{"id":79,"date":"2021-04-28T23:14:50","date_gmt":"2021-04-28T23:14:50","guid":{"rendered":"https:\/\/moralesmendozar.com\/dir\/?p=79"},"modified":"2021-04-28T23:25:38","modified_gmt":"2021-04-28T23:25:38","slug":"maximization","status":"publish","type":"post","link":"https:\/\/moralesmendozar.com\/dir\/2021\/04\/28\/maximization\/","title":{"rendered":"Maximization"},"content":{"rendered":"\n

Hello! Today I want to talk about maximization, and I will explain why.<\/p>\n\n\n\n

As part of my research, I have been working with networks and with a relatively simple maximization problem, in order to study how information flows across agents, namely, how can high-level information agents affect low-level information agents, as well as if there is any externality of heterogenous communities. This could be relevant because of policy implications, such as affordable rent ideas: allowing (or forcing) the existence of cheaper living units in rich neighborhoods. The main driver would be network effects.<\/p>\n\n\n\n

Now, I want to talk about the maximization problem I faced, which in my opinion is usually overlooked in the textbooks, and that is not trivial how to correctly formulate. Basically I had a problem like the one described in the following picture. Suppose you know $x_2$, but neither $x_1$ or $x_3$, and you need to maximize your choice of investment with that information. How do you do this? Suppose $p_1$ represents the probability that $x_1 = F$, and that $p_3$ represents the probability that $x_3 = F$. Furthermore, suppose the problem is symmetric, to make it simple.<\/p>\n\n\n\n

\"Decision<\/figure><\/div>\n\n\n\n

One way to do this would be to solve<\/p>\n\n\n\n

$$ \\max_{decision1} p_1 ( p_3 * f(decision_1 |x_1 = F) + (1-p_3) * f(decision_1|x_1 = F) ) + (1-p_1) ( p_3 * f(decision_1 |x_1 = T) + (1-p_3)* f(decision_1 |x_1 = T)),$$<\/p>\n\n\n\n

but another way to solve it would be to do instead:<\/p>\n\n\n\n

$$ \\max_{decision1} p_3 ( p_1 * f(decision_1 |x_3 = F) + (1-p_1) * f(decision_1|x_3 = F) ) + (1-p_3) ( p_1 * f(decision_1 |x_3 = T) + (1-p_1)* f(decision_1 |x_3 = T)),$$<\/p>\n\n\n\n

Perhaps you find the solution trivial, but it’s not. Think about it this way: which variable should you condition on? If you frame it like that, it seems obvious that one should condition on $x_1$, and that turns out to be the right answer. I write about this because this was a problem that got me stuck for a while. It is the kind of things that one learns at school without really asking or wondering why, but that faced with the problem, it becomes clear.<\/p>\n\n\n\n

I hope you find this useful. I finish by including a picture of a recent hike, this time in Mineral del Chico.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Hello! Today I want to talk about maximization, and I will explain why. As part of my research, I have been working with networks and with a relatively simple maximization problem, in order to study how information flows across agents, namely, how can high-level information agents affect low-level information agents, as well as if there ….  Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/posts\/79"}],"collection":[{"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/comments?post=79"}],"version-history":[{"count":5,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/posts\/79\/revisions"}],"predecessor-version":[{"id":89,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/posts\/79\/revisions\/89"}],"wp:attachment":[{"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/media?parent=79"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/categories?post=79"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/moralesmendozar.com\/dir\/wp-json\/wp\/v2\/tags?post=79"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}