{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T14:11:20Z","timestamp":1774966280814,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2014,8,21]],"date-time":"2014-08-21T00:00:00Z","timestamp":1408579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"DFG (German Research Council)","award":["BR4164\/1-1"],"award-info":[{"award-number":["BR4164\/1-1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Bounded rationality concerns the study of decision makers with limited information processing resources. Previously, the free energy difference functional has been suggested to model bounded rational decision making, as it provides a natural trade-off between an energy or utility function that is to be optimized and information processing costs that are measured by entropic search costs. The main question of this article is how the information-theoretic free energy model relates to simple \u03b5-optimality models of bounded rational decision making, where the decision maker is satisfied with any action in an \u03b5-neighborhood of the optimal utility. We find that the stochastic policies that optimize the free energy trade-off comply with the notion of \u03b5-optimality. Moreover, this optimality criterion even holds when the environment is adversarial. We conclude that the study of bounded rationality based on \u03b5-optimality criteria that abstract away from the particulars of the information processing constraints is compatible with the information-theoretic free energy model of bounded rationality.<\/jats:p>","DOI":"10.3390\/e16084662","type":"journal-article","created":{"date-parts":[[2014,8,21]],"date-time":"2014-08-21T11:00:23Z","timestamp":1408618823000},"page":"4662-4676","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Information-Theoretic Bounded Rationality and \u03b5-Optimality"],"prefix":"10.3390","volume":"16","author":[{"given":"Daniel","family":"Braun","sequence":"first","affiliation":[{"name":"Max Planck Institute for Biological Cybernetics, Max Planck Institute for Intelligent Systems, Spemannstrasse 38, T\u00fcbingen 72076, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro","family":"Ortega","sequence":"additional","affiliation":[{"name":"GRASP Laboratory, Electrical and Systems Engineering Department, University of Pennsylvania, Philadelphia, PA 19104, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0140525X07000581","article-title":"A Framework for the Unification of the Behavioral Sciences","volume":"30","author":"Gintis","year":"2006","journal-title":"Behav. Brain Sci"},{"key":"ref_2","unstructured":"Russell, S., and Norvig, P. (1995). Artificial Intelligence: A Modern Approach, Prentice-Hall. [1st ed.]."},{"key":"ref_3","unstructured":"Kreps, D.M. (1988). Notes on the Theory of Choice, Westview Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.tics.2008.04.010","article-title":"Decision making, movement planning and statistical decision theory","volume":"12","author":"Trommershauser","year":"2008","journal-title":"Trends Cogn. Sci"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Braun, D.A., Nagengast, A.J., and Wolpert, D. (2011). Risk-sensitivity in sensorimotor control. Front. Hum. Neurosci, 5.","DOI":"10.3389\/fnhum.2011.00001"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1016\/j.conb.2012.05.003","article-title":"Motor control is decision-making","volume":"22","author":"Wolpert","year":"2012","journal-title":"Curr. Opin. Neurobiol"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Fishburn, P. (1982). The Foundations of Expected Utility, D. Reidel Publishing.","DOI":"10.1007\/978-94-017-3329-8"},{"key":"ref_8","unstructured":"Neumann, J.V., and Morgenstern, O. (1944). Theory of Games and Economic Behavior, Princeton University Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1037\/h0042769","article-title":"Rational choice and the structure of the environment","volume":"63","author":"Simon","year":"1956","journal-title":"Psychol. Rev"},{"key":"ref_10","unstructured":"McGuire, C.B., and Radner, R. (1972). Decision and Organization, North Holland Pub. Co."},{"key":"ref_11","unstructured":"Simon, H. (1984). Models of Bounded Rationality, MIT Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1006\/game.1997.0585","article-title":"Rationality and Bounded Rationality","volume":"21","author":"Aumann","year":"1997","journal-title":"Games Econ. Behav"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rubinstein, A. (1998). Modeling bounded rationality, MIT Press.","DOI":"10.7551\/mitpress\/4702.001.0001"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1257\/000282803322655392","article-title":"Maps of Bounded Rationality: Psychology for Behavioral Economics","volume":"93","author":"Kahneman","year":"2003","journal-title":"Am. Econ. Rev"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1006\/game.1995.1023","article-title":"Quantal Response Equilibria for Normal Form Games","volume":"10","author":"McKelvey","year":"1995","journal-title":"Games Econ. Behav"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1023\/A:1009905800005","article-title":"Quantal Response Equilibria for Extensive Form Games","volume":"1","author":"Mckelvey","year":"1998","journal-title":"Exp. Econ"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Braha, D., Minai, A.A., and Bar-Yam, Y. (2006). Complex Engineered Systems, Springer.","DOI":"10.1007\/3-540-32834-3"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Spiegler, R. (2011). Bounded Rationality and Industrial Organization, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780195398717.001.0001"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1093\/jopart\/mug028","article-title":"Bounded Rationality Political Science: Lessons from Public Administration and Public Policy","volume":"13","author":"Jones","year":"2003","journal-title":"J. Public Adm. Res. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gigerenzer, G., and Selten, R. (2001). Bounded rationality: The adaptive toolbox, MIT Press.","DOI":"10.7551\/mitpress\/1654.001.0001"},{"key":"ref_21","unstructured":"Camerer, C. (2003). Behavioral Game Theory: Experiments in Strategic Interaction, Princeton University Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1037\/a0017187","article-title":"Rational adaptation under task and processing constraints: implications for testing theories of cognition and action","volume":"116","author":"Howes","year":"2009","journal-title":"Psychol. Rev"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1111\/j.1756-8765.2010.01125.x","article-title":"Identifying Optimum Performance Trade-Offs Using a Cognitively Bounded Rational Analysis Model of Discretionary Task Interleaving","volume":"3","author":"Janssen","year":"2011","journal-title":"Top. Cogn. Sci"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Lewis, R., Howes, A., and Singh, S. (2014). Computational rationality: Linking mechanism and behavior through bounded utility maximization. Top. Cogn. Sci, in press.","DOI":"10.1111\/tops.12086"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"42","DOI":"10.2307\/136022","article-title":"Information Processing and Bounded Rationality: A Survey","volume":"28","author":"Lipman","year":"1995","journal-title":"Can. J. Econ"},{"key":"ref_26","unstructured":"Russell, S. (1995, January 20\u201325). Rationality and Intelligence. Montreal, Canada."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1613\/jair.133","article-title":"Provably bounded-optimal agents","volume":"3","author":"Russell","year":"1995","journal-title":"J. Artif. Intell. Res"},{"key":"ref_28","unstructured":"Glimcher, P., Fehr, E., Camerer, C., and Poldrack, R. (2008). Neuroeconomics: Decision Making and the Brain, Elsevier Science."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Friston, K., Schwartenbeck, P., Fitzgerald, T., Moutoussis, M., Behrens, T., and Dolan, R.J. (2013). The anatomy of choice: Active inference and agency. Front. Hum. Neurosci, 7.","DOI":"10.3389\/fnhum.2013.00598"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Moss, S., and Rae, J. (1992). Artificial Intelligence and Economic Analysis: Prospects and Problems, Edward Elgar Publishing.","DOI":"10.4337\/9781782541769"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ortega, P., and Braun, D. (2010, January 5\u20138). A conversion between utility and information. Lugano, Switzerland.","DOI":"10.2991\/agi.2010.10"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Schmidhuber, J., Th\u00f3risson, K.R., and Looks, M. (2011). Artificial General Intelligence, Springer.","DOI":"10.1007\/978-3-642-22887-2"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Braun, D.A., Ortega, P.A., Theodorou, E., and Schaal, S. (2011, January 11\u201315). Path integral control and bounded rationality. Paris, France.","DOI":"10.1109\/ADPRL.2011.5967366"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ortega, P.A., and Braun, D.A. (2013). Thermodynamics as a theory of decision-making with information-processing costs. Proc. R. Soc. A, 469.","DOI":"10.1098\/rspa.2012.0683"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"036102","DOI":"10.1103\/PhysRevE.85.036102","article-title":"Hysteresis effects of changing parameters of noncooperative games","volume":"85","author":"Wolpert","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_36","unstructured":"Luce, R. (1959). Individual choice behavior, Wiley."},{"key":"ref_37","unstructured":"Zarembka, P. (1974). Frontiers in Econometrics, Academic Press."},{"key":"ref_38","unstructured":"Meginnis, J. (1976). 1976 Proceedings of the American Statistical Association, Business and Economic Statistics Section, American Statistical Association."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1006\/game.1993.1021","article-title":"Learning mixed equilibria","volume":"5","author":"Fudenberg","year":"1993","journal-title":"Games Econ. Behav"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sutton, R., and Barto, A. (1998). Reinforcement Learning: An Introduction, MIT Press.","DOI":"10.1109\/TNN.1998.712192"},{"key":"ref_41","unstructured":"Luce, R. (2000). Utility of gains and losses: Measurement-theoretical and experimental approaches, Erlbaum."},{"key":"ref_42","unstructured":"Train, K. (2009). Discrete Choice Methods with Simulation, Cambridge University Press. [2nd ed.]."},{"key":"ref_43","unstructured":"Toussaint, M., Harmeling, S., and Storkey, A. (2006). Probabilistic inference for solving (PO)MDPs, University of Edinburgh."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1613\/jair.3062","article-title":"A minimum relative entropy principle for learning and acting","volume":"38","author":"Ortega","year":"2010","journal-title":"J. Artif. Intell. Res"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1038\/nrn2787","article-title":"The free-energy principle: A unified brain theory?","volume":"11","author":"Friston","year":"2010","journal-title":"Nat. Rev. Neurosci"},{"key":"ref_46","unstructured":"Vassilis, H.T. (2011). Perception-reason-action cycle: Models, algorithms and systems, Springer."},{"key":"ref_47","first-page":"1","article-title":"Optimal control as a graphical model inference problem","volume":"1","author":"Kappen","year":"2012","journal-title":"Mach. Learn"},{"key":"ref_48","unstructured":"Vijayakumar, S., Rawlik, K., and Toussaint, M. (2012, January 9\u201313). On Stochastic Optimal Control and Reinforcement Learning by Approximate Inference. Sydney, Australia."},{"key":"ref_49","unstructured":"Ortega, P.A., and Braun, D.A. (July, January 30). Free Energy and the Generalized Optimality Equations for Sequential Decision Making. Edinburgh, Scotland."},{"key":"ref_50","first-page":"269","article-title":"Generalized Thompson sampling for sequential decision-making and causal inference","volume":"5","author":"Ortega","year":"2014","journal-title":"Complex Adap. Syst. Model"},{"key":"ref_51","unstructured":"Ortega, P.A., Braun, D.A., and Tishby, N. (June, January 31). Monte Carlo Methods for Exact & Efficient Solution of the Generalized Optimality Equations. Hong Kong, China."},{"key":"ref_52","unstructured":"Auer, P., Cesa-Bianchi, N., Freund, Y., and Schapire, R.E. (1995, January 23\u201325). Gambling in a rigged casino: The adversarial multi-armed bandit problem. Milwaukee, WI, USA."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1006\/jcss.1997.1504","article-title":"A Decision-theoretic Generalization of On-line Learning and an Application to Boosting","volume":"55","author":"Freund","year":"1997","journal-title":"J. Comput. Syst. Sci"},{"key":"ref_54","unstructured":"Feynman, R.P. (1996). The Feynman Lectures on Computation, Addison-Wesley."},{"key":"ref_55","unstructured":"Fudenberg, D., and Levine, D. (1998). The Theory of Learning in Games, MIT Press."},{"key":"ref_56","unstructured":"Noam, N., Roughgarden, T., \u00c9va, T., and Vazirani, V. (2007). Algorithmic Game Theory, Cambridge University Press."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1016\/0165-1889(94)00819-4","article-title":"Consistency and cautious fictitious play","volume":"19","author":"Fudenberg","year":"1995","journal-title":"J. Econ. Dyn. 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