zbMATH — the first resource for mathematics

On the structure and diversity of rational beliefs. (English) Zbl 0811.90022
Summary: The paper proposes that the theory of expectations be reformulated under the assumption that agents do not know the structural relations (such as equilibrium prices) of the economy. Instead, we postulate that they can observe past data of the economy and form probability beliefs based on the data generated by the economy. Using past data agents can compute relative frequencies and the basic assumption of the theory is that the system which generates the data is stable in the sense that the empirically computed relative frequencies converge. It is then shown that the limit of these relative frequencies induce a probability on the space of infinite sequences of the observables in the economy. This probability is stationary. A belief of an agent is a probability on the space of infinite sequences of the observable variables in the economy. Such a probability represents the “theory” or “hypothesis” of the agent about the mechanism which generates the data. A belief is said to be compatible with the data if under the proposed probability belief the economy would generate the same limit of the relative frequencies as computed from the real data. A theory which is “compatible with the data” is a theory which cannot be rejected by the data. A belief is said to be a Rational Belief if it is (i) compatible with the data and (ii) satisfies a certain technical condition. The main theorem provides a characterization of all rational beliefs.

91B44 Economics of information
91E40 Memory and learning in psychology
62C10 Bayesian problems; characterization of Bayes procedures
Full Text: DOI
[1] Ash, R.B.: Real analysis and probability, New York: Academic Press, 1972 · Zbl 0249.28001
[2] Aumann, R.J.: Agreeing to disagree. Ann. Stat.4, 1236-1239 (1976) · Zbl 0379.62003 · doi:10.1214/aos/1176343654
[3] Aumann, R.J.: Correlated equilibrium as an expression of Bayesian rationality. Econometrica55, 1-18 (1987) · Zbl 0633.90094 · doi:10.2307/1911154
[4] Billingsley, P.: Convergence of probability measures. New York: Wiley 1968 · Zbl 0172.21201
[5] Blackwell, D., Dubins, L.: Merging of opinions with increasing information. Ann. Math. Stat.33, 882-886 (1962) · Zbl 0109.35704 · doi:10.1214/aoms/1177704456
[6] Blackwell, D., Dubins, L.: On existence and non-existence of proper, regular, conditional distributions. Ann. Probab.3, 741-752 (1975) · Zbl 0348.60003 · doi:10.1214/aop/1176996261
[7] Choquet, G.: Lectures on analysis, vol.II. New York: Benjamin, W.A. 1969 · Zbl 0181.39602
[8] Diaconis, P., Freedman, D.: On the consistency of Bayes estimates. Ann. Stat.14, 1-26 (1986) · Zbl 0595.62022 · doi:10.1214/aos/1176349830
[9] Dowker, Y.N.: Finite and sigma-finite invariant measures. Ann. Math.54, 595-608 (1951) · Zbl 0044.04806 · doi:10.2307/1969491
[10] Dowker, Y.N.: On measurable transformations in finite measure spaces. Ann. Probab.8, 962-973
[11] Feldman, M.: Bayesian learning and convergence to rational expectations. J. Math. Econ.16, 297-313 (1987) · Zbl 0637.90023 · doi:10.1016/0304-4068(87)90015-2
[12] Feldman, M.: On the generic nonconvergence of Bayesian actions and beliefs. Econ. Theory1, 301-321 (1991) · Zbl 0803.62004 · doi:10.1007/BF01229311
[13] Fontana, R.J., Gray, R.M., Kieffer, J.C.: Asymptotically mean stationary channels. I.E.E.E. Trans. Inform. Theory27, 308-316 (1981) · Zbl 0483.94010 · doi:10.1109/TIT.1981.1056348
[14] Gray, R.M., Kieffer, J.C.: Asymptotically mean stationary measures. Ann. Probabil.8, 962-973 (1980) · Zbl 0447.28014 · doi:10.1214/aop/1176994624
[15] Gray, R.M.: Probability, random processes, and ergodic properties. New York: Springer-Verlag 1988 · Zbl 0644.60001
[16] Harrison, M., Kreps, D.: Martingales and arbitrage in multiperiod security markets. J. Econ. Theory20, 381-408 (1979) · Zbl 0431.90019 · doi:10.1016/0022-0531(79)90043-7
[17] Harsanyi, J. C.: Games with incomplete information played by ?Bayesian? players. Parts I, II, III. Manage. Sci.14, 159-182; 320-334; 486-502 (1967-1968) · Zbl 0207.51102 · doi:10.1287/mnsc.14.3.159
[18] Kalai, E., Lehrer, E.: Rational learning leads to Nash equilibrium. Discussion Paper, Northwestern University 1991
[19] Kieffer, J.C., Rahe, M.: Markov channels are asymptotically mean stationary. Siam J. Math. Anal.12, 293-305 (1981) · Zbl 0475.94008 · doi:10.1137/0512027
[20] Kurz, M.: On the structure and diversity of rational beliefs. Working Paper, Stanford University 1990 (revised, February 1993)
[21] Kurz, M.: On rational belief equilibria. Econ. Theory4, 859-876 (1994) · Zbl 0811.90021 · doi:10.1007/BF01213816
[22] Nerlove, M., Grether, D.M., Carvalho, J.L.: Analysis of economic time series. New York: Academic Press (1979) · Zbl 0473.62077
[23] Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco: Holden-Day CA. (1965) · Zbl 0137.11301
[24] Nyarko, Y.: Bayesian learning leads to correlated equilibria in normal form games. Econ. Theory4, 821-841 (1994) · Zbl 0811.90135 · doi:10.1007/BF01213814
[25] Nyarko, Y.: Bayesian learning without common priors and convergence to Nash equilibria. New York University · Zbl 0911.90371
[26] Parthasarathy, K.R.: Probability measures on metric spaces. New York, London: Academic Press 1967 · Zbl 0153.19101
[27] Phelps, R.R.: Integral representation for elements of convex sets. In: Bartle, R.G. (ed.) Studies in functional analysis, vol. 21, pp. 115-157 MAA Studies in Mathematics 1980 · Zbl 0453.46014
[28] Rechard, O.W.: Invariant measures for many-one transformation. Duke J. Math.23, 447-448 (1956) · Zbl 0070.28001 · doi:10.1215/S0012-7094-56-02344-4
[29] Royden, H.L.: Real analysis, 3rd edition. New York, London: MacMillan (1988) · Zbl 0704.26006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.