# Basic Game Theory

let $x$ be a generic act in a given set $F$ of feasible acts and let $f(x)$ be an index associated to (or appraising) $x$; then find those $x^{0}$ in F which yield the maximum (or minimum) index, i.e., $f(x^{0})$ greater than or equal to $f(x)$ for all $x$ in $F$.

I am not understanding what it means for $f$ of $x$ to be an 'index to or appraising $x$'

I am also confused about the notation used for $x^{0}$ -is it the case the $x^{0}$ is just an indication of a single act? -is $x^{0}$ a part of a the set $x^{0}$,$x^{1}$,$x^{2}$...$x^{n}$?

I am also confused about how to interpret $f(x^{0})$ greater than or equal to $f(x)$ -will the clarification for my second question help me interpret $f(x^{0})$ -what is the difference between $f(x^{0})$ and $f(x)$

I apologize for my lack of knowledge... I am having a lot of problems interpreting the different symbols that are presented in books on probability, statistics and game theory. I keep running into notation involving infinite sets and complex functions... is there a book that can provide an elementary introduction to infinite sets and/or complex functions?

I have not began to take a calculus class and I have only began a pre-calculus class for my first semester in college. I was hoping to get ahead by studying topics such as, probability statistics and game theory.

-
f is a function from F to the set of real numbers. We have to find the elements x of F at which f(x) is maximum or minimum. – Shahab Jul 18 '11 at 17:42

I'm pretty sure $x^0$ is just meant to be an arbitrary variable denoting a member of the set $F$. (Using a superscript like that feels strange to me, though; I'm more used to seeing $x_0$ used that way.) All the superscript 0 is supposed to do is indicate that $x^0$ is not the same variable as plain $x$. The author could just as well have called it $y$ or $x'$ or whatever.

When I see notation like that used, there's usually a connotation that the super/subscript 0 indicates a constant: we are to find one fixed $x^0$ which satisfies the criterion $f(x^0) \ge f(x)$ for all $x$ in $F$. Here, plain $x$ is a bound variable which has no definite value outside the scope of that statement, while $x^0$ is an unbound constant which is hereby defined and available for use later.

Essentially, we're "picking out one of the $x$'s" as special and assigning it the label "0". (Note, though, that in the exercise you quoted, $x^0$ may not actually be uniquely defined.)

Of course, if there was need to single out another element of $F$, an author might well call if $x^1$ (or $x_1$, in the notation I'm more used to) and there could conceivably also be an $x^2$ (or $x_2$) and so on. But merely defining $x^0$ doesn't really imply that any other sub/superscripted $x$'s would have to be defined too; the 0 is just a label, not part of a sequence.

As for "let $f(x)$ be an index associated to (or appraising) $x$", that's presumably just a funny way of saying that $f$ maps (or "associates") each $x$ to a number, which is presumably supposed to somehow "appraise" the action $x$ (such that the best action maps to the highest number). "Index" here is basically just another word for "number".

-
"All the superscript 0 is supposed to do is indicate that x^0 is not the same variable as plain x. The author could just as well have called it y or x′ or whatever." This helped a lot. So it is the case that the author is setting up the elements of F as x^0,x^1,x^2... x^n being any act along the set F? – Derek Jul 18 '11 at 18:41
I don't think the author is trying to say that $F = \{x^0, x^1, \ldots, x^n\}$ or anything like that. He's just using $x$ to stand for any member of the set $F$ (which could be anything), and $x^0$ for a particular one of them (except, of course, that there might be more than one $x^0$ satisfying $f(x^0) \ge f(x)$ for all $x$ in $F$, if $f$ has several maxima). – Ilmari Karonen Jul 18 '11 at 21:12
That makes sense. Thank you for the clarification. – Derek Jul 19 '11 at 2:05

"An index associated to (or appraising)" is not a mathematical term. It's just saying $f$ is some function of $x$ that we want to maximize (or minimize). $x^0$ is just a particular value of $x$ that you want to find. The condition says that the value of $f$ at $x^0$ is the greatest possible value, i.e. it is greater than or equal to the values of $f$ at all possible $x$.

-
Thank you both. That gave me a lot more clarification. Does my knowledge of the notation for functions and infinite sets come with me study of calculus or is that something that is studied separately, apart from calculus or game theory? – Derek Jul 18 '11 at 17:55
You ought to study calculus to develop sufficient mathematical maturity. Otherwise unless F happens to be (a special kind of) a subset of real numbers, you dont need calculus to answer questions related to maximum and minimum value of f. – Shahab Jul 18 '11 at 18:05
Ok. Thank you very much for your help. – Derek Jul 18 '11 at 18:42