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The graphical form is an alternate compact representation of a game using the interaction among participants. Consider a game with players with strategies each. We will represent the players as nodes in a graph in which each player has a utility function that depends only on him and his neighbors. As the utility function depends on fewer other ...
Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games. [124] Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms , especially online ...
To better understand the game tree, it can be thought of as a technique for analyzing adversarial games, which determine the actions that player takes to win the game. In game theory, a game tree is a directed graph whose nodes are positions in a game (e.g., the arrangement of the pieces in a board game) and whose edges are moves (e.g., to move ...
Game-tree complexity of a game is the number of leaf nodes in the smallest full-width decision tree that establishes the value of the initial position. [1] A full-width tree includes all nodes at each depth. This is an estimate of the number of positions one would have to evaluate in a minimax search to determine the value of the initial position.
The graph contains an additional node s (the supply node). Each edge in the graph has a cost. The cost of each coalition S is the minimum cost of a spanning tree connecting all nodes in S to the supply node s. The value of S is minus the cost of S. Thus, a MCST game can be represented by O(n 2) values. [5]
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In game theory, fictitious play is a learning rule first introduced by George W. Brown. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies.
Further analysis of the game can be made using graph theory by considering the board as a collection of vertices and edges and examining the paths to each vertex that lies on the ground (which should be considered as a distinguished vertex — it does no harm to identify all the ground points together — rather than as a line on the graph).