{"id":580,"date":"2010-01-20T09:33:59","date_gmt":"2010-01-20T14:33:59","guid":{"rendered":"http:\/\/2d823b65bb.nxcli.io\/?p=580"},"modified":"2018-12-30T12:00:40","modified_gmt":"2018-12-30T17:00:40","slug":"duel-curious-mathematical-puzzle","status":"publish","type":"post","link":"https:\/\/www.robweir.com\/blog\/2010\/01\/duel-curious-mathematical-puzzle.html","title":{"rendered":"The Duel: A curious mathematical puzzle"},"content":{"rendered":"<p>Captain Galaxy and Commander Glarcon are locked in mortal combat.\u00a0\u00a0 Each mans a battle tank armed with N photonic missiles which move at the speed of light. \u00a0 They move toward each other at constant velocity=v on a 1-dimensional track, unable to stop or reverse direction.\u00a0 Assume v &lt;&lt; c.\u00a0 The probability of scoring a kill with a missile is described by a function f(d) which monotonically increases from 0 to 1 as the distance between the tanks decreases from infinity to 0.\u00a0 If the distance closes to 0 and no missiles are fired, both tanks are destroyed in the collision.\u00a0\u00a0\u00a0 Assume each combatant attempts to maximize their own probability of survival.<\/p>\n<p><a href=\"https:\/\/2d823b65bb.nxcli.io\/blog\/wp-content\/uploads\/2010\/01\/duel.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-584 alignnone\" title=\"The Duel\" src=\"https:\/\/2d823b65bb.nxcli.io\/blog\/wp-content\/uploads\/2010\/01\/duel.png\" alt=\"\" width=\"557\" height=\"127\" srcset=\"https:\/\/www.robweir.com\/blog\/wp-content\/uploads\/2010\/01\/duel.png 557w, https:\/\/www.robweir.com\/blog\/wp-content\/uploads\/2010\/01\/duel-300x68.png 300w\" sizes=\"auto, (max-width: 557px) 100vw, 557px\" \/><\/a><\/p>\n<p>Note that this is not strictly a zero-sum game, since it is possible for neither player to survive.\u00a0 But it is impossible for both to survive.<\/p>\n<p>The state of the game is thus described by three variables:<\/p>\n<ul>\n<li>d=distance between the players<\/li>\n<li>N1= number of own missiles remaining<\/li>\n<li>N2= number of opponent&#8217;s missiles remaining<\/li>\n<\/ul>\n<p>A strategy S(d,N1,N2) would describe a combatant&#8217; actions (shoot or don&#8217;t shoot) for all possible states.<\/p>\n<ol>\n<li>If each player has exactly one missile what is the optimal strategy?\u00a0 Clearly, if the first player shoots and misses, the 2nd will win by waiting for d to approach 0 and then make a last minute shot.<\/li>\n<li>What if each player has exactly two missiles?<\/li>\n<li>What if each player has N missiles?<\/li>\n<\/ol>\n<p>It may simplify the problem to assume f(d) is proportionate to 1\/d or 1\/d^2 and then solve the general case.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Captain Galaxy and Commander Glarcon are locked in mortal combat.\u00a0\u00a0 Each mans a battle tank armed with N photonic missiles which move at the speed of light. \u00a0 They move toward each other at constant velocity=v on a 1-dimensional track, unable to stop or reverse direction.\u00a0 Assume v &lt;&lt; c.\u00a0 The probability of scoring a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[178],"tags":[108,180,181,179],"class_list":{"0":"post-580","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-puzzles","7":"tag-add-new-tag","8":"tag-game-theory","9":"tag-math","10":"tag-probability","11":"entry"},"_links":{"self":[{"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/posts\/580","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/comments?post=580"}],"version-history":[{"count":8,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/posts\/580\/revisions"}],"predecessor-version":[{"id":2687,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/posts\/580\/revisions\/2687"}],"wp:attachment":[{"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/media?parent=580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/categories?post=580"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.robweir.com\/blog\/wp-json\/wp\/v2\/tags?post=580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}