The paper analyzes the interactions between monetary and fiscal policies. Its emphasis is on a monetary union; one in which (some of) the governments are excessively ambitious. In contrast to conventional games, our novel game theoretic framework allows for stochastic timing of policy actions. The fact that moves occur with some ex-ante probability distribution (rather than certainty every period) enables us to model various degrees of fiscal rigidity and indiscipline that are heterogeneous across the member countries. We examine a number of specifications in discrete and continuous time, such as the widely-used Calvo (1983) timing, as well as a fully general probability distribution of the timing of policy actions. We derive the necessary and sufficient degree of monetary commitment that eliminates socially inferior (subgame perfect Nash) equilibria. This degree is shown to be increasing in (i) the degree of fiscal rigidity of each member country, (ii) their relative economic size, (iii) the structure of the economy (that determines eg inflation and output variability costs), and (iv) the degree of the central bankers impatience. Interestingly, such a strong monetary commitment - interpretable as a sufficiently explicit numerical inflation target - does not only ensure high credibility of the central bank, but it also indirectly disciplines the fiscal policymaker(s). As such, it leads to an improvement in monetary-fiscal policy cooperation and outcomes of both policies. We conclude by calibrating the model with European Monetary Union data. This exercise aims at providing some quantitative predictions regarding the required explicitness of the European Central Banks commitment to an inflation target.