1 When are Two Games the Same ?

1 Logic and games: historical encounters Logic and games have had several encounters in history. One major factor in the emergence of logic in Antiquity was a systematic interest in the rules of argumentation, which have a clear game-like structure. Games were mathematized for the first time, one might say, in the17th century: when Pascal and Huygens based their idea of probability on betting odds. The logic connection for this came later, e.g., with the Dutch Book theorems of the 1950s. But that same decade also saw some basic results inspired by Game Theory in the modern sense, such as the Gale-Stewart theorem stating the determinacy of closed infinite games (extending an old result of Zermelo's), that became important in descriptive set theory. Such connections have multiplied in recent decades, witness the variety of topics at the biannual TARK and LOFT conferences. This time, the focus is rather on the role of knowledge in action, obviously a shared concern between logic and game theory, including deep issues of rationality, counterfactual reasoning, communication, and cooperation. These contacts between logic and game theory do not all fit the same mold. There are uses of logic as a formalism for analysing and sharpening up game-theoretical argumentation, and there is logic as a general meta-theory of games. But there is also logic as a 'producer' of specific games for logical purposes (argumentation and proof, evaluation of sentences, model comparison, model construction), which then makes it a 'consumer' of game theory! Such different, and changing relationships have nothing strange, of course, in real life: the same human being can be both in our care, and our best advisor. 2 Our purpose: the logical structure of games We want to step back from any particular detailed concern, and look at games with the eyes of a logician. What is a game? This broad question translates into several more detailed ones, witness our section headings below. First, what kind of analysis would one expect? For a start, it seems obvious that games are a special kind of processes, which involve both concrete action and cognitive action. Thus, the world can change, but there is also information flow. But then, the fundamental question is: what kind of processes? A typical foundational approach should ask here for the invariances defining the field of enquiry.