# Quasi-magical thinking and superrationality for Bayesian agents

As part of our objective and subjective rationality model, we want a focal agent to learn the probability that others will cooperate given that the focal agent cooperates ($p$) or defects ($q$). In a previous post we saw how to derive point estimates for $p$ and $q$ (and learnt that they are the maximum likelihood estimates):

$p_0 = \frac{n_{CC} + 1}{n_{CC} + n_{CD} + 2}$, and $q_0 = \frac{n_{DC} + 1}{n_{DC} + n_{DD} + 2}$

where $n_{XY}$ is the number of times Alice displayed behavior $X$ and saw Bob display behavior $Y$. In the above equations, a number like $n_{CD}$ is interpreted by Alice as “the number of times I cooperated and Bob ‘responded’ with a defection”. I put ‘responded’ in quotations because Bob cannot actually condition his behavior on Alice’s action. Note that in this view, Alice is placing herself in a special position of actor, and observing Bob’s behavior in response to her actions; she is failing to put herself in Bob’s shoes. Instead, she can realize that Bob would be interested in doing the same sort of sampling, and interpret $n_{CD}$ more neutrally as “number of times agent 1 cooperates and agent 2 defects”, in this case she will see that for Bob, the equivalent quantity is $n_{DC}$.

With this simple theory of mind that realizes that Bob is also capable of agency, the proper interpretation for $p_\mathrm{ToM}$ ($q_\mathrm{ToM}$; I am introducing subscripts to avoid overloading notation) becomes “how often an agent cooperates given that the other agent cooperates (defects)”. The proper estimate becomes:

$p_\mathrm{ToM} = \frac{2n_{CC} + 1}{2n_{CC} + n_{CD} + n_{DC} + 2}$, and $q_\mathrm{ToM} = \frac{n_{CD} + n_{DC} + 1}{n_{CD} + n_{DC} + 2n_{DD} + 2}$

Note that Alice is now double counting the symmetric events $CC$ and $DD$. If the real proportion of cooperation in the population is $r$ then in the limit of $n \rightarrow \infty$ the theory-of-mind estimates still converge to the rational $p_\mathrm{ToM}^* = r$ and $q_\mathrm{ToM}^* = r$.

What if the agent employs quasi-magical thinking, i.e. know that their actions do not affect the decision of others but behave as if they do? If we follow Masel (2007) then quasi-magical thinking is equivalent to Alice updating her beliefs not only with the result of Bob’s action, but also her simulation of what she would have done. As an example, everytime she sees a $CC$ or $CD$ event, she also counts as $CC$ event since she would have cooperated in that case. This gives us the estimates:

$p_{1/2} = \frac{2 n_{CC} + n_{CD} + 1}{ 2(n_{CC} + n_{CD}) + 2}$, and $q_{1/2} = \frac{ n_{DC} + 1}{ 2(n_{DC} + n_{DD}) + 2}$

As before, we look at the limit of $n \rightarrow \infty$ with true proportion of cooperation $r$:

$p_{1/2}^* = \frac{r}{2} + \frac{1}{2}$ and $q_{1/2}^* = \frac{r}{2}$

In other words, regardless of the true probability of cooperation, this quasi-magical thinking agent will believe that her partner is 50% more likely to cooperate with her if she cooperates.

I chose the subscript $1/2$ suggestively. In a more general treatment, Alice can choose by how much she weighs her simulated self-action against Bob’s real action. If we let $\alpha$ be a measure of self-absorption or the weight she attributes to herself and $1 - \alpha$ to Bob then the estimates become:

$p_\alpha = \frac{n_{CC} + \alpha n_{CD} + 1}{ n_{CC} + n_{CD} + 2}$, and $q_\alpha = \frac{ (1 - \alpha)n_{DC} + 1}{ n_{DC} + n_{DD} + 2}$

With the limit of large $n$ given by:

$p_\alpha^* = (1 - \alpha)r + \alpha$ and $q_\alpha^* = (1 - \alpha)r$

In words, an $\alpha$-self-absorbed quasi-magical thinking agent will believe her partner to be $\alpha$ more likely to cooperate with her if she cooperates regardless of the real proportion of cooperation $r$. If we consider a completely self-absrobed agent ($\alpha = 1$) then we recover Hofstadter’s superrationality: an agent so dogmatic that he assumes that every other agents will always arrive at the same exact conclusion as him for what behavior to pursue.

If Alice suffers from the quasi-magical delusions, but otherwise acts rationally on her beliefs then we can calculate how she will behave. Suppose that she is $\alpha$-self-absrobed, thinks that she is playing the cooperate-defect game $\begin{pmatrix} 1 & U \\ V & 0 \end{pmatrix}$, and is out to maximize her expected utility. She will cooperate if:

$\alpha + (1 - \alpha)(r + (1 - r)U) > (1 - \alpha)rV$

A superrational agent $\alpha = 1$ will always cooperate. For other agents, cooperation will be risk dominant (equivalent to $r = 1/2$) strategy when:

$1 + 2 \frac{\alpha}{1 - \alpha} > V - U$

Note that by setting tag-mutation rate $\nu = \frac{\alpha}{1 - \alpha}$ we recover the condition for cooperation in set structured populations. Can we think of set-structured populations in cognitive terms as a type of quasi-magical thinking? A group-think delusion or just a mathematical coincidence?

Masel, J. (2007). A Bayesian model of quasi-magical thinking can explain observed cooperation in the public good game Journal of Economic Behavior & Organization, 64 (2), 216-231 DOI: 10.1016/j.jebo.2005.07.003