text:
\section{Introduction}

\begin{figure}
    \centering
    \begin{subfigure}{0.45\textwidth}
        \centering
        \includestandalone{figs/RPS}
        \caption{Rock-Paper-Scissors.}
        \label{fig:RPS}
    \end{subfigure}
    \hfill
    \begin{subfigure}{0.45\textwidth}
        \centering
        \includestandalone{figs/MP}
        \caption{Matching Pennies}
        \label{fig:MP}
    \end{subfigure}
    \caption{The fundamental graphs (Definition~\ref{def: fundamental graph}) of a symmetric game, Rock-Paper-Scissors (\ref{fig:RPS}), and a non-symmetric game, Matching Pennies (\ref{fig:MP}). Both games are zero-sum.}
    \label{fig:fundamental graphs}
\end{figure}

To quote \citet{hofbauer1998evolutionary}, ``a major task of game theory [is] to describe the dynamical outcome of model games described by strategies, payoffs and adaptive mechanisms." The best-studied adaptive mechanism
is the \emph{replicator dynamic}~\cite{taylor1978evolutionary,schuster1983replicator,hofbauer2003evolutionary,weibull1997evolutionary,sandholm2010population}. Since its introduction, this model has been extensively analysed by biologists, mathematicians, economists and computer scientists~\cite{hofbauer2003evolutionary,sandholm2010population,papadimitriou2016nash,kleinberg2011beyond}.
In much the same way that the Nash equilibrium~\cite{nash1951non} plays a central role in classical game theory, the replicator dynamic is the centrepiece of evolutionary game theory~\cite{weibull1997evolutionary,hofbauer2003evolutionary,sandholm2010population}.

Describing the `dynamical outcome' of games under the replicator dynamic involves answering a fundamental question: \emph{to which strategy profiles do we converge over time?}

A strategy profile to which all nearby profiles converge is called \emph{asymptotically stable}. Asymptotically stable profiles are `dynamical outcomes' of the game, but unfortunately they do not generally exist. In fact, we know that if such a profile exists it must be a \emph{strict Nash equilibrium}~\cite{weibull1997evolutionary,sandholm2010population}. Instead, we generically converge to a \emph{set} of points.
The set-valued generalisation of an asymptotically stable point is an \emph{attractor}~\cite{strogatz2018nonlinear,robinson1998dynamical}. Attractors are sets of points which are \emph{invariant} (points inside the set remain there for all time), \emph{asymptotically stable} (trajectories starting from nearby points converge to the set) and \emph{minimal} (they do not contain a smaller set with the first two properties).

Attractors are fundamental objects in dynamical systems theory, and by extension, the attractors of the replicator dynamic are fundamental objects in evolutionary game theory~\cite{zeeman1980population,benaim2012perturbations,hofbauer2003evolutionary,hofbauer2009time,weibull1997evolutionary,akin1984evolutionary}. Yet despite decades of research, \textbf{the attractors of the replicator dynamic have not been characterised}, even in zero-sum games, arguably the best-studied special case~\cite{akin1984evolutionary,sato2002chaos,hofbauer1996evolutionary,hofbauer1998evolutionary,hofbauer2003evolutionary,sandholm2010population,piliouras2014optimization,papadimitriou2016nash,bailey2018multiplicative}. We summarise the previous work on this problem in Section~\ref{sec: related work}. \textbf{In this paper we characterise the attractor of the replicator dynamic in zero-sum games}, and our solution is surprisingly elegant. We establish that an attractor exists, is unique, and attracts all points on the interior of the game. This attractor depends only on players' discrete preferences over their strategies, captured in an object we call the \emph{fundamental graph} (Definition~\ref{def: fundamental graph}) of the game.

The fundamental graph of a game is the key tool behind our results. The fundamental graph is a directed graph, whose nodes are the profiles of the game. There is an arc between profiles if they are \emph{comparable}, meaning that they differ in the strategy of a single player, with the arcs directed toward the preferred profile for that player. The fundamental graph is a combinatorial representation of the game, which captures precisely the order in which players prefer their strategies, given the strategies of the other players.

The games studied in evolutionary game theory come in two distinct flavours: single- and multiple-population.
The multiple-population case corresponds to standard games in classical game theory: each population corresponds to a \emph{player}, and the distribution of types in the population corresponds to a \emph{mixed strategy} for that player---a probability distribution over their pure strategies. An example is Matching Pennies~\cite{sandholm2010population}, where two players choose Heads or Tails, where one player wins if the coins match and the other wins if the coins mismatch. Importantly, just as in classical game theory, the payoff to a type in the population depends only on the other populations. On two-player zero-sum games, the fundamental graph coincides with an existing concept called the \emph{response graph}~\cite{papadimitriou2019game}. Figure~\ref{fig:MP} depicts the fundamental graph of Matching Pennies. In the single-population case\footnote{Single-population games are often called simply \emph{population games}~\cite{sandholm2010population}, or sometimes \emph{matrix games}~\cite{andrade2021learning}.} the payoff to a type depends on the proportion of types within the one population. In classical game theory, this corresponds to a two-player \emph{symmetric game}, where both players have the same payoffs and play the same strategy. A zero-sum example is the Rock-Paper-Scissors game. In these games profiles are strategies, and all profiles are comparable, so the fundamental graph is a tournament describing which strategies defeat which others. As an example, in Rock-Paper-Scissors, this graph is a 3-cycle between Rock, Paper and Scissors (Figure~\ref{fig:RPS}).

The replicator dynamic has distinct properties in the single- and multiple-population cases. Perhaps counter-intuitively, the single-population case is much more difficult to solve. A recent result~\cite{andrade2021learning} found that the replicator in a single-population game can approximate \emph{any} continuously differentiable dynamical system, suggesting that proving general results on single-population games is unlikely. Conversely, general properties of games under the multiple-population replicator dynamic are known, with one of the most important being \emph{volume preservation}, first established by \citet{akin1984evolutionary}. Specifically, after a differentiable change of variables on the interior of the strategy space, the volume of measurable sets is preserved under the replicator flow. Single-population games are not generally volume-preserving in this way. However, single-population \emph{zero-sum} games are volume-preserving~\cite{sandholm2010population}, and this is the case which we study in this paper. Volume conservation has an important consequence for our goal: asymptotically stable sets, (and thus attractors) cannot exist in the interior---instead they must intersect the boundary~\cite{akin1984evolutionary,weibull1997evolutionary}. This result connects the attractors of the replicator with the structure of the fundamental graph~\cite{biggar2022replicator}.

One aspect of our results is particularly important to emphasise. The replicator dynamic is defined in terms of payoffs, which represent the economic notion of `utility'. There has been long debate in economics and game theory on the merits of this utility-centred approach, beginning even before \citet{von2007theory}. Indeed, we find in this book the following quote: ``one may take the attitude that the only `natural' datum in this domain is the relation `greater,' i.e. the concept of preference." Even though the evolutionary approach---through its use of differential equations---seems to be taking us farther from the discrete structure of the preferences, our result shows that the long-run outcomes of the replicator dynamic in a zero-sum game depends solely on preferences, and not cardinal utility values. This has three important consequences. From a practical perspective, it (1) lends our predictions robustness (our models do not need precise knowlegde of utility to make predictions) and (2) allows us to compute the attractor efficiently from the fundamental graph (Section~\ref{sec: computing the attractor}). Finally, (3) from a philosophical perspective, this result supports the premise of the quote above, that discrete preferences form the fundamental structure of a game~\cite{biggar2022twoplayer}.

\subsection{Contributions}

The main result of the paper (Theorem~\ref{attractor characterisation}) states that under the (one- or two-population) replicator, every zero-sum game has a unique global attractor.
This attractor is defined by the fundamental graph. The fundamental graph decomposes into a collection of \emph{strongly connected components}, which are sets of nodes where there are paths in both directions between any pair of nodes in the set. A \emph{sink strongly connected component} (which we abbreviate to \emph{sink component}) is a strongly connected component with no arcs from a node inside the component to a node outside. The fundamental graphs of zero-sum games always have a \emph{unique} sink component~\cite{biggar2022twoplayer}\footnote{Non-zero-sum games, such as the $2\times 2$ Coordination game~\cite{papadimitriou2016nash}, may have fundamental graphs with multiple sink components, and thus can have multiple attractors under the replicator dynamic.}. We prove that the attractor is precisely the \emph{content} (Definition~\ref{def: content}, see~\citet{biggar2022replicator}) of the fundamental graph's unique sink component.
The sink component is a set of profiles; its content, and thus the attractor, is the set of \emph{mixed} profiles whose support contains only profiles in this component. The content of the sink component is an invariant set, and \citet{biggar2022replicator} have established that it is a lower bound for any attractors, so it remains only for us to show that in a zero-sum game it is asymptotically stable, for which we use a Lyapunov argument. The proof separates into the one- and two-population cases, reflecting the differences between the definition of the replicator and the associated fundamental graphs. The one-population case is straightforward. The two-population case is more complex, and involves embedding the two-population zero-sum game into a one-population zero-sum game we call the \emph{symmetrised game} (Definition~\ref{def:symised game}). To our knowledge, this symmetrised zero-sum game is a novel construction.

In Section~\ref{sec: nash} we discuss the geometry of this attractor. In one-population games, this attractor is always a face of the strategy simplex. In two-population games, this attractor can have more interesting structure; it is always a simply-connected CW complex~\cite{hatcher2002algebraic}, but is not necessarily a simplex, or even convex. More specifically, we extend a result from~\cite{biggar2022twoplayer} to show that the attractor is \emph{step-shaped} (Theorem~\ref{step shaped}), and we give a non-trivial example. Secondly, we connect our result to Nash equilibria, proving that (1) the Nash equilibrium of a two-player zero-sum game is always contained in a subgame with a strongly connected fundamental graph, and (2) that subgame is contained within this attractor. The former result establishes a novel relationship between the fundamental graph and Nash equilibria. Thirdly we discuss \emph{chain recurrence}~\cite{alongi2007recurrence}, proving that the attractor is also a \emph{sink chain component}~\cite{papadimitriou2019game}, and so our results give the first characterisation of the sink chain components of zero-sum games (see Section~\ref{sec: related work} and~\ref{sec: chain recurrence}). Chain recurrence in zero-sum games was thought~\cite{papadimitriou2016nash,papadimitriou2018nash} to relate to Nash equilibria; we find instead that it is a property of the fundamental graph. Finally, in Section~\ref{sec: computing the attractor} we discuss how to compute the sink component, and thus the attractor, in quasilinear\footnote{Specifically, given a game defined by an $n\times n$ matrix, we compute the sink component in $O(n^2\log n)$ time.} time.

\section{Related work} \label{sec: related work}

Since its inception, long-run stability of strategy profiles has been a core topic in evolutionary game theory~\cite{smith1973logic}. Early work was especially concerned with the relationship between classical solution concepts, such as Nash equilibria---particularly \emph{evolutionarily stable strategies}---and dynamics~\cite{hofbauer1998evolutionary}. The replicator dynamic originates in \cite{taylor1978evolutionary}, and was named by \citet{schuster1983replicator}. An early study of its attractors was done by~\citet{zeeman1980population}, who identified when evolutionarily stable strategies are attractors, and suggested that the qualitative behaviour of the replicator can be split into a finite number of classes; indeed our results show that in zero-sum games the qualitative behaviour is defined by the fundamental graph, of which there are only a finite number. \citet{akin1984evolutionary} studied zero-sum games under the replicator, proving the crucial \emph{volume conservation} property, analysed in depth by \citet{hofbauer1996evolutionary}. Further work established many key convergence properties for the replicator; for instance, that strict equilibria are attractors, and that dominated strategies vanish in the long run. See the textbooks~\cite{hofbauer1998evolutionary,weibull1997evolutionary,sandholm2010population}. Of particular interest is the results of~\citet{ritzberger1995evolutionary}, who showed that when a subgame is closed under ``weakly better responses", then it is an attractor under the replicator. The fundamental graph of a game is defined by the weakly better responses, and our results determine that the attractor is the content of the minimal set of (pure) profiles which is closed under weakly better responses. As we show in Section~\ref{subsec: geometry}, the sink component may not be a subgame of the game.

In the multiple-population case, what we call the fundamental graph coincides with an existing concept called the \emph{response graph}.
The response graph and its sink components have appeared in the game theory for some time~\cite{fabrikant2008complexity, kleinberg2011beyond, candogan2011flows, candogan2013dynamics}, but it is only more recently that they have become the subject of focused study. \citet{biggar2022twoplayer} characterise the response graphs of zero-sum games and show that the sink component is always unique in these games. \citet{papadimitriou2019game} present the sink components of the response graph as the core of their novel evolutionary solution concept of \emph{Markov-Conley chains}, based on the notion of \emph{chain recurrence}.

Despite the replicator's intuitive definition, the actual behaviour can be chaotic, even in simple zero-sum games~\cite{sato2002chaos}. Chaotic behaviour presents a significant conceptual challenge for predicting dynamical systems. To address this problem, \citet{papadimitriou2016nash,papadimitriou2018nash,papadimitriou2019game} proposed that we make use of \citeauthor{conley1978isolated}'s Fundamental Theorem of Dynamical Systems. This result is based on a notion called \emph{chain recurrence}. A chain recurrent point in a dynamical system is one which returns to itself in finite time, allowing for a finite number of infinitesimal `jumps'. Chain recurrence is the terminus of a sequence of concepts attempting to formalise the concept of points which `return to themselves', originating with \citet{H1881}. \citet{alongi2007recurrence} present a detailed treatment of the history and theoretical motivation behind chain recurrence. Chain recurrent points are grouped in connected components called \emph{chain components}. Again allowing for infinitesimal `jumps', points in one chain component can reach another; this defines a partial order on the chain components. \citet{papadimitriou2019game} propose that the \emph{sink} chain components, those that are minimal in this order, are the right solution concept for evolutionary game theory, capturing both stability (nearby points converge there) and recurrence (points remain there once reached). We discuss chain recurrence in Section~\ref{sec: chain recurrence}. Importantly, sink chain components are a slight generalisation of attractors (see Section~\ref{sec: chain recurrence}), and so our results in this paper additionally present the first characterisation of the sink chain components of zero-sum games.

While this concept applies to all game dynamics, \citet{papadimitriou2019game} use the replicator as the motivating example, and they link its chain recurrence to the response graph. Specifically, they suggested that the sink components of the graph are a good surrogate for the sink chain components. A number of follow-up papers have worked to identify the sink chain components of the replicator. \citet{omidshafiei2019alpha} proved that all attractors of the replicator dynamic contain sink components, and used this to motivate their proposed ranking methodology for strategies called $\alpha$-rank. \citet{papadimitriou2019game} conjectured, and \citet{biggar2022replicator} proved, that sink chain components exist in all games under the replicator. \cite{piliouras2014optimization,mertikopoulos2018cycles} established that in zero-sum games with interior Nash equilibria, all mixed profiles are recurrent under the replicator, so the sink chain component is the whole game. They showed further that if all equilibria are not interior, then all interior profiles converge to the smallest subgame containing all equilibria, which is called the \emph{essential subgame}. \citet{boone2019darwin} extended this to a class of games including both symmetric and non-symmetric zero-sum games. Perhaps counter-intuitively, these results do not provide a characterisation of the attractors (or sink chain components) of zero-sum games. While the essential subgame is contained within the attractor (Lemma~\ref{nash in attractor}), the essential subgame itself is not necessarily asymptotically stable---some nearby points move far away from the set before returning. We discuss this point in Section~\ref{sec: nash}, and give an example of a game where the attractor is strictly larger than the essential subgame. We show in fact that the recurrence of games with interior equilibria is a property of their fundamental graphs, rather than the location of the equilibria (Section~\ref{sec: chain recurrence}).

Most close to our work, \citet{biggar2022replicator} introduced the content of the sink components of the fundamental graph, and proved that this is a lower bound for the attractors of the replicator \emph{in any game}. They conjectured that this is also always an upper bound, that is, the attractors of the replicator are precisely the content of the sink components. By showing that this set is an attractor, we prove this conjecture in zero-sum games.

\section{Preliminaries} \label{sec: preliminaries}

In game theory, a game is defined by a triple consisting of the \emph{players}, \emph{strategy sets} for each player, and \emph{payoffs}. A combination of strategies for each player is called a \emph{strategy profile} or simply a profile, and for each profile there is a real-valued payoff to each player. In this paper we focus on two-player games, where we denote the players by the integers 1 and 2 and their strategy sets by $S_1$ and $S_2$. The strategy names are simply labels, so we assume $S_1 = [n] := \{i\in \mathbb{N}_0 |\ i < n\}$ and $S_2 = [m] := \{i\in \mathbb{N}_0 |\ i < m\}$. The profiles are the pairs $S_1\times S_2$. We call this an $n\times m$ game. An $n\times m$ game is represented by a pair $(A,B)$ of matrices with $A\in\mathbb{R}^{n\times m}$
 and $B\in \mathbb{R}^{m\times n}$ where the payoff in the profile $(s_1,s_2)$ is $A_{s_1,s_2}$ to player 1 and $B_{s_2,s_1}$ to player 2. A two-player game is \emph{zero-sum} if $A = -B^T$. In that case, we represent it by a single matrix $M$, assumed to be the payoffs to the first player, with the second player receiving payoffs of $-M^T$.
A \emph{subgame} of a game is formed by choosing subsets $T_1\subseteq S_1$ and $T_2\subseteq S_2$ of each player's strategy sets and restricting the game to the profiles in $T_1\times T_2$. We typically represent a subgame by its product set of profiles $T_1\times T_2$.

A \emph{mixed strategy} is a distribution over a player's strategies, and a \emph{mixed profile} is an assignment of a mixed strategy to each player. We sometimes refer to a profile as a \emph{pure profile} to distinguish it from a mixed profile. If $x$ is a mixed strategy for player $i$ and $s$ is a strategy for that player, we write $x_s$ for the $s$-entry of $x$. In two-player games, we will denote mixed strategies by pairs $(x,y)$ where $x$ and $y$ are distributions over the first and second player's strategies, respectively.
The \emph{support} of a mixed strategy is the set of strategies which have non-zero probability density. The define the \emph{support} of a mixed profile to be the set of profiles whose strategies are in the support of each player's mixed strategy. As distributions over a finite set, mixed strategies can be naturally embedded in the standard probability simplex in Euclidean space, by choosing some arbitrary ordering of the strategies in $S_1$ and $S_2$. We denote these spaces by $\Delta(S_1)$ and $\Delta(S_2)$. The set of mixed profiles is the product $\Delta(S_1)\times \Delta(S_2)$, which we call this the \emph{strategy space} of the game. It also has a natural embedding in Euclidean space, so we can talk about geometric properties of sets of mixed profiles

The payoffs of a game extends naturally to mixed profiles. In a game $(A,B)$, the \emph{expected payoff} of a mixed profile $(x,y)$ is $y^T A x$ to player 1 and $x^T B y $ to player 2.

\subsection{One-population games}

One-population games consist of a single set of strategies, denoted $S$, with $|S| = n$ where $|\cdot|$ returns the cardinality of its argument. `Strategy profiles' are simply individual strategies, and mixed profiles are distributions over $S$. The payoffs are defined by a matrix $M\in\mathbb{R}^{n\times n}$. The associated game is exactly the two-population \emph{symmetric} game $(M,M)$, where both players always play the same mixed strategy. For either player, the payoff for playing $s$ given the other player chose $t$ is $M_{s,t}$, and the expected payoff for the mixed profile $(x,x)$ is $x^TMx$ for both players.

A one-population game $M$ is zero-sum if $(M,M)$ defines a zero-sum game, that is $M = -M^T$. This is equivalent to requiring that $M$ is \emph{anti-symmetric}. Thus, every anti-symmetric real matrix defines a one-population zero-sum game in a natural way.

\subsection{Fundamental graphs}

Two profiles are \emph{$i$-comparable} if they differ only in the strategy of player $i$; they are \emph{comparable} if they are $i$-comparable for some player $i$. If two profiles are comparable, then there is exactly one $i$ such that they are $i$-comparable.
In a one-population game, profiles are strategies, and we define all profiles to be comparable. The strategic behaviour of the game is stored in the \emph{payoff differences} between comparable profiles~\cite{candogan2011flows}. We store this in a matrix we call the \emph{weight matrix} $W$ of the game.

\begin{defn}
    Let $M$ be a zero-sum game, and let $p$ and $q$ be comparable profiles. If $M$ is one-population, then profiles and strategies are the same, and we define $W_{p,q}$ to be the same as $M_{p,q}$. If $M$ is two-population, then $p=(p_1,p_2)$ and $q=(q_1,q_2)$ and
    \[
    W_{p,q} =
    \begin{cases}
    M_{p_1,p_2} - M_{q_1,q_2} & \text{$p$ and $q$ are $1$-comparable} \\
    M_{q_1,q_2} - M_{p_1,p_2} & \text{$p$ and $q$ are $2$-comparable}
    \end{cases}
    \]
    If $p$ and $q$ are not comparable, then $W_{p,q}$ \emph{is undefined}.
\end{defn}
We deliberately leave the payoff differences between incomparable profiles undefined, so that it is clear to the reader that we will only reference $W$ when the associated profiles are comparable. Note that for any comparable profiles $p$ and $q$, $W_{p,q} = -W_{q,p}$.
\begin{defn} \label{def: fundamental graph}
    Let $M$ be a zero-sum game. The \emph{fundamental graph} of $M$ is the graph whose nodes are the profiles of the game and where there is an arc $\arc{p}{q}$ between profiles $p$ and $q$ if and only if they are comparable and $W_{p,q} < 0$ (equivalently, $W_{q,p} \geq 0$).
\end{defn}
While the definition is the same for one- and two-population games, the graphs are not the same, because the weight matrix is defined differently. In one-population zero-sum games all profiles are comparable, so the fundamental graph is a \emph{tournament} (a directed graph with an arc between every pair of nodes). In two-population games the fundamental graph is the game's response graph~\cite{biggar2022twoplayer}, which is never a tournament because not all profiles are comparable. We think of the entries in the weight matrix as being weights on the associated arc, as in $\arc[|W_{p,q}|]{p}{q}$.
\subsection{Dynamical Systems} \label{sec: dynamical systems}
The replicator dynamic is a continuous-time dynamical system~\cite{sandholm2010population,hofbauer2003evolutionary}, defined by an ordinary differential equation. We first present the version for a two-population game $(A,B)$. In the following, $(x,y)$ is a mixed profile, with $s\in S_1$ and $t\in S_2$.
\begin{align*}
    \dot x_s &= x_s ((Ay)_s - x^T A y) \\
    \dot y_t &= y_t ((B x)_t - y^T B x) \\
\end{align*}
In the zero-sum case $(M,-M^T)$ this becomes
\begin{align*}
    \dot x_s &= x_s ((M y)_s - x^T M y) \\
    \dot y_t &= -y_t ((M^T x)_t - x^T M y)
\end{align*}
In a one-population game, the replicator is defined a similar way:
\begin{align*}
    \dot x_s &= x_s ((M x)_s - x^T M x)
\end{align*}
If the one-population game is zero-sum, then $x^T M x = 0$ because $M$ is anti-symmetric, and so this reduces to
\begin{align} \label{one pop replicator}
    \dot x_s &= x_s (M x)_s
\end{align}

The solutions to this equation define a \emph{flow}~\cite{sandholm2010population} on the strategy space of a game, which is a function $\phi:X\times \mathbb{R}\to X$ that is a continuous group action of the reals on $X$. We call this the \emph{replicator flow}. The forward orbit of the flow from a given point is called a \emph{trajectory} of the system. A set of points $Y$ is called \emph{invariant} under $\phi$ if $\phi(Y,t) = Y$ for any $t\in\mathbb{R}$. We are interested in sets that are both invariant and \emph{asymptotically stable}.

\begin{defn}[\citet{sandholm2010population}] \label{def: attractor}
Let $A$ be a compact set under a flow $\phi$ on a compact space $X$. If there is a neighbourhood $U$ of $A$ such that \[
\lim_{t\to\infty} \sup_{x\in U} \inf_{y \in A} \mathbf{d}(\phi(x,t), y) = 0
\]
where $\mathbf{d}$ is a metric, then we say $A$ is \emph{asymptotically stable}. If $A$ is also invariant, we call it an \emph{attracting set}. If $A$ is an attracting set, and no smaller attracting set is contained within it, we call it an \emph{attractor}. Attracting sets of the time-reversed flow $\phi^{-1}$ we call \emph{repelling sets}, and attractors of the reversed flow we call \emph{repellors}.
\end{defn}
There are some differences in terminology in the literature to be wary of. What we call attracting sets are sometimes called attractors (e.g.~\cite{conley1978isolated,sandholm2010population,biggar2022replicator}), in which case what we call an attractor is a \emph{minimal attractor}. An important result is the following:
\begin{lem}[\citet{sandholm2010population}, Proposition~9.B.1] \label{lem: trapping regions}
If there exists an open set $B$ with $\phi(\cl(B),\tau) \subset B $ for some $\tau>0$, then $A = \bigcap_{t\geq 0} \phi(B,t)$ is an attracting set. $B$ is called a \emph{weak trapping region}, and every attracting set has a weak trapping region.
\end{lem}
Attracting sets and repelling sets are linked via these regions.
\begin{lem}[\citet{sandholm2010population}]
Let $A$ be an attracting set, with $B$ a trapping region for $A$. Then $A^* := \bigcap_{t\leq 0} \phi(X\setminus B,t)$ is a repelling set, which we call the \emph{dual repelling set of $A$}.
\end{lem}
The set of points which approach an attractor $A$ in the limit $t\to\infty$ is called the attractor's \emph{basin of attraction}. We call an attractor \emph{global} if its basin of attraction includes all points in $\intr(X)$.

\section{The attractor of the replicator}

In this section we characterise the attractor of the replicator in (one- and two-population) zero-sum games (Theorem~\ref{attractor characterisation}). The crucial definition is that of the \emph{content} of a set of profiles.

\begin{defn}[\citet{biggar2022replicator}] \label{def: content}
Let $H$ be a collection of profiles in a game. The \emph{content} of $H$, denoted $\content(H)$, is the set of all mixed profiles $x$ where all profiles in the support of $x$ are in $H$.
\end{defn}

The content is a union of subgames, and it is well-known \cite{sandholm2010population} that every subgame is an invariant set under the replicator, so the content of any set of profiles is an invariant set (Lemma~\ref{lem: invariance}). We show now that the content of the sink component is a global attractor.

\begin{thm}[The attractor of the replicator] \label{attractor characterisation}
In a (one- or two-population) zero-sum game, the content of the unique sink component of the fundamental graph is the unique global attractor of the replicator dynamic.
\end{thm}

The proof of Theorem~\ref{attractor characterisation} requires showing that (1) the sink component is unique (Lemma~\ref{lem: uniqueness}), (2) the content is an invariant set (Lemma~\ref{lem: invariance}), (3) the content is an asymptotically stable global attractor (Proposition~\ref{upper bound}), and (4) showing that no other set can be an attractor, because every asymptotically stable set contains the content (Lemma~\ref{lem: lower bound}). The bulk of (1), (2) and (4) follow from established results in the literature. The main novel contribution of the proof is showing that the content is asymptotically stable and attracts all points in the interior of the game (Proposition~\ref{upper bound}).

We will make use of the following notation. Throughout, $M$ denotes our (one- or two-population) game and $H$ denotes its unique (Lemma~\ref{lem: uniqueness}) sink component. Mixed profiles are generally denote by $x$, pure profiles by $p$ and $q$, or $h$ when they are in $H$. Two notations are particularly important. Firstly, in one-population games, $x$ defines a distribution over strategies, and strategies are profiles, so $x$ defines a distribution over profiles. We write $x_p$ for the probability mass on $p$. In the two-population case $x$ is a pair $(x_1,x_2)$ of distributions over strategies, which again implicitly defines a distribution over profiles. Mimicking the one-population case, we write
\begin{equation} \label{x sub p}
x_p := {x_1}_{p_1} {x_2}_{p_2}
\end{equation}
to denote the probability mass on $p = (p_1,p_2)$. Secondly, given a set of profiles $H$ in a one- or two-population game, we write $x_H$ for the total probability mass on $H$, that is,
\begin{equation} \label{x sub H}
x_H := \sum_{h\in H} x_h
\end{equation}
\begin{lem}[Uniqueness] \label{lem: uniqueness}
    The fundamental graph of a zero-sum game has a unique sink component.
\end{lem}
\begin{proof}
    In the two-player (non-symmetric) case, the fundamental graph is the response graph, and the result follows from Theorem~4.10 of \cite{biggar2022twoplayer}. In the one-player (symmetric) case, the fundamental graph is a tournament, and all tournaments have one sink component, by the following argument. Suppose for contradiction that $a$ and $b$ are nodes in two distinct sink components of a tournament, so there are no paths from $a$ to $b$ or vice versa. But as a tournament, either the arc $\arc{a}{b}$ or $\arc{b}{a}$ exists, contradicting the previous statement.
\end{proof}

\begin{lem}[Invariance] \label{lem: invariance}
    The content of a set $H$ of profiles is an invariant set.
\end{lem}
\begin{proof}
    Observe that if $x\in \content(H)$, then the support of $x$ is contained in $H$, and because all mixed profiles in the subgame $\Delta(\supp(x))$ have the same support, $\Delta(\supp(x))\subseteq\content(H)$. It follows that $\content(H)$ is a union of subgames. By Theorem~5.4.7 of \cite{sandholm2010population}, all subgames are invariant sets under the replicator, and unions of invariant sets are invariant.
\end{proof}

\begin{lem}[$\content(H)$ is a lower bound] \label{lem: lower bound}
    Let $A$ be an attracting set under the replicator in a (one- or two-population) zero-sum game $M$. Then $\content(H)\subseteq A$, where $H$ is the sink component of the fundamental graph of the game.
\end{lem}
\begin{proof}
    By Theorems~9.1.2 and 9.1.6 of \cite{sandholm2010population}, no asymptotically stable set can exist in the interior of the strategy space of a one- or two-population zero-sum game. Subgames have the same properties as the whole game under the replicator so the same is true of all subgames. Dually, no repelling set can exist in the interior of any subgame. The remainder of the proof follows that of Theorem~5.2 of~\citet{biggar2022replicator}.

    (\emph{Claim}: every attracting set contains a profile.)
    This follows by induction, using the fact that the replicator dynamic on a subgame has the same properties as on the whole game. In the whole game, an asymptotically stable set intersects the boundary. This intersection with the boundary must also be asymptotically stable in any subgame it intersects on the boundary, and so it intersects the boundary of this smaller subgame, and so on. We conclude that such a set contains a pure profile, the smallest possible subgame.

    (\emph{Claim}: every attracting set contains all profiles in $H$.)
    An arc $\arc{p}{q}$ of the fundamental graph is also a subgame, where only the profile $p$ and $q$ are in the support. The (one- or two-population) replicator reduces to $\dot x_p = x_p(1-x_p) W_{q,p}$ on this subgame, where $W_{q,p} \geq 0$ (for the player for which these profiles are comparable). If $p$ is contained in an asymptotically stable set, then $ q$ must also be contained in this set, because points near $p$ move to $q$ along this arc. We know that asymptotically stable sets contain a pure profile---by this argument we deduce that they contain all pure profiles reachable from that one in the fundamental graph. Such a set of profiles always contains the sink component $H$.

    (\emph{Claim}: every attracting set contains $\content(H)$.)
    Let $Y$ be a subgame, where the pure profiles in $Y$ are in an attracting set $A$. If $Y$ is a pure profile, then all mixed profiles in $Y$ are in the set, trivially. Now suppose for induction that all points on the boundary of $Y$ are in $A$. Suppose for contradiction that there is a point $x\in\intr(Y)$ that is not in $A$. The set $A\cap Y$ is attracting in $Y$, and the boundary is contained in $A\cap Y$, but this means that the dual repelling set of $A\cap Y$ is contained in the interior of $Y$, but no such sets can be contained in the interior. Hence all of $Y$ is within $A$. By induction on subgames, we find that all of $\content(H)$ is within every attracting set.
\end{proof}

\begin{prop} \label{upper bound}
    Let $H$ be the sink component of the fundamental graph of a zero-sum game. Then $\content(H)$ is asymptotically stable under the replicator, and its basin of attraction is all profiles whose support intersects $H$.
\end{prop}
\begin{proof}
    We first prove the general statement, deferring the proofs of two technical lemmas (\ref{lem: symmetric lyapunov} and \ref{lem: two-pop lyapunov}), which we prove afterwards. We use a Lyapunov argument, a standard mechanism for demonstrating that a set is asymptotically stable~\cite{strogatz2018nonlinear}. We construct a scalar function, called a Lyapunov function, which is zero on the set and strictly decreasing in a neighbourhood of the set. Our choice of function is straightforward: $1 - x_H := 1 - \sum_{h\in H} x_h$ (equation~\ref{x sub H}).

    First observe that $1 - x_H = 0$ if and only if all of the profiles in the support of $x$ are in $H$, which is the definition of $\content(H)$, and so $1-x_H = 0$ if and only if $x\in\content(H)$. Otherwise, $1-x_H > 0$. \emph{We will show in Lemmas~\ref{lem: symmetric lyapunov} and \ref{lem: two-pop lyapunov} that under the one- and two-population replicator respectively, $\dot x_H > 0$ whenever $x_H\in (0,1)$, that is, the support of $x$ intersects $H$.}

    Let $\zeta \in (0,1)$, and define $H_\zeta = \{ x\in X\ | 1 -x_H < \zeta\}$. The probability mass $x_H$ on $H$ is a uniformly continuous function of the mixed profile $x$, and so $H_\zeta$ is an open neighbourhood of $\content(H)$. Note that the closure of $H_\zeta$ is $\cl(H_\zeta) = \{ x\in X\ | 1 -x_H \leq \zeta\}$. By Lemmas~\ref{lem: symmetric lyapunov} and \ref{lem: two-pop lyapunov}, $1 - x_H$ is strictly decreasing for each $x$ in $\cl(H_\zeta)$ with $x_H\in (0,1)$, and so after any time period $t>0$, $\phi(\cl(H_\zeta), t) \subset H_\zeta$. By Lemma~\ref{lem: trapping regions}, the intersection $A = \cap_{t\geq 0} \phi(\cl(H_\zeta), t)$ is an attracting set which must contain $\content(H)$. Because $\phi(\cl(H_\zeta), t) \subset H_\zeta$ holds for any $\zeta\in (0,1)$ we conclude that $1-x_H = 0$ for every $x\in A$, and so $A$ is exactly $\content(H)$, as required.
\end{proof}

We have now laid out the key structure of the proof, and we need only prove Lemmas~\ref{lem: symmetric lyapunov} and \ref{lem: two-pop lyapunov}. Unlike previous parts of the proof, these cases differ for one- and two-population games. The one-population case (Lemma~\ref{lem: symmetric lyapunov}) is straightforward, and we complete it below. For two-population games we must first embed our two-population zero-sum game $M$ in a one-population game which we call the \emph{symmetrised game}, denoted $\mathcal{S}_M$. We will define this and develop the appropriate theory in Section~\ref{subsec: symmetry}.

\begin{lem} \label{dot H lemma}
Let $M$ be one-population, and $H$ the sink component of its fundamental graph. Then, under the replicator,
\begin{align}
\dot x_H = \sum_{q\not\in H}\sum_{h\in H} x_q x_h M_{h,q}
\end{align}
\end{lem}
\begin{proof}
Note that $\dot x_H = \frac{\mathrm{d}}{\mathrm{d} t} \sum_{h\in H} x_h = \sum_{h\in H} \frac{\mathrm{d}}{\mathrm{d} t} x_h = \sum_{h\in H} x_h (M x)_h$ (equation~\ref{x sub H}). As $M$ is anti-symmetric,  $x_h x_q M_{h,q} + x_q x_h M_{q,h} = 0$ if $q\in H$. Hence,
\[
\dot x_H = \sum_{q\in S_1\times S_2}\sum_{h\in H} x_q x_h M_{h,q} = \sum_{q\not\in H}\sum_{h\in H} x_q x_h M_{h,q}.
\]
\end{proof}
\begin{lem} \label{lem: symmetric lyapunov}
    In a one-population zero-sum game $M$, under the replicator, if $x_H\in (0,1)$ then
\[
\dot x_H > 0,
\]
\end{lem}
\begin{proof}
    By Lemma~\ref{dot H lemma},
    \[
\dot x_H = \sum_{q\not\in H}\sum_{h\in H} x_q x_h M_{h,q}.
\]
For any $q\notin H$ and $h\in H$, as $H$ is a sink connected component, there is an arc $\arc{q}{h}$ in the fundamental graph. That is, $M_{h,q} > 0$. Hence, all summands are nonnegative. Moreover, due to the fact that $x_H \in (0,1)$, there exist a $q\not\in H$ and an $h\in H$ such that $x_qx_h > 0$. Thus, $\dot x_H > 0$.
\end{proof}

\subsection{Symmetrising a two-population game} \label{subsec: symmetry}

Our goal is to construct a one-population zero-sum game $\mathcal{S}_M$, which we call the \emph{symmetrised game}, into which we can embed the flow of the two-population replicator dynamic on a zero-sum game $(M, -M^T)$. The strategy space of $\mathcal{S}_M$ is $S_1\times S_2$. That is, the \emph{strategy profiles} of the two-player game become the \emph{strategies} of the symmetrised game. Mixed profiles also have a natural interpretation: the product distribution define by the mixed profiles, by $x_p := {x_1}_{p_1} {x_2}_{p_2}$ (equation~\ref{x sub p}). From Section~\ref{sec: preliminaries}, we know each single-population zero-sum game is defined by an anti-symmetric matrix representing the payoff differences between any pair of strategies, which in this case are profiles of the original two-population game. For profiles $p$ and $q$ that are comparable in the original two-population game, there is an obvious choice of payoff difference: the weight $W_{p,q}$. We construct the symmetrised game $\mathcal{S}_M$ by extending the relation given by $W$ over all pairs of profiles.

\begin{defn}\label{def:symised game}
Let $(M,-M^T)$ be a zero-sum game, with $M\in \mathbb{R}^{n\times m}$. The \emph{symmetrised game}, written $\mathcal{S}_M$ is defined as the following $\mathbb{R}^{(nm)\times (nm)}$ matrix, which we index by profiles $p = (p_1,p_2)$ and $q = (q_1,q_2)$:
\[
(\mathcal{S}_M)_{p,q} = M_{p_1,q_2} - M_{q_1,p_2}.
\]
\end{defn}
We may omit the subscript $M$ where it is clear from context. When constructing $\mathcal{S}$, we desired two properties: (1) that it be anti-symmetric (and so define a single-population zero-sum game) and (2) that it extends $W$ from comparable pairs of profiles to all profiles. We now verify that $\mathcal{S}$ satisfies both requirements.
\begin{lem}
$\mathcal{S}$ is anti-symmetric.
\end{lem}
\begin{proof}
Let $p = (p_1,p_2)$ and $q = (q_1,q_2)$ be profiles. $\mathcal{S}_{p,q} = M_{p_1,q_2} - M_{q_1,p_2} = -(M_{q_1,p_2} - M_{p_1,q_2}) = -\mathcal{S}_{q,p}$.
\end{proof}
\begin{lem}
If $p$ and $q$ are comparable then $\mathcal{S}_{p,q} = W_{p,q}$.
\end{lem}
\begin{proof}
Let $p = (p_1,p_2)$ and $q = (q_1,q_2)$. Suppose they are 1-comparable, so $p_2 = q_2$. Then $\mathcal{S}_{p,q} = M_{p_1,q_2} - M_{q_1,p_2} = M_{p_1,p_2} - M_{q_1,q_2} = W_{p,q}$. Similarly, if $p_1 = q_1$, then $\mathcal{S}_{p,q} = M_{p_1,q_2} - M_{q_1,p_2} = M_{q_1,q_2} - M_{p_1,p_2} = W_{p,q}$.
\end{proof}
Finally, we can also characterise $\mathcal{S}$ in terms of the arc weight matrix $W$.
\begin{lem} \label{lemma: symmetry weights}
Let $p = (p_1,p_2)$ and $q = (q_1,q_2)$ be profiles. Then the following holds:
\[
\mathcal{S}_{p,q} = W_{(p_1,q_2),p} + W_{(q_1,p_2),p} = W_{q,(p_1,q_2)} + W_{q,(q_1,p_2)}.
\]
\end{lem}
\begin{proof}
    \begin{align*}
        W_{p,(p_1,q_2)} + W_{p,(q_1,p_2)} &= (M_{p_1,q_2}-M_{p_1,p_2}) + (M_{p_1,p_2} - M_{q_1,p_2}) \\
        &= M_{p_1,q_2} - M_{q_1,p_2} = \mathcal{S}_{p,q}
    \end{align*}
    and
    \begin{align*}
        W_{(p_1,q_2),q} + W_{(q_1,p_2),q} &= (M_{p_1,q_2} -M_{q_1,q_2}) + (M_{q_1,q_2} - M_{q_1,p_2}) \\
        &= M_{p_1,q_2} - M_{q_1,p_2} = \mathcal{S}_{p,q}
    \end{align*}
\end{proof}

At this point, it is not clear why this definition is the `right' embedding of the two-population game into a one-population game. The following lemma justifies our choice, by showing that the replicator flow on the two-population game is exactly the one-population replicator flow on the symmetrised game, restricted to the subspace of product distributions.

\begin{lem} \label{symmetric replicator}
Let $(M,-M^T)$ be a two-player zero-sum game under the replicator dynamic. Let $(x,y)$ be a mixed profile and $p = (p_1,p_2)$ a pure profile. Write $z_p := x_{p_1}y_{p_2}$ as in equation~\ref{x sub p}. Then
\[ \dot z_p = z_p (\mathcal{S}_M z)_p. \]
\end{lem}
\begin{proof}
The two-population replicator dynamic (written for player 1, the player 2 case is similar) is equivalent to
\begin{align*}
    \dot x_s &= x_s ((M y)_s - x^T M y) \\
    &= x_s \sum_{t\in S_1} x_t ((M y)_s - (M y)_t) \\
    &= x_s\sum_{t \in S_1} x_t \sum_{r\in S_2}  y_r \left (M_{s,r} - M_{t,r} \right ) \\
    &= x_s \sum_{t\in S_1} \sum_{r \in S_2} x_t y_r \left (M_{s,r} - M_{t,r} \right ) \\
    &= x_s\sum_{p=(p_1,p_2)\in S_1\times S_2} z_p \left (M_{s,p_2} - M_{p_1,p_2} \right ).
\end{align*}

Now we observe that for $p = (p_1,p_2)$,
\begin{align*}
    \dot z_p &= \frac{\mathrm{d}}{\mathrm{d} t} (x_{p_1}y_{p_2}) \\
    &= (x_{p_1}y_{p_2}) (\frac{\dot x_{p_1}}{x_{p_1}} + \frac{\dot y_{p_2}}{y_{p_2}}) \quad \text{(product rule)} \\
    &= z_p \left(\sum_{q\in S_1\times S_2} z_q (M_{p_1,q_2} - M_{q_1,q_2}) + \sum_{q\in S_1\times S_2} z_q (M_{q_1,q_2} -M_{q_1,p_2})\right ) \quad \text{(by above)}\\
    &= z_p \sum_{q\in S_1\times S_2} z_q \left (M_{p_1,q_2} - M_{q_1,q_2} + M_{q_1,q_2} - M_{q_1,p_2} \right )\\

    &=  z_p \sum_{q\in S_1\times S_2} z_q \mathcal{S}_{p,q} \\
    &=  z_p (\mathcal{S} z)_p.
\end{align*}
\end{proof}

Notice that this expression is exactly the definition of the one-population replicator \eqref{one pop replicator} on the symmetric zero-sum game $\mathcal{S}_M$.
We can now complete the proof of Theorem~\ref{attractor characterisation} on two-population games by showing again that $1-x_H$ is decreasing in $x_H\in (0,1)$, as in Lemma~\ref{lem: symmetric lyapunov}, making use of the symmetrised game.

\begin{lem} \label{lem: two-pop lyapunov}
    In a two-population zero-sum game $M$, under the replicator, if $x_H\in (0,1)$ then
\[
\dot x_H > 0,
\]
\end{lem}
\begin{proof}
By Lemma~\ref{symmetric replicator}, we know that for a mixed profile $x = (x_1,x_2)$, and $h = (h_1,h_2)$ in $H$, $\dot x_h = \frac{\mathrm{d}}{\mathrm{d} t} ({x_1}_{h_1}{x_2}_{h_2})= x_h (\mathcal{S} x)_h $. The symmetrised game $\mathcal{S}$ is one-population, so we can use Lemma~\ref{dot H lemma}, giving
    \[
\dot x_H = \sum_{q\not\in H}\sum_{h\in H} x_q x_h \mathcal{S}_{h,q}.
\]
Now pick some $q\not\in H$ and $h\in H$ with $x_qx_h > 0$. Since $x_H \in (0,1)$, at least one such pair exist. We will show that the sum above is strictly positive. Firstly, observe that if $q$ and $h$ are comparable, the arc $\arc{q}{h}$ of the fundamental graph goes from $q$ to $h$ (and not vice versa), and so $\mathcal{S}_{h,q} = W_{h,q} > 0$. Otherwise, suppose  that $q = (q_1,q_2)$ and $h = (h_1,h_1)$ are not comparable, and let $a = (q_1,h_2)$ and $b = (h_1,q_2)$. We have the following three cases, visually summarised in Figure~\ref{fig:cases}:
\begin{figure}
    \centering
    \includestandalone{figs/cases}
    \caption{The three cases in the proof of Lemma~\ref{lem: two-pop lyapunov}. The enclosed region is the sink component $H$. In each case, $q\not\in H$ and $h\in H$.}
    \label{fig:cases}
\end{figure}
\begin{enumerate}
    \item $a,b\in H$. Then the arcs $\arc{q}{b}$ and $\arc{q}{a}$ are directed towards $a$ and $b$ because $q$ is outside the sink component. By Lemma~\ref{lemma: symmetry weights}, $\mathcal{S}_{h,q} = W_{q,a} + W_{q,b} > 0$.

    \item $a,b\not\in H$.
    Then the arcs $\arc{a}{h}$ and $\arc{b}{h}$ are directed towards $h$ because $a$ and $b$ are outside the sink component. By Lemma~\ref{lemma: symmetry weights}, $\mathcal{S}_{h,q} = W_{a,h} + W_{b,h} > 0$.

    \item $a\in H$, $b\not\in H$ ($b\in H$, $a\not\in H$ is identical). In this case the sum $\sum_{q\not\in H}\sum_{h\in H} x_q x_h \mathcal{S}_{h,q}$ includes the terms $x_qx_h \mathcal{S}_{h,q}$ and $x_bx_a \mathcal{S}_{a,b}$. However, $x_qx_h = q_1q_2h_1h_2 = x_bx_a$, and so $x_qx_h \mathcal{S}_{h,q} + x_bx_a \mathcal{S}_{a,b} = x_qx_h (\mathcal{S}_{h,q} + \mathcal{S}_{b,a}) = x_qx_h(W_{q,a} + W_{q,b} + W_{b,q} + W_{b,h}) = x_qx_h(W_{q,a} + W_{b,h})$ by Lemma~\ref{lemma: symmetry weights}. Since $a$ and $h$ are inside the sink component and $q$ and $b$ are outside, and the arcs $\arc{q}{a}$ and $\arc{b}{h}$ must be directed into the component, we conclude that $W_{q,a} + W_{b,h}$ is positive and so the sum of these terms is positive.
\end{enumerate}

Overall, we conclude that $\dot x_H > 0$.
\end{proof}

This completes the proof of  Theorem~\ref{attractor characterisation}.

\section{Attractor Geometry, Nash Equilibria and Chain Recurrence} \label{sec: nash}

In this section we discuss some immediate consequences of our characterisation of the attractor of a zero-sum game. In the first part we give a description of the geometric properties of the attractor, demonstrating that it is ``step-shaped" (Theorem~\ref{step shaped}) and thus \emph{simply-connected} (it has no holes), but not always convex. In the second part we discuss the relationship with Nash equilibria, proving that all Nash equilibria are within the attractor and are contained in a strongly connected subgame of the fundamental graph. Thirdly we discuss chain recurrence. Our results give the first complete characterisation of sink chain components in zero-sum games, and clarify the relationship between chain recurrence and equilibria.

\subsection{Attractor geometry} \label{subsec: geometry}

We have characterised the attractor of a zero-sum game, but what are the geometric properties of this set? In symmetric games, the answer is straightforward: the attractor is a simplex, because the content of a set of pure profiles is the sub-simplex spanned by those profiles. In the non-symmetric case, the attractor has more interesting and complex structure.

The content of the sink component of a two-player game is a union of connected subgames. Topologically, this is a space that can be formed by gluing together products of simplices, so is a finite CW complex~\cite{hatcher2002algebraic}. However, it need not itself be a subgame.

\begin{thm} \label{step shaped}
    In a two-player zero-sum game, the unique sink component $H$ of the fundamental graph is \emph{step-shaped}, that is, there exist subsets
    \begin{align*}
        s_1 &\subset s_2 \subset \dots \subset s_n \subseteq S_1 \\
        t_1 &\subset t_2 \subset \dots \subset t_n \subseteq S_2
    \end{align*}
    such that
    \[ H  = \bigcup_{i=1}^n (s_i \times t_{n-i+1})\]
    Additionally,
    \[ \content(H)  = \bigcup_{i=1}^n (\Delta (s_i) \times \Delta(t_{n-i+1}))\]
\end{thm}
\begin{proof}
    Let $H$ be the sink component. If $A = A_1\times A_2 \subseteq H$ is a subgame, with no other subgame $B$ such that $A\subset B \subseteq H$, we call $A$ \emph{maximal}. Now let $A_1\times A_2$ and $B_1\times B_2$ be distinct maximal subgames inside $H$. The subgame $(A_1\cup B_1) \times (A_2\cup B_2)$ contains both $A_1\times A_2$ and $B_1\times B_2$---because these are maximal, $(A_1\cup B_1) \times (A_2\cup B_2)$ cannot be within $H$. Thus there is a profile $(r,c)\not\in H$ with $r\in A_1\cup B_1$ and $c\in A_2\cup B_2$. Assume without loss of generality that $r\in A_1$. If $c\in A_2$ then $(r,c)\in A_1\times A_2 \subseteq H$, hence $c\in B_2\setminus A_2$. By identical reasoning, as $c\in B_2$ we must have $r\in A_1\setminus B_1$.

    (\emph{Claim}: $B_1\subset A_1$ and $A_2\subset B_2$.)

    Theorem~4.10 of \cite{biggar2022twoplayer} asserts that for any pairs $(a,b)\in H$ and $(c,d)\in H$ at least one of $(a,d)$ or $(c,b)$ must be in $H$. Choose any strategies $s\in B_1$ and $t\in A_2$. Then $(s,c)\in B_1\times B_2\in H$ and $(r,t)\in A_1\times A_2\in H$, and so at least one of $(s,t)$ or $(r,c)$ is in $H$ by this theorem~\cite{biggar2022twoplayer}. We know $(r,c)\not\in H$, so we conclude that $(s,t)\in H$. Hence $B_1 \subseteq A_1\cap B_1$ and $A_2\subseteq A_2\cap B_2$ which implies that $B_1\subset A_1$ and $A_2\subset B_2$. The inclusion is strict because $A_1\times A_2$ and $B_1\times B_2$ are maximal.

    If there are $n$ maximal subgames of the form $s_i\times t_i$, then for any pair $s_i\times t_i$ and $s_j\times t_j$ we have  (without loss of generality) $s_i\subset s_j$ and $t_j\subset t_i$. That is, the strategy sets $s_i$ for player 1 and $t_i$ for player 2 are totally ordered in reverse orders. As $H$ is the union of all maximal subgames, we conclude that $H = \cup_{i=1}^n (s_i\times t_i)$ with $i > j$ implying that $s_i\subset s_j$ and $t_j\subset t_i$.

    Finally, we characterise $\content(H)$. Let $x\in \content(H)$. Then the support of $x$ is some subgame $a\times b$ contained in $H$, so is contained within some maximal subgame $s_i\times t_i$, and so $x\in \Delta(s_i)\times \Delta(t_i)$. Conversely, if $x\in \Delta(s_i)\times \Delta(t_i)$ for some maximal subgame $s_i\times t_i$, then the support of $x$ is contained in $s_i\times t_i$, so is in $H$, so $x\in \content(H)$.
\end{proof}
The reason for the name ``step-shaped" becomes apparent when the theorem is expressed visually, as in Figure~\ref{fig:steps}. The theorem asserts that we can order the strategies of both players such that the sink component because shaped like a set of steps. There may be only one `step'---in that case the sink component is exactly a subgame. The simplest zero-sum game with an attractor demonstrating non-trivial structure (meaning, in this case, not a subgame) is the $3\times 3$ ``Outer Diamond game"~\cite{biggar2022twoplayer}, depicted in Figure~\ref{fig:outer diamond}. The attractor of this game is the union of two three-dimensional subgames, shown shaded in Figure~\ref{fig:outer diamond}. Using this step-shaped property, we can easily prove that the content of the sink component is simply connected~\cite{hatcher2002algebraic}. More specifically, it is a \emph{star domain}, that is there exists a point from the line segments to all other points are contained within the set.

\begin{figure}
    \centering
    \includestandalone{figs/steps}
    \caption{A visual interpretation of Theorem~\ref{step shaped}. The shaded region is the attractor $\content(H)$, composed of a union of subgames of the form $\Delta(s_i) \times \Delta(t_{n-i+1})$. The resultant set looks like a staircase.}
    \label{fig:steps}
\end{figure}
\begin{figure}
    \centering
    \includestandalone{figs/outer_diamond}
    \caption{A payoff matrix $M$. The fundamental graph of the associated zero-sum game $(M,-M)$ is called the ``Outer Diamond graph" in \cite{biggar2022twoplayer,biggar2022replicator}. This fundamental graph demonstrates how the content of the sink component of zero-sum games, and thus the attractor, can be non-convex.}
    \label{fig:outer diamond}
\end{figure}

\begin{lem}
    Let $H$ be the sink component of a zero-sum game. Then $\content(H) = \bigcup_{i=1}^n \Delta (s_i) \times \Delta(t_{n-i+1})$ is a star domain. Specifically, for any mixed profile $x\in \Delta (s_1)\times \Delta(t_1)$ and $y\in\content(H)$, the line segment between $x$ and $y$ is contained in $\content(H)$.
\end{lem}
\begin{proof}
    Let $x$ be a point in $\Delta (s_1)\times \Delta(t_1)$, and let $y$ be any point in $\content(H)$. Let $L$ be the line segment between them, that is, $L = \{ \theta x + (1-\theta) y\ |\ \theta \in [0,1]\}$. Fix any $\theta\in [0,1]$. The support of $\theta x + (1-\theta) y$ is the union of the support of $y$ and the support of $x$. The support of $x$ is contained in $s_1\times t_1$, and by Theorem~\ref{step shaped}, there exists some $i$ such that the support of $y$ is contained in $s_i\times t_{n-i+1}$. This is a superset of $s_1\times t_1$, so the support of $\theta x + (1-\theta) y$ is contained in $s_i\times t_{n-i+1}$, and all such points are inside $\content(H)$ by Theorem~\ref{step shaped}.
\end{proof}

\subsection{Nash equilibria}

In a two-player zero-sum game, the set of Nash equilibria is a connected convex set, which is generically a single profile. These equilibria live in a unique subgame, called the \emph{essential subgame}~\cite{mertikopoulos2018cycles} which is the largest subgame with a Nash equilibrium in its interior.
In this section we discuss two lemmas describing the relationship between the attractor of the replicator and Nash equilibria. The first is not so surprising: the essential subgame is contained within the unique attractor.
\begin{lem} \label{nash in attractor}
    In a zero-sum game, all Nash equilibria are contained within the attractor of the replicator.
\end{lem}
\begin{proof}
    Assume the attractor is not the whole game, in which case it is on the boundary, by Theorem~\ref{attractor characterisation}, all interior points are in the basin of attraction, and by \cite{piliouras2014optimization}, trajectories starting from interior points converge to the essential subgame in the limit, and so the essential subgame must be within the attractor.
\end{proof}
Though the essential subgame is contained within the attractor, it may be smaller than the whole attractor. As an example, the Nash equilibrium of the Outer Diamond game (Figure~\ref{fig:outer diamond}) has support $\{2,3\}\times \{1,2\}$, that is, the subgame consisting of the second two rows of $M$ and the first two columns. A result of \cite{piliouras2014optimization} showed that when the essential subgame of a zero-sum game is on the boundary, all interior trajectories converge to this subgame under the replicator. This does not contradict our findings, because when this subgame is smaller than the sink component (such as in the Outer Diamond game) it is not asymptotically stable, because this convergence is not uniform. There is no neighbourhood where all points approach the subgame.

The next result is more surprising: the subgraph of the fundamental graph induced by the essential subgame is always strongly connected. Again, the Outer Diamond game is an example: the $2\times 2$ subgame $\{2,3\}\times \{1,2\}$ containing the equilibrium is strongly connected. This fact is surprising because, though we use the replicator for the proof, the result is \emph{independent of the replicator}, and only relates equilibria and the fundamental graph.

\begin{lem} \label{strongly connected nash}
    In every two-player zero-sum game, any subgame that is the support of a Nash equilibrium has a strongly connected fundamental graph.
\end{lem}
\begin{proof}
    Let $x$ be a point in the interior of a game which does not have a strongly connected fundamental graph. This means there is a sink component $H$ whose content is an attractor (Theorem~\ref{attractor characterisation}). The point $x$ is in the basin of attraction of $\content(H)$, so $x$ converges to $\content(H)$, but Nash equilibria are fixed points under the replicator~\cite{sandholm2010population}, so $x$ cannot be a Nash equilibrium because it is not fixed.
\end{proof}

\subsection{Chain recurrence} \label{sec: chain recurrence}

 Our findings in the previous section have important consequences for the state-of-the-art understanding of \emph{chain recurrence} of the replicator dynamic in zero-sum games. We discussed chain recurrence in the introduction, and here we give a formal definition, along the lines of \cite{alongi2007recurrence}.
 \begin{defn}[Chain Recurrence]
Let $\phi$ be the flow of a dynamical system on a compact metric space $X$. Let $x$ and $y$ be points in $X$. There is an \emph{$(\epsilon,T)$-chain} from $x$ to $y$ if there is a finite sequence of points $x_1,x_2,\dots,x_n$ with $x=x_1$ and $y=x_n$, and times $t_1,\dots,t_n \in [T,\infty)$ such that $\mathbf{d}(\phi(x_i, t_i), x_{i+1}) < \epsilon$. If there is an $(\epsilon,T)$-chain from $x$ to $y$ for \emph{all} $\epsilon > 0$ and $T>0$ we say there is a \emph{pseudo-orbit} from $x$ to $y$. \textbf{A point is called \emph{chain recurrent} if it has a pseudo-orbit to itself.}
\end{defn}

Two points are \emph{chain equivalent} if there are pseudo-orbits between them in both directions, and equivalent chain recurrent points are grouped in connected components called \emph{chain components}. Reachability under pseudo-orbits provides an ordering on the chain components, and sink chain components are those which are minimal in this order. Chain recurrence~\cite{papadimitriou2016nash,papadimitriou2018nash} and particularly sink chain components~\cite{papadimitriou2019game} lie at the core of a new proposed approach to evolutionary game theory.

While sink chain components were recently proven to always exist under the replicator~\cite{biggar2022replicator}, they have only been characterised in potential games~\cite{papadimitriou2016nash,papadimitriou2018nash} and $2\times n$ strict\footnote{Games where the payoffs between comparable profiles are not equal.} games~\cite{biggar2022replicator}. In zero-sum games, \citet{papadimitriou2016nash,papadimitriou2018nash} proved that when the game has an interior equilibrium, all profiles are chain recurrent, and so the sink chain component is the whole game. We show now that attractors are always sink chain components when they exist, so our results give the complete characterisation of the sink chain components of zero-sum games.

\begin{lem}[Attractors are sink chain components]
    In any flow, every attractor is a sink chain component.
\end{lem}
\begin{proof}
    We first show that all points in the attractor are chain recurrent. Attracting sets are closed under intersection~\cite{kalies2021lattice,biggar2022replicator}, so an attractor cannot overlap any other attracting set---that would contradict the minimality of the attractor. \citeauthor{conley1978isolated} showed that points are chain recurrent if, for each attracting set $A$, the point is contained in either $A$ or its dual repelling set $A^*$. The attractor is compact, invariant, and no attracting set overlaps it, so an attracting or repelling set must contain all points in the attractor. Hence all points are chain recurrent.

    Pseudo-orbits cannot leave attracting sets \cite{akin1984evolutionary}. Consequently, no point outside the attractor is chain equivalent to a point inside it, and all points in the attractor are chain equivalent, so it is a chain component. It is a \emph{sink} chain component because no pseudo-orbits leave the set.
\end{proof}
 The proof~\cite{papadimitriou2018nash} that zero-sum games with interior equilibria are chain recurrent relies critically on the fact that an interior equilibrium exists, which appears to suggest that the presence of equilibria is closely connected to chain recurrence. Later, \citet{biggar2022replicator} proved that whenever a subgame of a game has a strongly connected fundamental graph, all profiles are chain recurrent, regardless of whether an interior Nash equilibrium exists. They additionally noted that no cases were known of a Nash equilibrium in the interior of a zero-sum game which was not strongly connected. We resolve this question in Lemma~\ref{recurrence and connectedness}: a game has all profiles chain recurrent \emph{if and only if} it is strongly connected (Lemma~\ref{recurrence and connectedness}). The fact that the subgame containing the equilibrium is chain recurrent \emph{is a consequence of graph structure}, and not of the equilibria. We summarise this in Figure~\ref{fig:chain recurrence}.

\begin{lem} \label{recurrence and connectedness}
    In a two-player zero-sum game, all profiles are chain recurrent under the replicator dynamic if and only if the fundamental graph is strongly connected.
\end{lem}
\begin{proof}
    If the fundamental graph is strongly connected, then all profiles are chain recurrent, by Corollary~5.3 of \cite{biggar2022replicator}. For the converse, observe that if the fundamental graph is not strongly connected, the sink component $H$ contains some but not all profiles, and so $\content(H)$ is an attractor which does not contain the whole game (Corollary~\ref{attractor characterisation}). Consider a point $x\in\intr(X)$. This point is not in $\content(H)$, but is in its basin of attraction, so is also not in its \emph{dual repellor}~\cite{alongi2007recurrence}. Consequently $x$ cannot be chain recurrent (Theorem~3.4.27 of \citet{alongi2007recurrence}).
\end{proof}

\begin{figure}
    \centering
    \includestandalone{figs/relations}
    \caption{The relationship between strong connectedness of the fundamental graph, chain recurrence, and Nash equilibria, as described in Lemmas~\ref{strongly connected nash} and \ref{recurrence and connectedness}}
    \label{fig:chain recurrence}
\end{figure}

\section{Computing the attractor} \label{sec: computing the attractor}

Because the attractor is the content of the sink component of the fundamental graph, identifying the attractor of a given game only requires computing the the sink component.
In general, computing the sink component of a graph can be done using Tarjan's algorithm~\cite{tarjan1972depth} in time $O(n+m)$ where $n$ is the number of nodes and $m$ is the number of arcs. Therefore, we can compute the attractor by constructing the fundamental graph and using Tarjan's algorithm to find the sink component.

In one-population games, this method is essentially optimal, because reading the payoff matrix input takes $O(n^2)$ operations, and the fundamental graph can be constructed in $O(n^2)$ steps and has $O(n^2)$ arcs, so our algorithm is $O(n^2)$ overall. However, in the case of an $n\times n$ two-population game, the fundamental graph has $O(n^3)$ arcs, and so this method instead takes $O(n^3)$ operations.
In this section we show how that we can improve the two-population result to $O(n^2\log n)$.

\subsection{Computing the sink component}

First, observe that if we ignore the orientations of the arcs, each node in the fundamental graph of an $n\times n$ game has degree $2(n-1)$, giving a total of $n^2(n-1)\in O(n^3)$ arcs. We will reduce the complexity by constructing another graph, called the \emph{reduced fundamental graph}. There is a path between two nodes in this graph if and only if there is such a path in the fundamental graph (Lemma~\ref{lem: reduced response}), so the sink component is the same. However, the reduced fundamental graph has only $O(n^2)$ arcs. Applying Tarjan's algorithm to this graph finds the sink component in $O(n^2)$ time. We can construct it using only $O(n^2\log n)$ operations.
\begin{defn}
    Let $(A,B)$ be a two-player game. The \emph{reduced fundamental graph} of the game is a subgraph of the fundamental graph which has an arc $\arc{p}{q}$ between $i$-comparable profiles $p$ and $q$ if $W_{p,q} < 0$ and there is no other $i$-comparable profile $r$ with $W_{p,r} < 0$ and $W_{r,q} < 0$.
\end{defn}

The reduced fundamental graph is constructed by taking the transitive reduction of the fundamental graph for each fixed strategy of one player, hence the name. We construct this graph $G$ from $(A,B)$ as follows. We first set the node set to be $S_1\times S_2$. Fix a strategy $s\in S_1$ for player 1. Then we consider the $s$-column of player 2's payoff matrix $B$, and we sort the entries by payoff, giving an order $t_1,t_2,\dots,t_n\in S_2$ on player 2's strategies, where $B_{t_1,s} \leq B_{t_2,s} \leq \dots\leq B_{t_n,s}$. We then add the following $n-1$ arcs to the graph $G$:
\[
\begin{tikzcd} (t_1,s) \ar[r,-Latex] & (t_2,s) \ar[r,-Latex] & \dots \ar[r,-Latex] & (t_n,s) \end{tikzcd}
\]
We repeat this process for every fixed $s\in S_1$, and then for every fixed $t\in S_2$. Each step requires reading and sorting $n$ entries of the matrix, which takes $O(n\log n)$ steps, and this is repeated $2n$ times, giving $O(n^2\log n)$ steps overall\footnote{If the payoffs are assumed to be integers in a fixed range, we can use \emph{counting sort}~\cite{cormen2022introduction} to sort the $n$ items in $O(n)$ time, reducing the overall complexity of the algorithm to $O(n^2)$.}. Finally, we show that this graph achieves our goal:

\begin{lem} \label{lem: reduced response}
    The reduced fundamental graph has $O(n^2)$ arcs, and there is a path between nodes $p$ and $q$ in the reduced fundamental graph if and only if there is such a path in the fundamental graph.
\end{lem}
\begin{proof}
    For each strategy of one player, we added $n-1$ arcs to the fundamental graph, giving a total of $2n(n-1)$ arcs in the overall graph, as required.

    The reduced fundamental graph is a subgraph of the fundamental graph, so we need only show that if there is a path between $p$ and $q$ in the fundamental graph, there is also a path between them in the reduced fundamental graph.

    Consider an arc $\arc{(a,s)}{(b,s)}$ between 1-comparable profiles. Then there is a path
    \[
    \begin{tikzcd} (a,s) \ar[r,-Latex] & (\alpha_1,s) \ar[r,-Latex] & \dots \ar[r,-Latex] & (\alpha_n,s) \ar[r,-Latex] & (b,s)\end{tikzcd}
    \]
    between $(a,s)$ and $(b,s)$ in the reduced fundamental graph. The 2-comparable case is identical. More generally, if we have a path $p,v_1,v_2,\dots,v_n,q$ between profiles $p$ and $q$ in the fundamental graph, we can replace each $i$-comparable arc by a path of $i$-comparable profiles in the reduced fundamental graph, which when concatenated form a path between $p$ and $q$.
\end{proof}

\section{Conclusions and Future Work}

In this paper we gave the first characterisation of the unique attractor of the replicator in zero-sum games, thereby describing the long-term behaviour of the dynamic. As a secondary result, we have demonstrated the importance of the fundamental graph as a tool for analysing game dynamics. In particular, in the long run, only the \emph{preferences} dictates the outcome of the game. This surprising conclusion has potentially significant consequences for the theory of utility. In game theory---especially evolutionary game theory---precise modelling of utilities is difficult; our results show that modelling only the preferences for each player is sufficient to characterise the attractor of the replicator dynamic.

An important goal for future research is to characterise the attractors of the replicator in \emph{all} games, not just zero-sum ones. Currently, the attractors of the replicator have been characterised in only a few games, such as $2\times n$ games and potential games. See \citet{biggar2022replicator} for a summary. Our paper adds zero-sum games to this list of solved classes. Some individual games, such as the Asymmetric Cyclic Matching Pennies game from \cite{kleinberg2011beyond}, have also had their attractors characterised. Remarkably, in all cases that we know of where an attractor has been characterised in the multiple-population\footnote{The single-population replicator dynamic, when not zero-sum, may have attractors that are not the content of sink components. For instance, interior mixed profiles may be asymptotically stable. See Theorem~8.5.2 of \cite{sandholm2010population}.} replicator dynamic, the attractors are \emph{always exactly the content of the sink components}. We conjecture that this is true in all games.

\begin{conj}
    In any game, the attractors of the multiple-population replicator dynamic are exactly the content of the sink components of the fundamental graph.
\end{conj}

A similar conjecture is presented by \citet{biggar2022replicator}, stated for sink chain components rather than attractors. In non-zero-sum games, the content of the sink components may not have the same properties as those we established in this paper. For instance, there may be more than one (such as the $2\times 2$ Coordination game~\cite{papadimitriou2016nash}), and the content may not be simply-connected (such as Asymmetric Cyclic Matching Pennies game~\cite{kleinberg2011beyond}). We believe that further study of the fundamental graph and its sink components will lead to greater understanding of this foundational problem in evolutionary game theory.

\bibliographystyle{ACM-Reference-Format}

redpajama_set_name: RedPajamaArXiv


text:
They Played the Game
Memories from 47 Major Leaguers
Norman L. Macht
Noted baseball historian Norman L. Macht brings together a wide‑ranging collection of baseball voices from the Deadball Era through the 1970s, including nine Hall of Famers, who take the reader onto the field, into the dugouts and clubhouses, and inside the minds of both players and managers. These engaging, wide-ranging oral histories bring surprising revelations—both highlights and lowlights—about their careers, as they revisit their personal mental scrapbooks of the days when they played the game.
Not all of baseball's best stories are told by its biggest stars, especially when the stories are about those stars. Many of the storytellers you'll meet in They Played the Game are unknown to today's fans: the Red Sox's Charlie Wagner talks about what it was like to be Ted Williams's roommate in Williams's rookie year; the Dodgers' John Roseboro recounts his strategy when catching for Don Drysdale and Sandy Koufax; former Yankee Mark Koenig recalls batting ahead of Babe Ruth in the lineup, and sometimes staying out too late with him; John Francis Daley talks about batting against Walter Johnson; Carmen Hill describes pitching against Babe Ruth in the 1927 World Series.
Norman L. Macht is the author of more than thirty books, including Connie Mack and the Early Years of Baseball (Nebraska, 2007); Connie Mack: The Turbulent and Triumphant Years, 1915–1931 (Nebraska, 2012); and The Grand Old Man of Baseball: Connie Mack in His Final Years, 1932–1956 (Nebraska, 2015); as well as numerous biographies for middle school readers including Cy Young, Babe Ruth, and Lou Gehrig. For more information about the author visit NormanMacht.com.
"[An] excellent oral history for fans with a taste for the game's past."—Wes Lukowsky, Booklist
"Wow, 47 pros whose memories cover playing days from 1912 to 1981. Wonderful mix of legends, good players, and those who had 'a cup of coffee.'"—John Vorperian, SABR Lajoie Chapter newsletter
"No matter how long his career lasted or how he compiled his statistics, a baseball player is always eager to share his experiences. Author Norman Macht has collected anecdotes from 47 former players who played from the Deadball Era of the early 20th century up to the 1970's in this fast-paced book."—Lance Smith, Guy Who Reviews Sports Books
"Noted baseball historian Norman L. Macht brings together a wide-ranging collection of baseball voices from the Deadball Era in the 1910s through the 1970s, including nine Hall of Famers, in the new book They Played The Game. They take the reader onto the field, into the dugouts and clubhouses, and inside the minds of both players and managers. These engaging, wide-ranging oral histories bring surprising revelations, both highlights and lowlights, about their careers, as they revisit their personal mental scrapbooks of the days when they played the game."—Jason Schott, Brooklyn Digest
"A fine collection of player memories. . . . Macht allows 47 players to take a wistful, candid and, in some cases, critical look at their baseball past. . . . Macht's mixture of player interviews works well."—Bob D'Angelo, Sports Bookie
"Interview books like this are like peanuts. Once you start reading, you can't stop. . . . Macht's collection is one that baseball fans won't want to miss."—John Rossi, Inside Game
From the interviews:
"They talk about those Yankee teams with all those hitters and pitchers. But we were the biggest bunch of red asses; we got on each other. . . . Only Joe DiMaggio didn't have to say anything. He just had to look at you."—Gene Woodling
"Drysdale and Koufax, who are throwing 90-plus on the black part of the plate and using the fastball to move batters back off the plate when we get ahead in the count—I defy somebody to get a hit. It's just not possible."—Johnny Roseboro
"Do I think we should have won some pennants during Leo's [Durocher] years in Chicago? Absolutely. We had the best talent in baseball and we didn't win. I don't know why. If we had won in '69, we probably would have won the next two or three years. But there was a stigma attached to not winning that year."—Don Kessinger
"Casey Stengel would never give you a direct answer to a question. If an interviewer asked him one question, he'd get four answers. And if you had four questions to ask, you'd never get past the first one."—George "Highpockets" Kelly
"Most of the guys who succeed in sports and businesses and life's accomplishments get a little dig in the ass going and God damn it pushing all the time. They want to be the best, they want to do things, they want to be remembered, they want to be on top of the heap. No question about it. And Lefty Grove was one of them son of a bitches."—Ted Williams
"Honesty has gotten more managers fired than incompetence."—Pat Corrales
1. Joe Adcock, 1950–1966
2. Richie Ashburn, 1948–1962
3. Elden Auker, 1933–1942
4. Dick Bartell, 1927–1946
5. Ray Berres, 1934–1945
6. Bill Bruton, 1953–1964
7. Ralph "Putsy" Caballero, 1944–1952
8. Jimmy Cooney, 1917–1928
9. Johnny Cooney, 1921–1928
10. Tony Cuccinello, 1930–1945
11. John Francis Daley, 1912
12. Joe DeMaestri, 1951–1961
13. Woody English, 1927–1938
14. Ferris Fain, 1947–1955
15. William "Dutch" Fehring, 1934
16. Dave "Boo" Ferriss, 1945–1948
17. Harry Gumbert, 1935–1950
18. Harvey Haddix, 1952–1965
19. Carmen Hill, 1915–1930
20. Sid Hudson, 1940–1954
21. Travis Jackson, 1922–1936
22. George "High Pockets" Kelly, 1920–1930
23. Don Kessinger, 1964–1979
24. Mark Koenig, 1925–1936
25. Ted Lyons, 1923–1946
26. Mike Marshall, 1967–1981
27. Barney McCosky, 1939–1963
28. Gil McDougald, 1951–1960
29. Sam Mele, 1947–1956
30. Wilmer "Vinegar Bend" Mizell, 1952–1962
31. Rocky Nelson, 1949–1961
32. Hal Newhouser, 1940–1954
33. Bill Nicholson, 1936–1953
34. Mickey Owen, 1937–1954
35. Mel Parnell, 1947–1956
36. Claude Passeau, 1936–1947
37. George Pipgras, 1923–1934
38. Johnny Roseboro, 1957–1973
39. Hal W. Smith, 1955–1964
40. Billy Sullivan Jr., 1931–1947
41. Bobby Thomson, 1946–1960
42. Bob Turley, 1953–1963
43. Broadway Charlie Wagner, 1938–1946
44. Monte Weaver, 1931–1939
45. Ted Williams, 1939–1960
46. Gene Woodling, 1943–1962
47. Don Zimmer, 1954–1965
Visit the author's website.
The Grand Old Man of Baseball
Connie Mack and the Early Years of Baseball
Foreword by Connie Mack III
Connie Mack
Football's Last Iron Men
Uncle Robbie
Jack Kavanagh and Norman Macht
Macho Row
William C. Kashatus
Robert C. Trumpbour and Kenneth Womack
Foreword by Mickey Herskowitz
From the Dugouts to the Trenches
Jim Leeke

redpajama_set_name: RedPajamaCommonCrawl


text:
Home > Center Affiliates > Michelle Steward
Michelle Steward
Research Affiliate & Advisor
Associate Professor of Marketing
Wake Forest University School of Business
stewarmd@wfu.edu | 336-758-4426
Dr. Michelle D. Steward (Ph.D., Arizona State University) is an Associate Professor at the School of Business at Wake Forest University. Michelle's research interests center on issues involving business-to-business marketing, and how internal working relationships drive superior performance. Within this research domain, she examines the interrelationships among salespeople, supply managers and their internal customers.
Michelle's research has been published in journals such as the Journal of the Academy of Marketing Science, Journal of Business Research, Journal of International Marketing, Journal of Business-to-Business Marketing, Omega: The International Journal of Management Science, Journal of Marketing Education, Journal of Marketing Theory & Practice, Business Horizons and Marketing Management Journal.
Michelle has received both the Faculty Scholarship Award and the T.B. Rose Fellowship in Business for Instructional Innovation at Wake Forest University's School of Business, and has been internationally recognized with her co-authors as the winners of the S. Tamer Cavusgil Award for the "most significant contribution to the advancement of the practice of international marketing management." In addition to teaching in the United States, Michelle has lived abroad and worked for the Ministry of Education in the Republic of South Korea. Michelle is an Associate Editor for the Journal of Marketing Education.


text:
Scheduling sex may actually be great for your relationship | Well+Good
The Busy-As-Hell Couple That Schedules Sex Together, Stays Together, Says One Pro
Mary Grace Garis
Photo: Getty Images/Cavan Images
Scheduling sex is some truly divisive territory. At best, it reads as a power move for folks who identify as Mirandas. At worst, is sounds like some horrifying Type A behavior that signals a lack of luster in your relationship. But if we're getting real for a second, what it may just be is a saving grace for the couple who keeps falling asleep after a never-ending work day.
Now, have I ever scheduled sex personally? Weeeeell, my partner reads everything I write, as do my ever-so-supportive my parent. So I'll just say no. But I've started pitching weekend getaways because our Google Cal is getting crowded with baby showers, weddings, and a platter of un-sexy family obligations. When errands and work and real life in general start to usurp so much time and energy that the minuscule amount remaining is only enough to lift the remote and flip on Queer Eye, it's legitimately tough to remember to prioritize intimacy with your loved one. But does this personal status put me and my relationship on the fast track to the most monotonous reality ever? Apparently not, says one pro. Phew.
"Scheduling sex is a way to show yourself and your partner that your sex life is valuable to you, and that you're willing to make an active effort to create the time and space for intimacy." —Vanessa Marin, sex therapist
"I'm a huge fan of scheduled sex," says Vanessa Marin, sex therapist and creator of The Passion Project, an online course for couples. "I know that the idea freaks a lot of people out. But the reality is that our lives are full and busy, and we need to schedule the things that are important to us. You plan things like family functions, special dates, and nights out with friends. Why should sex be any different?"
Excellent point! After all, we are busy, we are tired, and many of us are burnt out, all of which necessitates acts of self care and downtime to compensate for how effing busy we are. And when you're feeling burnt out, you (me) don't necessarily feel sexy. You (I) just feel like curling up with The Marvelous Mrs. Maisel and a weighted blanket and tell your partner, "Not tonight, bb." That may sound a little bit depressing (because it is a little bit depressing) and yet it doesn't have to indicative of a dying relationship.
"Scheduling sex is a way to show yourself and your partner that your sex life is valuable to you, and that you're willing to make an active effort to create the time and space for intimacy," Marin says.
So yeah, it may not be ideal that you don't have unlimited sex time like you did back in college, or whenever. But making the effort to book some boinking time shows that the passion is still there, somewhere between, oh, 7:30 and 8:15 p.m. But if you get a "No" on the Google Cal event invite? Well, that might warrant some concern.
BTW, if you're looking to change up your sex routine, you may want to get a sex pillow, STAT. And if you're into spicing up things via text, enjoy this quick guide to sexting like an erotica author.
Tags: Relationship Tips, Sex Advice


Summary/Abstract: Carsharing is becoming very popular in the western world as an effective method to contain rising prices of fuel and vehicles. It was brought about as a possible solution to the changing condition of growing economies and population where there is an increased need for travel but there is also a need for expense economizing. Due to its role in reducing fuel consumption, controlling pollution and traffic congestion, there is an increased usage of carsharing services as an alternative transportation option in big cities. Today, worldwide several different carsharing methods are being used.This paper examines the current situation of carsharing in the world and two initiatives for starting carsharing programs in Croatia. The main goal of this paper is to define what carsharing is, how successful it is in the world and if there is potential for it in Croatia. The most developed carsharing regions in the world are Western Europe and North America. In the last several years Asia, Australia and South America have started carsharing projects and have the biggest growth in members and vehicles. Carsharing is still not recognized in Croatia. There were a few initiatives for starting the program but they failed. In order to determine the possibility and potential of carsharing in Croatia it is necessary to conduct further market research.


Our wrap maxi dress is a must have piece for your Neon Gypsy wardrobe!
Wrap across front, v-neck, ties through the waist and around. Adjust to fit your waist and bust.
Uber flattering on multiple body shapes!
3/4 sleeve, maxi length with elongated side hems and a higher cut back hem.
Larger busts size up, alternatively this can be styled with a slip dress underneath and tie create and deep v-neck and ruched at the front – see photos!Just pair with one of our rayon jersey tank dresses underneath.
This is always a popular style and best seller.
NB: This print may have some print imperfections throughout, this is part of the printing process for this fabric and not considered a fault. Please be aware when purchasing.
Bridie is wearing Size S-M, she is 175cm Tall.


The view on the south side of Little Trees Hill, the highest point on the Gog Magog Hills, looking over towards Linton Water Tower and beyond, is quite splendid and rather special. So too is the ground lying beneath you as you stand there.
There is very little soil on this chalky hillside which makes it the ideal environment for chalk grassland and flowers to flourish, a habitat now very rare and one we have to nurture and cherish. Chalk grasslands have been in decline in the UK for the last 50 years. We have the opportunity to restore a truly wildlife rich habitat – up to 40 species of flowering plants can be found in one square metre of lowland calcareous grassland – it has even been called the European equivalent of the tropical rainforest. Along with a rich flora, butterflies and insects are returning to this south facing slope of Magog Down.
The previous heavy footfall across the top of the hill led the Governors to the decision to close the path, in order to allow the flora to recover. Additionally the area has been incorporated into a paddock, where sheep are grazed. This is the best management for chalk grassland.
Now that the flora is recovering, this paddock will be open for access [from later in February] but ONLY when there are no sheep grazing.
It will ALWAYS remain closed to dogs.
the area remains one for quiet recreation and for young children to play on grass free of dogs faeces.
The paddock will have to be shut whenever sheep are expected to graze so PLEASE respect the notices on the gates. There will be no through route with entrance and exit by one OR other of the gates on Shepherds's walk.
We appreciate your help and cooperation in keeping Magog Down a special place, particularly this rare hilltop.


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Kelly Services is ready to help you with your staffing needs.
Simply complete this form and a consultant will contact you to confirm your order.
Alternatively, you can contact our office directly. Click here for the contact details of the Hungarian Kelly office.


As the first female physician to have an obstetrics and gynecology practice in eastern Louisville, Dr. Terry's vision was to establish an atmosphere dedicated to women supporting women. Today, Women First has 11 board-certified physicians specializing in obstetrics and gynecology, and every patient has a Care Team dedicated to their health care. Comprised of a physician, nurse practitioner and/or physician assistant, nurses, clinical staff and others, our Care Teams work with you to understand your individual health concerns, medical history and personal challenges to better help you receive focused attention and care.
Championing women to take charge of their health care and themselves, we offer a full line of comprehensive services and have implemented leading technologies to offer the most advanced options in women's health care. 3D Tomosynthesis mammography, minimally invasive surgeries including da Vinci® robotics, in-office diagnostics and medical procedures, electronic communications and medical records, online check-in and more are just a few of the innovations we provide to meet your needs.
Our Women First Care includes annual wellness exams and mammograms; bone density testing; breast, cervical, uterine, ovarian and colon screenings; pelvic health addressing prolapse and urinary incontinence; plus help for abnormal bleeding, ovarian cysts, menopause, osteoporosis, and more with comprehensive diagnosis and treatment including surgery. Family Planning and Sexual Health services provide guidance for contraception, preconceptual health, and permanent birth control, while Maternity Care offers obstetrics services through pregnancy, delivery, and postpartum care.
With many of our physicians having been acknowledged as Top Docs and Top Surgeons in the Louisville area through the years, we are committed to providing exceptional medical services and the most advanced options in women's health with empathy and compassionate care—because of who we are caring for.
We love caring for you! Women…first, in all we do.


Direct mail is one of the most trusted and effective marketing mediums you can use. Whether you're promoting a product, service or brand, either B2B or B2C.
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The foundation of successful direct mail campaigns is the quality of the list. When you are using the right list you can use it to develop personalized & targeted marketing materials. This will help you establish and build long term relationships.
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You've done the work of establishing a relationship with your customers & leads, next you need to continue to interact with them in a positive and consistent manner. This can be through ongoing direct mail pieces, emails or call – whatever works best for your business model.
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When it comes to direct mail campaigns you don't need to go it alone. Micro Mailing Canada has been helping businesses develop and grow their direct mail campaigns for years. Visit them today to see how they can help you spend your direct mail campaign dollars effectively.


Weoley Castle's adult gaming centre, Cashino re-opened with a real bang on Saturday afternoon (October 7th) – with live performances from Malc Evans, two £50 cash draws, goodie bags and free bingo.
Cashino, which operates 174 gaming centres across the country, has enjoyed significant success in Weoley Castle. As such the decision was made earlier this year to refurbish the premises, and offer customers an even better entertainment experience. A part of the new offering includes Bingo Plus.
To celebrate the opening of the newly renovated venue, customers and passers-by were treated to goodie bags, special discounts and live entertainment throughout the day, including a special performance from Malc Evans.


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