From game theory to players theory

post by Victor Porton (victor-porton) · 2023-09-15T06:23:52.656Z · LW · GW · 0 comments

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Leading mathematicians (I won't name names here) say that there is no formula for subjective probability (e.g. "I think, there is 20% of tomorrow rain", or "I think that Navier-Stokes conjecture is true with 90% probability"). There is a number and no formula? that's funny and untrue.

The subjective probability certainly make no sense in (the usual) probability theory and math statistics, because the (objective) probability (or expected value) by definition are calculated from multiple cases of an event, not from a single event. (So 20% of tomorrow rain makes no sense in probability theory: either no rain (0%) or there is rain (100%). Well, we could define the rain probability tomorrow quantum-mechanically, but that makes no practical sense, because it is too hard to calculate.

I created a mathematical framework, where subjective opinions of players are taken into account. So, we have a transition from game theory to players theory.

Let’s fix a (probabilistic or not) game (in the sense of game theory) or more generally a probabilistic distribution of games (being in some way “averaged” to be considered as a single game), a hypothesis (a logical statement) a “doubtful” player (a function that makes probabilistic decisions for the game) dependent on boolean variable (called trueness of our hypothesis) for a “side” (e.g. for white or black in chess) of the game. Let’s “factor” the game by restricting the side to only these decisions that the doubtful player can make. The guessed probability is defined as the probability that given (fixed!) player (a probabilistic function that makes decisions for our side of the game) will choose the variant in which our hypothesis is true.

A variation of this is when the doubtful player’s input is mediated by a function that takes on input logical statements instead of a boolean value. (The above is easy to rewrite in this case.)

More generally guessed probability can be generalized to guessing real number (or any measure space) variables by replacing the doubtful player by a real number function (or a function from our measure space) and defining the guessed value as the probability distribution of player’s decisions in the “factored” by restricting to guessed player decisions game.

A special case of the above are perceived or guessed value of an economical asset in an economical game (a game about becoming richer).

The guessed value of assets of a person is a scientific way to measure success. (It’s a well known fact, however denied by many pseudoscientists, that measuring success by money is often irrelevant.)

In economical games it’s often relevant to consider the entire market (a set of players) as our relevant player. So we obtain a kind of market value.

Remember what whiskered red-neckers tend to say: "How much an outcome is true, is how much I am going to pay for it". Contrary to bad image of a red-necker, it is mathematically accurate. We can define subjective probability of an event as the amount we could bet for some event, if our purpose is to win a game where our bet is dependent on the event. E.g. having subjective probability of the tomorrow rain 20% may mean that somebody (the "subject" or the player) is willing to bet $20 for tomorrow rain and $80 for tomorrow no rain, if he is required to pay $100 in total.

Let put it into exact mathematical terms: Let we have a game whose players' purpose is to increase some real number. Then the subjective probability of an event is the subjective value of the probability that maximizes profit. (Here I didn't think it out well, please help to formulate this "bet" theory exactly. It is clear that this should work only for small bets, because big bets become unlinear (e.g. I want to preserve $10 more than to win $100), so the definition should somehow incorporate derivative. Really small bets also may not work, e.g. $0.001 is not worth to bet for, so the player will behave arbitrarily. So, before getting the derivative, sometimes we need to make the game smoother, eliminating unlinear dependencies on small values.)

Maybe, we should publish it in a peer-reviewed journal? But I don't have expertise in game theory to refer to it adequately. So, I welcome possible co-authors.

An open question: How to formalize the "red-necker's" saying "If you don't know anything about the outcome, it is 50%".

We are still to develop "subjective probability theory" (e.g. what is Bayesian in our logic?)

Further research is highly perspective. Particularly it defines scientific (not necessarily monetary) values of scientific hypotheses and is useful in research planning.

I think, in our times of near-nuclear-war changing the paradigm from game theory to something new (e.g. this players theory) will be good for mankind. At the other side, players theory tries to "break into" human brains, what may be considered unethical.

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