Shannon switching game

The Shannon switching game is an abstract strategy game for two players, invented by American mathematician and electrical engineer Claude Shannon, the "father of information theory" some time before 1951. Two players take turns coloring the edges of an arbitrary graph. One player has the goal of connecting two distinguished vertices by a path of edges of their color. The other player aims to prevent this by using their color instead (or, equivalently, by erasing edges). The game is commonly played on a rectangular grid; this special case of the game was independently invented by American mathematician David Gale in the late 1950s and is known as Gale or Bridg-It.

The game is played on a finite graph with two special nodes, A and B. Each edge of the graph can be either colored or removed. The two players are called Short and Cut, and alternate moves. On Cut 's turn, he deletes from the graph a non-colored edge of his choice. On Short 's turn, he colors any edge still in the graph. If Cut manages to turn the graph into one where A and B are no longer connected, he wins. If Short manages to create a colored path from A to B, he wins. The game always terminates after a finite number of moves, and one of the two players has to win. Either Short, Cut, or the player moving first is guaranteed the existence of a winning strategy on any given graph.

The Short and Cut games are a duality; that is, the game can be restated so that both players have the same goal: to secure a certain edge set with distinguished edge e. Short tries to secure the edge set that with e makes up a circuit, while Cut tries to secure an edge set that with e makes up a cutset, the minimal set of edges that connect two subgraphs.

Versions of the Shannon switching game played on a directed graph and an oriented matroid have been described for theoretical purposes; but no corresponding commercial games have been published.

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