Synchronic information, knowledge and common knowledge in extensive games.pdfVIP

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Synchronic information, knowledge and common knowledge in extensive games.pdf

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Synchronic information, knowledge and common knowledge in extensive games

1SYNCHRONIC INFORMATION, KNOWLEDGE AND COMMON KNOWLEDGE IN EXTENSIVE GAMES Pierpaolo Battigalli and Giacomo Bonanno * 1. Introduction The language of extensive games is complex and rich. It allows one to express such notions as the order of moves, the information a player has when it is her turn to move, etc. It is not, however, a sufficiently rich language in the sense that there are meaningful and natural statements that one can make (about a given extensive game) whose truth cannot be decided without making the language richer. We shall give two examples. Consider first the extensive form of Figure 1. Restricting attention to the class of extensive games defined by von Neumann and Morgenstern with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of “maximum information” we are able to provide an alternative characterization of the meet of the information partitions. 2 Common knowledge in extensive games 1 1 2 t 0 t 1 t 2 t 3 t 5 3 1 t 4 2 t 7t 6 t 15 t 14 t 13 t 12 t 11 t 10 t 9 t 8 3 t 16 t 17 t 18 t 19 t 20 t 21 t 22 t 23 a FIGURE 1 Consider the following statement: ? If node t 2 is reached, then every player knows that player 1 chose action a, but it is not common knowledge among the players that player 1 chose a ?. Is this statement true for the extensive game of Figure 1? If one adopts the standard semantics for knowledge and common knowledge 1 then, in order to answer this question, one needs th