Ex Ante versus Interim rationality and the existence of bubbles

Tirole (1982) is commonly interpreted as proving that bubbles are impossible with finitely many rational traders with common priors. We study a simple variation of his model in which bubbles can occur, even though traders have common priors and even though it is common knowledge that the asset has no fundamental value at all. In the equilibria we construct agents purchase the asset at successively higher prices (in expectation) until the bubble "bursts" and no subsequent trade occurs. In equilibrium, each trader has a finite "truncation date" and the date at which the bubble bursts is a function of these. Since no trader knows everyone's truncation date, none knows when the bubble will burst. As we show, these random truncations can arise from extrinsic uncertainty (i.e. sunspots) or intrinsic uncertainty (such as uncertainty regarding the initial wealth of other traders). There are two key differences between our model and Tirole's which enable us to use this device to construct equilibrium bubble. First, Tirole requires ex ante optimality, while we only require every trader's strategy to be optimal conditional on his information (specifically, on his truncation date)--i.e., interim optimal. Since each trader knows his information before he actually trades, this would seem to be the relevant definition of optimality. Second, Tirole considers rational expectations equilibria, while we analyze a demand submission game.