Many important collective bodies make decisions by weighted majority voting. Examples are the Electoral College in the US, the International Monetary Fund, the European Union Council of Ministers and any legislature with disciplined political parties. An important question in this setting is how the distribution of votes affects payoffs. If one party has 80 seats, is it twice as powerful as a party with 40 seats? A possible reason why power may deviate from the number of seats is the presence of ideological constraints: a small centrist party may be more influential than a larger extremist party. A second reason is related to the distribution of seats itself. Suppose there are three parties, with 40, 40 and 20 seats respectively, and 51 votes are needed for a majority. Even though the first two parties have more seats than the third one, any two parties have a majority, hence there is no practical difference between the three parties.
In this Nottingham School of Economics working paper, Maria Montero provides a necessary and sufficient condition for payoffs to be proportional to weights in the absence of ideological constraints. Parties are modelled as unitary actors that bargain over the division of a resource (eg. ministerial portfolios) by majority rule. The author shows that expected equilibrium payoffs are proportional to voting weights if and only if the set of winning coalitions with the smallest total weight is sufficiently rich; in particular, each party must be present in at least one such coalition. The author also shows that some counterintuitive results are predicted by this model as equilibrium phenomena: it is possible for two parties to have the same expected payoff even though one of them has more alternative coalitions available, a valuable party may be demanding very little to be in the government in comparison with others and yet may receive no proposals, and oversized coalitions may form.
CeDEx Discussion Paper 2016-02, Proportional payoffs in legislative bargaining with weighted voting: a characterization by Maria Montero
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Posted on Friday 4th March 2016