Model rests on standard rational-agent assumptions in dynamic trading games; no new entities postulated.

free parameters (1)
  • κ
    Per-period cost of signal acquisition, varied parametrically to identify thresholds.
axioms (2)
  • domain assumption Traders acquire signals optimally each period in a dynamic game with partially informed agents.
    Core setup extending Ostrovsky 2012.
  • domain assumption Separability is defined with respect to the traders' information structure.
    Inherited from the 2012 reference.

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Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

    work page

  2. [2]

    Generic existence of completely revealing equilibria for economies with uncertainty when prices convey information

    Beth Allen. Generic existence of completely revealing equilibria for economies with uncertainty when prices convey information. Econometrica, pages 1173--1199, 1981

    work page 1981

  3. [3]

    Information aggregation in auctions with costly information

    Alp E Atakan and Mehmet Ekmekci. Information aggregation in auctions with costly information. Working paper, 2023

    work page 2023

  4. [4]

    Swift, Emile Servan-Schreiber, Philip Tetlock, Lyle Ungar, and Barbara Mellers

    Pavel Atanasov, Phillip Rescober, Eric Stone, Samuel A. Swift, Emile Servan-Schreiber, Philip Tetlock, Lyle Ungar, and Barbara Mellers. Distilling the wisdom of crowds: Prediction markets vs. prediction polls. Management Science, 63 0 (3): 0 691--706, 2017

    work page 2017

  5. [5]

    Two reasons to make aggregated probability forecasts more extreme

    Jonathan Baron, Barbara A Mellers, Philip E Tetlock, Eric Stone, and Lyle H Ungar. Two reasons to make aggregated probability forecasts more extreme. Decision Analysis, 11 0 (2): 0 133--145, 2014

    work page 2014

  6. [6]

    Berg, Forrest D

    Joyce E. Berg, Forrest D. Nelson, and Thomas A. Rietz. Prediction market accuracy in the long run. International Journal of Forecasting, 24 0 (2): 0 285--300, 2008

    work page 2008

  7. [7]

    Comparison of experiments

    David Blackwell. Comparison of experiments. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 93--102. University of California Press, 1951

    work page 1951

  8. [8]

    The cost of optimally-acquired information

    Alexander W Bloedel and Weijie Zhong. The cost of optimally-acquired information. Unpublished Manuscript, 2020

    work page 2020

  9. [9]

    Evaluating replicability of laboratory experiments in economics

    Colin F Camerer, Anna Dreber, Eskil Forsell, Teck-Hua Ho, J \"u rgen Huber, Magnus Johannesson, Michael Kirchler, Johan Almenberg, Adam Altmejd, Taizan Chan, et al. Evaluating replicability of laboratory experiments in economics. Science, 351 0 (6280): 0 1433--1436, 2016

    work page 2016

  10. [10]

    Revealed preference, rational inattention, and costly information acquisition

    Andrew Caplin and Mark Dean. Revealed preference, rational inattention, and costly information acquisition. American Economic Review, 105 0 (7): 0 2183--2203, 2015

    work page 2015

  11. [11]

    A testable theory of imperfect perception

    Andrew Caplin and Daniel Martin. A testable theory of imperfect perception. The Economic Journal, 125 0 (582): 0 184--202, 2015

    work page 2015

  12. [12]

    Rational inattention, optimal consideration sets, and stochastic choice

    Andrew Caplin, Mark Dean, and John Leahy. Rational inattention, optimal consideration sets, and stochastic choice. The Review of Economic Studies, 86 0 (3): 0 1061--1094, 2019

    work page 2019

  13. [13]

    Rationally inattentive behavior: Characterizing and generalizing shannon entropy

    Andrew Caplin, Mark Dean, and John Leahy. Rationally inattentive behavior: Characterizing and generalizing shannon entropy. Journal of Political Economy, 130 0 (6): 0 1676--1715, 2022

    work page 2022

  14. [14]

    Pennock, Robin D

    Yiling Chen, Stanko Dimitrov, Rahul Sami, Daniel M Reeves, David M. Pennock, Robin D. Hanson, Lance Fortnow, and Rica Gonen. Gaming prediction markets: Equilibrium strategies with a market maker. Algorithmica, 58 0 (4): 0 930--69, 2010

    work page 2010

  15. [15]

    Designing informative securities

    Yiling Chen, Mike Ruberry, and Jennifer Wortman Vaughan. Designing informative securities. Mimeo, 2012

    work page 2012

  16. [16]

    When do security markets aggregate dispersed information? Management Science, 2022

    Brice Corgnet, Cary Deck, Mark DeSantis, Kyle Hampton, and Erik O Kimbrough. When do security markets aggregate dispersed information? Management Science, 2022

    work page 2022

  17. [17]

    Corporate prediction markets: Evidence from G oogle, F ord, and F irm X

    Bo Cowgill and Eric Zitzewitz. Corporate prediction markets: Evidence from G oogle, F ord, and F irm X . The Review of Economic Studies, 82 0 (4): 0 1309--41, 2015

    work page 2015

  18. [18]

    Are markets more accurate than polls? the surprising informational value of ``just asking''

    Jason Dana, Pavel Atanasov, Philip Tetlock, and Barbara Mellers. Are markets more accurate than polls? the surprising informational value of ``just asking''. Judgment and Decision Making, 14 0 (2): 0 135--147, 2019

    work page 2019

  19. [19]

    Reaching a consensus

    Morris H DeGroot. Reaching a consensus. Journal of the American Statistical association, 69 0 (345): 0 118--121, 1974

    work page 1974

  20. [20]

    Aggregation, determinacy, and informational efficiency for a class of economies with asymmetric information

    Peter DeMarzo and Costis Skiadas. Aggregation, determinacy, and informational efficiency for a class of economies with asymmetric information. Journal of Economic Theory, 80 0 (1): 0 123--52, 1998

    work page 1998

  21. [21]

    On the uniqueness of fully informative rational expectations equilibria

    Peter DeMarzo and Costis Skiadas. On the uniqueness of fully informative rational expectations equilibria. Economic Theory, 13 0 (1): 0 1--24, 1999

    work page 1999

  22. [22]

    Posterior separable cost of information

    Tommaso Denti. Posterior separable cost of information. American Economic Review, 112 0 (10): 0 3215--59, 2022

    work page 2022

  23. [23]

    Experimental cost of information

    Tommaso Denti, Massimo Marinacci, and Aldo Rustichini. Experimental cost of information. American Economic Review, 112 0 (9): 0 3106--3123, 2022

    work page 2022

  24. [24]

    Non-myopic strategies in prediction markets

    Stanko Dimitrov and Rahul Sami. Non-myopic strategies in prediction markets. In Proceedings of the 9th ACM Conference on Electronic Commerce, pages 200--9, 2008

    work page 2008

  25. [25]

    Where has all the data gone? The Review of Financial Studies, 35 0 (7): 0 3101--3138, 2022

    Maryam Farboodi, Adrien Matray, Laura Veldkamp, and Venky Venkateswaran. Where has all the data gone? The Review of Financial Studies, 35 0 (7): 0 3101--3138, 2022

    work page 2022

  26. [26]

    An axiomatic characterization of cfmms and equivalence to prediction markets

    Rafael Frongillo, Maneesha Papireddygari, and Bo Waggoner. An axiomatic characterization of cfmms and equivalence to prediction markets. arXiv preprint arXiv:2302.00196, 2023

    work page arXiv 2023

  27. [27]

    Subgame-perfect equilibria of finite-and infinite-horizon games

    Drew Fudenberg and David Levine. Subgame-perfect equilibria of finite-and infinite-horizon games. Journal of Economic Theory, 31 0 (2): 0 251--68, 1983

    work page 1983

  28. [28]

    Drew Fudenberg and David K. Levine. Limit games and limit equilibria. Journal of Economic Theory, 38 0 (2): 0 261--79, 1986

    work page 1986

  29. [29]

    Syntactic foundations for unawareness of theorems

    Spyros Galanis. Syntactic foundations for unawareness of theorems. Theory and Decision, 71 0 (4): 0 593--614, 2011

    work page 2011

  30. [30]

    Unawareness of theorems

    Spyros Galanis. Unawareness of theorems. Economic Theory, 52 0 (1): 0 41--73, 2013

    work page 2013

  31. [31]

    Dynamic consistency, valuable information and subjective beliefs

    Spyros Galanis. Dynamic consistency, valuable information and subjective beliefs. Economic Theory, 71: 0 1467--1791, 2021

    work page 2021

  32. [32]

    Updating awareness and information aggregation

    Spyros Galanis and Stelios Kotronis. Updating awareness and information aggregation. B.E. Journal of Theoretical Economics, 2021

    work page 2021

  33. [33]

    Ioannou, and Stelios Kotronis

    Spyros Galanis, Christos A. Ioannou, and Stelios Kotronis. Information aggregation under ambiguity: Theory and experimental evidence. Review of Economic Studies, 2024

    work page 2024

  34. [34]

    We can't disagree forever

    John Geanakoplos and Heraklis Polemarchakis. We can't disagree forever. Journal of Economic Theory, 28: 0 192--200, 1982

    work page 1982

  35. [35]

    Naive learning in social networks and the wisdom of crowds

    Benjamin Golub and Matthew O Jackson. Naive learning in social networks and the wisdom of crowds. American Economic Journal: Microeconomics, 2 0 (1): 0 112--149, 2010

    work page 2010

  36. [36]

    On the efficiency of competitive stock markets where trades have diverse information

    Sanford Grossman. On the efficiency of competitive stock markets where trades have diverse information. The Journal of Finance, 31 0 (2): 0 573--85, 1976

    work page 1976

  37. [37]

    Combinatorial information market design

    Robin Hanson. Combinatorial information market design. Information Systems Frontiers, 5 0 (1): 0 107--19, 2003

    work page 2003

  38. [38]

    Logarithmic market scoring rules for modular combinatorial information aggregation

    Robin Hanson. Logarithmic market scoring rules for modular combinatorial information aggregation. Journal of Prediction Markets, 1 0 (1): 0 3--15, 2007

    work page 2007

  39. [39]

    The use of knowledge in society

    Friedrich August Hayek. The use of knowledge in society. The American Economic Review, pages 519--30, 1945

    work page 1945

  40. [40]

    Neighborhood-based information costs

    Benjamin H \'e bert and Michael Woodford. Neighborhood-based information costs. American Economic Review, 111 0 (10): 0 3225--3255, 2021

    work page 2021

  41. [41]

    Rational inattention when decisions take time

    Benjamin H \'e bert and Michael Woodford. Rational inattention when decisions take time. Journal of Economic Theory, 208, 2023

    work page 2023

  42. [42]

    Optimally biased expertise

    Pavel Ilinov, Andrei Matveenko, Maxim Senkov, and Egor Starkov. Optimally biased expertise. CERGE-EI Working Paper Series, 0 (736), 2024

    work page 2024

  43. [43]

    Bloated disclosures: Can chatgpt help investors process financial information? arXiv preprint arXiv:2306.10224, 2023

    Alex Kim, Maximilian Muhn, and Valeri Nikolaev. Bloated disclosures: Can chatgpt help investors process financial information? arXiv preprint arXiv:2306.10224, 2023

    work page arXiv 2023

  44. [44]

    Information aggregation with costly information and random ordering: Experimental evidence

    Carlo Kraemer, Markus N \"o th, and Martin Weber. Information aggregation with costly information and random ordering: Experimental evidence. Journal of Economic Behavior & Organization, 59 0 (3): 0 423--432, 2006

    work page 2006

  45. [45]

    Albert S. Kyle. Continuous auctions and insider trading. Econometrica, 53 0 (6): 0 1315--35, 1985

    work page 1985

  46. [46]

    The efficient market hypothesis and insider trading on the stock market

    Jean-Jacques Laffont and Eric S Maskin. The efficient market hypothesis and insider trading on the stock market. Journal of Political Economy, 98 0 (1): 0 70--93, 1990

    work page 1990

  47. [47]

    The evolution of stock market efficiency over time: A survey of the empirical literature

    Kian-Ping Lim and Robert Brooks. The evolution of stock market efficiency over time: A survey of the empirical literature. Journal of Economic Surveys, 25 0 (1): 0 69--108, 2011

    work page 2011

  48. [48]

    Rational inattention: A review

    Bartosz Ma \'c kowiak, Filip Mat e jka, and Mirko Wiederholt. Rational inattention: A review. Journal of Economic Literature, 61 0 (1): 0 226--273, 2023

    work page 2023

  49. [49]

    Rational inattention to discrete choices: A new foundation for the multinomial logit model

    Filip Mat e jka and Alisdair McKay. Rational inattention to discrete choices: A new foundation for the multinomial logit model. American Economic Review, 105 0 (1): 0 272--298, 2015

    work page 2015

  50. [50]

    McKelvey and Talbot Page

    Richard D. McKelvey and Talbot Page. Public and private information: An experimental study of information pooling. Econometrica, 58 0 (6): 0 1321--39, 1990

    work page 1990

  51. [51]

    Uncertainty, information acquisition and price swings in asset markets

    Antonio Mele and Francesco Sangiorgi. Uncertainty, information acquisition and price swings in asset markets. The Review of Economic Studies, pages 1533--1567, 2015

    work page 2015

  52. [52]

    Information aggregation in dynamic markets with strategic traders

    Michael Ostrovsky. Information aggregation in dynamic markets with strategic traders. Econometrica, 80 0 (6): 0 2595--2647, 2012

    work page 2012

  53. [53]

    An experimental analysis of information acquisition in prediction markets

    Lionel Page and Christoph Siemroth. An experimental analysis of information acquisition in prediction markets. Games and Economic Behavior, 101: 0 354--378, 2017

    work page 2017

  54. [54]

    How much information is incorporated into financial asset prices? experimental evidence

    Lionel Page and Christoph Siemroth. How much information is incorporated into financial asset prices? experimental evidence. The Review of Financial Studies, 34 0 (9): 0 4412--4449, 2021

    work page 2021

  55. [55]

    ( I n) efficiency in information acquisition and aggregation through prices

    Alessandro Pavan, Savitar Sundaresan, and Xavier Vives. ( I n) efficiency in information acquisition and aggregation through prices. Working paper, 2022

    work page 2022

  56. [56]

    Can chatgpt assist in picking stocks? Finance Research Letters, 2023

    Matthias Pelster and Joel Val. Can chatgpt assist in picking stocks? Finance Research Letters, 2023

    work page 2023

  57. [57]

    Rational expectations equilibrium: Generic existence and the information revealed by prices

    Roy Radner. Rational expectations equilibrium: Generic existence and the information revealed by prices. Econometrica, 47 0 (3): 0 655--78, 1979

    work page 1979

  58. [58]

    Axioms for constant function market makers

    Jan Christoph Schlegel, Mateusz Kwa \'s nicki, and Akaki Mamageishvili. Axioms for constant function market makers. Available at SSRN, 2022

    work page 2022

  59. [59]

    Christopher A. Sims. Implications of rational inattention . Journal of Monetary Economics, 50 0 (3): 0 665--690, April 2003

    work page 2003

  60. [60]

    Prediction markets for economic forecasting

    Erik Snowberg, Justin Wolfers, and Eric Zitzewitz. Prediction markets for economic forecasting. In Handbook of economic forecasting, volume 2, pages 657--687. Elsevier, 2013

    work page 2013

  61. [61]

    How spooks are turning to superforecasting in the cosmic bazaar

    The Economist . How spooks are turning to superforecasting in the cosmic bazaar. https://www.economist.com/science-and-technology/2021/04/15/how-spooks-are-turning-to-superforecasting-in-the-cosmic-bazaar, 2021. Accessed: 04-17-2021

    work page 2021

  62. [62]

    Robust scoring rules

    Elias Tsakas. Robust scoring rules. Theoretical Economics, 15 0 (3): 0 955--987, 2020

    work page 2020

  63. [63]

    Possible generalization of boltzmann-gibbs statistics

    Constantino Tsallis. Possible generalization of boltzmann-gibbs statistics. Journal of statistical physics, 52: 0 479--487, 1988

    work page 1988

  64. [64]

    Prediction markets

    Justin Wolfers and Eric Zitzewitz. Prediction markets. Journal of Economic Perspectives, 18 0 (2): 0 107--26, 2004

    work page 2004