Candidates would greatly reduce their effort if they factored in their actions' probability of decisiveness. Individual voters have low probability of decisiveness, but Presidential candidates have a lot. One wrong word by a candidate can ruin his chances of victory. Both politicians maximize their chances of winning if they both offer the platform they expect the median voter wants. Thus, if they both care solely about winning, platforms converge.
Uncertainty only leads to divergence if politicians care about policy as well as winning.
Then they will accept a greater risk of losing the election in order to pursue their preferred policies. Assume that transactions costs of political bargaining are zero. With zero transactions costs, the Mean Voter Theorem holds, not the Median Voter Theorem. Under these assumptions, political bargaining leads the most efficient policies to be adopted. And it works well for some small issues like smoking.