Politicians prefer to use vague words and outline several policies in their electoral promises, which is termed as "political ambiguity." It is reasonable to interpret political ambiguity as a lottery, that is, a probability distribution on policies, in the following manner: Candidates announce a lottery, and a voter chooses a candidate who announces a better lottery for themselves. However, it is still difficult to explain why politicians prefer to use vague promises (i.e., a lottery) based on a rational-choice model.

To understand the difficulties in this regard, we should begin by locating a Condorcet winner that allows political ambiguity. Black (1948) shows that if we suppose that voters can choose only a single policy, a unique Condorcet winner is at the median policy (i.e., the median voter's preferred policy). On the other hand, my work supposes a lottery instead of a single policy, which is called a Condorcet winning lottery: A lottery that defeats or ties with all other lotteries in pairwise elections. In summary, the unique Condorcet winning lottery is the median policy when voters are risk-averse or risk-neutral, while its existence is not ensured when voters are risk-loving.