{"id":615,"date":"2008-03-19T21:16:34","date_gmt":"2008-03-19T21:16:34","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2008\/03\/19\/introducing-game-theory\/"},"modified":"2008-03-19T21:16:34","modified_gmt":"2008-03-19T21:16:34","slug":"introducing-game-theory","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2008\/03\/19\/introducing-game-theory\/","title":{"rendered":"Introducing Game Theory"},"content":{"rendered":"

Lots of people wanted game theory, so game theory it is. The logical first question: what is<\/em> game theory?<\/p>\n

Game theory is typical of math. What mathematicians like to do is reduce
\nthings to fundamental abstract structures or systems, and understand them in
\nterms of the abstraction. So game theory studies an abstraction of games – and
\nbecause of the level of abstraction, it turns out be be applicable to a wide
\nvariety of things besides what you might typically think of as games.<\/p>\n

Game theory starts with the fundamental idea of a game. What is
\na game?<\/p>\n

<\/p>\n

A game consists of a system with at least two agents (that is,
\nentities that can perform actions), who each have their own
\ninterests which they are trying to satisfy. They have a situation,
\nwhich consists of the set of things that they have, and the things that
\nthey want. They have a set of rules, which describes what actions they
\ncan take in the situation, and when they can take them.<\/p>\n

Game theory looks at that, and tries to understand how the
\nagents will interact. You can study it from many different points
\nof view, and with many different objectives.<\/p>\n

For example, you can look at it from a very “pure” perspective, and
\ntry to find optimal strategies for a particular kind of game. For example,
\nif you analyze the game of Nim, you can find that there’s a guaranteed
\nwinning strategy for player two. (We’ll look at Nim, and why it’s always winnable by player two in a later post.)<\/p>\n

You can also use game theory for things that seem very non-game like.
\nBasically, any situation in which you have multiple interacting agents with
\ndistinct, possibly conflicting goals, you can analyze it using game
\ntheory.<\/p>\n

A classic example of this is the prisoner’s dilemma<\/em>. In the
\nprisoners dilemma, you have two criminals who’ve been arrested for a murder.
\nThe two criminals (now prisoners) are the agents. The police know that they
\ndid it; but they don’t have enough evidence to convict them of murder, only of
\na lesser charge. So the police want to get the prisoners to rat on each other.
\nYou end up with the following situation:<\/p>\n

    \n
  1. If neither prisoner rats on the other, they’ll both get off with a very light sentence of 6 months in jail. <\/li>\n
  2. If one prisoners rats on the other, that prisoner goes free,
    \nand the other one gets a life sentence.<\/li>\n
  3. If both prisoners rat on each other, they each get 10 years in
    \njail.<\/li>\n<\/ol>\n

    If you look at this in the abstract, it’s obvious what the prisoners
    \nshould do. If they both keep their mouths shut, then they’ll both get off
    \neasy. But if you look at it in terms of cost\/benefit for one<\/em> of them,
    \nyou get a very different answer: for a prisoner acting only in his own self-interest, the correct choice is to rat on his partner. Game theory
    \nlooks at it from that latter point of view: each agent is concerned
    \nonly<\/em> with maximizing the benefit\/minimizing the penalty for
    \nthemselves.<\/p>\n

    To see why that works, just look at it from of one of the prisoners. We’ll call him Adam, and we’ll call his partner Bert. Adam doesn’t know what Bert is going to do. So he looks at the options.<\/p>\n

    If Bert keeps his mouth shut, Adam has two choices: he can keep quiet, or
    \nhe can rat on Bert. If he keeps quiet, he’ll go to jail for six months. If he
    \nrats, he’ll walk. Clearly, in this case, ratting on Bert is the best choice
    \nfor Adam.<\/p>\n

    If Bert rats, Adam has the same two choices. If he keeps quiet,
    \nhe goes to jail for life. If he rats, then he goes to jail for ten years. Once again, clearly, the better choice for Adam is to rat.<\/p>\n

    So each prisoner maximizes his own benefits by testifying, even though
    \nthat means that they’ll both wind up spending 10 years in jail, when they
    \ncould have gotten off with 1 year each by cooperating. By trying to always
    \nmaximize their own benefit, both suffer.<\/li>\n

    Game theory works out a kind of categorization of games, based on
    \nhow many agents, whether they can cooperate or communicate, whether
    \nthey move simultaneously or take turns, etc. It also categorizes
    \nbased on the kinds of benefit\/penalty relationships that define the
    \npossible outcomes of the game. For games of different forms, it
    \ndescribes strategies, equilibriums, and tipping points in those games.<\/p>\n

    It turns out to be an incredibly useful framework. It’s used in computer
    \nscience for things like protocol design; it’s used in economics for models of
    \nmarkets; it’s used in negotiation studies; it’s used in sociology. The basic
    \nidea of multiple parties acting in their own interest is a fundamental
    \nframework for understanding almost anything involving multiple people or
    \nmultiple interacting processes of any kind.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Lots of people wanted game theory, so game theory it is. The logical first question: what is game theory? Game theory is typical of math. What mathematicians like to do is reduce things to fundamental abstract structures or systems, and understand them in terms of the abstraction. So game theory studies an abstraction of games […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[88],"tags":[],"class_list":["post-615","post","type-post","status-publish","format-standard","hentry","category-game-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9V","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/615","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=615"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/615\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=615"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=615"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}