Group Voting

Designing fair voting systems is hard. In fact, Arrow’s impossibility theorem tells us that certain desirable features of a ranked voting system simply cannot coexist.

Even in unranked “choose a single winner” systems, setting them up to be fair is sometimes unintuitive. Consider the case of several groups deciding amongst themselves which group is the “winner”. Let \(G\) be the set of groups it’s possible to vote for and \(p_{g_n}\) the probability that group \(g_n\) ends up the winner. Then, naively,

\[p_{g_n} = \frac{1}{|G|}\]

Of course, in real situations we might have something closer to “everyone votes for their own group”, which just means that the largest group will win. Even in the best case where each group is the exact same size, we’ll end up with an \(n\)-way tie.

\[p_{g_n} = \begin{cases} 1 &\text{if } |g_n| = \max_{g \in G} |g| \\ 0 &\text{if } |g_n| < \max_{g \in G} |g| \end{cases}\]

One obvious way to manage this is to forbid voters from voting for their own group. However, this actually disadvantages the largest groups, since it takes their extra votes and spreads them out among all other groups, but not their own.

\[P_{g_n} = \frac{\frac{\left(\sum_{g \in G} |g|\right) - |g_n|}{\sum_{g \in G} |g|}}{|G| - 1}\]

In words, this is the fraction of votes belonging to all groups except \(g_n\), divided by the number of all groups except \(g_n\). Consider this mapping from groups to votes:

A 1
B 2
C 3
D 4
E 5

In this simple case, we’ve pushed the win probability of any given group away from the baseline of 20%, in inverse proportion to how big that group is relative to the others.

A 23.33% +3.33%
B 21.67% +1.67%
C 20.00% -0.00%
D 18.33% -1.67%
E 16.67% -3.33%

We might further counteract this by slightly devaluing the votes of some groups relative to others, but that comes with all the problems inherent in telling some individuals their votes are worth less simply because of external facts (which may be completely outside their control), like how many other people belong to that group. Not to mention that “punishing popularity” is the opposite of the intention of most voting.

Of course, in the real world things are even more complex. Bigger groups might have other advantages like more power, higher likelihood of containing extreme individuals, farther reach, preexisting notoriety, etc. This is all just to say that designing truly fair voting systems can’t be done naively, even when those systems seem relatively simple.