Answers.Body: <p>Yes, there is at least one probabilistic version of minimax, which is called <a href="https://en.wikipedia.org/wiki/Expectiminimax" rel="nofollow noreferrer">expectiminimax</a>. In <a href="https://en.wikipedia.org/wiki/Expectiminimax" rel="nofollow noreferrer">expectiminimax</a>, in addition to <em>min</em> and <em>max</em> nodes, there are also <em>chance</em> nodes, which perform a weighted sum of the successors, so the probabilities associated with chance nodes must be known. Given that <a href="https://en.wikipedia.org/wiki/Expectiminimax" rel="nofollow noreferrer">expectiminimax</a> assumes the existence of random events (represented by the chance nodes), the decisions are thus based on <a href="https://en.wikipedia.org/wiki/Expected_value" rel="nofollow noreferrer">expected values</a>. </p>
  
  <p>Section 5.5 of the book <a href="https://www.cin.ufpe.br/~tfl2/artificial-intelligence-modern-approach.9780131038059.25368.pdf" rel="nofollow noreferrer">Artificial Intelligence: A Modern Approach</a> provides a description of the expectiminimax algorithm, which was introduced by <a href="http://www.aiai.ed.ac.uk/~dm/" rel="nofollow noreferrer">Donald Michie</a> in <a href="https://www.researchgate.net/publication/240013423_Game-playing_and_game-learning_automata" rel="nofollow noreferrer">Game-playing and game-learning automata</a> (1966). The paper <a href="http://www.inf.u-szeged.hu/actacybernetica/edb/vol18n2/pdf/Melko_2007_ActaCybernetica.pdf" rel="nofollow noreferrer">Optimal strategy in games with chance nodes</a> (2007) also gives a decent description of the expectiminimax algorithm.</p>
  
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Answers.CreationDate: 2019-12-20T15:41:12.423Z
Answers.LastActivityDate: 2019-12-20T15:56:27.007Z
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Body: <p>How would a probabilistic version of minimax work?</p>
  
  <p>For example, we may choose a move that could result in a very bad outcome, but that outcome might just be extremely unlikely so we might think it would be worth the risk.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-20T13:20:10.250Z
Comments.Text: You might want to look up 'Markov decision process' and how it is used in reinforcement learning. This is probabilistic minimax in essence.
Comments.UserDisplayName: Lustwelpintje
Tags: <minimax><expectiminimax>
Title: Is there a probabilistic version of minimax?
UserDisplayName: zooby
ThreadText: # Is there a probabilistic version of minimax?
  
  (Asked by: username_0 on 2019-12-20)
  
  How would a probabilistic version of minimax work?
  
  For example, we may choose a move that could result in a very bad outcome, but that outcome might just be extremely unlikely so we might think it would be worth the risk.
  
  **Tags:** minimax, expectiminimax
  > **Comments:**
  > **username_1**: You might want to look up 'Markov decision process' and how it is used in reinforcement learning. This is probabilistic minimax in essence.  
  
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  ## 1 answers
  
  ### Answer by username_2 on 2019-12-20. Score: 1 | Accepted answer
  
  Yes, there is at least one probabilistic version of minimax, which is called [expectiminimax](https://en.wikipedia.org/wiki/Expectiminimax). In [expectiminimax](https://en.wikipedia.org/wiki/Expectiminimax), in addition to *min* and *max* nodes, there are also *chance* nodes, which perform a weighted sum of the successors, so the probabilities associated with chance nodes must be known. Given that [expectiminimax](https://en.wikipedia.org/wiki/Expectiminimax) assumes the existence of random events (represented by the chance nodes), the decisions are thus based on [expected values](https://en.wikipedia.org/wiki/Expected_value).
  
  Section 5.5 of the book [Artificial Intelligence: A Modern Approach](https://www.cin.ufpe.br/~tfl2/artificial-intelligence-modern-approach.9780131038059.25368.pdf) provides a description of the expectiminimax algorithm, which was introduced by [Donald Michie](http://www.aiai.ed.ac.uk/~dm/) in [Game-playing and game-learning automata](https://www.researchgate.net/publication/240013423_Game-playing_and_game-learning_automata) (1966). The paper [Optimal strategy in games with chance nodes](http://www.inf.u-szeged.hu/actacybernetica/edb/vol18n2/pdf/Melko_2007_ActaCybernetica.pdf) (2007) also gives a decent description of the expectiminimax algorithm.
  
  
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