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The intermediate value theorem and decision-making in psychology and economics: an expositional consolidation. (English) Zbl 1501.91045


MSC:

91B06 Decision theory

References:

[1] Grabiner, J.V.; Cauchy and Bolzano: Tradition and transformation in the history of mathematics; Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen: Cambridge, MA, USA 1984; Volume Volume 4 ,105-124.
[2] Grabiner, J.V.; Is mathematical truth time-dependent?; Am. Math. Mon.: 1974; Volume 81 ,354-365. · Zbl 0284.01013
[3] Bolzano, B.; ; Rein Analytisches Beweis des Lehrsatzes dass Zwischen je Zwey Werthen, Die ein Entgegengesetzetes Resultat Gewähren, Wenigsten eine Reelle Wurzel der Gleichung Liege: Prague, Czech Republic 1817; .
[4] Cauchy, A.-L.; ; Cours d’analyse de l’École Royale Polytechnique: Paris, France 1821; .
[5] Vrahatis, M.N.; Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros; Topol. Its Appl.: 2020; Volume 275 ,107036. · Zbl 1512.55004
[6] Barany, M.J.; Stuck in the middle: Cauchy’s intermediate value theorem and the history of analytic rigor; Not. Am. Math. Soc.: 2013; Volume 60 ,1334-1338. · Zbl 1322.26001
[7] Grabiner, J.V.; ; The Origins of Cauchy’s Rigorous Calculus: Cambridge, MA, USA 1981; . · Zbl 0517.01002
[8] Barany, M.J.; God, king, and geometry: Revisiting the introduction to Cauchy’s Cours d’analyse; Hist. Math.: 2011; Volume 38 ,368-388. · Zbl 1221.01065
[9] Laugwitz, D.; Infinitely small quantities in Cauchy’s textbooks; Hist. Math.: 1987; Volume 14 ,258-274. · Zbl 0638.01012
[10] Rowe, C.; Note on a pair of properties which characterize continuous functions; Bull. Am. Math. Soc.: 1926; Volume 32 ,285-287. · JFM 52.0241.04
[11] Fine, N.; Continuity from Darboux property and closedness of g−1(r); Am. Math. Mon.: 1966; Volume 73 ,782.
[12] Velleman, D.J.; Characterizing continuity; Am. Math. Mon.: 1997; Volume 104 ,318-322. · Zbl 0871.26002
[13] Yannakakis, M.; Equilibria, fixed points, and complexity classes; Comput. Sci. Rev.: 2009; Volume 3 ,71-85. · Zbl 1302.68143
[14] Morales, C.H.; A Bolzano’s theorem in the new millennium; Nonlinear Anal. Theory Methods Appl.: 2002; Volume 51 ,679-691. · Zbl 1009.47042
[15] Turzański, M.; The Bolzano-Poincaré-Miranda theorem-discrete version; Topol. Its Appl.: 2012; Volume 159 ,3130-3135. · Zbl 1333.54051
[16] Eilenberg, S.; Ordered topological spaces; Am. Math.: 1941; Volume 63 ,39-45. · Zbl 0024.19203
[17] Rosenthal, A.; On the continuity of functions of several variables; Math. Z.: 1955; Volume 63 ,31-38. · Zbl 0064.30003
[18] Khan, M.A.; Uyanik, M.; Topological connectedness and behavioral assumptions on preferences: A two-way relationship; Econ. Theory: 2021; Volume 71 ,411-460. · Zbl 1479.91086
[19] Chinn, W.G.; Steenrod, N.E.; ; First Concepts of Topology: The Geometry of Mappings of Segments, Curves, Circles, and Disks: Washington, DC, USA 1966; Volume Volume 18 . · Zbl 0201.55303
[20] Uyanik, M.; Khan, M.A.; The continuity postulate in economic theory: A deconstruction and an integration; J. Math. Econ.: 2022; ,102704. · Zbl 1497.91116
[21] Ghosh, A.; Khan, M.A.; Uyanik, M.; Continuity postulates and solvability axioms in economic theory and in mathematical psychology: A consolidation of the theory of individual choice; Theory Decis.: 2022; .
[22] Schmeidler, D.; A condition for the completeness of partial preference relations; Econometrica: 1971; Volume 39 ,403-404. · Zbl 0217.26703
[23] Herstein, I.N.; Milnor, J.; An axiomatic approach to measurable utility; Econometrica: 1953; Volume 21 ,291-297. · Zbl 0050.36705
[24] Galaabaatar, T.; Khan, M.A.; Uyanik, M.; Completeness and transitivity of preferences on mixture sets; Math. Soc. Sci.: 2019; Volume 99 ,49-62. · Zbl 1426.91086
[25] Giarlotta, A.; Watson, S.; A bi-preference interplay between transitivity and completeness: Reformulating and extending Schmeidler’s theorem; J. Math. Psychol.: 2020; Volume 96 ,102354. · Zbl 1442.91030
[26] Khan, M.A.; Uyanik, M.; On an extension of a theorem of Eilenberg and a characterization of topological connectedness; Topol. Its Appl.: 2020; Volume 273 ,107-117. · Zbl 1437.54019
[27] Khan, M.A.; Uyanik, M.; Binary relations in mathematical economics: On continuity, additivity and monotonicity postulates in Eilenberg, Villegas and DeGroot; Positivity and Its Applications: Basel, Switzerland 2021; ,229-250. · Zbl 1471.91375
[28] Uyanik, M.; Khan, M.A.; On the consistency and the decisiveness of the double-minded decision-maker; Econ. Lett.: 2019; Volume 185 ,108657. · Zbl 1425.91126
[29] Marschak, J.; Rational behavior, uncertain prospects, and measurable utility; Econometrica: 1950; Volume 18 ,111-141. · Zbl 0036.22001
[30] Nash, J.F.; Non-cooperative games; Ann. Math.: 1951; Volume 54 ,286-295. · Zbl 0045.08202
[31] Bleichrodt, H.; Moscati, C.L.I.; Wakker, P.P.; Nash was a first to axiomatize expected utility; Theory Decis.: 2016; Volume 81 ,309-312. · Zbl 1378.91007
[32] Fishburn, P.; Wakker, P.; The invention of the independence condition for preferences; Manag. Sci.: 1995; Volume 41 ,1130-1144. · Zbl 0843.90011
[33] Luce, R.D.; Tukey, J.W.; Simultaneous conjoint measurement: A new type of fundamental measurement; J. Math. Psychol.: 1964; Volume 1 ,1-27. · Zbl 0166.42201
[34] Suppes, P.; A set of independent axioms for extensive quantities; Port. Math.: 1951; Volume 10 ,163-172. · Zbl 0044.17102
[35] Krantz, D.; Luce, D.; Suppes, P.; Tversky, A.; ; Foundations of Measurement, Volume I: Additive and Polynomial Representations: New York, NY, USA 1971; . · Zbl 0232.02040
[36] Abdellaoui, M.; Wakker, P.; Savage for dummies and experts; J. Econ. Theory: 2020; Volume 186 ,104991. · Zbl 1432.91057
[37] Karni, E.; Axiomatic foundations of expected utility and subjective probability; Handbook of the Economics of Risk and Uncertainty: Amsterdam, The Netherlands 2014; Volume Volume 1 ,1-39.
[38] Karni, E.; Maccheroni, F.; Marinacci, M.; Ambiguity and nonexpected utility; Handbook of Game Theory with Economic Applications: Amsterdam, The Netherlands 2015; Volume Volume 4 ,901-947.
[39] Karni, E.; Schmeidler, D.; Utility theory with uncertainty; Handbook of Mathematical Economics: Amsterdam, The Netherlands 1991; Volume Volume 4 ,1763-1831. · Zbl 1001.91022
[40] Moscati, I.; ; Measuring Utility: Oxford, UK 2016; .
[41] Thurstone, L.L.; The nature of general intelligence and ability (III); Br. J. Psychol.: 1924; Volume 14 ,243.
[42] Thurstone, L.L.; The vectors of mind; Psychol. Rev.: 1934; Volume 41 ,1-32.
[43] Sen, A.; ; Collective Choice and Social Welfare: An Expanded Edition: Cambridge, MA, USA 2017; .
[44] Lax, P.D.; Change of variables in multiple integrals; Am. Math. Mon.: 1999; Volume 106 ,497-501. · Zbl 1007.26006
[45] Wold, H.; A synthesis of pure demand analysis, I-III; Scand. Actuar. J.: 1943-1944; Volume 26-27 ,69-263. · Zbl 0063.08307
[46] Wold, H.; Jureen, L.; ; Demand Analysis: New York, NY, USA 1953; . · Zbl 0050.36802
[47] Debreu, G.; A social equilibrium existence theorem; Proc. Natl. Acad. Sci. USA: 1952; Volume 38 ,886-893. · Zbl 0047.38804
[48] Nash, J.F.; Equilibrium points in n-person games; Proc. Natl. Acad. Sci. USA: 1950; Volume 36 ,48-49. · Zbl 0036.01104
[49] Burmeister, E.; Dobell, A.R.; ; Mathematical Theories of Economic Growth: New York, NY, USA 1970; . · Zbl 0238.90009
[50] Arrow, K.J.; Hahn, F.H.; ; General Competitive Analysis: Amsterdam, The Netherlands 1971; . · Zbl 0311.90001
[51] Arrow, K.J.; Debreu, G.; Existence of an equilibrium for a competitive economy; Econometrica: 1954; Volume 22 ,265-290. · Zbl 0055.38007
[52] McKenzie, L.W.; On equilibrium in Graham’s model of world trade and other competitive systems; Econometrica: 1954; Volume 22 ,147-161. · Zbl 0055.13702
[53] Uzawa, H.; Walras’ existence theorem and Brouwer fixed-point theorem; Econ. Stud. Q.: 1962; Volume 13 ,59-62.
[54] Le, T.; Pham, C.L.N.-S.; Saglam, C.; Direct proofs of the existence of equilibrium, the Gale-Nikaido-Debreu lemma and the fixed point theorems using Sperner’s lemma; Hal Work.: 2020; .
[55] Khan, M.A.; McLean, R.P.; Uyanik, M.; The KKM lemma and the Fan-Browder theorems: Equivalences and some circular tours; Linear Nonlinear Anal.: 2021; Volume 7 ,33-62. · Zbl 1542.47085
[56] Khan, M.A.; On the Finding of an Equilibrium: Düppe-Weintraub and the Problem of Scientific Credit; J. Econ. Lit.: 2020; Volume 1 ,1-50.
[57] Hahn, H.; Rosenthal, A.; ; Set Functions: Albuquerque, NM, USA 1948; . · Zbl 0033.05301
[58] Mizel, V.J.; Martin, A.D.; A representation theorem for certain nonlinear functionals; Arch. Ration. Anal.: 1964; Volume 15 ,353-367. · Zbl 0131.33001
[59] Sohrab, H.H.; ; Basic Real Analysis: New York, NY, USA 2014; . · Zbl 1308.26006
[60] Oman, G.; The converse of the intermediate value theorem: From Conway to Cantor to cosets and beyond; Mo. J. Math. Sci.: 2014; Volume 26 ,134-150. · Zbl 1311.26002
[61] Fierro, R.; Martinez, C.; Morales, C.H.; The aftermath of the intermediate value theorem; Fixed Point Theory Appl.: 2004; Volume 2004 ,1-8. · Zbl 1090.47051
[62] Milgrom, P.; Roberts, J.; Comparing equilibria; Am. Econ. Rev.: 1994; Volume 84 ,441-459.
[63] Guillerme, J.; Intermediate value theorems and fixed point theorems for semi-continuous functions in product spaces; Proc. Am. Math. Soc.: 1995; Volume 123 ,2119-2122. · Zbl 0835.47041
[64] Wu, Z.; A note on fixed point theorems for semi-continuous correspondences on [0, 1]; Proc. Am. Math. Soc.: 1998; Volume 126 ,3061-3064. · Zbl 0902.47046
[65] Debreu, G.; ; Theory of Value: An Axiomatic Analysis of Economic Equilibrium: Yale, CT, USA 1959; . · Zbl 0193.20205
[66] Ricci, R.G.; A note on a Tarski type fixed-point theorem; Int. J. Game Theory: 2021; Volume 50 ,751-758. · Zbl 1471.91011
[67] Amir, R.; Castro, L.D.; Nash equilibrium in games with quasi-monotonic best-responses; J. Econ. Theory: 2017; Volume 172 ,220-246. · Zbl 1414.91011
[68] Willard, S.; ; General Topology: New York, NY, USA 1970; . · Zbl 0205.26601
[69] Mongin, P.; A note on mixture sets in decision theory; Decis. Econ. Financ.: 2001; Volume 24 ,59-69. · Zbl 1019.91013
[70] Munkres, J.R.; ; Topology: Hoboken, NJ, USA 2000; . · Zbl 0951.54001
[71] Liapounoff, A.; Sur les fonctions-vecteurs complétement additives; Izv. Ross. Akad. Nauk. Seriya Mat.: 1940; Volume 4 ,465-478. · JFM 66.0219.02
[72] Loeb, P.A.; Rashid, S.; Lyapunov’s Theorem; The New Palgrave Dictionary of Economics: London, UK 2016; ,1-4.
[73] Khan, M.A.; Sagara, N.; Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces; Ill. J. Math.: 2013; Volume 57 ,145-169. · Zbl 1298.28027
[74] Ross, D.A.; An elementary proof of Lyapunov’s theorem; Am. Math. Mon.: 2005; Volume 112 ,651-653. · Zbl 1122.28002
[75] Nyman, K.L.; Su, F.E.; A Borsuk-Ulam equivalent that directly implies Sperner’s lemma; Am. Math. Mon.: 2013; Volume 120 ,346-354. · Zbl 1280.55001
[76] Su, F.E.; Borsuk-Ulam implies Brouwer: A direct construction; Am. Math. Mon.: 1997; Volume 104 ,855-859. · Zbl 0884.54028
[77] Volovikov, A.Y.; Borsuk-Ulam Implies Brouwer: A Direct Construction Revisited; Am. Math. Mon.: 2008; Volume 115 ,553-556. · Zbl 1158.55004
[78] Kulpa, W.; The Poincaré-Miranda theorem; Am. Math. Mon.: 1997; Volume 104 ,545-550. · Zbl 0891.47040
[79] Matoušek, J.; ; Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry: Berlin, Germany 2003; . · Zbl 1016.05001
[80] Vrahatis, M.N.; Generalization of the Bolzano theorem for simplices; Topol. Its Appl.: 2016; Volume 202 ,40-46. · Zbl 1347.55004
[81] Milgrom, P.; Mollner, J.; Equilibrium selection in auctions and high stakes games; Econometrica: 2018; Volume 86 ,219-261. · Zbl 1419.91338
[82] Khan, M.A.; McLean, R.P.; Uyanik, M.; ; On a Generalization of the Poincaré-Miranda Theorem: 2022; .
[83] Szymańska-Debowska, K.; Resonant problem for some second-order differential equation on the half-line; Electron. Differ. Equ.: 2007; Volume 2007 ,1-9. · Zbl 1141.34314
[84] Szymańska-Debowska, K.; On a generalization of the Miranda Theorem and its application to boundary value problems; J. Differ. Equ.: 2015; Volume 258 ,2686-2700. · Zbl 1336.47056
[85] Wakker, P.; The algebraic versus the topological approach to additive representations; J. Math. Psychol.: 1988; Volume 32 ,421-435. · Zbl 0663.92024
[86] Banerjee, K.; Mitra, T.; On Wold’s approach to representation of preferences; J. Math. Econ.: 2018; Volume 79 ,65-74. · Zbl 1418.91190
[87] Nash, J.F.; The bargaining problem; Econometrica: 1950; Volume 18 ,155-162. · Zbl 1202.91122
[88] von Neumann, J.; Morgenstern, O.; ; Theory of Games and Economic Behavior: Princeton, NJ, USA 1947; . · Zbl 1241.91002
[89] Fishburn, P.C.; ; The Foundations of Expected Utility: Boston, MA, USA 1982; . · Zbl 0497.90001
[90] Luce, R.D.; Narens, L.; Measurement: The theory of numerical assignments; Psychol. Bull.: 1986; Volume 99 ,166-180.
[91] Suppes, P.; The measurement of belief; J. R. Stat. Soc.: 1974; Volume 36 ,160-175. · Zbl 0287.60007
[92] Ciesielski, K.C.; Miller, D.; A continuous tale on continuous and separately continuous functions; Real Anal.: 2016; Volume 41 ,19-54. · Zbl 1388.26010
[93] Genocchi, A.; Peano, G.; ; Calcolo Differentiale e Principii di Calcolo: Torino, ON, USA 1884; . · JFM 16.0223.01
[94] Kruse, R.; Deely, J.; Joint continuity of monotonic functions; Am. Math. Mon.: 1969; Volume 76 ,74-76. · Zbl 0172.33304
[95] Young, W.; A note on monotone functions; Q. J. Pure Appl. Math.: 1910; Volume 41 ,79-87. · JFM 40.0438.01
[96] Yokoyama, T.; Continuity conditions of preference ordering; Osaka Econ. Pap.: 1956; Volume 4 ,39-45.
[97] Dubra, J.; Continuity and completeness under risk; Math. Soc. Sci.: 2011; Volume 61 ,80-81. · Zbl 1208.91036
[98] Dubra, J.; Maccheroni, F.; Ok, E.A.; Expected utility theory without the completeness axiom; J. Econ. Theory: 2004; Volume 115 ,118-133. · Zbl 1062.91025
[99] Gilboa, I.; Maccheroni, F.; Marinacci, M.; Schmeidler, D.; Objective and subjective rationality in a multiple prior model; Econometrica: 2010; Volume 78 ,755-770. · Zbl 1229.91103
[100] Karni, E.; Safra, Z.; Continuity, completeness, betweenness and cone-monotonicity; Math. Soc. Sci.: 2015; Volume 74 ,68-72. · Zbl 1310.91058
[101] Bergstrom, T.C.; Parks, R.P.; Rader, T.; Preferences which have open graphs; J. Math. Econ.: 1976; Volume 3 ,265-268. · Zbl 0387.90009
[102] Deshpandé, J.V.; On continuity of a partial order; Proc. Am. Math. Soc.: 1968; Volume 19 ,383-386. · Zbl 0155.03401
[103] Schmeidler, D.; Competitive equilibria in markets with a continuum of traders and incomplete preferences; Econometrica: 1969; Volume 37 ,578-585. · Zbl 0184.45201
[104] Shafer, W.; The nontransitive consumer; Econometrica: 1974; Volume 42 ,913-919. · Zbl 0291.90007
[105] Ward, L.; Partially ordered topological spaces; Proc. Am. Math. Soc.: 1954; Volume 5 ,144-161. · Zbl 0055.16101
[106] Gerasimou, G.; Consumer theory with bounded rational preferences; J. Math. Econ.: 2010; Volume 46 ,708-714. · Zbl 1232.91424
[107] Gerasimou, G.; On continuity of incomplete preferences; Soc. Choice Welf.: 2013; Volume 41 ,157-167. · Zbl 1288.91050
[108] Gerasimou, G.; (Hemi)continuity of additive preference preorders; J. Math. Econ.: 2015; Volume 58 ,79-81. · Zbl 1319.91062
[109] Luce, D.; Semiorders and a theory of utility discrimination; Econometrica: 1956; Volume 24 ,178-191. · Zbl 0071.14006
[110] Giarlotta, A.; Watson, S.; Universal semiorders; J. Math. Psychol.: 2016; Volume 73 ,80-93. · Zbl 1396.91112
[111] Özbay, E.Y.; Filiz, E.; A representation for intransitive indifference relations; Math. Soc. Sci.: 2005; Volume 50 ,202-214. · Zbl 1115.91011
[112] Giarlotta, A.; New trends in preference, utility, and choice: From a mono-approach to a multi-approach; New Perspectives in Multiple Criteria Decision Making: Berlin/Heidelberg, Germany 2019; ,3-80.
[113] Giarlotta, A.; Greco, S.; Necessary and possible preference structures; J. Math. Econ.: 2013; Volume 49 ,163-172. · Zbl 1271.91045
[114] Giarlotta, A.; Watson, S.; Necessary and possible indifferences; J. Math. Psychol.: 2017; Volume 81 ,98-109. · Zbl 1397.91160
[115] Chipman, J.; Consumption theory without transitive indifference; Preferences, Utility and Demand: A Minnesota Symposium: New York, NY, USA 1971; ,224-253. · Zbl 0293.90003
[116] Gorman, W.M.; Preference, revealed preference, and indifference; Preferences, Utility and Demand: A Minnesota Symposium: New York, NY, USA 1971; ,81-113. · Zbl 0272.90006
[117] Huang, X.-C.; From intermediate value theorem to chaos; Math. Mag.: 1992; Volume 65 ,91-103. · Zbl 0824.26002
[118] Li, T.-Y.; Yorke, J.A.; Period three implies chaos; Am. Math. Mon.: 1975; Volume 82 ,985-992. · Zbl 0351.92021
[119] Deng, L.; Khan, M.A.; On growing through cycles: Matsuyama’s M-map and Li-Yorke chaos; J. Math. Econ.: 2018; Volume 74 ,46-55. · Zbl 1388.91129
[120] Du, B.-S.; A simple proof of Sharkovsky’s theorem revisited; Am. Math. Mon.: 2007; Volume 114 ,152-155. · Zbl 1120.37020
[121] Du, B.-S.; An interesting application of the intermediate value theorem: A simple proof of Sharkovsky’s theorem and the towers of periodic points; arXiv: 2017; .
[122] Lucas, R.; Prescott, E.; Equilibrium search and unemployment; J. Political Econ.: 1974; Volume 7 ,188-209.
[123] Stokey, N.L.; Lucas, R.; ; Recursive Methods in Economic Dynamics: Cambridge, MA, USA 1989; . · Zbl 0774.90018
[124] Hildenbrand, W.; An exposition of Wald’s existence proof; Karl Menger: Berlin/Heidelberg, Germany 1998; ,51-61.
[125] Arrow, K.J.; Lectures on the Theory of Competitive Equilibrium. UC San Diego: Department of Economics, UCSD; ; .
[126] Pröhl, E.; Existence and Uniqueness of Recursive Equilibria with Aggregate and Idiosyncratic Risk. Available at SSRN 3250651; 2021; .
[127] Rockafellar, R.T.; ; Convex Analysis: New York, NY, USA 1970; . · Zbl 0193.18401
[128] McManus, M.; Equilibrium, numbers and size in Cournot oligopoly; Bull. Econ. Res.: 1964; Volume 16 ,68-75.
[129] Roberts, J.; Sonnenschein, H.; On the existence of Cournot equilbrium without concave profit functions; J. Econ. Theory: 1976; Volume 13 ,112-117. · Zbl 0341.90011
[130] Khan, M.A.; Schlee, E.E.; Money-metric complementarity and price-dependent normality with nonordered preferences; Proceedings of the SAET Meetings: ; .
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