{"id":630,"date":"2022-10-05T01:46:23","date_gmt":"2022-10-05T01:46:23","guid":{"rendered":"http:\/\/morganredfield.com\/blog\/?p=630"},"modified":"2022-10-07T01:44:08","modified_gmt":"2022-10-07T01:44:08","slug":"the-paradox-of-the-two-envelopes","status":"publish","type":"post","link":"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/","title":{"rendered":"The Paradox of the Two Envelopes"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This problem comes from Chapter 4 of <a href=\"https:\/\/www.goodreads.com\/en\/book\/show\/11729796-an-introduction-to-decision-theory\" data-type=\"URL\" data-id=\"https:\/\/www.goodreads.com\/en\/book\/show\/11729796-an-introduction-to-decision-theory\">An Introduction to Decision Theory.<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose I show you two envelopes. I tell you that they both have money in them. One has twice as much as the other. You can choose one and keep it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Obviously you want the one that has more money. Knowing only what I&#8217;ve told you already, both envelopes seem equally good to you.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s say you choose the left envelope. I give it to you, but before you can open it I stop you and tell you this:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You&#8217;ve chosen one envelope. Do you want to open this one or switch to the other one?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You think about this, and come up with the following argument:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">I chose the left envelope, so let&#8217;s assume it has L dollars in it. I know that the other envelope has L\/2 dollars with 50% probability and 2L dollars with 50% probability. That makes the expected value of switching envelopes 1\/2<em>(L\/2) + 1\/2<\/em>(2L), or 5\/4 * L. So it does make sense for you to switch to the right envelope.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The decision matrix for this might look something like:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>R = L\/2<\/th><th>R = 2L<\/th><\/tr><\/thead><tbody><tr><td>Keep L<\/td><td>L<\/td><td>L<\/td><\/tr><tr><td>Switch R<\/td><td>L\/2<\/td><td>2L<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">But just as you go to open the right envelope, I ask you if you want to switch to the left. By the same argument you just made, the value of the left envelope is either R\/2 or 2R and it does make sense to switch.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Every time you&#8217;re just about to open an envelope, you convince yourself to switch. Back and forth forever.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The Real Answer<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Obviously the original reasoning doesn&#8217;t make sense, but teasing out why is tricky. In order to understand why it doesn&#8217;t make sense to switch, we need to think a bit more about how the problem is set up.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here&#8217;s what we know:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>L is the value of money in the left envelope<\/li><li>R is the value of money in the right envelope<\/li><li>R + L = V, a constant value that never changes after you&#8217;re presented with the envelopes<\/li><li>L = 2R or L = R\/2<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">There are actually two ways that I could have set up the envelopes before offering them to you.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Problem setup 1:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>choose the value in the left envelope<\/li><li>flip a coin<\/li><li>if tails, the right envelope will have half the value of the left<\/li><li>if heads, the right envelope will have twice the value<\/li><\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Problem setup 2:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>choose the total value of both envelopes<\/li><li>flip a coin<\/li><li>if tails, the left envelope has 1\/3 of the total value<\/li><li>if heads, the left envelope has 2\/3 of the total value<\/li><\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Both these formulations result in the same presentation of the problem, but they need different analysis to understand.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Problem Setup 1: L = 20<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s make things concrete to make it easier to follow. In problem setup one we start by choosing the value in the left envelope, so let&#8217;s set it to $20. Now the game tree for the problem looks like this:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Leq20.png\"><img loading=\"lazy\" decoding=\"async\" width=\"732\" height=\"441\" src=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Leq20.png\" alt=\"\" class=\"wp-image-631\" srcset=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Leq20.png 732w, https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Leq20-300x181.png 300w\" sizes=\"auto, (max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 984px) 61vw, (max-width: 1362px) 45vw, 600px\" \/><\/a><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">We&#8217;re showing the true game state here, but when you actually choose the Left envelope you don&#8217;t know what world you&#8217;re in. That means that, once you choose the left envelope, you could be in either of the &#8220;have 20&#8221; states shown in the game tree (marked in green). Let&#8217;s look at the decision matrix:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>R = L\/2<\/th><th>R = 2L<\/th><\/tr><\/thead><tbody><tr><td>Keep L<\/td><td>20<\/td><td>20<\/td><\/tr><tr><td>Switch R<\/td><td>10<\/td><td>40<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">In this case, the outcomes are which one of the two green-marked worlds you&#8217;re actually in. At the time you&#8217;re making this choice, you don&#8217;t know what L is. We&#8217;re filling in numbers from the &#8220;god-perspective&#8221; here.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now the math here matches what we saw at first. The benefit of switching is 10\/2 + 40\/2 = 25. The benefit of staying is 20. So we should switch.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Once you switch, you&#8217;re offered the same option. But in this case the decision matrix looks different. You don&#8217;t know which of the orange-marked R states you&#8217;re in, but they&#8217;re both different values:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>L = R\/2<\/th><th>L = 2R<\/th><\/tr><\/thead><tbody><tr><td>Keep R<\/td><td>40<\/td><td>10<\/td><\/tr><tr><td>Switch L<\/td><td>20<\/td><td>20<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Now it makes sense to keep R! The reason is that the &#8220;R&#8221; value is different depending on which world you&#8217;re in. That wasn&#8217;t true of L because of how the envelopes were filled originally.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With this problem setup, it makes sense to switch the first time. It doesn&#8217;t make sense to switch after that. There&#8217;s no infinite switching, and no paradox.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In fact, it only makes sense to switch the first time because we &#8220;know&#8221; that the Left envelope has a fixed amount of money in it (the $20 we started the problem with).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If instead we didn&#8217;t know if the fixed amount was in the left or the right, we would be equally happy keeping or switching. That&#8217;s because both options would give us a 50\/50 shot at 5\/4 the fixed value or the fixed value.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Problem Setup 2: V = 60<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In the second setup, the questioner choose the total value of both envelopes instead of choosing how much is in one envelope. Let&#8217;s set the total value to be $60. That means that one envelope will have $20 and the other will have $40, but we don&#8217;t know which is which.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The game tree for this problem is different than the last one:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png\"><img loading=\"lazy\" decoding=\"async\" width=\"732\" height=\"408\" src=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png\" alt=\"\" class=\"wp-image-632\" srcset=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png 732w, https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60-300x167.png 300w\" sizes=\"auto, (max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 984px) 61vw, (max-width: 1362px) 45vw, 600px\" \/><\/a><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Now assume that I choose the Left envelope, just like before. In this case I don&#8217;t know which of the two possible worlds I&#8217;m in. So I don&#8217;t know if I have $20 or $40. This makes the decision matrix look like this:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>R = L\/2<\/th><th>R = 2L<\/th><\/tr><\/thead><tbody><tr><td>Keep L<\/td><td>40<\/td><td>20<\/td><\/tr><tr><td>Switch R<\/td><td>20<\/td><td>40<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">In this case it&#8217;s very clear that the situation is symmetric from the start. The &#8220;L value&#8221; is different depending on whether you&#8217;re in the &#8220;R = L\/2&#8221; or &#8220;R = 2L&#8221; world. There&#8217;s no reason to switch even once.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The fundamental mistake<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The fundamental mistake that originally led to the idea of a paradox is this: we assumed that the amount of money in the left envelope that you chose was fixed.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is true that once you choose an envelope, there&#8217;s a certain unchanging amount of money in it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It&#8217;s also true that the other, unchosen, envelope either has twice or half the amount of money in it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is <em>not true<\/em> that the amount of money in your envelope is the same regardless of whether the other envelope has double or half the amount. You just don&#8217;t know which world you&#8217;re in (or which branch of the game tree). The expected value calculation we did at first used a fixed value for what was in your current envelope. The key is that if you&#8217;re making assumptions about what could be in the other envelope, then you need to update your assumption for what&#8217;s in your current envelope to match.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Resolving the two letter paradox<\/p>\n","protected":false},"author":1,"featured_media":632,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-630","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Paradox of the Two Envelopes - Theory and Practice<\/title>\n<meta name=\"description\" content=\"Resolving the two envelope paradox\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Resolving the two envelope paradox\" \/>\n<meta property=\"og:description\" content=\"Resolving the two envelope paradox\" \/>\n<meta property=\"og:url\" content=\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\" \/>\n<meta property=\"og:site_name\" content=\"Theory and Practice\" \/>\n<meta property=\"article:published_time\" content=\"2022-10-05T01:46:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-10-07T01:44:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png\" \/>\n\t<meta property=\"og:image:width\" content=\"732\" \/>\n\t<meta property=\"og:image:height\" content=\"408\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"morgan\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Resolving the two envelope paradox\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"morgan\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\"},\"author\":{\"name\":\"morgan\",\"@id\":\"https:\/\/morganredfield.com\/blog\/#\/schema\/person\/15fce334735dd60ce7bb5712c44a62dd\"},\"headline\":\"The Paradox of the Two Envelopes\",\"datePublished\":\"2022-10-05T01:46:23+00:00\",\"dateModified\":\"2022-10-07T01:44:08+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\"},\"wordCount\":1148,\"publisher\":{\"@id\":\"https:\/\/morganredfield.com\/blog\/#\/schema\/person\/15fce334735dd60ce7bb5712c44a62dd\"},\"image\":{\"@id\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/morganredfield.com\/blog\/wp-content\/uploads\/2022\/10\/twoLettersParadox_Veq60.png\",\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\",\"url\":\"https:\/\/morganredfield.com\/blog\/2022\/10\/the-paradox-of-the-two-envelopes\/\",\"name\":\"The Paradox of the Two Envelopes - 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