Anyway, the mathematical analysis reveals that, if cooperation evolutionarily prevails over "honest" cheating (where cheaters don't try to hide their cheating), then a new strategy can invade that tactically deceives cooperators (by hiding or misrepresenting their behavior). But this can only happen if cooperators aren't good at recognizing rare cheaters. In that case, you'd expect a mixed population of cooperators and deceptive cheaters. The equilibrium ratio of cooperators to deceptive cheaters depends on how difficult it is to deceive relative to how good cooperators are at recognizing both cheaters and deception. The more difficult deception is and the easier recognizing it is, the greater the ratio of cooperators to deceptive cheaters.
What's really interesting about the mathematical result is that if cooperators are terrible at catching cheaters, then "honest" cheaters can invade the mixed population of cooperators and deceivers because they don't pay the cost of deception that deceivers do, but they still reap all the benefits of cheating. In that case, cooperation prevails. Hurray. But if cooperators are good at catching cheaters, it pays to deceive ... at least for rare deceptive cheaters. I'm pretty sure that these mathematical results make sense and you might guess at them without doing any calculus. That's not a mark against the models. Instead, it's helpful when a mathematical model with explicit, formal assumptions confirms our intuition, which derives from implicit assumptions and informal logic.
The authors argue that their mathematical model implies a positive correlation between the number of cooperative strategies and the number of deceptive strategies in a species. Actually, their model implies that, under some very specific circumstances, we'd expect a positive correlation between the frequency of cooperators and the frequency of defectors. That said, it's not too much of a logical leap.
To examine this prediction, the authors did a comparative analysis of species in the order Primates (to which we belong). The data compiled the presence or absence of different types of cooperative and deceptive behaviors. They used a method called independent contrasts to examine the relationship between cooperativeness and deception, controlling for the phylogenetic relationships among species and for the research effort into a particular species (because more research yields more observations of different types of behaviors). Here is are scatterplots of the independent contrasts with best fitting lines through the points.
What's fascinating about the empirical results is that there is no statistically significant relationship between neocortex size (a measure of cognitive capacity) and deception rate when controlling for cooperativeness. This goes against the grain of the Machiavellian intelligence hypothesis, which argues that there should be a positive correlation between deception rate and neocortex size.
But is the non-significance of neocortex size simply due to a collinearity problem? A collinearity problem happens when you fit a regression in which two of the predictor variables are highly correlated. The effect of collinearity is that it inflates the confidence intervals of your regression coefficients (which measure the relationship between the outcome variable and the predictors). Wider confidence intervals mean larger p-values and lower statistical significance. The number of cooperative behaviors in a species and its neocortex size might be correlated. Indeed, R.I.M. Dunbar's classic study found that group size is correlated with neocortex size in primates, and group size is a problematic but still useful proxy for social complexity.
And this is why journals need to allow more room for the methods section: because we should never penalize scientists for doing collinearity diagnostics.