Counterfactuals in wonderland

In a well-known, controversial paper, Professor Robert J. Aumann (1995) gave a formal presentation and a proof of the claim, that in games with perfect information, common knowledge of rationality implies that the backward induction outcome is reached. What keeps the players in Aumann’s model from deviating from the backward induction path is the following. Each player knows that if he deviates from the backward induction path, then all players will follow the backward induction path that starts at the node that follows the deviation. This new backward induction path is, by definition, worse for the would-be deviating player than the original backward induction path, which is precisely the reason why the player does not deviate. Why would the players follow the new backward induction path after a deviation which, of course, violates the common knowledge of rationality assumption? Because common knowledge of rationality implies that players know (indeed commonly know) that even if they deviate from the backward induction path, common knowledge of rationality will still be preserved for the rest of the game. In his paper Aumann persistently emphasizes the inevitable role of counterfactuals in game theory and decision theory. His whole argument hinges on what players know would have happened had things that wouldn’t and couldn’t have happened in fact happened. This little note tries to trace the literary and cultural sources of the main ideas in Aumann’s article. There is more than just a hunch, there is textual evidence, that Professor Aumann wanted, at least subliminally, to lead us to his sources, by planting a very obvious clue in his paper. One of the most striking examples in Aumann’s paper is a two-player game played by Ann and Bob. These names were chosen on purpose in alphabetical order to belittle their significance, as it were. But we are not deceived. Restricting names to those starting with A and B is a minor imposition that still leaves a lot of latitude.