The paper discusses certain problems raised recently by B. Peleg, about the existence of equilibria and `nice' equilibria for various classes of voting games. First it is shown that under practically every non-dictatorial voting procedure, and for every sincere preference profile, one can find a Nash equilibrium with nice properties. Secondly, a Paretian and anonymous decision procedure is constructed, under which one can always find equilibria with certain attractive properties. Lastly, it is shown that if the number of alternatives is large enough, then for some sincere preference profile, no equilibrium may exist under many decision procedures.