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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 ...
succinct versions of many graph problems, with graphs represented as Boolean circuits, [43] ordered binary decision diagrams [44] or other related representations: s-t reachability problem for succinct graphs. This is essentially the same as the simplest plan existence problem in automated planning and scheduling. planarity of succinct graphs
The first player unable to extend the path loses. An illustration of the game (containing some cities in Michigan) is shown in the figure below. In a generalized geography (GG) game, we replace the graph of city names with an arbitrary directed graph. The following graph is an example of a generalized geography game.
The problem of finding a (possibly mixed) Nash equilibrium in a graphical game is PPAD-complete. [5] Finding a correlated equilibrium of a graphical game can be done in polynomial time, and for a graph with a bounded treewidth, this is also true for finding an optimal correlated equilibrium. [2]
For solving game trees with proof-number search family of algorithms, game trees are to be mapped to and–or trees. MAX-nodes (i.e. maximizing player to move) are represented as OR nodes, MIN-nodes map to AND nodes. The mapping is possible, when the search is done with only a binary goal, which usually is "player to move wins the game".
The ingredients of a stochastic game are: a finite set of players ; a state space (either a finite set or a measurable space (,)); for each player , an action set (either a finite set or a measurable space (,)); a transition probability from , where = is the action profiles, to , where (,) is the probability that the next state is in given the current state and the current action profile ; and ...
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Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms ...