{"id":326,"date":"2011-03-01T06:47:27","date_gmt":"2011-02-28T22:47:27","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=326"},"modified":"2024-02-16T21:13:46","modified_gmt":"2024-02-16T13:13:46","slug":"dinosaur-dodger-tips-and-strategy","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/03\/dinosaur-dodger-tips-and-strategy\/","title":{"rendered":"Dinosaur Dodger Tips and Strategy"},"content":{"rendered":"<p>I recently uploaded a game I call &#8220;<a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/dinosaur-dodger.html\">Dinosaur Dodger<\/a>&#8220;. It&#8217;s based on an interesting paradox I read about on <a href=\"http:\/\/www.thebigquestions.com\/2010\/05\/20\/the-absent-minded-driver\/\" target=\"_blank\" rel=\"noopener\">this blog<\/a>, by an economist who has authored a number of <a href=\"http:\/\/www.amazon.com\/gp\/redirect.html?ie=UTF8&amp;location=http%3A%2F%2Fwww.amazon.com%2Fs%3Fie%3DUTF8%26redirect%3Dtrue%26ref_%3Da9_sc_1%26keywords%3Dsteve%2520landsburg%2520economics%26qid%3D1298896287%26rh%3Di%253Aaps%252Ck%253Asteve%2520landsburg%2520economics&amp;tag=dmmgfk-20&amp;linkCode=ur2&amp;camp=1789&amp;creative=390957\" target=\"_blank\" rel=\"noopener\">good<\/a> <a href=\"http:\/\/www.amazon.com\/gp\/product\/0029177766?ie=UTF8&amp;tag=dmmgfk-20&amp;linkCode=as2&amp;camp=1789&amp;creative=390957&amp;creativeASIN=0029177766\" target=\"_blank\" rel=\"noopener\">books<\/a>. The paradox is called the &#8220;Paradox of the Absent Minded Driver&#8221;. It goes like this : imagine a driver, driving home along a highway. They need to take the second exit to get off, but for some reason they can&#8217;t recall which exit they are at when they get to an exit.<\/p>\n<figure style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" title=\"The Absent Minded Driver paradox, and Dinosaur Didger, involve finding the best strategy to get home on this track\" src=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/images\/Dinosaur-Dodger-Map.png\" alt=\"The Absent Minded Driver paradox, and Dinosaur Didger, involve finding the best strategy to get home on this track\" width=\"600\" height=\"212\" \/><figcaption class=\"wp-caption-text\">The Absent Minded Driver paradox, and Dinosaur Didger, involve finding the best strategy to get home on this track<\/figcaption><\/figure>\n<p><!--more-->Since the two exits are identical, and since the driver can never recall, when they reach an exit, whether they&#8217;ve already passed one, the only way to have any chance of getting home is to choose randomly whether or not to turn.<\/p>\n<p>If their probability of going straight is P, then, to get home,<\/p>\n<ul>\n<li>At the first exit, they must go straight &#8211; they have a probability of P of doing that.<\/li>\n<li>At the second exit, they must turn &#8211; they have a probability of 1-P of doing that.<\/li>\n<li>Therefore, their chance of getting home is P x (1-P) = P &#8211; P<sup>2<\/sup><\/li>\n<\/ul>\n<p>To make this chance as big as possible, the driver decides to choose P=1\/2. A coin flip at each intersection. That&#8217;s still only a 1\/4 chance of getting home, but any other P is worse. For example, if they choose either P=1 (always go straight) or P=0 (always turn), their chance of getting home is zero &#8211; in the first case because they never turn, in the second case because they always turn too soon.<\/p>\n<p>Anyway, suppose the driver gets his coin ready, and starts to drive. Then he gets to an intersection, and thinks : &#8220;hang on, this is more likely to be the first intersection &#8211; I always reach the first intersection on every trip, but I only reach the second intersection on half my trips&#8221;. In fact, there&#8217;s a 1\/3 chance he&#8217;s at the second intersection, and a 2\/3 chance he&#8217;s at the first. So he says &#8220;All right, I&#8217;ll go straight with probability 3\/5, and only turn with probability 2\/5&#8221;. That way, his chance of getting home is 22\/75 &#8211; almost 30% instead of 25%.<\/p>\n<p>The paradox is this. He already knew, when he was starting his trip, that he would reach the first intersection &#8211; so arriving there gives him no new information! How could he come to a different logical conclusion?<\/p>\n<p>Now <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/dinosaur-dodger.html\">Dinosaur Dodger<\/a> is not based precisely on this paradox. Instead of one driver driving home with one strategy, there&#8217;s an explorer on a jungle track, receiving advice from a different player at each turn-off. This is quite a different conundrum from the Absent-Minded Driver Paradox. Since you are not alone in advising the explorer, your best strategy in Dinosaur Dodger depends on what everyone else is doing. Unfortunately, you don&#8217;t know exactly what that is &#8211; though perhaps you can get some clue from the <a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/dino-dodger-high-scores.html\">high scores table<\/a>.<\/p>\n<p>Suppose that, <em>on average<\/em>, the other drivers are going straight with probability Q. If you decide to go straight with probability P, then your chance of getting the explorer back home is (P + Q &#8211; 2PQ)\/(2-Q). If you can guess Q, then you can substitute Q and different values of P into that formula, to see how to get the most explorers home. I&#8217;ll give some examples so you can test if you are working this out properly.<\/p>\n<ul>\n<li>Suppose you think that Q=1\/4, and you try P=1\/4. Then the explorer gets home with probability (1\/4 + 1\/4 &#8211; 2 x 1\/4 x 1\/4) \/ (2 &#8211; 1\/4) = 3\/14, or only about 21%.<\/li>\n<li>On the other hand, if you think Q=1\/4 and you try P=3\/4, then the explorer gets home with probability (1\/4 + 3\/4 &#8211; 2 x 1\/4 x 3\/4) \/ (2 &#8211; 1\/4), which is 5\/14, or about 36%. Clearly, <strong><em>if<\/em><\/strong> you think Q=1\/4, it is better to choose P=3\/4 than P=1\/4.<\/li>\n<\/ul>\n<p>If you think Q=1\/2, then, amazingly, it doesn&#8217;t matter what you choose! This is because, when Q=1\/2, the explorer gets home with probability (P + 1\/2 &#8211; P)\/(3\/2), which is 1\/3 no matter what P is!<\/p>\n<p>Hmm&#8230; that&#8217;s probably more than enough advice from me about the best strategy to use in Dinosaur Dodger. I wouldn&#8217;t want to spoil the game! And I&#8217;ve given enough information already for a clever spark to figure out the best strategy all by him or herself.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I recently uploaded a game I call &#8220;Dinosaur Dodger&#8220;. It&#8217;s based on an interesting paradox I read about on this blog, by an economist who has authored a number of good books. The paradox is called the &#8220;Paradox of the Absent Minded Driver&#8221;. It goes like this : imagine a driver, driving home along a &hellip; <a href=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/03\/dinosaur-dodger-tips-and-strategy\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Dinosaur Dodger Tips and Strategy<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22],"tags":[218,217,219,17,108],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/326"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=326"}],"version-history":[{"count":2,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/326\/revisions"}],"predecessor-version":[{"id":1394,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/326\/revisions\/1394"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}