In many strategic interactions of interest, the participants are uncertain about each others' characteristics. They are playing a game of incomplete information. Therefore one may have to consider agents' beliefs about each others' preferences or strategic options, beliefs about beliefs about preferences and strategic options, and so on. This infinite beliefs hierarchy is a potential threat to tractability. For that reason, in nearly all the literature on games of incomplete information, the issue is finessed by restricting uncertainty to fundamentals, and otherwise replacing beliefs with knowledge. This approach has been tremendously successful in applications of game theory to auctions, bargaining, industrial organization, macroeconomics etc. Nevertheless, it is a shortcut, and in this paper we construct a simple environment in which this shortcut is invalid. Moreover, we demonstrate that by acknowledging the more fundamental nature of uncertainty in games, one can obtain novel theoretical insights and open the perspective on interesting empirical regularities. In prior work, Blume and Gneezy [2000], we considered a cognitively simple pie chart game played in to rounds: In the first round, two players simultaneously and independently choose a sector from a pie chart with three identical sectors. The first round choices are recorded with identical markers on the pie chart. In the second round, player choose again, this time with the benefit of knowing the first-round choices. Assuming complete information and common knowledge, the unique optimal attainable strategy (OAS) dictates in the absence of a first-round match to choose the previously unchosen sector. However, in the data we see the choice of the previously unchosen sector only in 63.7% of the observations without a first-round match. This is better than would be predicted by chance alone, but not fully explained by OAS with complete information and common knowledge. If we maintain OAS, this raises the question whether we are primarily dealing with a failure of cognition or whether a substantial fraction of the data is explained by a failure of common knowledge of cognition. In the first case the ``optimal sector" is not chosen by a substantial number of subjects simply because they do not recognize that it is the unique distinct sector. In the latter case there are again some players who do not recognize the distinction of the optimal sector, but also others who recognize the distinction themselves but are not sufficiently confident that others recognize it as well. This paper provides an answer to the question of whether failure to coordinate in games of this kind is due to a failure of cognition or of common knowledge. We show that both factors play a role. Not all players identify a distinct sector, and players who do are more likely to choose the distinct sector in play with a cognitively identical player. We use two ways of establishing the latter fact: One way has players play with themselves, the other permits players to select the game. The latter way is of additional interest because it uses cognitive sorting as means for reducing strategic uncertainty. This is a novel form of forward induction in which a player signals cognitive understanding by choosing a cognitively more demanding game that yields higher rewards if players recognize its structure and have common knowledge of the structure. We show that such cognitive sorting does occur.

Co-author Andreas Blume