{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:07:26Z","timestamp":1760238446262,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,3]],"date-time":"2020-08-03T00:00:00Z","timestamp":1596412800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Disagreement is an essential element of science and life in general. The language of probabilities and statistics is often used to describe disagreements quantitatively. In practice, however, we want much more than that. We want disagreements to be resolved. This leaves us with a substantial knowledge gap, which is often perceived as a lack of practical intuition regarding probabilistic and statistical concepts. Here, we propose to address disagreements using the methods of financial economics. In particular, we show how a large class of disagreements can be transformed into investment opportunities. The expected financial performance of such investments quantifies the amount of disagreement in a tangible way. This provides intuition for statistical concepts such as the R\u00e9nyi divergence, which becomes connected to the financial performance of optimized investments. Investment optimization takes into account individual opinions as well as attitudes towards risk. The result is a market-like social mechanism by which funds flow naturally to support a more accurate view. Such social mechanisms can help us with difficult disagreements (e.g., financial arguments concerning the future climate). In terms of scientific validation, we used the findings of independent neurophysiological experiments as well as our own research on the equity premium.<\/jats:p>","DOI":"10.3390\/e22080860","type":"journal-article","created":{"date-parts":[[2020,8,3]],"date-time":"2020-08-03T06:16:47Z","timestamp":1596435407000},"page":"860","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Economics of Disagreement\u2014Financial Intuition for the R\u00e9nyi Divergence"],"prefix":"10.3390","volume":"22","author":[{"given":"Andrei N.","family":"Soklakov","sequence":"first","affiliation":[{"name":"Strategic Development, Deutsche Bank, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_2","unstructured":"R\u00e9nyi, A. (1961). On measures of entropy and information. Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability, University of California Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1002\/j.1538-7305.1956.tb03809.x","article-title":"A New Interpretation of Information Rate","volume":"2","author":"Kelly","year":"1956","journal-title":"Bell Syst. Tech. J."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"MacLean, L.C., Thorp, E.O., and Ziemba, W.T. (2011). The Kelly Capital Growth Investment Criterion, World Scientific.","DOI":"10.1142\/9789814293501"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2493","DOI":"10.1073\/pnas.68.10.2493","article-title":"The Fallacy of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling","volume":"68","author":"Samuelson","year":"1971","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0378-4266(79)90023-2","article-title":"Why we should not make mean log of wealth big though years to act are long","volume":"3","author":"Samuelson","year":"1979","journal-title":"J. Bank. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.physa.2006.01.012","article-title":"Interpretations of R\u00e9nyi entropies and divergences","volume":"365","year":"2006","journal-title":"Physica A"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnwald, P.D. (2007). The Minimum Description Length Principle, The MIT Press.","DOI":"10.7551\/mitpress\/4643.001.0001"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Shayevitz, O. (August, January 31). On R\u00e9nyi measures and hypothesis testing. Proceedings of the 2011 IEEE International Symposium on Information Theory Proceedings, St. Petersburg, Russia.","DOI":"10.1109\/ISIT.2011.6034266"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1038\/nature05852","article-title":"Probabilistic reasoning by neurons","volume":"447","author":"Yang","year":"2007","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/j.neuron.2015.01.007","article-title":"A neural implementation of Wald\u2019s sequential probability ratio test","volume":"85","author":"Kira","year":"2015","journal-title":"Neuron"},{"key":"ref_12","unstructured":"The actual returns are random but, in the long run, Bob can hope to achieve the expected growth of Figure 1B almost surely by simple repetition, i.e., by reinvesting the actual returns from one run of the game into the next. This is, of course, assuming that Bob\u2019s belief b turns out to be correct. In general, the long-term realized returns will follow an instance of Equation (11) where p is chosen to coincide with the correct (i.e., the realized) distribution."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Soklakov, A. (2015). Model Risk Analysis via Investment Structuring. arXiv.","DOI":"10.2139\/ssrn.2639369"},{"key":"ref_14","unstructured":"von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behaviour, Princeton University Press."},{"key":"ref_15","first-page":"81","article-title":"Elasticity theory of structuring","volume":"December","author":"Soklakov","year":"2016","journal-title":"Risk"},{"key":"ref_16","first-page":"175","article-title":"Specimen Theoriae Nova de Mensura Sortis","volume":"5","author":"Bernoulli","year":"1738","journal-title":"Commentarii Academiae Scientiarum Imperialis Petropolitanae"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mehra, R. (2008). Handbook of the Equity Risk Premoum, Elsevier.","DOI":"10.1016\/B978-044450899-7.50004-X"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Even a single strategy can have multiple (simultaneous) implementations within the neocortex (see Hawkins, J., Lewis, M., Klukas, M., Purdy, S.; Ahmad, S. A framework for intelligence and cortical function based on grid cells in the neocortex. Front. Neural Circuits 2019, 12, 121).","DOI":"10.3389\/fncir.2018.00121"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Damodaran, A. (2015, April 16). Equity Risk Premiums (ERP): Determinants, Estimation and Implications\u2014The 2014 Edition SSRN-id2409198, updated annually since 2008. Available online: http:\/\/people.stern.nyu.edu\/adamodar\/.","DOI":"10.2139\/ssrn.2409198"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0304-3932(85)90061-3","article-title":"The equity premium: A puzzle","volume":"15","author":"Mehra","year":"1985","journal-title":"J. Monet. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1007\/BF00271862","article-title":"Memory related motor planning activity in posterior parietal cortex of macaque","volume":"70","author":"Gnadt","year":"1988","journal-title":"Exp. Brain Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1574","DOI":"10.1152\/jn.1997.78.3.1574","article-title":"Responces of intraparietal neurons to saccadic targets and visual distractors","volume":"78","author":"Platt","year":"1997","journal-title":"J. Neurophysiol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1146\/annurev.neuro.22.1.319","article-title":"Space and attention in parietal cortex","volume":"22","author":"Colby","year":"1999","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3656","DOI":"10.1523\/JNEUROSCI.5074-05.2006","article-title":"Activity in the lateral intraparietal area predicts the goal and latency of saccades in a free-viewing visual search task","volume":"26","author":"Ipta","year":"2006","journal-title":"J. Neurosci."},{"key":"ref_25","first-page":"115","article-title":"Bayesian lessons for payout structuring","volume":"September","author":"Soklakov","year":"2011","journal-title":"Risk"},{"key":"ref_26","first-page":"85","article-title":"Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten","volume":"8","year":"1963","journal-title":"Magyar. Tud. Akad. Mat. Kutato Int. Kozl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1143\/JPSJ.18.328","article-title":"Markov processes and the H-theorem","volume":"18","author":"Morimoto","year":"1963","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1111\/j.2517-6161.1966.tb00626.x","article-title":"A general class of coefficients of divergence of one distribution from another","volume":"28","author":"Ali","year":"1966","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bleuler, C., Lapidoth, A., and Pfister, C. (2019). Gambling and R\u00e9nyi divergence. arXiv.","DOI":"10.1109\/ISIT.2019.8849800"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Bleuler, C., Lapidoth, A., and Pfister, C. (2020). Conditional R\u00e9nyi divergences and horse betting. Entropy, 22.","DOI":"10.3390\/e22030316"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/860\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:53:53Z","timestamp":1760176433000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/860"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,3]]},"references-count":30,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["e22080860"],"URL":"https:\/\/doi.org\/10.3390\/e22080860","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,8,3]]}}}