Monetary and fiscal policies interact in many ways. Recently, the stance of fiscal policy in a number of countries (including the EU and the US) has raised concerns about risks for the outcomes of monetary policy. Our paper first shows that these concerns are justified since - under an ‘ambitious’ fiscal policymaker - inflation bias and lack of monetary policy credibility may obtain in equilibrium, even if the central banker is fully independent, patient, and ‘responsible’. To reach a possible solution the paper proposes a novel asynchronous game theoretic framework that generalizes the standard commitment concept. Most importantly, it allows for concurrent and partial commitment, ie both policies may be committed at the same time, and may do so with varying degrees. It is demonstrated that the undesirable scenario can be prevented if monetary commitment is sufficiently strong relative to fiscal commitment. Interestingly, such strong monetary commitment can not only resist fiscal pressure, but also ‘discipline’ an ambitious fiscal policymaker and achieve socially desirable outcomes for both policies. We then extend the setting to the European monetary union case with a common central bank and many heterogeneous fiscal policymakers and show that these findings carry over. The policy implication therefore follows: by more explicitly committing to a numerical (long-run) inflation target, the ECB, the Fed, and others would not only ensure their credibility, but also indirectly induce a reduction in the size of the budget deficit and debt. The paper concludes by showing that all our predictions are empirically supported.