# Time dynamics of probability (from a game theoretic point of view)

Many commenters on this question about a property of a random 2x2 matrix seem to assume that there is no such thing as a random 2x2 matrix. We are talking here about a matrix with integer or rational entries, so the students will write down some matrix, and since we cannot predict it, why not admit that this matrix will be random from the perspective of the teacher.

From a game theoretic perspective, just because the teacher doesn't know the strategy used by a student to generate a random matrix, it doesn't follow that there is no such strategy. But the reasoning is probably more that the process by which a student generates a random matrix will only be able to generate a finite amount of entropy, but there are 2x2 matrices with integer entries which contain more entropy than any given finite amount of entropy.

In Evaluating gambles using dynamics, the authors argue that especially from a game theoretic perspective, time dynamics is of prime importance. This time dynamics also seems to provide a solution to the above issues with the finite entropy. As soon as there are multiple rounds, and the strategy of a student for generating the random 2x2 matrix may change between rounds, he can use strategies which generate more entropy than any given finite amount of entropy in later rounds, and thereby overcome this apparent limit.

That quoted paper from 2014 seriously investigates the importance of time dynamics for probability in a game theoretic context. But I wonder whether the importance of time dynamics for probability hasn't already been investigated much earlier than that. You might argue that all I have to do for resolving this question is to look at the reference section of the quoted article, but most references there are newer than the year 2000 (it is a recent paper after all), and in addition it is an economic paper, which is more likely to cite other economics papers as opposed to philosophical papers.

• I am having hard time understanding the question. Markov chains ("time dynamics of probability"?) predate game theory, but both are mathematical fields. What kind of philosophical connection are you looking for? – Conifold Apr 12 '17 at 19:21
• @Conifold I am just looking for understanding of probability theory, mainly in the context of game theory. (And since probability theory was invented in the context of gambles, I simply disagree that Markov chains would predate game theory.) How can one understand the St. Petersburg paradox, for example. So a paper using Markov chains to resolve the St. Petersburg paradox and determine the fair price for that gamble would answer my question. But a paper which studies non time-homogeneous Markov chains, but doesn't use this time dynamics to explain something, would not. – Thomas Klimpel Apr 12 '17 at 20:01
• Ok, but this still sounds like a history of math question to me. And even then, what is the scope? If you consider analysis of gambles "game theory" then it goes at least as far back as Pascal-Fermat Correspondence of 1654. Surely you can't expect a list of references from there onwards? – Conifold Apr 12 '17 at 23:16
• @Conifold I am not looking for a list of references, just for some pre 2000 reference that seriously considers the "importance of time dynamics". And "importance" here is to be understood in explaining at least something, for example why so many people get the Monty hall problem wrong, why so many people believe that there is no fair price for the St. Petersburg gamble, or why so many people believe that there is no random 2x2 matrix (with rational entries). – Thomas Klimpel Apr 13 '17 at 7:30