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Endogenous clustering and analogy-based expectation equilibrium

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that when players group games into analogy classes to minimize prediction error, an equilibrium over strategies and partitions always exists if players can randomize over partitions.

desk verdict A genuinely new idea—endogenous analogy partitions via clustering—with a clean existence theorem, but the flagship monitoring application is built on a false proposition and needs major revision. read the letter →

arxiv 2505.13022 v1 pith:XSKIOVIN submitted 2025-05-19 econ.TH

Philippe Jehiel , Giacomo Weber This is my paper · ORCID
classification econ.TH MSC 91A1091A2662H30
keywords Analogy-BasedExpectationEquilibriumPrototypetheoryK-meansclusteringEndogenouscategorizationCoarsebeliefsmultiplicitySelf-attractivevsself-repellingpartitionsBoundedrationality
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 what happens when players cannot keep separate expectations for every possible game and instead group games into a small number of analogy classes, chosen to minimize how wrong their opponent predictions are. The authors define clustered analogy-based expectation equilibria and prove that, in any finite two-player normal-form environment, an equilibrium always exists once players are allowed to randomize over which analogy partition they use. They also show that pure clustered equilibria can fail, and that environments can be self-attractive (many partitions are self-sustaining) or self-repelling (players must mix over partitions). Applications to beauty contests, employer-worker monitoring, and Bertrand/Cournot duopolies suggest that this endogeneity of categories is a new channel for equilibrium multiplicity and for heterogeneous beliefs within a society.

What carries the argument

The load-bearing object is the clustered distributional analogy-based expectation equilibrium, built from two fixed-point requirements. On the strategic side, each player best-responds to coarse beliefs $\beta_i(\alpha)$ about the opponent's play in each analogy class $\alpha$, and $\beta_i(\alpha)$ is the probability-weighted mean of the opponent's actual play in $\alpha$. On the clustering side, an analogy partition is locally clustered if every game is assigned to the class whose mean is nearest, and globally clustered if it minimizes total prediction error; the mean is the optimal prototype because both distances are Bregman divergences, for which the conditional mean is the best predictor. Theorem 1 composes the best-response correspondence and the global-clustering correspondence into a single upper-hemicontinuous, convex-valued mapping and applies Kakutani's fixed point theorem.

What would settle it

Run the monitoring game with unequal majority probabilities—say $p_a=0.4$, $p_b=0.3$, $p_c=0.3$—and pick $\nu^*=0.45$, which lies inside $(p_c/(p_a+p_c),\, p_b/(p_b+p_c))$. Compute the global clustering costs of the two two-element partitions $\{a,c\}$ and $\{b,c\}$ under the strategies described in Proposition 5; if the costs are not equal, no CD-ABEE can put positive probability on both, which would contradict the paper's monitoring conclusion.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: in finite environments, with prediction error measured by squared Euclidean distance or Kullback-Leibler divergence, there always exists a locally and a globally clustered distributional analogy-based expectation equilibrium—a profile of mixed strategies and a distribution over analogy partitions such that each strategy is a best response to coarse beliefs given its partition, and every partition in the support minimizes the opponent-prediction error given the aggregate play. The same endogeneity that makes pure clustered equilibria fail in some three-matching-pennies environments is what the distributional extension absorbs: when the data being clustered are themselves produced by the clustering, a single partition may be self-defeating, and only mixing over partitions can be self-consistent. The paper then classifies environments by whether analogy partitions are self-attractive, sustaining many equilibria (beauty contests with strong coordination, strategic complements), or self-repelling, forcing mixing and heterogeneous beliefs (the monitoring game, strategic substitutes).

Load-bearing premise

The monitoring-game claim that heterogeneous partitions must coexist relies on the two majority worker types being exactly equally likely; if $p_a \neq p_b$, the two candidate partitions cannot both minimize the employer's prediction error, and the described mixed-partition equilibrium disappears.

Editorial extensions

If this is right

  • In any finite two-player game environment with finitely many actions, a clustered distributional ABEE exists for both squared Euclidean and KL divergence, so endogenous analogy classes are consistent with equilibrium existence.
  • Some finite environments (three matching pennies games with two classes for one player and three for the other) admit no pure clustered ABEE; equilibrium then requires players in the same role to use different analogy partitions despite seeing the same data.
  • Environments with self-attractive partitions—beauty contests with a high coordination motive and strategic complements—can support many partitions and correspondingly many equilibrium behaviors, a new channel for cross-society differences.
  • Environments with self-repelling partitions—the monitoring game and strategic substitutes—force mixing over partitions; in the monitoring game this produces polarized beliefs about a minority type, with some employers treating it like the always-shirking type and others like the always-working type.

Reading between the lines

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

  • A testable extension: in a laboratory beauty contest with the coordination weight $r$ near 1, different groups facing identical fundamentals should settle on different category boundaries and different action distributions, whereas with $r$ near 0 they should converge to the variance-minimizing equal-split partition.
  • The self-attractive/self-repelling dichotomy suggests a policy lever: interventions that change the interaction parameter could shift a market between multiple stable categorization regimes and regimes where categorization heterogeneity is unavoidable.
  • The paper's continuum applications (beauty contest and linear best-response families) fall outside the finite-environment existence theorem; they are handled by direct construction, so a general existence theorem for continuum games remains an open extension, not established here.
  • If clustering were allowed to use additional attributes such as the player's own payoff structure or the state, the equilibrium set and the self-attractive/self-repelling classification would in general change; the paper notes this but does not develop it.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper introduces a solution concept for two-player normal-form games in which each player partitions the set of games into K analogy classes so as to minimize prediction error about the opponent's behavior, and strategies form an analogy-based expectation equilibrium given those partitions. The authors define locally and globally clustered ABEE and a distributional extension (CD-ABEE) that allows mixing over partitions. They prove (Theorem 1) that a CD-ABEE always exists in finite environments for squared Euclidean or KL divergence, and provide two learning models whose steady states correspond to these equilibria. Applications include beauty contests (self-attractive partitions, many equilibria), a monitoring game (self-repelling partitions, mixing over partitions), and linear best-response games with strategic complements/substitutes. The paper concludes with a discussion of belief heterogeneity and equilibrium multiplicity.

Significance. The conceptual contribution is valuable: endogenizing analogy partitions via clustering connects behavioral game theory with prototype theory and K-means, and the distinction between self-attractive and self-repelling partitions offers a new channel for multiplicity and heterogeneity. The existence proof for CD-ABEE is a genuine technical contribution and appears sound, and the learning foundations give the solution concept behavioral content. However, the monitoring application's main result (Proposition 5) is false for generic parameter values, and the proof of Proposition 1 in the appendix contains an error. These problems are fixable but currently preclude acceptance.

major comments (3)
  1. [Section 3.2, Proposition 5] The profile described in Proposition 5 cannot be a globally clustered CD-ABEE when p_a differs from p_b. With d the squared Euclidean distance and q defined as Pr(e=1|c), the global clustering costs of the two candidate partitions are V(ac)=p_a p_c q^2/(p_a+p_c) and V(bc)=p_b p_c (1-q)^2/(p_b+p_c). Both partitions can belong to the support of the equilibrium distribution only if V(ac)=V(bc), which forces q/(1-q) = sqrt(p_b(p_a+p_c)/(p_a(p_b+p_c))); this ratio equals 1 (so q=1/2) only when p_a=p_b. The proposition assumes only p_c no larger than p_a and p_b, so for generic type frequencies the proposed q=1/2 makes the two partitions unequal, violating Definition 5. In addition, the employer's prescribed actions require (p_a+p_c(1-q))/(p_a+p_c) at least nu* in class ac and p_c(1-q)/(p_b+p_c) below nu* in class bc, which are not implied by the stated bounds on nu*. For example, with p_a=0.01, p_b=0.98, p_c=0.01 and nu*=0.9, the cost-equalizing q about 0.585 gives expected low effort in ac about 0.708, below 0.9, so the employer strictly prefers D in ac. Thus the uniqueness claim and the characterization in the monitoring application fail for generic parameter values; the statement and proof need to be revised.
  2. [Appendix, proof of Proposition 1] The case analysis appears to contain an error. The proof states that 'if sigma1(x1)=D, then sigma2(x1)=R', but in the matching pennies game of Example 1, if Row plays D, Column's payoff is 1 from L and 0 from R, so Column's best response is L, not R. This invalidates the subsequent inference that beta_L is at least 1/2 and the conclusion that any ABEE requires beta_L = 1/(2+x1). The non-existence claim may well be true, but the proof as written does not establish it.
  3. [Section 3.2, Proposition 4] The proof's first case asserts that from beta_{e=1} at most p_c/(p_a+p_c) and p_c/(p_a+p_c) < nu* it follows that the employer chooses C. Since the employer chooses C only when the expected probability of low effort is at least nu*, i.e., beta_{e=1} at most 1-nu*, this inference requires p_c/(p_a+p_c) at most 1-nu*, which is not among the assumptions. For nu* greater than 1/2 the stated bound does not suffice. The contradiction may be recoverable by a subcase analysis, but the proof as given is incomplete.
minor comments (4)
  1. [Definition 5] The name 'Clustered Distributional Analogy-Based Expected Equilibrium' should be 'Expectation Equilibrium' for consistency with the ABEE terminology used elsewhere in the paper.
  2. [Appendix, proof of Theorem 1] The domain of the GC correspondence is written as 'Sigma-bar times ((Delta A_i)^{K_i|K_i|} times (Delta A_j)^{K_j|K_j|})', which appears to be a typographical error; it should presumably be the product of the relevant simplex spaces for each of the |K_i| and |K_j| partitions.
  3. [Appendix, proof of Proposition 1] The expression 'probability 2/2+x1' should read '2/(2+x1)'.
  4. [Section 2.6, Learning model 2] The claim that steady states correspond to locally clustered CD-ABEE is asserted rather than proved; the text says an explicit study 'would require further work.' Since this is one of the two learning foundations for the solution concept, a formal statement or a clearer caveat would be desirable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equilibrium concept is a fixed point by design, and the main results are proven from the stated definitions rather than imported from self-citations.

full rationale

The paper's central object, CD-ABEE, is explicitly defined as a fixed point: strategies are ABEE given partitions and partitions solve the clustering problem given strategies. Theorem 1 proves existence by Kakutani on BR×GC, which is a substantive existence argument and not an identity; the proof verifies non-emptiness, upper hemi-continuity and convex-valuedness of the correspondence. The applications derive behavior from these equilibrium conditions; no parameter is fitted to a subset of data and then relabeled as a prediction. In the monitoring application, the c-worker's mixing is chosen to satisfy the clustering-cost equalization required by global clustered equilibrium; that is an equilibrium condition, not an exogenous fitted input. The learning-foundation section characterizes steady states as CD-ABEE; this is a fixed-point equivalence by construction, but it is presented as a characterization of the dynamics rather than as an independent empirical prediction. The only self-references are to Jehiel (2005) for the underlying ABEE concept (which is also defined in the paper) and to the authors' own working paper for fuller characterizations and a three-value case; these are background pointers and are not used to prove the paper's central existence, uniqueness, or classification results. No load-bearing step reduces by equation to its own inputs, so there is no circularity to report. A separate internal-consistency concern about the parameter range in Proposition 5 would be a correctness issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The framework introduces no new physical entities. All assumptions are behavioral modeling choices standard in the ABEE literature. No free parameters are fitted to data; the mixing probabilities in the monitoring application are equilibrium-determined.

assumptions (4)
  • domain assumption Players cluster opponent's behavior to minimize prediction error using a Bregman divergence (squared Euclidean or KL), with the conditional mean as prototype.
    Definition 2 and Eq. 1; motivated by psychology and machine learning, but it is an assumption about player behavior.
  • domain assumption The number of analogy classes K_i is fixed exogenously.
    Section 2.2; memory constraint from Miller (1956).
  • domain assumption In distributional ABEE, players best respond to aggregate opponent behavior that averages over the opponent's analogy partitions, and partitions are drawn independently across players.
    Section 2.5, Eqs. (2) and (3); formalizes a random assignment assumption.
  • standard math Kakutani's fixed point theorem and standard results on Bregman divergences hold.
    Proof of Theorem 1 and Online Appendix B.

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Pith. "Pith review of Endogenous clustering and analogy-based expectation equilibrium." pith.science (2026). https://pith.science/paper/XSKIOVIN

@misc{pith2026250513022,
  author       = {Pith},
  title        = {Pith review of: Endogenous clustering and analogy-based expectation equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSKIOVIN}},
  note         = {Machine review of arXiv:2505.13022}
}
read the original abstract

Normal-form two-player games are categorized by players into K analogy classes so as to minimize the prediction error about the behavior of the opponent. This results in Clustered Analogy-Based Expectation Equilibria in which strategies are analogy-based expectation equilibria given the analogy partitions and analogy partitions minimize the prediction errors given the strategies. We distinguish between environments with self-repelling analogy partitions in which some mixing over partitions is required and environments with self-attractive partitions in which several analogy partitions can arise, thereby suggesting new channels of belief heterogeneity and equilibrium multiplicity. Various economic applications are discussed.

Figures

Figures reproduced from arXiv: 2505.13022 by the authors.

Figure 1
Figure 1. Local Clustering for K = 4 and µ-sequence such that µk = k 4 , k = 0, 1, . . . , 4. (a) Increasing NE and ABEE functions with A = 1,B = 1 and C = 0.8 (b) Decreasing NE and ABEE functions with A = 4.1,B = −4 and C = 0.6 To present our result, we introduce the notion of equidistant-expectations sequence µ0, µ1, . . . , µK defined so that for any µk, with k ̸= 0, 1, the Euclidean distance be￾tween µk and the mean value… view at source ↗
Figure 2
Figure 2. Local Clustering for K = 4 and µ-sequence such that µk = k 4 , k = 0, 1, . . . , 4. (a) Increasing NE and ABEE functions with A = 1.5,B = 1 and C = 0.9 (b) Decreasing NE and ABEE functions with A = 1.5,B = −4 and C = 0.9 Proposition 9 In the strategic substitutes environment, whenever B ̸= −AC, there are no symmetric interval analogy partitions that are locally clustered with respect to the induced ABEE. Our environ… view at source ↗

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Al-Najjar N. and M. Pai (2014): ``Coarse Decision Making and Overfitting,'' Journal of Economic Theory, 150, 467--86

    work page 2014

  2. [2]

    Anderson J. R. (1991): ``The adaptive nature of human categorization,'' Psychological Review, 98(3), 409--429

    work page 1991

  3. [3]

    Gilboa, A

    Aragones E., I. Gilboa, A. Postlewaite and D. Schmeidler (2005): ``Fact-Free Learning,'' American Economic Review, 95(5), 1355--1368

    work page 2005

  4. [4]

    Guo and H

    Banerjee A., X. Guo and H. Wang (2005): ``On the optimality of conditional expectation as a Bregman predictor'', IEEE Transactions on Information Theory, 51(7), 2664-2669

    work page 2005

  5. [5]

    (1987): ``Comportamento razionale ed equilibrio nei giochi e nelle situazioni sociali'', unpublished undergraduate dissertation, Bocconi University, Milano

    Battigalli P. (1987): ``Comportamento razionale ed equilibrio nei giochi e nelle situazioni sociali'', unpublished undergraduate dissertation, Bocconi University, Milano

    work page 1987

  6. [6]

    Border, K. C. (1985): Fixed point theorems with applications to economics and game theory. Cambridge university press

    work page 1985

  7. [7]

    Esponda I. and D. Pouzo (2016): ``Berk-Nash Equilibrium: A Framework for Modeling Agents With Misspecified Models,'' Econometrica, 84 (3), 1093--1130

    work page 2016

  8. [8]

    Eyster E. and M. Rabin (2005): ``Cursed Equilibrium,'' Econometrica, 73(5), 1623--1672

    work page 2005

Show all 46 references
  1. [9]

    Fryer R. and M. Jackson (2008): ``A Categorical Model of Cognition and Biased Decision-Making,'' Contributions in Theoretical Economics, 8(1)

    2008

  2. [10]

    Fudenberg D. and D. Kreps (1993): ``Learning Mixed Equilibria,'' Games and Economic Behavior, 5 (3), 320--367

    1993

  3. [11]

    Fudenberg D. and G. Lanzani (2023): ``Which misperceptions persist?,'' Theoretical Economics, 18 (3), 1271--1315

    2023

  4. [12]

    Lanzani and P

    Fudenberg D., G. Lanzani and P. Strack (2024): ``Selective memory equilibrium'', Journal of Political Economy, 132 (12), 3978--4020

    2024

  5. [13]

    LiCalzi and M

    Gibbons R., M. LiCalzi and M. Warglien (2021): ``What situation is this? Shared frames and collective performance,'' Strategy Science 6 (2), 124--140

    2021

  6. [14]

    Harsanyi, J. C. (1973): ``Games with randomly disturbed payoffs: A new rationale for mixed--strategy equilibrium points.'' International journal of game theory, 2(1), 1--23

    1973

  7. [15]

    (1982): ``Moral

    Holmstr\" o m B. (1982): ``Moral

    1982

  8. [16]

    Jakobsen A. M. (2020): ``A Model of Complex Contracts.'', American Economic Review, 110 (5): 1243--73

    2020

  9. [17]

    (1995): ``Limited horizon forecast in repeated alternate games,'' Journal of Economic Theory, 67 (2), 497--519

    Jehiel P. (1995): ``Limited horizon forecast in repeated alternate games,'' Journal of Economic Theory, 67 (2), 497--519

    1995

  10. [18]

    (2005): ``Analogy-based expectation equilibrium,'' Journal of Economic Theory 123, 81--104

    Jehiel P. (2005): ``Analogy-based expectation equilibrium,'' Journal of Economic Theory 123, 81--104

    2005

  11. [19]

    (2022): ``Analogy-based expectation equilibrium and related concepts: Theory, applications, and beyond,'' Prepared for the twelfth World Congress of the Econometric Society

    Jehiel P. (2022): ``Analogy-based expectation equilibrium and related concepts: Theory, applications, and beyond,'' Prepared for the twelfth World Congress of the Econometric Society

    2022

  12. [20]

    Jehiel P. and F. Koessler (2008): ``Revisiting Bayesian games with analogy-based expectations,'' Games and Economic Behavior, 62, 533--557

    2008

  13. [21]

    Jehiel P. and E. Mohlin (2023): ``Categorization in Games: A Bias-Variance Perspective,'' WP ffhalshs04154272f

    2023

  14. [22]

    Jehiel P. and D. Samet (2007): ``Valuation equilibrium,'' Theoretical Economics, 2 (2), 163--185

    2007

  15. [23]

    Jehiel P. and G. Weber (2023): ``Calibrated Clustering and Analogy-Based Expectation Equilibrium,'' PSE Working Papers halshs-04154234, HAL

    2023

  16. [24]

    (1936): 'The General Theory of Employment, Interest and Money, Macmillan, Cambridge University Press

    Keynes J.M. (1936): 'The General Theory of Employment, Interest and Money, Macmillan, Cambridge University Press

    1936

  17. [25]

    (1957): ``Least squares quantization in PCM,'' Bell Laboratories Technical Memorandum, later published as: Lloyd, S.P

    Lloyd S.P. (1957): ``Least squares quantization in PCM,'' Bell Laboratories Technical Memorandum, later published as: Lloyd, S.P. (1982): ``Least squares quantization in PCM,'' IEEE Transactions on Information Theory, 28(2), 129--137

    1957

  18. [26]

    (1967): ``Classification and analysis of multivariate observations,'' In 5th Berkeley Symp

    MacQueen J. (1967): ``Classification and analysis of multivariate observations,'' In 5th Berkeley Symp. Math. Statist. Probability, 281-297

    1967

  19. [27]

    (2012): ``Learning Across Games,'' Games and Economic Behavior, 74(2), 601--619

    Mengel, F. (2012): ``Learning Across Games,'' Games and Economic Behavior, 74(2), 601--619

    2012

  20. [28]

    Medin, D. L. and M. M. Schaffer (1978): ``Context theory of classification learning,'' Psychological Review, 85(3), 207--238

    1978

  21. [29]

    Miller G. A. (1956): ``The magical number seven, plus or minus two: Some limits on our capacity for processing information,'' Psychological Review, 63(2), 81--97

    1956

  22. [30]

    (1965): 'Topology from the Differential Viewpoint', University Press of Virginia

    Milnor J. (1965): 'Topology from the Differential Viewpoint', University Press of Virginia

    1965

  23. [31]

    (2014): ``Optimal Categorization,'' Journal of Economic Theory, 152, 356--381

    Mohlin E. (2014): ``Optimal Categorization,'' Journal of Economic Theory, 152, 356--381

    2014

  24. [32]

    Myerson, R. and P. Reny (2020): ``Perfect Conditional -Equilibria of Multi-Stage Games With Infinite Sets of Signals and Actions,'' Econometrica, 88 (2), 495--531

    2020

  25. [33]

    (1985): ``Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma'', Economics letters, 19 (3), 227--229

    Neyman A. (1985): ``Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma'', Economics letters, 19 (3), 227--229

    1985

  26. [34]

    Nosofsky, R. M. (1986): ``Attention, similarity, and the identification-categorization relationship,'' Journal of Experimental Psychology: General, 115(1), 39--57

    1986

  27. [35]

    (1991): Learning in mis-specified models and the possibility of cycles, Journal of Economic Theory,, 55(2), 416--427

    Nyarko, Y. (1991): Learning in mis-specified models and the possibility of cycles, Journal of Economic Theory,, 55(2), 416--427

    1991

  28. [36]

    Osborne M. and A. Rubinstein (1998): ``Games with Procedurally Rational Players,'' American Economic Review,, 88(4), 834--47

    1998

  29. [37]

    (2011): ``Prior symmetry, categorization and similarity-based reasoning,'' Journal of Economic Theory, 146(1), 111--140

    P e ski M. (2011): ``Prior symmetry, categorization and similarity-based reasoning,'' Journal of Economic Theory, 146(1), 111--140

    2011

  30. [38]

    Posner, M. I. and S.W. Keele (1968): ``On the genesis of abstract ideas,'' Journal of Experimental Psychology, 77(3, Pt.1), 353--363

    1968

  31. [39]

    Reed, S. K. (1972): ``Pattern recognition and categorization,'' Cognitive Psychology, 3(3), 382--407

    1972

  32. [40]

    Rosch, E. and B.B. Lloyd (Eds.). (1978): Cognition and categorization. Lawrence Erlbaum

    1978

  33. [41]

    (1986): ``Finite automata play the repeated prisoner's dilemma'', Journal of economic theory, 39 (1), 83-96

    Rubinstein A. (1986): ``Finite automata play the repeated prisoner's dilemma'', Journal of economic theory, 39 (1), 83-96

    1986

  34. [42]

    Samuelson, L. (2001). ``Analogies, adaptation, and anomalies,'' Journal of Economic Theory, 97(2), 320--366

    2001

  35. [43]

    (2003): ``Implications of rational inattention,'' Journal of Monetary Economics, 50(3), 665--690

    Sims C.A. (2003): ``Implications of rational inattention,'' Journal of Monetary Economics, 50(3), 665--690

    2003

  36. [44]

    (2016): ``Bayesian Networks and Boundedly Rational Expectations,'' Quarterly Journal of Economics, 131, 1243--1290

    Spiegler R. (2016): ``Bayesian Networks and Boundedly Rational Expectations,'' Quarterly Journal of Economics, 131, 1243--1290

    2016

  37. [45]

    (1957): ``Sur la division des corps materiels en parties,'' Bulletin L'Academie Polonaise des Science, 4, 801--804

    Steinhaus H. (1957): ``Sur la division des corps materiels en parties,'' Bulletin L'Academie Polonaise des Science, 4, 801--804

    1957

  38. [46]

    (1999): Oligopoly pricing: old ideas and new tools, MIT press

    Vives X. (1999): Oligopoly pricing: old ideas and new tools, MIT press

    1999

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