Suppose a country with 'E' electorates and 'V' voters in each electorate, were to hold an election. Each vote is independent of all others, and has a 50% chance of being for party A and a 50% chance of being for party B. Let 'E' and 'V' both be odd, and the party with a majority of votes in a particular electorate win that seat, and the party which wins the majority of seats win the election overall.
What is the probability, as a function of E and V, that the party which wins the election overall loses the popular vote?
For instance, I can calculate as follows with E=V=3: There are 9 voters, so 2^9 total possibilities. For a party to win the election overall but lose the popular vote, it must win exactly 2 votes in 2 seats, and 0 votes in the third seat. The number of ways this could happen is 2 (parties which could win overall) * 3 (seats in which the winning party could get 0 votes) * 3 (electors which could vote against the winning party in the first seat won by them) * 3 (electors which could vote against the winning party in the second seat won by them). Thus the overall probability is 27/256.
Is there a general formula for calculating this probability as a function of E and V?
This seems like a somewhat difficult question to answer analytically, but numerically I get the following results when increasing E and V together. In the case of ties within a district, I split the electoral votes between the two parties. In terms of deciding the results of the election, I break the tie arbitrarily in favor of one of the two parties (consistently).
Interestingly enough, the results seem to converge around 20%. The zig-zag pattern accounts for a local effect which suggests that the even E/V numbers are more robust to different election decision styles.