{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:24:01Z","timestamp":1772252641248,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T00:00:00Z","timestamp":1540339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We show that a Bayesian game where the type space of each agent is a bounded set of m-dimensional vectors with non-negative components and the utility of each agent depends linearly on its own type only is equivalent to a simultaneous competition in m basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the     ( m \u2212 1 )    -dimensional simplex. This class of m-dimensional Bayesian games, via their equivalence with uniform multigames, can model decision making in multi-environments in a variety of circumstances, including decision making in multi-markets and decision making when there are both material and social utilities for agents as in the Prisoner\u2019s Dilemma and the Trust Game. We show that, if a uniform multigame in which the action set of each agent consists of one Nash equilibrium inducing action per basic game has a pure ex post Nash equilibrium on the boundary of its type profile space, then it has a pure ex post Nash equilibrium on the whole type profile space. We then develop an algorithm, linear in the number of types of the agents in such a multigame, which tests if a pure ex post Nash equilibrium on the vertices of the type profile space can be extended to a pure ex post Nash equilibrium on the boundary of its type profile space in which case we obtain a pure ex post Nash equilibrium for the multigame.<\/jats:p>","DOI":"10.3390\/g9040085","type":"journal-article","created":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T10:40:48Z","timestamp":1540377648000},"page":"85","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments"],"prefix":"10.3390","volume":"9","author":[{"given":"Abbas","family":"Edalat","sequence":"first","affiliation":[{"name":"Department of Computing, Imperial College London, London SW7 2RH, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4147-3181","authenticated-orcid":false,"given":"Samira","family":"Hossein Ghorban","sequence":"additional","affiliation":[{"name":"School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Lavasani Av., P.O. Box 19395-5746, Tehran, Iran"}]},{"given":"Ali","family":"Ghoroghi","sequence":"additional","affiliation":[{"name":"Department of Computing, Imperial College London, London SW7 2RH, UK"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Krishna, V., and Perry, M. (1998). Efficient Mechanism Design, Department of Economics, The Pennsylvania State University. Manuscript.","DOI":"10.2139\/ssrn.64934"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lucier, B., and Borodin, A. (2010, January 17\u201319). Price of anarchy for greedy auctions. Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX, USA.","DOI":"10.1137\/1.9781611973075.46"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1561\/0400000045","article-title":"Bayesian mechanism design","volume":"8","author":"Hartline","year":"2013","journal-title":"Found. 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