Cheating robot games

Cheating Robot games are two-player, perfect information games which formalize ideas present in two different areas of combinatorial games. The games are based on simultaneous-play but where one player (Right=Robot) has reflexes fast enough both to see where the opponent will play and to respond. This changes a simultaneous game into a sequential game but which is not covered by the theory of normal-play games. A position in a Cheating Robot game is a finite, complete bipartite graph where the edges are labeled. The vertices represent moves, and the edge labels are the subsequent positions resulting from the choice of vertices. In a position, Left chooses a vertex first, Right second, and then play moves to the new position corresponding to the edge label. This is a 'round'. This paper will only consider games in which, from any position, every sequence of moves terminates after a finite number of rounds. In a terminal game, the player with a move is the winner, if there are no moves the game is a draw. The basic theory and properties are developed, including showing that there is an equivalence relation and partial order on the games. Whilst there are no inverses in the class of all games, we show that there is a sub-class, simple hot games, in which the integers have inverses. In this sub-class, the optimal strategies are obtained by the solutions to a minimum-weight matching problem on a graph whose number of vertices equals the number of summands in the disjunctive sum.