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--- chess:programming:minimax_algorithm [2021/10/30 21:59] – created peter
+++ chess:programming:minimax_algorithm [2021/10/30 22:40] (current) – peter
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 ====== Chess - Programming - Minimax Algorithm ======
-Minmax considers all the possible ways the game can go and returns the best value for that move, assuming the opponent also plays optimally.
+**Minimax** is a kind of backtracking algorithm that is used in decision making and game theory to find the optimal move for a player, assuming that your opponent also plays optimally.
+  * In Minimax the two players are called maximizer and minimizer.
+  * The maximizer tries to get the highest score possible while the minimizer tries to do the opposite and get the lowest score possible.
+Every board state has a value associated with it.
+  * In a given state if the maximizer has upper hand then, the score of the board will tend to be some positive value.
+  * If the minimizer has the upper hand in that board state then it will tend to be some negative value.
+  * The values of the board are calculated by some heuristics which are unique for every type of game.
+Since this is a backtracking based algorithm, it tries all possible moves, then backtracks and makes a decision.
 ----
 [[Chess:Programming:Minimax Algorithm:A program to illustrate the Minimax Algorithm|A program to illustrate the Minimax Algorithm]]
+[[Chess:Programming:Minimax Algorithm:A Tic-Tac-Toe example program to illustrate the Minimax Algorithm|A Tic-Tac-Toe example program to illustrate the Minimax Algorithm]]
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     return bestVal
 </code>
+----
+===== Example =====
+Consider a game which has 4 final states and paths to reach final state are from root to 4 leaves of a perfect binary tree as shown below.
+  * Assume you are the maximizing player and you get the first chance to move, i.e., you are at the root and your opponent at next level.
+  * Which move you would make as a maximizing player considering that your opponent also plays optimally?
+{{:chess:programming:minmax.png?400|}}
+Since this is a backtracking based algorithm, it tries all possible moves, then backtracks and makes a decision.
+  * Maximizer goes LEFT:
+    * It is now the minimizers turn.
+    * The minimizer now has a choice between 3 and 5.
+    * Being the minimizer it will definitely choose the least among both, that is 3.
+  * Maximizer goes RIGHT:
+    * It is now the minimizers turn.
+    * The minimizer now has a choice between 2 and 9.
+    * He will choose 2 as it is the least among the two values.
+Being the maximizer you would choose the larger value that is 3.
+  * Hence the optimal move for the maximizer is to go LEFT and the optimal value is 3.
+Now the game tree looks like below:
+{{:chess:programming:minmax1.png?400|}}
+The above tree shows two possible scores when maximizer makes left and right moves.
+<WRAP info>
+**NOTE:**  Even though there is a value of 9 on the right subtree, the minimizer will never pick that.
+  * We must always assume that our opponent plays optimally.
+</WRAP>