Editing Game tree (section)

==Solving game trees==

===Deterministic algorithm version===

[[Image:Arbitrary-gametree-solved.svg|thumb|right|400px|An arbitrary game tree that has been fully colored]]

With a complete game tree, it is possible to "solve" the game – that is to say, find a sequence of moves that either the first or second player can follow that will guarantee the best possible outcome for that player (usually a win or a tie).  The [[deterministic algorithm]] (which is generally called [[backward induction]] or [[retrograde analysis]]) can be described recursively as follows.

# Color the final ply of the game tree so that all wins for player 1 are colored one way (Blue in the diagram), all wins for player 2 are colored another way (Red in the diagram), and all ties are colored a third way (Grey in the diagram). 
# Look at the next ply up.  If there exists a node colored opposite as the current player, color this node for that player as well.  If all immediately lower nodes are colored for the same player, color this node for the same player as well.  Otherwise, color this node a tie.
# Repeat for each ply, moving upwards, until all nodes are colored.  The color of the root node will determine the nature of the game.

The diagram shows a game tree for an arbitrary game, colored using the above algorithm.

It is usually possible to solve a game (in this technical sense of "solve") using only a subset of the game tree, since in many games a move need not be analyzed if there is another move that is better for the same player (for example [[alpha-beta pruning]] can be used in many deterministic games).

Any subtree that can be used to solve the game is known as a '''decision tree''', and the sizes of decision trees of various shapes are used as measures of [[game complexity]].<ref name="Allis1994">{{cite book | author = Victor Allis | author-link = Victor Allis | year = 1994 | title = Searching for Solutions in Games and Artificial Intelligence | publisher = Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands | isbn = 90-900748-8-0 | url = http://fragrieu.free.fr/SearchingForSolutions.pdf}}</ref>

===Randomized algorithms version===
[[Randomized algorithm]]s can be used in solving game trees. There are two main advantages in this type of implementation: speed and practicality. Whereas a deterministic version of solving game trees can be done in {{math|''Ο''(''n'')}}, the following randomized algorithm has an expected run time of {{math|''θ''(''n''<sup>0.792</sup>)}} if every node in the game tree has degree 2. Moreover, it is practical because randomized algorithms are capable of "foiling an enemy", meaning an opponent cannot beat the system of game trees by knowing the algorithm used to solve the game tree because the order of solving is random.

The following is an implementation of randomized game tree solution algorithm:<ref name="Roche2013">{{cite book | author = Daniel Roche | author-link = Daniel Roche | year = 2013 | title = SI486D: Randomness in Computing, Game Trees Unit | publisher = United States Naval Academy, Computer Science Department | url = http://www.usna.edu/Users/cs/roche/courses/s13si486d/u03/ | access-date = 2013-04-29 | archive-date = 2021-05-08 | archive-url = https://web.archive.org/web/20210508074443/https://www.usna.edu/Users/cs/roche/courses/s13si486d/u03/ | url-status = dead }}</ref>

<syntaxhighlight lang="python">
def gt_eval_rand(u) -> bool:
    """Returns True if this node evaluates to a win, otherwise False"""
    if u.leaf:
        return u.win
    else:
        random_children = (gt_eval_rand(child) for child in random_order(u.children))
        if u.op == "OR":
            return any(random_children)
        if u.op == "AND":
            return all(random_children)
</syntaxhighlight>

The algorithm makes use of the idea of "[[Short-circuit evaluation|short-circuiting]]": if the root node is considered an "{{mono|OR}}" operator, then once one {{mono|True}} is found, the root is classified as {{mono|True}}; conversely, if the root node is considered an "{{mono|AND}}" operator then once one {{mono|False}} is found, the root is classified as {{mono|False}}.

<ref>{{Cite journal|last1=Pekař|first1=Libor|last2=Matušů|first2=Radek|last3=Andrla|first3=Jiří|last4=Litschmannová|first4=Martina|date=September 2020|title=Review of Kalah Game Research and the Proposition of a Novel Heuristic–Deterministic Algorithm Compared to Tree-Search Solutions and Human Decision-Making|journal=Informatics|language=en|volume=7|issue=3|pages=34|doi=10.3390/informatics7030034|doi-access=free|hdl=10084/142398|hdl-access=free}}</ref>{{clear}}