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DataPacRat comments on Open Thread, Apr. 20 - Apr. 26, 2015 - Less Wrong Discussion
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Seeking Phi in Game Theory
For a bit of fiction I'm thinking of... is there any aspect of game theory in which offering the number/symbol phi can indicate a threat to defect unless the other player cooperates?
So far, a quick bit of Googling has only turned up phi in relation to the Bargaining problem, at https://en.wikipedia.org/wiki/Bargaining_problem , and while I /could/ wrestle such an interpretation out of what's there, I'm hoping there's a less abstruse way to go about it.
Aha - at http://www.pnas.org/content/109/26/10409.full.pdf , zero-determinant strategies are defined by two factors, chi and phi, and at least when chi is 1, maximum phi results in the strategy of Tit-for-Tat, which is exactly what I was looking for.
Hm... is there a mathematical notation for the maximum of a variable, like |x| indicates the absolute value of x?
In addition, when chi is at the extortionate level of 3.5, then when phi is maxxed out, the odds of player X cooperating if player Y cooperated last turn are 66.6% or 50%, depending on whether player X also cooperated on the previous turn. Thus, in order to give player Y enough of an incentive to want to cooperate - providing odds of at least 50% of ending up with CC - player X may have their own incentive to set chi to something below 3.5. As it happens, there are two mathematical constants a little below that that might be chosen - the inverse Fibonacci constant, also called psi, about 3.36, or the more widely-known pi, 3.14.
Is)
not sufficient?
My goal is for a relatively simple, even iconic, image or logo, which can be easily interpreted regardless of the viewer's language. The symbol for Phi - a circle with a line through it - provides fodder for as much interpretation as I desire, from the overlapped 1 and 0 of binary to an ouroborus to an axis-and-equator to the Golden Ratio - and if a minimal modification can explicitly add the "maximum value of the variable denoted by this symbol", I'll be a happy little rat indeed.
This page suggests "∨".
(I've only ever seen that symbol meaning "or", but that's a kind of maximum.)
That's one of the two possibilities I've found over the past day. The other is ⊤ from https://en.wikipedia.org/wiki/Greatest_element .
(I wonder if presenting this idea to an actual mathematician would induce any wincing? Off to /r/math to find out...)
∨ is the mathematical symbol for "or" (in logic) -- my guess is that it may be derived from the fact that the initial letter of the Latin word for "or" is "v". There's a kinda convention that when you have a(n associative) binary operator, you use a bigger version of it to signify applying it to all the things in a sequence or set, so you'd want a larger one -- a bit like a capital "V".
⊤ is the mathematical symbol for the "top" element of a Boolean algebra; maybe more generally of a lattice. You wouldn't use it to mean "maximum of these things" in general.