Effective Parties as Collisions

We compute the effective number of parties, $N_2$, such that:

$$N_2 = \frac{1}{\sum_{i=1}^{N_0} p_i^2}$$

Where $N_0$ is the actual number of parties and $p$ their respective shares.

The squared term in the denominator is deliberate: the logic is one of collision. Rae (1968) sets it all out very well. If we were to take all voters, put them in a room, rearrange them at random, and then let them introduce themselves to one another, the probability that someone who voted for party $i$ would meet someone else who voted in the same way is, in the limit, $p_i^2$. By summing these probabilities across parties, we get the probability that any two voters, regardless of what party they vote for, will have voted for the same party. We can then take the reciprocal of this probability to convert back into units of parties.

Collision Model

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Expected N₂: 2.00

Observed N₂: –

Total: 0

Similar: 0

PlayRestart

The interactive above shows a simple agent-based model. On the right-hand side there are 200 ‘voters’ in a room, while the slider on the left-hand side changes the balance between the orange and purple ‘parties’. This determines the expected effective number of parties, $N_2 = \frac{1}{\Pr(\text{Orange})^2 + \Pr(\text{Purple})^2}$. Once the simulation starts, the voters move around the room at random, colliding into one another. As these collisions accumulate, the total number of collisions divided by the number of collisions between voters of the same colour converges on the effective number of parties.