Comparing the Power of Games on Graphs

The descriptive complexity of a problem is the complexity of describing the problem in some logical formalism. Essentially the only known technique for proving separation results in descriptive complexity is to make use of games on graphs played between two players, called the spoiler and the duplicator. There are two types of these games, which differ in the order in which the spoiler and the duplicator make various moves. In one of these games, the rules seem to be tilted towards favoring the duplicator. These seemingly more favorable rules make it easier to prove separation results, since separation results are proven by showing that the duplicator has a winning strategy. In this paper, the relationship between these games is investigaed. It is shown that in one sense, the two games are equivalent. Specifically, each family of graphs used in one game (the game with the seemingly more favorable rules for the duplicator) to prove a separation result can in principle be used in the other game to prove the same result. This answers an open question of Ajtai and the author from 1989. It is also shown that in another sense, the games are not equivalent, in that there are situations where the spoiler requires strictly more resources to win one game than the other game. This makes formal the informal statement that one game is easier for the duplicator to win.