Riordan and Sappington (JET, 1988) show that in an agency relationship in which the agent's type is correlated with a public ex post signal, the principal may attain first best (full surplus extraction and efficient output levels) if the agent is faced with a lottery such that each type is rewarded for one signal realization and punished equally for all the others. Gary-Bobo and Spiegel (RAND, 2006) show that this kind of lottery is most likely to be locally incentive compatible when the agent is protected by limited liability. In this paper we investigate how the principal should construct the lottery to attain not only local but also global incentive compatibility. We first assess that the main issue with global incentive compatibility rests with intermediate types being potentially attractive reports to both lower and higher types. We then show that a lottery including three levels of profit (rather than only two) is optimal in that it is most likely to be globally incentive compatible under limited liability, if local incentive constraints are strictly satisfied. We identify conditions under which first best is implemented. In a setting with three types and three signals we also pin down the optimal distortions when those conditions are violated. In particular, when the first-best allocation is locally but not globally incentive compatible, output distortions are induced but all surplus is retained from the agent.