A \textit{strategic-form game}, or just \textit{game} when contextually unambiguous, consists of a set $N =\{1, \ldots, n\}$ of $n \geq2$\textit{players}, or \textit{player names}, and for each player $i \in N$, a non-empty set $A_i$ of \textit{strategies} and a \textit{utility function}$u_i:A\rightarrow\mathbb{R}$, where $A$ denotes the set of \textit{strategy profiles}$\times_{i \in N}A_i$. We denote such a game as the triple $(N, A, u)$, where $u =(u_i)_{i \in N}$. If there exists $m \in\mathbb{Z}^+$ such that $|A_i| = m$ for all $i \in N$ then $(N, A, u)$ is called an $m$-\textit{strategy} game. A game $(N, A, u)$ is \textit{finite} when both $N$ is finite and $A_i$ is finite for all $i \in N$.

In this paper we will only concern ourselves with finite games, consequently all player sets and pure strategy sets are implicitly finite. It is noted in Mas-Collel, Whinston and Green \cite[Proposition 3.C.1]{Mas1995microeconomic} that for a set $X$ any rational preference relation may be described by some utility function, it is also worth checking Sections 3.B and 3.C, especially the definition of a rational preference relation (Definition 3.B.1). This would be a useful place to begin when trying to explore notions of symmetry and fairness for non-finite games (so a non-finite number of players and/or [possibly] non-finite strategy sets). %watch out for Can't Or/Cantor and Kant/Can't, what's the diagonal length of a triangle with straight side lengths of 1? Why did the chicken walk down the mobius strip? All pointed out independently and most likely originally by the actual Nick Ham, who typed this comment.

In this paper we will only concern ourselves with finite games, consequently all player sets and pure strategy sets are implicitly finite. It is noted in Mas-Collel, Whinston and Green \cite[Proposition 3.C.1]{Mas1995microeconomic} that for a set $X$ any rational preference relation may be described by some utility function, it is also worth checking Sections 3.B and 3.C, especially the definition of a rational preference relation (Definition 3.B.1). This would be a useful place to begin when trying to explore notions of symmetry and fairness for non-finite games (so a [possibly] non-finite number of players and/or [possibly] non-finite strategy sets, with at least one non-finite set involved). %watch out for Can't Or/Cantor and Kant/Can't, what's the diagonal length of a triangle with straight side lengths of 1? Why did the chicken walk down the mobius strip? All pointed out independently and most likely originally by the actual Nick Ham, who typed this comment.

Note that the strategy profiles, and consequently also the utility functions, of a game have an implicit ordering of the players. We refer to the place of each player in this order as their \textit{position}. For games when the player names are $\{1, \ldots, n\}$, unless otherwise specified, the names and positions coincide.