
The prisoner's dilemma does not violate classical probability laws
4/10/2015
Source(latest update : 20151005 11:27:35)
I have come across some papers and slides about quantum cognition, including:
 Cold and hot cognition: Quantum probability theory and realistic psychological modeling, by P. J. Corr
 Applications of quantum probability theory to dynamic decision making, by Busemeyer, Balakrishnan and Wang
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:
\(\Pr^\ast(A \textrm{ defects} \mid B \textrm{ defects}) > 0.5\)
\(\Pr^\ast(A \textrm{ defects} \mid B \textrm{ coop.}) > 0.5\)
\(\Pr(A \textrm{ defects}) < 0.5\)
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 socalled 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.