Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments
We show that a Bayesian game where the type space of each agent is a bounded set of <i>m</i>-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 <i>m</i> basic game...
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| Format: | Article |
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MDPI AG
2018-10-01
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| Series: | Games |
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| Online Access: | https://www.mdpi.com/2073-4336/9/4/85 |
| _version_ | 1818314049531674624 |
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| author | Abbas Edalat Samira Hossein Ghorban Ali Ghoroghi |
| author_facet | Abbas Edalat Samira Hossein Ghorban Ali Ghoroghi |
| author_sort | Abbas Edalat |
| collection | DOAJ |
| description | We show that a Bayesian game where the type space of each agent is a bounded set of <i>m</i>-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 <i>m</i> basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-dimensional simplex. This class of <i>m</i>-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’s 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. |
| first_indexed | 2024-12-13T08:43:28Z |
| format | Article |
| id | doaj.art-2988148179d04406b63e2e8b5000c458 |
| institution | Directory Open Access Journal |
| issn | 2073-4336 |
| language | English |
| last_indexed | 2024-12-13T08:43:28Z |
| publishDate | 2018-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Games |
| spelling | doaj.art-2988148179d04406b63e2e8b5000c4582022-12-21T23:53:29ZengMDPI AGGames2073-43362018-10-01948510.3390/g9040085g9040085Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-EnvironmentsAbbas Edalat0Samira Hossein Ghorban1Ali Ghoroghi2Department of Computing, Imperial College London, London SW7 2RH, UKSchool of Computer Science, Institute for Research in Fundamental Sciences (IPM), Lavasani Av., P.O. Box 19395-5746, Tehran, IranDepartment of Computing, Imperial College London, London SW7 2RH, UKWe show that a Bayesian game where the type space of each agent is a bounded set of <i>m</i>-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 <i>m</i> basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-dimensional simplex. This class of <i>m</i>-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’s 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.https://www.mdpi.com/2073-4336/9/4/85Multidimensional Bayesian gamemultigametype space partitionPrisoner’s DilemmaTrust Game |
| spellingShingle | Abbas Edalat Samira Hossein Ghorban Ali Ghoroghi Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments Games Multidimensional Bayesian game multigame type space partition Prisoner’s Dilemma Trust Game |
| title | Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments |
| title_full | Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments |
| title_fullStr | Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments |
| title_full_unstemmed | Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments |
| title_short | Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments |
| title_sort | ex post nash equilibrium in linear bayesian games for decision making in multi environments |
| topic | Multidimensional Bayesian game multigame type space partition Prisoner’s Dilemma Trust Game |
| url | https://www.mdpi.com/2073-4336/9/4/85 |
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