Implementable and ex-post IR rules in bilateral trading with discrete values

We study bargaining problems between one buyer and one seller for a single object when each set of valuations is finite. We derive a necessary and sufficient condition for interim-implementable trading rules, analogous to the result in Myerson and Satterthwaite (1983), and a necessary condition for mechanisms that are free from ex post regret on and off equilibrium path. Based on these two theorems, our contribution is three-fold. First, we illustrate the results by means of three examples. Second, in contrast to the continuous model, there always exist strictly positive probability mass functions such that ex post efficiency and interim-implementability is feasible. Third, we show the difference between interim- and ex post-implementation. In the class of threshold trading rules, interim-implementation only precludes inefficient trade, while ex post-implementation is equivalent to posted price implementation.