Part of the time spent conducting research for and typesetting this paper was supported by a Tasmania Graduate Research Scholarship (186). The author would especially like to express his gratitude towards:
\begin{enumerate}
\item Des FitzGerald, without whom the author would unlikely have been able to explore symmetric games for the thesis component of his honours year at the University of Tasmania, and who has been useful for advice since, particularly during times where the author has largely been dismissed and/or ignored by others;
\item Des FitzGerald, without whom the author would unlikely have been able to explore symmetric games for the thesis component of his honours year at the University of Tasmania, and who has been useful for advice since;
\item Jeremy Sumner for proof-reading early drafts with a number of useful suggestions;
\item Noah Stein who introduced the author to his doctoral work at MIT and patiently answered a number of questions from the author, many of which were rather stupid in hindsight. Note the author purposefully omitted Noah from the acknowledgements in previous versions of this paper uploaded to the arXiv due to the author's uncertainty about whether the author would be dismissed as a crank, especially prior to making the discussions on fairness much more concrete and precise;
\item Noah Stein who introduced the author to his doctoral work at MIT and patiently answered a number of questions from the author, many of which were rather stupid in hindsight. Note the author purposefully omitted Noah from the acknowledgements in previous versions of this paper uploaded to the arXiv due to the author's uncertainty about whether the author would be dismissed, especially prior to making the discussions on fairness more concrete and precise;
\item Asaf Plan who identified a mistake with the proof of Theorem \ref{basicsymequivthm}, which the author has since corrected (the author apologises that he forgot to acknowledge Asaf in at least one version uploaded to the arXiv since Asaf pointed the mistake out);
\item several anonymous referees whose useful comments have helped improve the exposition of this paper along with suggesting a number of relevant references; and
\item Martin Osborne, whose {\tt sgamevar}{\LaTeX} style file, see \cite{SGAMESTY}, was used for the example games throughout this paper.
\section{Morphisms Between Games}\label{sec:morphisms}
There are two important reasons why our simplifying assumption that players have the same strategy labels leaves our analysis incomplete. Our first reason is that relabelling the strategies for a standard symmetric game leads to a strategically equivalent game that may no longer be considered symmetric inside our label-dependent framework.
Ideally we want to be able to determine when two games merely differ by player and strategy labels without having to go through and check all possible rearrangements of the labels.
@@ -11,9 +11,9 @@ Under the theme of anonymity rather than fairness, Brandt et al. \cite{brandt200
A number of people have examined notions of symmetry which may not be captured inside our label-dependent framework, see for example Nash \cite{NashNCG}, Peleg et al. \cite{peleg1999canonical}, Sudh\"{o}lter et al. \cite{sudholter2000canonical} and Stein \cite{NoahXE}. In order to discuss and analyse such notions we will need to make a detour to examine morphisms between games, the complexity of which has been investigated by Gabarr\'{o} et al. \cite{IsoComplexity}. Inside what will later be referred to as our label-independent framework game automorphisms act on strategy profiles, which also allows players to have distinct strategy labels.
We begin in Section 2 by reviewing numerous mathematical concepts that will play an important role throughout our analysis. In Section 3 we survey various label-dependent notions of anonymity and fairness.
We begin in Section \ref{sec:background} by reviewing numerous mathematical concepts that will play an important role throughout our analysis. In Section \ref{sec:labdepnotions} we survey various label-dependent notions of anonymity and fairness.
In Section 4 we review game morphisms while showing that game bijections and game isomorphisms form groupoids, which appears to be missing from relevant literature, and introduce matchings as a convenient characterisation of strategy triviality.
In Section \ref{sec:morphisms} we review game morphisms while showing that game bijections and game isomorphisms form groupoids, which appears to be missing from relevant literature, and introduce matchings as a convenient characterisation of strategy triviality.
Finally, in Section 5 we survey various label-independent notions of fairness, discuss how to classify a given game, and outline how to construct and partially order parameterised symmetric games with numerous examples that range over various classes.
Finally, in Section \ref{sec:labindnotions} we survey various label-independent notions of fairness, discuss how to classify a given game, and outline how to construct and partially order parameterised symmetric games with numerous examples that range over various classes.
\section{Label-Dependent Notions of Symmetry}\label{sec:labdepnotions}
There are various ways to define a notion of symmetry, not all of which are distinct. In each case we need all players to have the same number of strategies, consequently all games are implicitly $m$-strategy games. It is often assumed when defining symmetric games that all players have the same strategy labels and any notion of symmetry will treat the same labels as equivalent. We shall refer to these as \textit{label-dependent} notions.
\subsection{Permutations Acting On Strategy Profiles}
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@@ -15,6 +15,8 @@ The author notes that our somewhat unintuitive notation has been chosen so that
\end{proof}
\end{lemma}
It might be worth explaining which bijections of $A$ we have from $\{s\mapsto\pi(s): \pi\in S_N\}$?? Same for $\{s\mapsto g(s): g \in\bij(\Gamma)\}$ later.
Since $\pi^{-1}(s)=(s_{\pi(i)})_{i \in N}$ for all $s \in A$, $s \mapsto\pi(s)$ and $s \mapsto\pi^{-1}(s)$ are dual to each other. Hence the dual results hold for $\pi^{-1}$.
\begin{lemma}
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@@ -49,7 +51,9 @@ This gives us $\pi(\sigma_i) = \pi(\sigma)_{\pi(i)} \in \Delta(A_{\pi(i)})$ and
\end{proof}
\end{proposition}
\begin{proposition}
Note that since $A_i = A_j$ for all $i, j \in N$, the following groupoids in Propositions \ref{prop:firstgroupoidprop} and \ref{prop:secondgroupoidprop} are all groups (or are they? the first yes but what about the second with regards to order/positions of the tuples?). (should they be rephrased, perhaps even with different notation?? what to denote the $A_i$'s as though? Maybe $A_0$? or $A'$ or $\overline{A}$ or $\underline{A}$ or do strategy profiles as $\mathcal{A}$ but would confuse people, as would using $\mathcal{A}= A^n$)
\begin{proposition}\label{prop:firstgroupoidprop}
$\set{\sigma_i \mapsto\pi(\sigma_i): i \in N, \pi\in S_N}$ is a subgroupoid of $\set{\bij(\Delta(A_i), \Delta(A_j)): i, j \in N}$.
\begin{proof}
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@@ -64,7 +68,7 @@ This gives us $\pi(\sigma_i) = \pi(\sigma)_{\pi(i)} \in \Delta(A_{\pi(i)})$ and