  We consider collective decision making when the society consists of groups
endowed with voting weights. Each group chooses an internal rule that specifies
the allocation of its weight to the 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. Our numerical computation for the US
Electoral College verifies its sensibility.
