An important, but largely unexplored, class of bargaining problems involve two negotiators, who send signals to a third party. Such problems are especially common in politics, where elected officials must worry about approval from voters and sovereign leaders in international negotiations must appeal to domestic politicians and voters. The signaling incentives of the negotiators significantly influence (i) the proposals that they offer and (ii) their decisions to accept of reject their opponent’s proposal. For instance, consider the case upon which we focus: Congress makes a take-it-or-leave-it offer (a bill) to the president, who either accepts or rejects (signs or vetoes) it. A third party, a set of voters, is uninformed about the president’s preferences; however, by observing the bill that Congress writes and the president’s veto decision, they can learn about these preferences. In our model the president wants to appear moderate to voters, while Congress wants him to appear extreme. As a consequence, Congress sometimes writes a bill that it knows the president will veto, instead of writing a compromising bill that they both prefer to the status quo. Thus, despite Congress and the president being completely informed, the presence of an uniformed third party causes the outcome to be Pareto inefficient. The model generates many empirical implications, and we test one of these - that the president’s approval should tend to drop after a veto. Data from 1956 to 1992 strongly support the hypothesis. A veto of a major bill causes the president’s approval to drop approximately two percentage points - approximately doubt the effect of a one-percentage- point drop in GDP.