We consider collective decision making when society consists of groups endowed with voting weights. Each group chooses an internal rule that specifies the allocation of its weight to alternatives as a function of its members' preferences. Under fairly general conditions, we show that the winner-take-all rule is a dominant strategy, while the equilibrium is Pareto dominated, highlighting the dilemma structure between optimality for each group and for the whole society. We also develop a technique for asymptotic analysis and show Pareto dominance of the proportional rule.