1. ### The prisoner's dilemma does not violate classical probability laws

4/10/2015
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(latest update : 2015-10-05 11:27:35)

I have come across some papers and slides about quantum cognition, including:

These texts and other ones claim that the prisoner’s dilemma violates the law of total probability.

Quoting Corr’s paper:

The literature shows: (1) knowing that one’s partner has defected leads to a higher probability of defection; (2) knowing that one’s partner has cooperated also leads to a higher probability of defection; and, most troubling for Classical Probability theory, (3) not knowing one’s partner’s decision leads to a higher probability of cooperation.

The slides by Busemeyer & al. provide some empirical data supporting this claim. The above quote has an obvious flaw, as I explain now.

I want to emphasize this is not an attempt to discredit quantum cognition. This story of violation of the TP law is older than quantum cognition, but this is where I found it. I asked on a stackexchange network about psychology whether this flaw were known, and then what is the purpose of quantum cognition about the prisoner’s dilemma. I got some reactions of persons falling into the flaw. That showed me that it is at least not known by everybody, and this helped me to achieve, I hope, a better explanation, which is the content of this blog post.

Consider two experiments:

• The “true” prisoner’s dilemma: each of the prisoners $A$ and $B$ has the choice to cooperate or defect, without knowing the other prisoner’s choice.

• The “faked” prisoner’s dilemma: each of the prisoners $A$ and $B$ has the choice to cooperate or defect, but $B$ is asked first, and $A$ knows $B$’s choice.

Consider the two probabilities $\Pr$ and $\Pr^\ast$ respectively corresponding to these two experiments.

The above quote claims that the literature shows:

1. $\Pr^\ast(A \textrm{ defects} \mid B \textrm{ defects}) > 0.5$

2. $\Pr^\ast(A \textrm{ defects} \mid B \textrm{ coop.}) > 0.5$

3. $\Pr(A \textrm{ defects}) < 0.5$

And it claims that this is troubling for Classical Probability theory. This is not troubling. But this is troubling if you assume $\Pr=\Pr^\ast$. And I believe this is the flaw in the reasoning. Indeed, the TP law shows that $\Pr(A \textrm{ defects})$ is a weighted average of $\Pr(A \textrm{ defects} \mid B \textrm{ defects})$ and $\Pr(A \textrm{ defects} \mid B \textrm{ coop.})$. Assuming $\Pr=\Pr^\ast$ implies that $\Pr(A \textrm{ defects})$, which is lower than $0.5$, would be a weighted average of two numbers higher than $0.5$, which is impossible.

Of course the two experiments are totally different, and there is no reason to assume $\Pr=\Pr^\ast$. The so-called violation of the TP actually shows that $\Pr\neq\Pr^\ast$.

I am afraid this misunderstanding of the conditional probability could have an origin in the name which is sometimes used to call it : $\Pr(A \textrm{ defects} \mid B \textrm{ defects})$ is the probability that $A \textrm{ defects}$ knowing that $B \textrm{ defects}$. And it looks like it can be misinterpreted as: the probability that $A \textrm{ defects}$ when $A$ knows that $B \textrm{ defects}$. And $A$ never knows whether $B$ defects in the “true” prisoner dilemma.

I think I am starting to have an idea about the purpose of quantum cognition, for this example of the prisoner’s dilemma. Roughly speaking, it aims to have a mathematical axiomatic system for the “cognitive state” of $A$, with a rule describing the behavior of this cognitive state when $A$ learns the choice of $B$. Quantum probability could provide such an axiomatic, according to psychologists working in that field. It would provide a mathematical model for the state of $A$ under ignorance of $B$’s choice and also how this state would move when $A$ learns $B$’s choice. This is something different than the purpose of modeling the issue of one of the two above experiments, which is achieved by classical probability theory.