\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, especially prior to making the discussions on fairness 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 the author 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.
\end{enumerate}
The author would also like to thank the people who check whether people are not being given an actual choice with things, being lied to, misled, held back for the wrong reasons, of if there are threats when accusations are made against people while almost everyone knows there are types of horrible abuse occurring around the globe and nothing is done about it when people are identified as one of the perpetrators, especially those voluntarily making threats, voluntarily helping carry them out, and/or voluntarily engaging in the wrong kind of abusing other people, animals and/or things. It is not okay for people to be stuck in abusive relationships, whether the abuse has ceased or not. The use of this work is gifted to people who are actually helping put an end to the global hostage crisis and helping with rescuing people from the wrong kinds of abuse.
\item a \textit{positive linear transformation} when there exists $\alpha\in\posreals$ and $\beta\in\reals$ such that $f(x)=\alpha{x}+ b$ for all $x \in X$.
\end{enumerate}
Note that the strictly increasing functions are a subgroup of the bijections from $\reals$ to $\reals$, and that the positive linear transformations are a subgroup of the strictly increasing functions. For proofs see \cite[Propositions ?? and ??]{ham2011honoursthesis}.
Note that the strictly increasing functions are a subgroup of the bijections from $\reals$ to $\reals$, and that the positive linear transformations are a subgroup of the strictly increasing functions. For proofs see \cite[Propositions 2.2.4 and 2.2.6]{ham2011honoursthesis}.
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.
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. 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.
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.
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@@ -201,7 +203,7 @@
\end{align*}
\end{enumerate}
Hence with $-n +\sum_{i \in N}|A_i|$ values are already specified the rest of $\sigma$ is uniquely determined. Consequently $\Delta(A)- f(\nabla(A))$ is non-empty. Note that we have said nothing about whether the values specified are independent from each other, it is irrelevant for this proof.
Hence with $-n +\sum_{i \in N}|A_i|$ values already specified the rest of $\sigma$ is uniquely determined. Consequently $\Delta(A)- f(\nabla(A))$ is non-empty. Note that we have said nothing about whether the values specified are independent from each other, it is irrelevant for this proof.
\end{proof}
\end{proposition}
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@@ -277,12 +279,12 @@
Given a $2$-player zero-sum game $\Gamma=(\{1, 2\}, A_1\times A_2, (u_1, u_2))$, the maximin and minimax values for each player are equal. Ie. $\underline{u}_i =\overline{u}_i$ for all $i \in\{1, 2\}$.
\begin{proof}
There is a number of proofs for this throughout the literature and on the internet, one shall be added and cited it later.
\end{proof}
\end{proposition}
\begin{proposition}
\begin{conjecture}
Given a zero-sum game $\Gamma=(N, A, u)$, each player's maximin value and minimax value are equal.
\begin{proof}
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@@ -290,9 +292,18 @@
Given a zero-sum game $\Gamma=(N, A, u)$, for each player $i \in N$ let $\Gamma' =(\{i, -i\}, A_i\times{A_{-i}}, (u^*_i, u^*_{-i}))$ be the $2$-player zero-sum game where $u^*_i = u_i$ and $u^*_{-i}=\Sigma_{j \in N-\{i\}}u_j$, ie. $u^*_{-i}(s)=\Sigma_{j \in N-\{i\}}u_j(s)$ for all $s \in A$.
It remains to show $\underline{u}_i =\underline{u}^*_i$ and $\overline{u}^*_i =\overline{u}_i$? Ie. need to prove/show/establish that:
For each $i \in N$, noting that it follows from Proposition \ref{prop:2pzsminmax=maxmin} that $\underline{u}^*_i =\overline{u}^*_i$, we have $\underline{u}_i =\underline{u}^*_i =\overline{u}^*_i =\overline{u}_i$.
\end{proof}
\end{proposition}
\end{conjecture}
We denote the subgroup relation as $\leq$, the group generated by a subset $H$ of a group $G$ as $\langle{H}\rangle$, the group of permutations on a non-empty set $X$ as $S_X$, and the subset of transpositions on $X$ as $T_X$. The reader is reminded that the permutations on $X$ are equivalent to the bijections from $X$ to itself, henceforth we will refer to them interchangeably.
@@ -12,10 +12,12 @@ The author notes that our somewhat unintuitive notation has been chosen so that
$s \mapsto\pi(s)$ is a left action of $S_N$ on $A$.
\begin{proof}
The identity permutation trivially acts as an identity so we need only establish associativity. For each $\pi, \tau\in S_N$, $s \in A$ and $i \in N$, $\bigl((\tau\circ\pi)(s)\bigr)_i = s_{(\tau\circ\pi)^{-1}(i)}= s_{\pi^{-1}(\tau^{-1}(i))}=\bigl(\pi(s)\bigr)_{\tau^{-1}(i)}=\Bigl(\tau\bigl(\pi(s)\bigr)\Bigr)_i$.
For each $s, s' \in A$. We have $\pi(s)=\pi(s')$ if and only if $s_{\pi^{-1}(i)}= s_{\pi^{-1}(i)}'$ for all $i \in N$ and for each $s \in A$, $\pi^{-1}(s)\in A$ and $\pi(\pi^{-1}(s))=(\pi\circ\pi^{-1})(s)= s$. Hence for each $s \in A$, $s \mapsto\pi(s)$ is both injective and surjective, ie. $s \mapsto\pi(s)\in\bij(A, A)$.
\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.
It might be worth explaining which bijections of $A$ we have using $\{s\mapsto\pi(s): \pi\in S_N\}$? Ie. $\bij(A)-\{s\mapsto\pi(s): \pi\in S_N\}$, though is trivially just the bijections where there's consistency with mapping strategies from one player to the same possibly other player. 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 $\pi^{-1}$.
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@@ -51,7 +53,7 @@ This gives us $\pi(\sigma_i) = \pi(\sigma)_{\pi(i)} \in \Delta(A_{\pi(i)})$ and
\end{proof}
\end{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 $\mathcal{A}= A^n$)
Note that when$A_i = A_j$ 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 $\set{\bij(\Delta(A_i), \Delta(A_j)): i, j \in N}$.
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@@ -228,15 +230,15 @@ Interestingly, fairness has not appeared much in the game theory literature. Her
\item\textit{fair} if $\underline{u}_i =\overline{u}_i =0$ for all $i \in N$;
\item\textit{standard strictly fair} if the
\item\textit{fully strictly fair} if
\item\textit{standard ordinally fair} if "there is a transitive subgroup of player permutations that preserve preferences over pure strategy profiles" (need to work out how to phrase this);
\item\textit{standard ordinally fair} if there is a transitive subgroup of player permutations that preserve preferences over pure strategy profiles (need to work out how to phrase this and establish that it is precise/concrete/unambiguous);
\item\textit{fully ordinally fair}
\item\textit{standard cardinally fair} if "there is a transitive subgroup of player permutations that preserve preferences over mixed strategy profiles" (if the players are indifferent between the positions of their opponents up to preserving preferences over the pure strategies when the strategy sets are matched up).
\item\textit{fully cardinally fair} if the players are indifferent between the positions of their opponents up to preserving preferences over the mixed strategy profiles when the strategy sets are matched up.
\item\textit{standard cardinally fair} if there is a transitive subgroup of player permutations that preserve preferences over mixed strategy profiles (need to work out how to phrase this and establish that it is precise/concrete/unambiguous, including if the players are indifferent between the positions of their opponents up to preserving preferences over the pure strategies when the strategy sets are matched up).
\item\textit{fully cardinally fair} if the players are indifferent between the positions of their opponents up to preserving preferences over the mixed strategy profiles when the strategy sets are matched up (need to work out how to phrase this and establish that it is precise/concrete/unambiguous).
\end{enumerate}
\end{definition}
\begin{definition}
(don't need this anymore) We shall refer to a game $\Gamma=(N, A, u)$ as \textit{maximin fair and minimax fair} if for each $i, j \in N$:
(don't need this anymore) We shall refer to a game $\Gamma=(N, A, u)$ as \textit{maximin fair} and \textit{minimax fair} if for each $i, j \in N$:
@@ -265,19 +267,11 @@ Interestingly, fairness has not appeared much in the game theory literature. Her
\end{proof}
\end{proposition}
Need to work out all the right statements for below..
\begin{proposition}
If a zero-sum game $\Gamma=(N, A, u)$ is standard symmetric then it is fair.
\begin{proof}
\end{proof}
\end{proposition}
\begin{proposition}
If a $2$-player zero-sum game is standard symmetric then it is fair.
\end{proposition}
Need to work out all the right statements for below.. Possible conjectures though:
\begin{enumerate}
\item If a $2$-player zero-sum game is standard symmetric then it is fair; and
\item If a zero-sum game $\Gamma=(N, A, u)$ is standard symmetric then it is fair.
\end{enumerate}
\subsection{Notions of Symmetry}\label{subsec:labeldepnotionsofsymmetry}
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@@ -374,7 +368,7 @@ Since $T_N \subseteq S_N$, it follows from Condition (v) in Theorem \ref{basicsy
We conclude this subsection by providing the reader with an accurate historical account of the mistake from \cite[Definition 7]{DMaskin} being identified. The mistake was first pointed out by the author with an edit on the 4\textsuperscript{th} of May 2011 to the Wikipedia page for symmetric games, which the author then revised on the 8\textsuperscript{th} of May 2011 due to not having a published reference for the author's claim that the definition is incorrect. Both of these edits are visible on the Wikipedia revision history for the symmetric games page \cite{WikiSGRV}. The mistake was also pointed out in the author's 2011 honours thesis \cite[Subsection 5.8]{ham2011honoursthesis}.
Upon contacting the authors from \cite{DMaskin} in 2018, the author received a response from Maskin confirming that they made a slight mistake, which unintentionally made the definition of symmetry stronger than intended. Maskin suggested the mistake did not affect their own results, but has had the unfortunate effect of possibly leading other researchers astray. Prior to 2011 \cite{DMaskin} had 949 citations, and as at December 2018 it has 1,374 citations, so the author feels it is a good idea for the mistake to be noted to hopefully avoid any researchers being lead astray in the future.
Upon contacting the authors from \cite{DMaskin} in 2018, the author received a response from Maskin suggesting that they made a slight mistake, unintentionally making the definition of symmetry given stronger than intended. Maskin suggested the mistake did not affect their own results, but has had the unfortunate effect of possibly leading other researchers astray. Prior to 2011 \cite{DMaskin} had 949 citations, and as at December 2018 it has 1,374 citations, so the author feels it is a good idea for the mistake to be noted to hopefully avoid any researchers being led astray in the future.
The mistake was also pointed out independently by Vester in his 2012 Masters thesis \cite[Appendix B]{vester2012symmetric}, who also proved the statement in Theorem \ref{DMprop}. Theorem \ref{DMprop} does not appear in the author's honours thesis, a proof was first released by the author publicly with the first revision of this paper uploaded to the arXiv November 2013, see \cite[Version 1]{ham2018arxivversion}. Hence credit goes to Vester for first releasing a proof publicly, see \cite[Theorem 32]{vester2012symmetric}.