In this article, we propose a polling accuracy measure for multi-party elections based on a generalization of Martin, Traugott, and Kennedy's two-party predictive accuracy index. Treating polls as random samples of a voting population, we first estimate an intercept only multinomial logit model to provide proportionate odds measures of each party's share of the vote, and thereby both unweighted and weighted averages of these values as a summary index for poll accuracy. We then propose measures for significance testing, and run a series of simulations to assess possible bias from the resulting folded normal distribution across different sample sizes, finding that bias is small even for polls with small samples. We apply our measure to the 2012 French presidential election polls to demonstrate its applicability in tracking overall polling performance across time and polling organizations. Finally, we demonstrate the practical value of our measure by using it as a dependent variable in an explanatory model of polling accuracy, testing the different possible sources of bias in the French data.