This paper studies a special class of differential information games with pre-play communication-games with ‘cheap play’. The authors consider problems in which there are several rounds of payoff-irrelevant publicly observable choice) or discussion) of actions, followed by a final round in which actions are binding and payoff relevant. A natural focal subset of equilibria of such games in one that consists of equilibria involving no regret. Such games were first studied by Green and Laffont, where a criterion called posterior implementability is introduced with the intention of identifying regret-free equilibria in games with cheap play. This is simply a restriction on the Bayesian equilibrium of the underlying one-shot game. If indeed such a restriction does characterize regret-freeness, then the analytics of such situations would be enormously simplified since one can ignore the messy extended-form of the cheap play game; merely examining the one-shot game is sufficient. The authors argue that regret-freeness of an equilibrium has a subtle distinction: regret-freeness in moves and regret-freeness in assessments. They show that the former causes the extended-form to be irrelevant; posterior implementability completely characterizes equilibria with regret-freeness in moves. The latter, on the other hand, does not yield a similar principle: the extended-form cannot be ignored.
