| This dissertation analyzes three decisions problems under uncertainty. Chapter one reconsiders the general moral hazard problem between a principal and an agent. This has, since Grossman and Hart (1983), been considered largely intractable. Applying techniques from lattice theory, first introduced into economics by Milgrom and Shannon (1994), we show that general results are in fact available. In particular, a wide range of comparative static conclusions can be drawn.; Chapter two considers a problem in which a drawer of political boundaries (a gerrymanderer) observes a noisy signal of voter preferences from a continuous distribution and creates N districts of equal size to maximize the expected number of districts which she wins. Under mild regularity conditions we show that "cracking" is never optimal---one's most ardent supporters should be grouped together. Moreover, for sufficiently precise signals the optimal solution involves creating a district which matches extreme "Republicans" with extreme "Democrats," then continuing to match toward the center of the signal distribution. This contrasts with standard intuitions for optimal gerrymandering. These involve concentrating one's extreme opponents in "unwinnable" districts ("packing") and spreading one's supporters evenly over "winnable" districts ("cracking"). These intuitions come from models with either no uncertainty about voter preferences, or in which there are only two voter types.; Chapter three considers optimal prize structures in tournaments. Within-firm job promotions, wage increases, bonuses, and CEO compensation are often interpreted as resulting from rank-order tournaments. They are prizes for top performers in rank-order tournaments set up to deal with moral hazard. We find that, because agents are risk averse, it is almost always optimal to punish poor performers rather than to reward top performers. This finding thus poses a puzzle for the theory of compensation. |
