{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:56:21Z","timestamp":1760234181536,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T00:00:00Z","timestamp":1618790400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP16K01408"],"award-info":[{"award-number":["JP16K01408"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The frequency of occurrence of step length in the migratory behaviour of various organisms, including humans, is characterized by the power law distribution. This pattern of behaviour is known as the L\u00e9vy walk, and the reason for this phenomenon has been investigated extensively. Especially in humans, one possibility might be that this pattern reflects the change in self-confidence in one\u2019s chosen behaviour. We used simulations to demonstrate that active assumptions cause changes in the confidence level in one\u2019s choice under a situation of lack of information. More specifically, we presented an algorithm that introduced the effects of learning and forgetting into Bayesian inference, and simulated an imitation game in which two decision-making agents incorporating the algorithm estimated each other\u2019s internal models. For forgetting without learning, each agents\u2019 confidence levels in their own estimation remained low owing to a lack of information about the counterpart, and the agents changed their hypotheses about the opponent frequently, and the frequency distribution of the duration of the hypotheses followed an exponential distribution for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and exponential distributions universally turned into power law distribution.<\/jats:p>","DOI":"10.3390\/sym13040718","type":"journal-article","created":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T21:59:49Z","timestamp":1618869589000},"page":"718","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8442-836X","authenticated-orcid":false,"given":"Shuji","family":"Shinohara","sequence":"first","affiliation":[{"name":"Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan"}]},{"given":"Nobuhito","family":"Manome","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan"},{"name":"SoftBank Robotics Group Corp, Department of Research and Development, Tokyo 105-0021, Japan"}]},{"given":"Yoshihiro","family":"Nakajima","sequence":"additional","affiliation":[{"name":"Graduate School of Economics, Osaka City University, Osaka 558-8585, Japan"}]},{"given":"Yukio Pegio","family":"Gunji","sequence":"additional","affiliation":[{"name":"Department of Intermedia Art and Science, School of Fundamental Science and Technology, Waseda University, Tokyo 169-8555, Japan"}]},{"given":"Toru","family":"Moriyama","sequence":"additional","affiliation":[{"name":"Faculty of Textile Science, Shinshu University, Ueda 386-8567, Japan"}]},{"given":"Hiroshi","family":"Okamoto","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan"}]},{"given":"Shunji","family":"Mitsuyoshi","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4691-6394","authenticated-orcid":false,"given":"Ung-il","family":"Chung","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1038\/nature11098","article-title":"Generalized L\u00e9vy walks and the role of chemokines in migration of effector CD8+ T cells","volume":"486","author":"Harris","year":"2012","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8396","DOI":"10.1038\/ncomms9396","article-title":"Swarming bacteria migrate by L\u00e9vy Walk","volume":"6","author":"Ariel","year":"2015","journal-title":"Nat. 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