Part of the time spent typesetting this paper was supported by a Tasmania Graduate Research Scholarship (186). The author would especially like to express his gratitude towards Jeremy Sumner for proof-reading early drafts and Des FitzGerald who allowed me to explore symmetric games for the thesis component of my honours year at the University of Tasmania and who has been useful for advice since then. The author also thanks Asaf Plan who identified a mistake with the proof of Theorem \ref{basicsymequivthm}, which the author has since corrected, and a number of anonymous referees who have given useful comments that have improved the exposition of this paper along with suggesting a number of relevant references.
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 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 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.
@@ -126,7 +126,7 @@ Our second reason is that there are weaker notions of fairness that cannot be ca
Game isomorphisms induce an equivalence relation where games in the same equivalence class have the same strategic structure. There is a finite number of ordinal equivalence classes for games with both a fixed number of players and fixed number of strategies for each of the players. Goforth and Robinson \cite{GoforthRobinson} counted 144 ordinal equivalence classes for the 2-player 2-strategy games.
\subsection{Bijections Acting on Strategy Profiles}
The bijections $S_{\Gamma}$ from a game to itself form a group that acts on the players and strategy profiles. In fact for an $m$-strategy game $S_{\Gamma}$ is isomorphic to the wreath product $S_N \wr S_M$ where $M =\{1, ..., m\}$, which may be seen by setting $A_i = M$ for all $i \in N$.
The bijections $S_{\Gamma}$ from a game to itself form a group that acts on the players and strategy profiles. In fact for an $m$-strategy game $S_{\Gamma}$ is isomorphic to the wreath product $S_N \wr S_M$ where $M =\{1, \ldots, m\}$, which may be seen by setting $A_i = M$ for all $i \in N$.
Given a game bijection $g =\bigl(\pi; (\tau_i)_{i \in N}\bigr)\in S_{\Gamma}$, we refer to $\pi$ as \textit{the player permutation used by $g$} and say that two game bijections $g, h \in S_{\Gamma}$\textit{have the same player permutation} if the player permutations used by $g$ and $h$ are identical.
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@@ -176,7 +176,7 @@ Hence strategy trivial subgroups have at most one bijection for each player perm
It follows that all paths from one player to another map the strategies in a canonical manner. Hence if $G$ is also player transitive then the strategy sets are matched such that they can be treated as the same set. We now introduce \textit{matchings} to formalise what is meant by the strategy sets being matched.
\begin{definition}
A \textit{matching of $A_1, ..., A_n$} is a relation $M \subseteq\times_{i \in N} A_i$ which is $i$-total and $i$-unique for all $i \in N$.
A \textit{matching of $A_1, \ldots, A_n$} is a relation $M \subseteq\times_{i \in N} A_i$ which is $i$-total and $i$-unique for all $i \in N$.
\end{definition}
\begin{example}\label{matchingeg}
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@@ -265,7 +265,7 @@ We denote by $M(n, m)$ the set of matchings for an $n$-player $m$-strategy game.
\end{enumerate}
\end{example}
There are a number of ways to count the number of matchings in $M(n, m)$. Below we present one, though note an alternative is to establish that $M(n, m)\cong\bij(A_1, A_2)\times...\times\bij(A_{n-1}, A_n)$. %\newpage
There are a number of ways to count the number of matchings in $M(n, m)$. Below we present one, though note an alternative is to establish that $M(n, m)\cong\bij(A_1, A_2)\times\ldots\times\bij(A_{n-1}, A_n)$. %\newpage
\begin{lemma}\label{matchinglemma1}
For each $n \geq2$: $M(n, 2)$ is a partition of $A$; and $|M(n, 2)| =2^{n-1}$.
\subsection{Notions of Fairness}\label{subsec:labelindepnotionsoffairness}
\subsection{Notions of Fairness (work in progress)}%\label{subsec:labelindepnotionsoffairness}
\begin{definition}
We shall refer to a game $\Gamma=(N, A, u)$ as:
\begin{enumerate}
\item\textit{strictly fair} if the
\item\textit{$n$-transitively strictly fair} if
\item\textit{ordinally fair} if "there is a transitive subgroup of bijections that preserve preferences over pure strategy profiles" (need to work out how to phrase this);
\item\textit{$n$-transitively ordinally fair}
\item\textit{cardinally fair} if "there is a transitive subgroup of bijections that preserve preferences over mixed strategy profiles".
\item\textit{$n$-transitively cardinally fair} if
\end{enumerate}
\end{definition}
\subsection{Notions of Symmetry}\label{subsec:labelindepnotionsofsymmetry}
Similar to our label-independent characterisations of our label-dependent notions of anonymity, Theorem \ref{strattrivmatchingthm} gives us the following label-independent characterisations of our label-dependent notions of fairness.
\begin{corollary}\label{indchar1}
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@@ -63,7 +77,7 @@ If all transitive subgroups of $S_N$ had regular subgroups then games with a tra
We will see in Example \ref{sixplayereg} that games which have a transitive subgroup $G$ isomorphic to $\overrightarrow{\Gamma}$ with $G_N =\{\id_{\Gamma}\}$ need not be standard symmetric.
So far we have considered many notions of symmetry, see Subsections \ref{subsec:labeldepnotionsofsymmetry} and \ref{subsec:labelindepnotionsoffairness}, which we have also been referring to as notions of fairness.
So far we have considered many notions of symmetry, see Subsections \ref{subsec:labeldepnotionsofsymmetry} and \ref{subsec:labelindepnotionsofsymmetry}, which we have also been referring to as notions of fairness.
When the author was explaining the similarities between symmetry and fairness in the context of games to James East, James posed the analogy of cake cutting to the author. Among numerous relevant topics within the area of fair division, there are several types of problems that have been studied, for example: