{"id":1215,"date":"2023-06-03T04:45:09","date_gmt":"2023-06-02T19:45:09","guid":{"rendered":"https:\/\/saraheee.com\/?p=1215"},"modified":"2023-07-02T22:52:24","modified_gmt":"2023-07-02T13:52:24","slug":"game-theory-8-chap14-bayesian-nash-equilibrium-example","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/06\/game-theory-8-chap14-bayesian-nash-equilibrium-example\/","title":{"rendered":"Game Theory 8 \u2013 chap14. Bayesian Nash equilibrium example"},"content":{"rendered":"<p><mark style=\"background-color:var(--global-color-10)\" class=\"has-inline-color\">EXAMPLE 4.7 (PUBLIC GOOD PROVISION, EXERCISE 2 ON PAGE 355)<\/mark>. <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Two players simultaneously and independently decide how much to contribute to a public good<\/li>\n\n\n\n<li>If player 1 contributes \\(x_{1}\\) and player 2 contributes \\(x_{2}\\), then the value of the public good is<\/li>\n<\/ul>\n\n\n\n<p>\\(2(x_{1} + x_{2} + x_{1}x_{2})\\)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Player 1 must pay a cost \\(x_{1}^{2}\\) of contributing \\(x_{1}\\). Player 2 must pay a cost \\(tx_{2}^{2}\\) of contributing \\(x_{2}\\), where the realization of t \u2208 {2, 3} is player 2&#8217;s private information. Player 1 only knows that each type is equally likely.<\/li>\n<\/ul>\n\n\n\n<p>Compute the Bayesian Nash equilibrium of this game<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><\/td><td>excludability<\/td><td>less excludable<\/td><\/tr><tr><td>rivalry<\/td><td>private goods<\/td><td>common resource<\/td><\/tr><tr><td>less rival<\/td><td>club goods<\/td><td>public goods<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-global-color-15-color\">when t = 2,<br>\\(2(x_{1} + x_{2} + x_{1}x_{2}) &#8211; 2x_{2}^{2}\\) \u21d2 (differentiate over \\(x_{2}\\)) \\(2 + 2x_{1} &#8211; 4x_{2}\\) = 0<br>\u2234 \\(x_{2}^{L} = \\frac{1+x_{1}}{2}\\)<\/mark><\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-global-color-15-color\">when t = 3,<br>\\(x_{3}^{H} = \\frac{1+x_{1}}{3}\\)<\/mark><\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-global-color-15-color\">player 2,<br>\\(\\frac{1}{2}(2(x_{1} + x_{2}^{L} + x_{1}x_{2}^{L})) + \\frac{1}{L}(2(x_{1} + x_{2}^{H} + x_{1}x_{2}^{H})) &#8211; x_{1}^{2}\\)<br>\u21d2 (differentiate over \\(x_{1}\\)) \\(1 + x_{2}^{L} + 1 + x_{2}^{H} &#8211; 2x_{1}\\) = 0<br>\\(x_{1} = 1 + \\frac{1}{2}(x_{2}^{L} + x_{2}^{H}\\)<\/mark><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x) = (1+x)\/2, f(x) = (1+x)\/3\nx-axis: x1, y-axis: x2<\/code><\/pre>\n\n\n\n<p><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Reference: Chang-Koo Chi, (30\/50) Game Theory and Applications 8 \u2013 \bBayesian Nash equilibrium example, Jul 10, 2020,&nbsp;<a href=\"https:\/\/youtu.be\/l3HqFpbyifw\" rel=\"noopener\">https:\/\/youtu.be\/l3HqFpbyifw<\/a><\/li>\n<\/ul>\n\n\n\n<p><\/p>","protected":false},"excerpt":{"rendered":"<p>Learn an example of a Bayesian Nash equilibrium.<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[74,76,4,73],"class_list":["post-1215","post","type-post","status-publish","format-standard","hentry","category-game-theory-and-applications","tag-bayesian-nash-equilibrium","tag-bne","tag-game-theory","tag-jun-3-2023"],"_links":{"self":[{"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/posts\/1215"}],"collection":[{"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/comments?post=1215"}],"version-history":[{"count":7,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/posts\/1215\/revisions"}],"predecessor-version":[{"id":1677,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/posts\/1215\/revisions\/1677"}],"wp:attachment":[{"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/media?parent=1215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/categories?post=1215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/saraheee.com\/ko\/wp-json\/wp\/v2\/tags?post=1215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}