# Biases in Mathematical Perception in Mother Jones

Lydia Nichols recently pointed me to an article in Mother Jones that had some mildly interesting flaws whose cognitive origins can be largely traced.

The main point of the article was to talk about gerrymandering, and the clear damage that highly successful gerrymandering–together with increased segregation along political lines–has done to our representational democracy.  And the article was, as far as I could tell, basically factually correct. However, the implications drawn had two neat errors.

The less interesting error made in the article was to take a mismatch between the proportions of partisanship in the popular vote and the representation in congress as evidence for gerrymandering or segregation. For instance, the article notes that "in Florida, Democrats won nearly half the House race votes but will fill about a third of the state’s congressional seats."  The idea is that in the absence of gerrymandering, representation would be proportional, so deviations from proportional representation are somehow nefarious. This line of reasoning is by no means unique to MoJo; it seems to be a quite common idea.

Common but wrong.  A mismatch between the proportion of voters for a particular party and that party’s congressional representation indicates gerrymandering only when either the two proportions lie on opposite sides of 50%, or when the majority party is underrepresented. The reason is straightforward: the US electoral system favors small majorities.  In a state with 52% support for some particular party, and a completely homogenous population with fair and uniform district boundaries, the majority party will win every district, garnering 100% of the representation.  Thus, over-representation of the majority is not, by itself, any indication of electoral impropriety beyond that built into every US. state constitution.

The less egregious but more cognitively interesting errors lie in the table that MoJo presents to indicate the unequal success republicans and democrats have enjoyed in gerrymandering. Pretending for the moment that the first error I pointed does not exist, and mismatches like those in the table accurately expose gerrymandering, there’s still something very interestingly fishy.  Notice that the  last two states in the table, Illinois and Maryland, actually favor Republicans. This is noted in the article, but the decision to align the table in terms of republicans and democrats instead of majority and minority party obscures the pattern it is supposed to reveal.

The table claims to show you the relative number of voters required to elect a representative, for each party.  In Ohio, for instance, about 75% of the representatives are republican, and only 25% are democrats; but just about as many people voted for each party.  If there were 4 voters and 4 candidates, then about 2 voters would have voted democrat, but only 1 representatives are D.  That’s a ratio of 2:1.  On the flip side, about 2 voters share 3 candidate for the republicans, for about 2:3.  3/2 is three times 1/2, so the graph shows 3 people, indicating  that there were about 3 times as many democrats per D representative as there were republicans per R.

Evidence of "gerrymandering"…or just evidence of "democracy"?

The math here is pretty simple: if d is the proportion of voters to representative for dems, and r for republications, then xd=r.  More explicitly, x(reps_d/voters_d)=(reps_r/voters_r).

So far, so good. The trouble is that this is implicitly proportional, and people have a very hard time thinking about proportions. As Berkeley educational psychologist Dor Abrahamson has persuasively shown (among other folks), we have a strong tendency to treat proportional relations in terms of subtractive differences, which distorts and flattens them.  In this case, we see Illinois and Maryland as cases of only moderate ‘bias’–less than a full person off.  In contrast, North Carolina is more than 2 people off–much more biased than Illinois, and even Maryland.  The trouble with seeing things this way is that it inaccurately portrays the underlying situation, which is really multiplicative.

Interestingly, the amazing cognitive psychologist Miriam Bassok has shown repeatedly that this particular case is especially difficult, because it deals wit people.  More precisely,  the graph (and the situation) demands that we treat people multiplicatively–that we multiply and divide one sort of person against another (note the equation above, in which we multiply and divided republican and democratic voters and representatives).  However, we have strong biases to treat people, along with other terms that common in discrete units and are of a common type, additively–as things we add and subtract.

We can get a sense for how these biases affect our perception of the situation if we simply rearrange the axes.  Instead of considering the ratio of voters per representative for republicans against democrats, we can consider the ratio for the majority vs. the minority. This yields the same graph for the first several states, but for the last two the colors (and parties) must be reversed.  For those, we ask how many  votes it took to elect a republican representative, for each vote it took to elect a democrat. The amended chart looks like this:

The same chart, corrected for one type of perceptual bias.

You can see that while Illinois is in the middle of the pack, Maryland is by far the most unequal state, in terms of mismatch between representation and votes.  Now, is this evidence of gerrymandering?  Not at all!  Remember that in a ‘perfect’ world, 50.1% of the votes would yield 100% of the rep’s, in the limit a ratio of infinity to 1.  Sampling noise limits this relationship, but the more partisan the state the less noise leads to upsets. Thus, the strong inequity in Maryland is nothing more than an indication of the strong partisan bias of the state.

None of this is to say that gerrymandering is a myth, or isn’t a problem, or that the republicans aren’t more successful and determined at it than the dems. All of those things are likely true, and good evidence for it comes from surveying the actual district maps.  But this kind of ‘evidence’ doesn’t indicate anything about gerrymandering.