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This paper examines competition in the standard one-dimensional Downsian model of two-candidate elections, but where one candidate (A) enjoys an advantage over the other candidate (D). Voters ’ preferences are Euclidean, but any voter will vote for candidate A over candidate D unless D is closer to her ideal point by some fixed distance $. The location of the median voter’s ideal point is uncertain, and its distribution is commonly known by both candidates. The candidates simulta-neously choose locations to maximize the probability of victory. Pure strategy equi-libria often fail to exist in this model, except under special conditions about $ and the distribution of the median ideal point. We solve for the essentially unique sym-metric mixed equilibrium with no-gaps, show that candidate A adopts more moderate policies than candidate D, and obtain some comparative statics results doi:10.1006jeth.2001.2821, available online a