# Spatial Parrondo games
with spatially dependent game $A$

Sung Chan Choi Department of Mathematics, University of Utah, 155 S. 1400 E.,
Salt Lake City, UT 84112, USA. e-mail<EMAIL_ADDRESS>
###### Abstract

Parrondo games with spatial dependence were introduced by Toral (2001) and
have been studied extensively. In Toral’s model, $N$ players are arranged in a
circle. The players play either game $A$ or game $B$. In game $A$, a randomly
chosen player wins or loses one unit according to the toss of a fair coin. In
game $B$, which depends on parameters $p_{0},p_{1},p_{2}\in[0,1]$, a randomly
chosen player, player $x$ say, wins or loses one unit according to the toss of
a $p_{m}$-coin, where $m\in\\{0,1,2\\}$ is the number of nearest neighbors of
player $x$ who won their most recent game. In this paper, we replace game $A$
by a spatially dependent game, which we call game $A^{\prime}$, introduced by
Xie et al. (2011). In game $A^{\prime}$, two nearest neighbors are chosen at
random, and one pays one unit to the other based on the toss of a fair coin.
Noting that game $A^{\prime}$ is fair, we say that the Parrondo effect occurs
if game $B$ is losing or fair and game $C^{\prime}$, determined by a random or
periodic sequence of games $A^{\prime}$ and $B$, is winning. We investigate
numerically the region in which the Parrondo effect appears. We give
sufficient conditions for the mean profit in game $C^{\prime}$ to converge as
$N\to\infty$. Finally, we compare the Parrondo region in the model of Xie et
al. with that in the model of Toral.

##  1 Introduction

The Parrondo effect, in which there is a reversal in direction in some system
parameter when two similar dynamics are combined, is the result of an
underlying nonlinearity. It was first described by Spanish physicist J. M. R.
Parrondo in 1996 in the context of games of chance: He showed that it is
possible to combine two fair or losing games, $A$ and $B$, to produce a
winning one, $C$. Here $C$ is the game obtained by playing games $A$ and $B$
in a random or periodic sequence. His motivation was to provide a discrete (in
time and space) version of the so-called flashing Brownian ratchet of Ajdari
and Prost [1]. Other versions of Parrondo’s games followed, including Toral’s
[2] spatially dependent games. These games were modified by Xie et al. [3],
and it is the goal of this paper to explore the latter games in greater depth
than was done by Ethier and Lee [4].

###  1.1 Toral’s spatially dependent games

Toral [2] introduced what he called cooperative Parrondo games with spatial
dependence. (We prefer the term spatially dependent Parrondo games so as to
avoid conflict with the field of cooperative game theory.) The games depend on
an integer parameter $N\geq 3$, the number of players, and four probability
parameters, $p_{0},p_{1},p_{2},p_{3}$. (This is a slight generalization of the
model described in the abstract.) The players are arranged in a circle and
labeled from 1 to $N$ (so that players 1 and $N$ are adjacent). At each turn,
a player is chosen at random to play. Suppose player $x$ is chosen. In game
$A$, he tosses a fair coin. In game $B$, he tosses a $p_{m}$-coin (i.e., a
coin whose probability of heads is $p_{m}$), where $m\in\\{0,1,2,3\\}$ depends
on the winning or losing status of his two nearest neighbors. A player’s
status as winner (1) or loser (0) is decided by the result of his most recent
game. Specifically,

$m=\begin{cases}0&\text{if $x-1$ and $x+1$ are both losers,}\\\ 1&\text{if
$x-1$ is a loser and $x+1$ is a winner,}\\\ 2&\text{if $x-1$ is a winner and
$x+1$ is a loser,}\\\ 3&\text{if $x-1$ and $x+1$ are both
winners,}\end{cases}$

where $N+1:=1$ and $0:=N$ because of the circular arrangement of players.
Player $x$ wins one unit with heads and loses one unit with tails. Replacing
$(p_{0}.p_{1},p_{2},p_{3})$ by $(p_{0},p_{1},p_{1},p_{2})$ gives the
3-parameter model described in the abstract.

These games have been studied in detail in a series of papers by Ethier and
Lee [5, 6, 7, 8]. For example, with Toral’s [2] choice of parameters, namely
$(p_{0},p_{1},p_{2},p_{3})=(1,0.16,0.16,0.7)$, one can compute the asymptotic
profit per turn to the set of $N$ players, for $3\leq N\leq 19$. For $N=5,6$
and $9\leq N\leq 19$, the Parrondo effect (where game $A$ is fair, game $B$ is
fair or losing, and the random mixture, game $C:=\frac{1}{2}A+\frac{1}{2}B$,
is winning) is present. In the cited papers, a strong law of large numbers and
a central limit theorem are obtained. In particular, the asymptotic cumulative
profits per turn exist and are the means in the SLLN. Further, it seems clear
that these means converges as $N\to\infty$. This has been proved under certain
conditions (see Ethier and Lee [7]).

###  1.2 The spatially dependent games of Xie et al.

Notice that Toral’s [2] game $A$ is not spatially dependent (i.e., the rules
of the game do not depend on the spatial structure of the players). Xie et al.
[3] proposed a modification of game $A$ that is spatially dependent as well as
being a fair game. To distinguish, we call that game $A^{\prime}$. As before,
the games depend on an integer parameter $N\geq 3$, the number of players, and
four probability parameters, $p_{0},p_{1},p_{2},p_{3}$. The players are
arranged in a circle and labeled from $1$ to $N$ (so that players $1$ and $N$
are adjacent). At each turn, a player is chosen at random to play. Suppose
player $x$ is chosen. In game $A^{\prime}$, he chooses one of his two nearest
neighbors at random and competes with that neighbor by tossing a fair coin.
The results is a transfer of one unit from one of the players to the other,
hence the wealth of the set of $N$ players is unchanged. Game $B$ is as
before. Player $x$ wins one unit with heads and loses one unit with tails.

These games were studied by Xie et al. [3], Li et al. [9], and Ethier and Lee
[4]. Only the random mixture case was treated, and convergence of the means
has not yet been addressed. Our aim in this paper is to fill in these gaps in
the literature. Further, we want to understand this model as well as Toral’s
model is understood.

We begin by establishing a strong law of large numbers and a central limit
theorem, especially in the periodic pattern case. We compute various means
numerically and use computer graphics to visualize the Parrondo region. Then
we address the issue of convergence of means, which involves certain
interacting particle systems. We then establish the convergence, both in the
random mixture setting and in the periodic pattern setting, on a large subset
of the parameter space.

##  2 SLLN/CLT for the games of Xie et al.

In this section, we restate the strong law of large numbers (SLLN) and the
central limit theorem (CLT) of Ethier and Lee [10], and we apply them to the
Parrondo games of Xie at al. [3].

###  2.1 SLLN and CLT

Ethier and Lee [10] proved an SLLN and a CLT for the Parrondo player’s
sequence of profits, motivated by the random mixture $C:=\gamma
A+(1-\gamma)B$. A subsequent version in the same paper treats the case of
periodic patterns.

Consider an irreducible aperiodic Markov chain $\\{X_{n}\\}_{n\geq 0}$ with
finite state space $\Sigma_{0}$. It evolves according to the one-step
transition matrix ${\bm{P}}=(P_{ij})_{i,j\in\Sigma_{0}}$. Let us denote its
unique stationary distribution by the row vector
${\bm{\pi}}=(\pi_{i})_{i\in\Sigma_{0}}$. Let
$w:\Sigma_{0}\times\Sigma_{0}\mapsto{\bf R}$ be an arbitrary function, which
we write as a matrix ${\bm{W}}=(w(i,j))_{i,j\in\Sigma_{0}}$ and refer to as
the payoff matrix. Define the sequences $\\{\xi_{n}\\}_{n\geq 1}$ and
$\\{S_{n}\\}_{n\geq 1}$ by

$\xi_{n}:=w(X_{n-1},X_{n}),\qquad n\geq 1,$

and

$S_{n}:=\xi_{1}+\cdots+\xi_{n},\qquad n\geq 1.$

Let ${\bm{\Pi}}$ denote the square matrix each of whose rows is ${\bm{\pi}}$,
and let ${\bm{Z}}:=({\bm{I}}-({\bm{P}}-{\bm{\Pi}}))^{-1}$ denote the
fundamental matrix. Denote by $\dot{\bm{P}}$ and $\ddot{\bm{P}}$ the Hadamard
(entrywise) products $\bm{P}\circ\bm{W}$ and $\bm{P}\circ\bm{W}\circ\bm{W}$
(so $\dot{P}_{ij}:=P_{ij}w(i,j)$ and $\ddot{P}_{ij}:=P_{ij}w(i,j)^{2}$). Let
$\bm{1}:=(1,1,\ldots,1)^{\textsf{T}}$ and define

$\mu:=\bm{\pi}\dot{\bm{P}}\bm{1}\quad{\rm
and}\quad\sigma^{2}:=\bm{\pi}\ddot{\bm{P}}\bm{1}-(\bm{\pi}\dot{\bm{P}}\bm{1})^{2}+2\bm{\pi}\dot{\bm{P}}(\bm{Z}-\bm{\Pi})\dot{\bm{P}}\bm{1}.$

######  Theorem 1 (Ethier and Lee [10]).

Under the above assumptions, and with the distribution of $X_{0}$ arbitrary,

$\frac{S_{n}}{n}\to\mu\;\;{\rm a.s.}$

and, if $\sigma^{2}>0$,

$\frac{S_{n}-n\mu}{\sqrt{n\sigma^{2}}}\to_{d}N(0,1).$

If $\mu=0$ and $\sigma^{2}>0$, then
$-\infty=\liminf_{n\to\infty}S_{n}<\limsup_{n\to\infty}S_{n}=\infty$ _a.s._

######  Example 2.

To illustrate this theorem, let us consider the original capital-dependent
Parrondo games (without a bias parameter). These are single-player games. In
game $A$, the player tosses a fair coin. In game $B$, the player tosses a
1/10-coin if capital is divisible by 3 and and a 3/4-coin otherwise. In either
case, the player wins one unit with heads and loses one unit with tails. The
underlying Markov chain corresponding to game $B$ has state space
$\Sigma_{0}:=\\{0,1,2\\}$ and one-step transition matrix

$\bm{P}_{B}:=\begin{pmatrix}0&1/10&9/10\\\ 1/4&0&3/4\\\
3/4&1/4&0\end{pmatrix}.$

Its unique stationary distribution is $\bm{\pi}_{B}=(1/13)(5,2,6)$. The payoff
matrix has the form

$\bm{W}:=\begin{pmatrix}0&1&-1\\\ -1&0&1\\\ 1&-1&0\end{pmatrix}.$

We find that

$\mu_{B}=\bm{\pi}_{B}\dot{\bm{P}}_{B}\bm{1}=0.$

The underlying Markov chain corresponding to game $A$ has the same state space
and one-step transition matrix

$\bm{P}_{A}:=\begin{pmatrix}0&1/2&1/2\\\ 1/2&0&1/2\\\ 1/2&1/2&0\end{pmatrix}$

with unique stationary distribution $\bm{\pi}_{A}=(1/3)(1,1,1)$. The payoff
matrix is the same, and we find that

$\mu_{A}=\bm{\pi}_{A}\dot{\bm{P}}_{A}\bm{1}=0,$

a result that is obvious without calculation. Finally, the underlying Markov
chain corresponding to game $C:=\frac{1}{2}A+\frac{1}{2}B$ has the same state
space and one-step transition matrix

$\bm{P}_{C}:=\frac{1}{2}(\bm{P}_{A}+\bm{P}_{B})=\begin{pmatrix}0&3/10&7/10\\\
3/8&0&5/8\\\ 5/8&3/8&0\end{pmatrix}$

with unique stationary distribution $\bm{\pi}_{C}=(1/709)(245,180,284)$. The
payoff matrix is the same, and we find that

$\mu_{C}=\bm{\pi}_{C}\dot{\bm{P}}_{C}\bm{1}=\frac{18}{709}\approx 0.0253879.$

This is perhaps the best-known example of Parrondo’s paradox, and the SLLN
justifies the conclusion: Two fair games combine to win.

We can also derive a CLT, which requires the fundamental matrix

$\bm{Z}_{B}:=(\bm{I}-(\bm{P}_{B}-\bm{\Pi}_{B}))^{-1}=\frac{1}{2197}\begin{pmatrix}1725&-38&510\\\
-95&1938&354\\\ 425&118&1654\end{pmatrix}.$

We find that

$\sigma^{2}_{B}=\bm{\pi}_{B}\ddot{\bm{P}}_{B}\bm{1}-(\bm{\pi}_{B}\dot{\bm{P}}_{B}\bm{1})^{2}+2\bm{\pi}_{B}\dot{\bm{P}}_{B}(\bm{Z}_{B}-\bm{\Pi}_{B})\dot{\bm{P}}_{B}\bm{1}=\bigg{(}\frac{9}{13}\bigg{)}^{2}\approx
0.479290.$

Similarly,

$\bm{Z}_{A}:=(\bm{I}-(\bm{P}_{A}-\bm{\Pi}_{A}))^{-1}=\frac{1}{9}\begin{pmatrix}7&1&1\\\
1&7&1\\\ 1&1&7\end{pmatrix},$

hence

$\sigma^{2}_{A}=\bm{\pi}_{A}\ddot{\bm{P}}_{A}\bm{1}-(\bm{\pi}_{A}\dot{\bm{P}}_{A}\bm{1})^{2}+2\bm{\pi}_{A}\dot{\bm{P}}_{A}(\bm{Z}_{A}-\bm{\Pi}_{A})\dot{\bm{P}}_{A}\bm{1}=1,$

as is obvious without the formula. Finally,

$\bm{Z}_{C}:=(\bm{I}-(\bm{P}_{C}-\bm{\Pi}_{C}))^{-1}=\frac{1}{502681}\begin{pmatrix}392265&22884&87532\\\
23585&408580&70516\\\ 80305&39900&382476\end{pmatrix},$

and we conclude that

$\sigma^{2}_{C}=\bm{\pi}_{C}\ddot{\bm{P}}_{C}\bm{1}-(\bm{\pi}_{C}\dot{\bm{P}}_{C}\bm{1})^{2}+2\bm{\pi}_{C}\dot{\bm{P}}_{C}(\bm{Z}_{C}-\bm{\Pi}_{C})\dot{\bm{P}}_{C}\bm{1}=\frac{311313105}{356400829}\approx
0.873492.$

In each case we have a CLT.

Next we turn to another SLLN and CLT of Ethier and Lee [10], this one
motivated by the case of periodic patterns.

Let $\bm{P}_{A}$ and $\bm{P}_{B}$ be one-step transition matrices for Markov
chains in a finite state space $\Sigma_{0}$. Fix integers $r,s\geq 1$. Assume
that $\bm{P}:=\bm{P}_{A}^{r}\bm{P}_{B}^{s}$, as well as all cyclic
permutations of $\bm{P}_{A}^{r}\bm{P}_{B}^{s}$, are ergodic, and let the row
vector $\bm{\pi}$ be the unique stationary distribution of $\bm{P}$. Let
$\bm{\Pi}$ be the square matrix each of whose rows is equal to $\bm{\pi}$, and
let ${\bm{Z}}:=({\bm{I}}-({\bm{P}}-{\bm{\Pi}}))^{-1}$ be the fundamental
matrix of $\bm{P}$. Given a real-valued function $w$ on
$\Sigma_{0}\times\Sigma_{0}$, define the payoff matrix
$\bm{W}:=(w(i,j))_{i,j\in\Sigma_{0}}$. Define
$\dot{\bm{P}}_{A}:=\bm{P}_{A}\circ\bm{W}$,
$\dot{\bm{P}}_{B}:=\bm{P}_{B}\circ\bm{W}$,
$\ddot{\bm{P}}_{A}:=\bm{P}_{A}\circ\bm{W}\circ\bm{W}$,
$\ddot{\bm{P}}_{B}:=\bm{P}_{B}\circ\bm{W}\circ\bm{W}$, where $\circ$ denotes
the Hadamard (entrywise) product. Let

$\mu_{[r,s]}:=\frac{1}{r+s}\bigg{[}\sum_{u=0}^{r-1}{\bm{\pi}}{\bm{P}}_{A}^{u}\dot{\bm{P}}_{A}\bm{1}+\sum_{v=0}^{s-1}{\bm{\pi}}{\bm{P}}_{A}^{r}{\bm{P}}_{B}^{v}\dot{\bm{P}}_{B}\bm{1}\bigg{]},$

and

$\displaystyle\sigma_{[r,s]}^{2}$
$\displaystyle=\frac{1}{r+s}\bigg{[}\sum_{u=0}^{r-1}[\bm{\pi}\bm{P}_{A}^{u}\ddot{\bm{P}}_{A}\bm{1}-(\bm{\pi}\bm{P}_{A}^{u}\dot{\bm{P}}_{A}\bm{1})^{2}]$
$\displaystyle\qquad\quad+\sum_{v=0}^{s-1}[\bm{\pi}\bm{P}_{A}^{r}\bm{P}_{B}^{v}\ddot{\bm{P}}_{B}\bm{1}-(\bm{\pi}\bm{P}_{A}^{r}\bm{P}_{B}^{v}\dot{\bm{P}}_{B}\bm{1})^{2}]$
$\displaystyle\qquad\quad{}+2\sum_{0\leq u<v\leq
r-1}\bm{\pi}\bm{P}_{A}^{u}\dot{\bm{P}}_{A}(\bm{P}_{A}^{v-u-1}-\bm{\Pi}\bm{P}_{A}^{v})\dot{\bm{P}}_{A}\bm{1}$
$\displaystyle\qquad\quad{}+2\sum_{u=0}^{r-1}\sum_{v=0}^{s-1}\bm{\pi}\bm{P}_{A}^{u}\dot{\bm{P}}_{A}(\bm{P}_{A}^{r-u-1}-\bm{\Pi}\bm{P}_{A}^{r})\bm{P}_{B}^{v}\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\qquad\quad{}+2\sum_{0\leq u<v\leq
s-1}\bm{\pi}\bm{P}_{A}^{r}\bm{P}_{B}^{u}\dot{\bm{P}}_{B}(\bm{P}_{B}^{v-u-1}-\bm{\Pi}\bm{P}_{A}^{r}\bm{P}_{B}^{v})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\qquad\quad{}+2\bigg{(}\sum_{u=0}^{r-1}\sum_{v=0}^{r-1}\bm{\pi}\bm{P}_{A}^{u}\dot{\bm{P}}_{A}\bm{P}_{A}^{r-u-1}\bm{P}_{B}^{s}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{v}\dot{\bm{P}}_{A}\bm{1}$
$\displaystyle\qquad\qquad\quad{}+\sum_{u=0}^{r-1}\sum_{v=0}^{s-1}\bm{\pi}\bm{P}_{A}^{u}\dot{\bm{P}}_{A}\bm{P}_{A}^{r-u-1}\bm{P}_{B}^{s}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{r}\bm{P}_{B}^{v}\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\qquad\qquad\quad{}+\sum_{u=0}^{s-1}\sum_{v=0}^{r-1}\bm{\pi}\bm{P}_{A}^{r}\bm{P}_{B}^{u}\dot{\bm{P}}_{B}\bm{P}_{B}^{s-u-1}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{v}\dot{\bm{P}}_{A}\bm{1}$
$\displaystyle\qquad\qquad\quad{}+\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\bm{\pi}\bm{P}_{A}^{r}\bm{P}_{B}^{u}\dot{\bm{P}}_{B}\bm{P}_{B}^{s-u-1}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{r}\bm{P}_{B}^{v}\dot{\bm{P}}_{B}\bm{1}\bigg{)}\bigg{]},$

where $\bm{1}$ denotes a column vector of $1$s with entries indexed by
$\Sigma_{0}$. Let $\\{X_{n}\\}_{n\geq 0}$ be a nonhomogeneous Markov chain in
$\Sigma_{0}$ with one-step transition matrices $\bm{P}_{A},\ldots,\bm{P}_{A}$
$(r\text{ times})$, $\bm{P}_{B},\ldots,\bm{P}_{B}$ $(s\text{ times})$,
$\bm{P}_{A},\ldots,\bm{P}_{A}$ $(r\text{ times})$,
$\bm{P}_{B},\ldots,\bm{P}_{B}$ $(s\text{ times})$, and so on. For each $n\geq
1$, define $\xi_{n}:=w(X_{n-1},X_{n})$ and $S_{n}:=\xi_{1}+\cdots+\xi_{n}$.

######  Theorem 3 (Ethier and Lee [10]).

Under the above assumptions, and with the distribution of $X_{0}$ arbitrary,

$\frac{S_{n}}{n}\to\mu_{[r,s]}\;\;{\rm a.s.}$

and, if $\sigma_{[r,s]}^{2}>0$, then

$\frac{S_{n}-n\mu_{[r,s]}}{\sqrt{n\sigma_{[r,s]}^{2}}}\to_{d}N(0,1)\text{ as
}n\to\infty.$

######  Example 4.

To illustrate this result, we consider the capital-dependent Parrondo games as
above, and we take $r=s=2$. Then

$\bm{P}=\bm{P}_{A}^{2}\bm{P}_{B}^{2}=\frac{1}{320}\begin{pmatrix}162&59&99\\\
151&58&111\\\ 111&47&162\end{pmatrix}.$

Its unique stationary distribution is $\bm{\pi}=(1/6357)(2783,1075,2499)$, and
the fundamental matrix is

$\bm{Z}=\frac{1}{525348837}\begin{pmatrix}569627023&10027235&-54305421\\\
22416463&532826915&-29894541\\\ -58953137&-14383645&598685619\end{pmatrix}.$

In this example, $\dot{\bm{P}}_{A}\bm{1}=\bm{0}$,
$\ddot{\bm{P}}_{A}=\bm{P}_{A}$, and $\ddot{\bm{P}}_{B}=\bm{P}_{B}$, and this
simplifies the mean and variance formulas considerably. Specifically, we have

$\mu_{[2,2]}=\frac{1}{4}\bm{\pi}\bm{P}_{A}^{2}(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$

and

$\displaystyle\sigma_{[2,2]}^{2}$
$\displaystyle=\frac{1}{4}\big{[}2+2-(\bm{\pi}\bm{P}_{A}^{2}\dot{\bm{P}}_{B}\bm{1})^{2}-(\bm{\pi}\bm{P}_{A}^{2}\bm{P}_{B}\dot{\bm{P}}_{B}\bm{1})^{2}$
$\displaystyle\quad+2\bm{\pi}\dot{\bm{P}}_{A}(\bm{P}_{A}-\bm{\Pi}\bm{P}_{A}^{2})(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\bm{P}_{A}\dot{\bm{P}}_{A}(\bm{I}-\bm{\Pi}\bm{P}_{A}^{2})(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\bm{P}_{A}^{2}\dot{\bm{P}}_{B}(\bm{I}-\bm{\Pi}\bm{P}_{A}^{2}\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\dot{\bm{P}}_{A}\bm{P}_{A}\bm{P}_{B}^{2}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{2}(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\bm{P}_{A}\dot{\bm{P}}_{A}\bm{P}_{B}^{2}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{2}(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\bm{P}_{A}^{2}\dot{\bm{P}}_{B}\bm{P}_{B}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{2}(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}$
$\displaystyle\quad+2\bm{\pi}\bm{P}_{A}^{2}\bm{P}_{B}\dot{\bm{P}}_{B}(\bm{Z}-\bm{\Pi})\bm{P}_{A}^{2}(\bm{I}+\bm{P}_{B})\dot{\bm{P}}_{B}\bm{1}\big{]}.$

We conclude that

$\mu_{[2,2]}=\frac{4}{163}\approx
0.0245399\quad\text{and}\quad\sigma_{[2,2]}^{2}=\frac{1923037543}{2195688729}\approx
0.875824.$

These numbers are consistent with Ethier and Lee [10].

###  2.2 Application to game $B$

The Markov chain formalized by Mihailović and Rajković [11] keeps track of the
status (loser or winner, 0 or 1) of each of the $N\geq 3$ players of game $B$.
Its state space is the product space

$\\{\eta=(\eta(1),\eta(2),\ldots,\eta(N)):\eta(x)\in\\{0,1\\}{\rm\ for\
}x=1,\ldots,N\\}=\\{0,1\\}^{N}$

with $2^{N}$ states. Let $m_{x}(\eta):=2\eta(x-1)+\eta(x+1)\in\\{0,1,2,3\\}$.
Of course $\eta(0):=\eta(N)$ and $\eta(N+1):=\eta(1)$ because of the circular
arrangement of players. Also, let $\eta_{x}$ be the element of $\\{0,1\\}^{N}$
equal to $\eta$ except at the $x$th coordinate. For example,
$\eta_{1}:=(1-\eta(1),\eta(2),\eta(3),\ldots,\eta(N))$.

The one-step transition matrix $\bm{P}_{B}$ for this Markov chain depends not
only on $N$ but on four parameters, $p_{0},p_{1},p_{2},p_{3}\in[0,1]$. It has
the form

$P_{B}(\eta,\eta_{x}):=\begin{cases}N^{-1}p_{m_{x}(\eta)}&\text{if
$\eta(x)=0$,}\\\ N^{-1}q_{m_{x}(\eta)}&\text{if $\eta(x)=1$,}\end{cases}\qquad
x=1,\ldots,N,\;\eta\in\\{0,1\\}^{N},$ (1)

and

$P_{B}(\eta,\eta):=N^{-1}\bigg{(}\sum_{x:\eta(x)=0}q_{m_{x}(\eta)}+\sum_{x:\eta(x)=1}p_{m_{x}(\eta)}\bigg{)},\qquad\eta\in\\{0,1\\}^{N},$
(2)

where $q_{m}:=1-p_{m}$ for $m=0,1,2,3$ and empty sums are 0. The Markov chain
is irreducible and aperiodic if $0<p_{m}<1$ for $m=0,1,2,3$. Under slightly
weaker assumptions (see Ethier and Lee [7]), the Markov chain is ergodic,
which suffices. For example, if $p_{0}$ is arbitrary and $0<p_{m}<1$ for
$m=1,2,3$, or if $0<p_{m}<1$ for $m=0,1,2$ and $p_{3}$ is arbitrary, then
ergodicity holds.

It appears at first glance that the theorem does not apply in the context of
game $B$ because the payoffs are not completely specified by the one-step
transitions of the Markov chain. Specifically, a transition from a state
$\eta$ to itself results whenever a loser loses or a winner wins, so the
transition does not determine the payoff.

Our original Markov chain has state space $\\{0,1\\}^{N}$ and its one-step
transition matrix $\bm{P}_{B}$ is given by (1) and (2). Assuming it is
ergodic, let $\bm{\pi}_{B}$ denote its unique stationary distribution. The
approach in Ethier and Lee [5] augments the state space, letting
$\Sigma^{*}:=\\{0,1\\}^{N}\times\\{1,2,\ldots,N\\}$ and keeping track not only
of the status of each player as described by $\eta\in\\{0,1\\}^{N}$ but also
of the label of the next player to play, say $x$. The new one-step transition
matrix $\bm{P}_{B}^{*}$ can be determined, as can its unique stationary
distribution $\bm{\pi}_{B}^{*}$, and the theorem applies.

However, there is a drawback to this approach, namely that it is not clear
that the variance parameter $(\sigma^{*})^{2}$ is the same as the original
one, $\sigma^{2}$. (It is easy to verify that $\mu^{*}=\mu$.) Therefore, we
take a different approach, namely the one used by Ethier and Lee [12] in their
study of two-dimensional spatial models.

Here a different augmentation of $\\{0,1\\}^{N}$ is more effective. We let
$\Sigma^{\circ}:=\\{0,1\\}^{N}\times\\{-1,1\\}$ and keep track not only of
$\eta\in\\{0,1\\}^{N}$ but also of the profit from the last game played, say
$s\in\\{-1,1\\}$. The new one-step transition matrix $\bm{P}_{B}^{\circ}$ has
the form, for every $(\eta,s)\in\Sigma^{\circ}$,

$P_{B}^{\circ}((\eta,s),(\eta_{x},1)):=\begin{cases}N^{-1}p_{m_{x}(\eta)}&\text{if
$\eta(x)=0$,}\\\ 0&\text{if $\eta(x)=1$,}\end{cases}$
$P_{B}^{\circ}((\eta,s),(\eta_{x},-1)):=\begin{cases}0&\text{if
$\eta(x)=0$,}\\\ N^{-1}q_{m_{x}(\eta)}&\text{if $\eta(x)=1$,}\end{cases}$

for $x=1,\ldots,N$, and

$P_{B}^{\circ}((\eta,s),(\eta,1)):=N^{-1}\sum_{x:\eta(x)=1}p_{m_{x}(\eta)},$
$P_{B}^{\circ}((\eta,s),(\eta,-1)):=N^{-1}\sum_{x:\eta(x)=0}q_{m_{x}(\eta)},$

where $q_{m}:=1-p_{m}$ for $m=0,1,2,3,4$ and
$m_{x}(\eta)=2\eta(x-1)+\eta(x+1)$. There are two inaccessible states,
$(\bm{0},1)$ and $(\bm{1},-1)$, but the Markov chain remains ergodic. Let
$\bm{\pi}_{B}^{\circ}$ denote the unique stationary distribution, which has
entry 0 at each of the two inaccessible states. The payoff function
$w^{\circ}$ can now be defined by

$w^{\circ}((\eta,s),(\eta_{x},t))=t\text{ if $\eta(x)=(1-t)/2$,}\qquad
w^{\circ}((\eta,s),(\eta,t))=t$

for all $(\eta,s)\in\Sigma^{\circ}$, $x=1,2,\ldots,N$, and $t\in\\{-1,1\\}$,
and $w^{\circ}=0$ otherwise. This allows us to define the matrix
$\bm{W}^{\circ}$ and then
$\dot{\bm{P}}_{B}^{\circ}:={\bm{P}}_{B}^{\circ}\circ\bm{W}^{\circ}$ and
$\ddot{\bm{P}}_{B}^{\circ}:={\bm{P}}_{B}^{\circ}\circ\bm{W}^{\circ}\circ\bm{W}^{\circ}$,
the Hadamard (or entrywise) products. Theorem 1 yields the following.

Let $0<p_{m}<1$ for $m=0,1,2$ or for $m=1,2,3$, so that the Markov chain with
one-step transition matrix $\bm{P}_{B}^{\circ}$ is ergodic, and let the row
vector $\bm{\pi}_{B}^{\circ}$ be its unique stationary distribution. Define

$\mu_{B}^{\circ}=\bm{\pi}_{B}^{\circ}\dot{\bm{P}}_{B}^{\circ}\bm{1},\qquad(\sigma_{B}^{\circ})^{2}=\bm{\pi}_{B}^{\circ}\ddot{\bm{P}}_{B}^{\circ}\bm{1}-(\bm{\pi}_{B}^{\circ}\dot{\bm{P}}_{B}^{\circ}\bm{1})^{2}+2\bm{\pi}_{B}^{\circ}\dot{\bm{P}}_{B}^{\circ}(\bm{Z}_{B}^{\circ}-\bm{1}\bm{\pi}_{B}^{\circ})\dot{\bm{P}}_{B}^{\circ}\bm{1}.$

where $\bm{1}$ denotes a column vector of $1$s with entries indexed by
$\Sigma_{B}^{\circ}$ and
$\bm{Z}_{B}^{\circ}:=(\bm{I}-(\bm{P}_{B}^{\circ}-\bm{1}\bm{\pi}_{B}^{\circ}))^{-1}$
is the fundamental matrix. (Notice that $\bm{1}\bm{\pi}_{B}^{\circ}$ is the
square matrix each of whose rows is equal to $\bm{\pi}_{B}^{\circ}$.) Let
$\\{X_{n}^{\circ}\\}_{n\geq 0}$ be a time-homogeneous Markov chain in
$\Sigma^{\circ}$ with one-step transition matrix $\bm{P}_{B}^{\circ}$, and let
the initial distribution be arbitrary. For each $n\geq 1$, define
$\xi_{n}:=w^{\circ}(X_{n-1}^{\circ},X_{n}^{\circ})$ and
$S_{n}:=\xi_{1}+\cdots+\xi_{n}$.

######  Theorem 5.

Under the above assumptions, and with the initial distribution arbitrary,

$\lim_{n\to\infty}\frac{S_{n}}{n}=\mu_{B}^{\circ}\;\;\emph{a.s.}$

and, if $(\sigma_{B}^{\circ})^{2}>0$, then

$\frac{S_{n}-n\mu_{B}^{\circ}}{\sqrt{n(\sigma_{B}^{\circ})^{2}}}\to_{d}N(0,1)\text{
as }n\to\infty.$

We next show that there is a simpler expression for this mean and variance.
Let us define

$\mu_{B}:=\bm{\pi}_{B}\dot{\bm{P}}_{B}\bm{1},\qquad\sigma_{B}^{2}:=\bm{\pi}_{B}\ddot{\bm{P}}_{B}\bm{1}-(\bm{\pi}_{B}\dot{\bm{P}}_{B}\bm{1})^{2}+2\bm{\pi}_{B}\dot{\bm{P}}_{B}(\bm{Z}_{B}-\bm{1}\bm{\pi}_{B})\dot{\bm{P}}_{B}\bm{1},\\\
$

where $\bm{1}$ is the column vector of 1s of the appropriate dimension,
$\dot{\bm{P}}_{B}$ is $\bm{P}_{B}$ with each $q_{m}$ replaced by $-q_{m}$, and
$\ddot{\bm{P}}_{B}=\bm{P}_{B}$. This “rule of thumb” for $\dot{\bm{P}}_{B}$
requires some caution: It must be applied before any simplifications to
$\bm{P}_{B}$ are made using $q_{m}=1-p_{m}$. Of course, $\bm{\pi}_{B}$ is the
unique stationary distribution, and $\bm{Z}_{B}$ is the fundamental matrix, of
$\bm{P}_{B}$.

######  Theorem 6.

$\mu_{B}^{\circ}=\mu_{B}$

and

$(\sigma_{B}^{\circ})^{2}=\sigma_{B}^{2}.$

######  Remark.

The proof is as in Ethier and Lee [12]. Let us explain its significance.
$\mu_{B}^{\circ}$ and $(\sigma_{B}^{\circ})^{2}$ are the mean and variance
that appear in the SLLN and the CLT. They are defined in terms of
$\bm{P}_{B}^{\circ}$, the augmented one-step transition matrix. $\mu_{B}$ and
$\sigma_{B}^{2}$ are defined analogously in terms of $\bm{P}_{B}$, the
original one-step transition matrix, using the rule of thumb.

###  2.3 Application to game $C^{\prime}:=\gamma A^{\prime}+(1-\gamma)B$

This case is not much different from the previous one. Notice that, if game
$A^{\prime}$ is played, the profit to the set of $N$ players is 0, since game
$A^{\prime}$ simply redistributes capital among the players. So we can use the
same augmentation of the state space as before, except that 0 is now a
possible value of the profit from the last game played. In other words,
$\Sigma^{\circ}:=\\{0,1\\}^{N}\times\\{-1,0,1\\}$. The transition
probabilities require some new notation. Let $\eta^{x,x\pm 1,\pm 1}$ be the
element of $\\{0,1\\}^{N}$ representing the players’ status after player $x$
plays player $x\pm 1$ and wins (1) or loses ($-1$). Of course player 0 is
player $N$ and player $N+1$ is player 1. E.g.,
$\eta^{1,2,-1}=(0,1,\eta(3),\ldots,\eta(N))$ (player 1 competes against player
2 and loses, leaving player 1 a loser and player 2 a winner, regardless of
their previous status). Then

$\displaystyle P_{C^{\prime}}^{\circ}((\eta,s),(\eta_{x},1))$
$\displaystyle=\begin{cases}(1-\gamma)N^{-1}p_{m_{x}(\eta)}&\text{if
$\eta(x)=0$,}\\\ 0&\text{if $\eta(x)=1$,}\end{cases}$ (3) $\displaystyle
P_{C^{\prime}}^{\circ}((\eta,s),(\eta_{x},-1))$
$\displaystyle=\begin{cases}0&\text{if $\eta(x)=0$,}\\\
(1-\gamma)N^{-1}q_{m_{x}(\eta)}&\text{if $\eta(x)=1$,}\end{cases}$ (4)
$\displaystyle P_{C^{\prime}}^{\circ}((\eta,s),(\eta^{x,x-1,-1},0))$
$\displaystyle=\gamma(4N)^{-1},$ (5) $\displaystyle
P_{C^{\prime}}^{\circ}((\eta,s),(\eta^{x,x-1,1},0))$
$\displaystyle=\gamma(4N)^{-1},$ (6) $\displaystyle
P_{C^{\prime}}^{\circ}((\eta,s),(\eta^{x,x+1,-1},0))$
$\displaystyle=\gamma(4N)^{-1},$ (7) $\displaystyle
P_{C^{\prime}}^{\circ}((\eta,s),(\eta^{x,x+1,1},0))$
$\displaystyle=\gamma(4N)^{-1},$ (8)

for $x=1,2,\ldots,N$, and

$\displaystyle P_{C^{\prime}}^{\circ}((\eta,s),(\eta,1))$
$\displaystyle=(1-\gamma)N^{-1}\sum_{x:\eta(x)=1}p_{m_{x}(\eta)},$ (9)
$\displaystyle P_{C^{\prime}}^{\circ}((\eta,s),(\eta,-1))$
$\displaystyle=(1-\gamma)N^{-1}\sum_{x:\eta(x)=0}q_{m_{x}(\eta)}.$ (10)

Of course, we could also define
$\bm{P}_{C^{\prime}}=\gamma\bm{P}_{A^{\prime}}+(1-\gamma)\bm{P}_{B}$. We
notice that Theorems 5 and 6 hold in this framework without change.

Let $0<p_{m}<1$ for $m=0,1,2$ or for $m=1,2,3$, so that the Markov chain with
one-step transition matrix
$\bm{P}_{C^{\prime}}^{\circ}:=\gamma\bm{P}_{A^{\prime}}^{\circ}+(1-\gamma)\bm{P}_{B}^{\circ}$
is ergodic, and let the row vector $\bm{\pi}_{C^{\prime}}^{\circ}$ be its
unique stationary distribution. Define

$\displaystyle\mu_{(\gamma,1-\gamma)^{\prime}}^{\circ}$
$\displaystyle=\bm{\pi}_{C^{\prime}}^{\circ}\dot{\bm{P}}_{C^{\prime}}^{\circ}\bm{1},$
$\displaystyle(\sigma_{(\gamma,1-\gamma)^{\prime}}^{\circ})^{2}$
$\displaystyle=\bm{\pi}_{C^{\prime}}^{\circ}\ddot{\bm{P}}_{C^{\prime}}^{\circ}\bm{1}-(\bm{\pi}_{C^{\prime}}^{\circ}\dot{\bm{P}}_{C^{\prime}}^{\circ}\bm{1})^{2}+2\bm{\pi}_{C^{\prime}}^{\circ}\dot{\bm{P}}_{C^{\prime}}^{\circ}(\bm{Z}_{C^{\prime}}^{\circ}-\bm{1}\bm{\pi}_{C^{\prime}}^{\circ})\dot{\bm{P}}_{C^{\prime}}^{\circ}\bm{1}.$

where $\bm{1}$ denotes a column vector of $1$s with entries indexed by
$\Sigma^{\circ}$ and
$\bm{Z}_{C^{\prime}}^{\circ}:=(\bm{I}-(\bm{P}_{C^{\prime}}^{\circ}-\bm{1}\bm{\pi}_{C^{\prime}}^{\circ}))^{-1}$
is the fundamental matrix. (Notice that $\bm{1}\bm{\pi}_{C^{\prime}}^{\circ}$
is the square matrix each of whose rows is equal to
$\bm{\pi}_{C^{\prime}}^{\circ}$.) Let $\\{X_{n}^{\circ}\\}_{n\geq 0}$ be a
time-homogeneous Markov chain in $\Sigma^{\circ}$ with one-step transition
matrix $\bm{P}_{C^{\prime}}^{\circ}$. For each $n\geq 1$, define
$\xi_{n}:=w^{\circ}(X_{n-1}^{\circ},X_{n}^{\circ})$ and
$S_{n}:=\xi_{1}+\cdots+\xi_{n}$.

######  Theorem 7.

Under the above assumptions, and with the distribution of $X_{0}$ arbitrary,

$\lim_{n\to\infty}\frac{S_{n}}{n}=\mu_{(\gamma,1-\gamma)^{\prime}}^{\circ}\;\;\emph{a.s.}$

and, if $(\sigma_{(\gamma,1-\gamma)^{\prime}}^{\circ})^{2}>0$, then

$\frac{S_{n}-n\mu_{(\gamma,1-\gamma)^{\prime}}^{\circ}}{\sqrt{n(\sigma_{(\gamma,1-\gamma)^{\prime}}^{\circ})^{2}}}\to_{d}N(0,1)\text{
as }n\to\infty.$

Let us define

$\displaystyle\mu_{(\gamma,1-\gamma)^{\prime}}$
$\displaystyle:=\bm{\pi}_{C^{\prime}}\dot{\bm{P}}_{C^{\prime}}\bm{1},$
$\displaystyle\sigma_{(\gamma,1-\gamma)^{\prime}}^{2}$
$\displaystyle:=\bm{\pi}_{C^{\prime}}\ddot{\bm{P}}_{C^{\prime}}\bm{1}-(\bm{\pi}_{C^{\prime}}\dot{\bm{P}}_{C^{\prime}}\bm{1})^{2}+2\bm{\pi}_{C^{\prime}}\dot{\bm{P}}_{C^{\prime}}(\bm{Z}_{C^{\prime}}-\bm{1}\bm{\pi}_{C^{\prime}})\dot{\bm{P}}_{C^{\prime}}\bm{1},$

where $\bm{1}$ is the column vector of 1s of the appropriate dimension, and
since $\dot{\bm{P}}_{A^{\prime}}$ can be defined to be $\bm{0}$,
$\dot{\bm{P}}_{C^{\prime}}$ is $(1-\gamma)\dot{\bm{P}}_{B}$ with each $q_{m}$
replaced by $-q_{m}$, and $\ddot{\bm{P}}_{C^{\prime}}=(1-\gamma)\bm{P}_{B}$.
This “rule of thumb” for $\dot{\bm{P}}_{C^{\prime}}$ requires some caution: It
must be applied before any simplifications to $\bm{P}_{C^{\prime}}$ are made
using $q_{m}=1-p_{m}$. Of course, $\bm{\pi}_{C^{\prime}}$ is the unique
stationary distribution, and $\bm{Z}_{C^{\prime}}$ is the fundamental matrix,
of $\bm{P}_{C^{\prime}}$. Notice that
$\dot{\bm{P}}_{A^{\prime}}^{\circ}=\bm{0}$, so
$\dot{\bm{P}}_{C^{\prime}}^{\circ}=(1-\gamma)\dot{\bm{P}}_{B}^{\circ}$ and
$\ddot{\bm{P}}_{C^{\prime}}^{\circ}=(1-\gamma)\ddot{\bm{P}}_{B}^{\circ}$.

######  Theorem 8.

$\mu_{(\gamma,1-\gamma)^{\prime}}^{\circ}=\mu_{(\gamma,1-\gamma)^{\prime}}$
(11)

and

$(\sigma_{(\gamma,1-\gamma)^{\prime}}^{\circ})^{2}=\sigma_{(\gamma,1-\gamma)^{\prime}}^{2}.$
(12)

###  2.4 Application to game $C^{\prime}:=(A^{\prime})^{r}B^{s}$

Next we need versions of the SLLN and the CLT suited to game
$C^{\prime}:=(A^{\prime})^{r}B^{s}$. The key result is Theorem 3.

For the same reason as before, the theorem does not apply directly to
$\bm{P}_{A^{\prime}}$ and $\bm{P}_{B}$. Therefore we again consider the Markov
chains in the augmented state space
$\Sigma^{\circ}:=\\{0,1\\}^{N}\times\\{-1,0,1\\}$ with one-step transition
matrix $\bm{P}_{A^{\prime}}^{\circ}$ and $\bm{P}_{B}^{\circ}$. The definitions
are as in (3)–(10) with $\gamma=1$ or $\gamma=0$. With $\bm{W}^{\circ}$ as
before, the theorem applies.

Fix $r,s\geq 1$. Assume that
$\bm{P}^{\circ}:=(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{s}$,
as well as all cyclic permutations of
$(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{s}$, are ergodic, and
let the row vector $\bm{\pi}^{\circ}$ be the unique stationary distribution of
$\bm{P}^{\circ}$. Let

$\mu_{[r,s]^{\prime}}^{\circ}:=\frac{1}{r+s}\sum_{v=0}^{s-1}\bm{\pi}^{\circ}(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{v}\dot{\bm{P}}_{B}^{\circ}\bm{1}$

and

$\displaystyle(\sigma_{[r,s]^{\prime}}^{\circ})^{2}$
$\displaystyle=\frac{1}{r+s}\bigg{\\{}s-\sum_{v=0}^{s-1}(\bm{\pi}^{\circ}(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{v}\dot{\bm{P}}_{B}^{\circ}\bm{1})^{2}$
$\displaystyle\qquad\;\;{}+2\bigg{[}\sum_{0\leq u<v\leq
s-1}\bm{\pi}^{\circ}(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{u}\dot{\bm{P}}_{B}^{\circ}((\bm{P}_{B}^{\circ})^{v-u-1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{}-\bm{1}\bm{\pi}^{\circ}(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{v})\dot{\bm{P}}_{B}^{\circ}\bm{1}$
$\displaystyle\qquad\;\;{}+\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\bm{\pi}^{\circ}(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{u}\dot{\bm{P}}_{B}^{\circ}(\bm{P}_{B}^{\circ})^{s-u-1}(\bm{Z}^{\circ}-\bm{1}\bm{\pi}^{\circ})(\bm{P}_{A^{\prime}}^{\circ})^{r}(\bm{P}_{B}^{\circ})^{v}\dot{\bm{P}}_{B}^{\circ}\bm{1}\bigg{]}\bigg{\\}}.$

Let $\\{X_{n}^{\circ}\\}_{n\geq 0}$ be a nonhomogeneous Markov chain in
$\Sigma^{\circ}$ with one-step transition matrices
$\bm{P}_{A^{\prime}}^{\circ},\ldots,\bm{P}_{A^{\prime}}^{\circ}$ $(r\text{
times})$, $\bm{P}_{B}^{\circ},\ldots,\bm{P}_{B}^{\circ}$ $(s\text{ times})$,
$\bm{P}_{A^{\prime}}^{\circ},\ldots,\bm{P}_{A^{\prime}}^{\circ}$ $(r\text{
times})$, $\bm{P}_{B}^{\circ},\ldots,\bm{P}_{B}^{\circ}$ $(s\text{ times})$,
and so on. For each $n\geq 1$, define
$\xi_{n}:=w^{\circ}(X_{n-1}^{\circ},X_{n}^{\circ})$ and
$S_{n}:=\xi_{1}+\cdots+\xi_{n}$.

######  Theorem 9.

Under the above assumptions, and with the distribution of $X_{0}$ arbitrary,

$\frac{S_{n}}{n}\to\mu_{[r,s]^{\prime}}^{\circ}\;\;\emph{a.s.}$

and, if $(\sigma_{[r,s]^{\prime}}^{\circ})^{2}>0$, then

$\frac{S_{n}-n\mu_{[r,s]^{\prime}}^{\circ}}{\sqrt{n(\sigma_{[r,s]^{\prime}}^{\circ})^{2}}}\to_{d}N(0,1)\text{
as }n\to\infty.$

Again there are simpler expressions for this mean and variance. We define
$\mu_{[r,s]^{\prime}}$ in terms of $\bm{\pi}$, $\bm{P}_{A^{\prime}}$,
$\bm{P}_{B}$, and $\dot{\bm{P}}_{B}$ in the same way that
$\mu_{[r,s]^{\prime}}^{\circ}$ was defined in terms of $\bm{\pi}^{\circ}$,
$\bm{P}_{A^{\prime}}^{\circ}$, $\bm{P}_{B}^{\circ}$, and
$\dot{\bm{P}}_{B}^{\circ}$. ($\dot{\bm{P}}_{B}$ is defined by the rule of
thumb.) Finally, $\sigma_{[r,s]^{\prime}}^{2}$ is defined analogously to
$(\sigma_{[r,s]^{\prime}}^{\circ})^{2}$.

######  Theorem 10.

$\mu_{[r,s]^{\prime}}^{\circ}=\mu_{[r,s]^{\prime}}$ (13)

and

$(\sigma_{[r,s]^{\prime}}^{\circ})^{2}=\sigma_{[r,s]^{\prime}}^{2}.$ (14)

###### Proof.

Eqs. (13) and (14) are proved in the same way as (11) and (12). ∎

##  3 Numerical computations

In this section, we compute various means numerically by using the reduced
state space and use computer graphics to visualize the Parrondo region of the
Parrondo games of Xie at al. [3].

###  3.1 State-space reduction

Let us begin by explaining what we mean by state-space reduction, which is an
important method for simplifying our computations.

In general, consider an equivalence relation $\sim$ on a finite set $E$. By
definition, $\sim$ is reflexive ($x\sim x$), symmetric ($x\sim y$ implies
$y\sim x$), and transitive ($x\sim y$ and $y\sim z$ imply $x\sim z$). It is
well known that an equivalence relation partitions the set $E$ into
equivalence classes. The set of all equivalence classes, called the quotient
set, will be denoted by $\bar{E}$. Let us write $[x]:=\\{y\in E:y\sim x\\}$
for the equivalence class containing $x$. Then $\bar{E}=\\{[x]:x\in E\\}$.

Now suppose $X_{0},X_{1},X_{2},\ldots$ is a (time-homogeneous) Markov chain in
$E$ with transition matrix $\bm{P}$. In particular, $P(x,y)={\rm
P}(X_{t+1}=y\mid X_{t}=x)$ for all $x,y\in E$ and $t=0,1,2,\ldots$. Under what
conditions on $\bm{P}$ is $[X_{0}],[X_{1}],[X_{2}],\ldots$ a Markov chain in
the “reduced” state space $\bar{E}$? A sufficient condition, apparently due to
Kemeny and Snell [13, p. 124], is that $\bm{P}$ be lumpable with respect to
$\sim$. By definition, this means that, for all $x,x^{\prime},y\in E$,

$x\sim
x^{\prime}\quad\text{implies}\quad\sum_{y^{\prime}\in[y]}P(x,y^{\prime})=\sum_{y^{\prime}\in[y]}P(x^{\prime},y^{\prime}).$
(15)

Moreover, if (15) holds, then the Markov chain
$[X_{0}],[X_{1}],[X_{2}],\ldots$ in $\bar{E}$ has transition matrix
$\bar{\bm{P}}$ given by

$\bar{P}([x],[y]):=\sum_{y^{\prime}\in[y]}P(x,y^{\prime}).$ (16)

Notice that (15) ensures that (16) is well defined.

For Parrondo games with one-dimensional spatial dependence, the state space,
assuming $N\geq 3$ players, is

$\\{\eta=(\eta(1),\eta(2),\ldots,\eta(N)):\eta(x)\in\\{0,1\\}{\rm\ for\
}x=1,2,\ldots,N\\}=\\{0,1\\}^{N},$

which has $2^{N}$ states. A state $\eta\in\\{0,1\\}^{N}$ describes the status
of each of the $N$ players, 0 for losers and 1 for winners. We can also think
of $\\{0,1\\}^{N}$ as the set of $N$-bit binary representations of the
integers $0,1,\ldots,2^{N}-1$, thereby giving a natural ordering to the
vectors in $\\{0,1\\}^{N}$.

Ethier and Lee [5] used the following equivalence relation on $\\{0,1\\}^{N}$:
$\eta\sim\zeta$ if and only if
$\zeta=\eta_{\sigma}:=(\eta(\sigma(1)),\ldots,\eta(\sigma(N)))$ for a
permutation $\sigma$ of $(1,2,\ldots,N)$ belonging to the cyclic group $G$ of
order $N$ of the rotations of the players. If, in addition, $p_{1}=p_{2}$, the
permutation $\sigma$ can belong to the dihedral group $G$ of order $2N$ of the
rotations and reflections of the players. They verified the lumpability
condition, with the result that the size of the state space was reduced by a
factor of nearly $N$ (or $2N$ if $p_{1}=p_{2}$) for large $N$. It should be
noted that a sufficient condition for the lumpability condition in this
setting is that, for every $\eta,\zeta\in\\{0,1\\}^{N}$,

$P(\eta_{\sigma},\zeta_{\sigma})=P(\eta,\zeta)\quad\text{for all $\sigma\in
G$}$

or for all $\sigma$ in a subset of $G$ that generates $G$.

To fully justify this, the following lemma is useful.

######  Lemma 11 (Ethier and Lee [6]).

Fix $N\geq 3$, let $G$ be a subgroup of the symmetric group $S_{N}$. Let
$\bm{P}$ be the one-step transition matrix for a Markov chain in
$\\{0,1\\}^{N}$ with a unique stationary distribution $\bm{\pi}$. Assume that

$P(\eta_{\sigma},\zeta_{\sigma})=P(\eta,\zeta),\qquad\sigma\in
G,\;\eta,\zeta\in\\{0,1\\}^{N}.$ (17)

Then $\pi(\eta_{\sigma})=\pi(\eta)$ for all $\sigma\in G$ and
$\eta\in\\{0,1\\}^{N}$.

Let us say that $\eta\in\\{0,1\\}^{N}$ is equivalent to
$\zeta\in\\{0,1\\}^{N}$ (written $\eta\sim\zeta$) if there exists $\sigma\in
G$ such that $\zeta=\eta_{\sigma}$, and let us denote the equivalence class
containing $\eta$ by $[\eta]$. Then, in addition, $\bm{P}$ induces a one-step
transition matrix $\bar{\bm{P}}$ for a Markov chain in the quotient set (i.e.,
the set of equivalence classes) $\bar{\Sigma}$ defined by the formula

$\bar{P}([\eta],[\zeta]):=\sum_{\zeta^{\prime}\in[\zeta]}P(\eta,\zeta^{\prime}),$

Furthermore, if $\bar{\bm{P}}$ has a unique stationary distribution
$\bar{\bm{\pi}}$, then the unique stationary distribution $\bm{\pi}$ is given
by $\pi(\eta)=\bar{\pi}([\eta])/|[\eta]|$, where $|[\eta]|$ denotes the
cardinality of the equivalence class $[\eta]$.

The lemma will apply to $\bm{P}_{A^{\prime}}$ and $\bm{P}_{B}$ (hence
$\bm{P}_{C^{\prime}}$) if we can verify (17) for $G$ being the cyclic group of
rotations or, if $p_{1}=p_{2}$, the dihedral group of rotations and
reflections.

The practical effect of this is that we can reduce the size of the state space
(namely, $2^{N}$) to what we will call its effective size, which is simply the
number of equivalence classes. For example, if $N=3$, there are eight states
and four equivalence classes, namely

$0=\\{000\\},\quad 1=\\{001,010,100\\},\quad 2=\\{011,101,110\\},\quad
3=\\{111\\}.$

Notice that we label equivalence classes by the number of 1s each element has.
If $N=4$, there are 16 states and six equivalence classes, namely

$\displaystyle 0$ $\displaystyle=\\{0000\\},$ $\displaystyle 1$
$\displaystyle=\\{0001,0010,0100,1000\\},$ $\displaystyle 2$
$\displaystyle=\\{0011,0110,1001,1100\\},$ $\displaystyle 2^{\prime}$
$\displaystyle=\\{0101,1010\\},$ $\displaystyle 3$
$\displaystyle=\\{0111,1011,1101,1110\\},$ $\displaystyle 4$
$\displaystyle=\\{1111\\}.$

In these two cases, it does not matter which of the two equivalence relations
we use; the result is the same.

The number of equivalence classes with $G$ being the group of cyclic
permutations follows the sequence A000031 in the The On-Line Encyclopedia of
Integer Sequences (Sloan [14]), described as the number of necklaces with $N$
beads of two colors when turning over is not allowed. There is an explicit
formula in terms of Euler’s phi-function. If $p_{1}=p_{2}$, we can reverse the
order of the players, and the number of equivalence classes with $G$ being the
dihedral group follows the sequence A000029 in the OEIS, described as the
number of necklaces with $N$ beads of two colors when turning over is allowed.
Again there is an explicit formula.

######  Example 12.

To illustrate our approach in a tractable case, we focus on the case $N=4$, as
did Xie et al. [3]. Here $\\{0,1\\}^{N}$ has 16 states, ordered as the 4-bit
binary representations of the number 0–15. First, $\bm{P}_{B}$ has the form as
in Eq. (12) of Xie et al. [3]. For example, the diagonal entries of
$4\bm{P}_{B}$ are

$\displaystyle d_{0}$ $\displaystyle:=4q_{0},$ $\displaystyle
d_{1}=d_{2}=d_{4}=d_{8}$ $\displaystyle:=p_{0}+q_{0}+q_{1}+q_{2},$
$\displaystyle d_{3}=d_{6}=d_{9}=d_{12}$
$\displaystyle:=p_{1}+p_{2}+q_{1}+q_{2},$ $\displaystyle d_{5}=d_{10}$
$\displaystyle:=2(p_{0}+q_{3}),$ $\displaystyle d_{7}=d_{11}=d_{13}=d_{14}$
$\displaystyle:=p_{1}+p_{2}+p_{3}+q_{3},$ $\displaystyle d_{15}$
$\displaystyle:=4p_{3},$

where $q_{m}:=1-p_{m}$ for $m=0,1,2,3$.

For the equivalence relation above, there are six equivalence classes, namely
$\\{0000\\}$, $\\{0001,0010,0100,1000\\}$, $\\{0011,0110,1001,1100\\}$,
$\\{0101,1010\\}$, $\\{0111,\break 1011,1101,1110\\}$, and $\\{1111\\}$.
Denoting the states by their decimal representations (0–15), the equivalence
classes are $\\{0\\}$, $\\{1,2,4,8\\}$, $\\{3,6,9,$ $12\\}$, $\\{5,10\\}$,
$\\{7,11,13,14\\}$, and $\\{15\\}$. It will be convenient to reorder the
states temporarily. Within each equivalence class, we order elements so that
each is a fixed rotation of the preceding one, that is, $\\{0000\\}$,
$\\{1000,0100,0010,0001\\}$, $\\{1100,0110,0011,\break 1001\\}$,
$\\{1010,0101\\}$, $\\{1110,0111,1011,1101\\}$, and $\\{1111\\}$, or
$\\{0\\}$, $\\{8,4,2,1\\}$, $\\{12,6,3,9\\}$, $\\{10,5\\}$,
$\\{14,7,11,13\\}$, and $\\{15\\}$. We now order states in this order: 0, 8,
4, 2, 1, 12, 6, 3, 9, 10, 5, 14, 7, 11, 13, 15, which leads to an alternative
form for the transition matrix shown in Figure 1

$\bm{P}_{B}:=\frac{1}{4}\left(\begin{array}[]{c|cccc|cccc|cc|cccc|c}d_{0}&p_{0}&p_{0}&p_{0}&p_{0}&0&0&0&0&0&0&0&0&0&0&0\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr q_{0}&d_{8}&0&0&0&p_{2}&0&0&p_{1}&p_{0}&0&0&0&0&0&0\\\
q_{0}&0&d_{4}&0&0&p_{1}&p_{2}&0&0&0&p_{0}&0&0&0&0&0\\\
q_{0}&0&0&d_{2}&0&0&p_{1}&p_{2}&0&p_{0}&0&0&0&0&0&0\\\
q_{0}&0&0&0&d_{1}&0&0&p_{1}&p_{2}&0&p_{0}&0&0&0&0&0\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&q_{2}&q_{1}&0&0&d_{12}&0&0&0&0&0&p_{2}&0&0&p_{1}&0\\\
0&0&q_{2}&q_{1}&0&0&d_{6}&0&0&0&0&p_{1}&p_{2}&0&0&0\\\
0&0&0&q_{2}&q_{1}&0&0&d_{3}&0&0&0&0&p_{1}&p_{2}&0&0\\\
0&q_{1}&0&0&q_{2}&0&0&0&d_{9}&0&0&0&0&p_{1}&p_{2}&0\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&q_{0}&0&q_{0}&0&0&0&0&0&d_{10}&0&p_{3}&0&p_{3}&0&0\\\
0&0&q_{0}&0&q_{0}&0&0&0&0&0&d_{5}&0&p_{3}&0&p_{3}&0\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&0&0&0&0&q_{2}&q_{1}&0&0&q_{3}&0&d_{14}&0&0&0&p_{3}\\\
0&0&0&0&0&0&q_{2}&q_{1}&0&0&q_{3}&0&d_{7}&0&0&p_{3}\\\
0&0&0&0&0&0&0&q_{2}&q_{1}&q_{3}&0&0&0&d_{11}&0&p_{3}\\\
0&0&0&0&0&q_{1}&0&0&q_{2}&0&q_{3}&0&0&0&d_{13}&p_{3}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&0&0&0&0&0&0&0&0&0&0&q_{3}&q_{3}&q_{3}&q_{3}&d_{15}\end{array}\right).$
Figure 1: The transition matrix $\bm{P}_{B}$ for the Markov chain describing
game $B$, with states ordered first by equivalence class and then in the order
$\\{0\\},\\{8,4,2,1\\},\\{12,6,3,9\\},\\{10,5\\},\\{14,7,11,13\\},\\{15\\}$.

The lumpability condition requires that, within each block, row sums be equal.
That this condition is met can be seen at a glance. Moreover, we can also see
that the sufficient condition (17) holds as well. Because of how we ordered
the states, this condition requires that each block be constant along each
diagonal parallel to the main diagonal (assuming periodic boundary
conditions).

We conclude that

$\bar{\bm{P}}_{B}=\frac{1}{4}\left(\begin{array}[]{cccccc}4q_{0}&4p_{0}&0&0&0&0\\\
q_{0}&p_{0}+q_{0}+q_{1}+q_{2}&p_{1}+p_{2}&p_{0}&0&0\\\
0&q_{1}+q_{2}&p_{1}+p_{2}+q_{1}+q_{2}&0&p_{1}+p_{2}&0\\\
0&2q_{0}&0&2(p_{0}+q_{3})&2p_{3}&0\\\
0&0&q_{1}+q_{2}&q_{3}&p_{1}+p_{2}+p_{3}+q_{3}&p_{3}\\\
0&0&0&0&4q_{3}&4p_{3}\end{array}\right).$

We turn next to game $A^{\prime}$. Again there are 16 states (namely, the
4-bit binary representations of the integers 0–15) and the transition matrix
can be easily evaluated. To verify the lumpability condition we reorder the
states and rewrite the matrix in block form as we did for $\bm{P}_{B}$. See
Figure 2.

$\bm{P}_{A^{\prime}}:=\frac{1}{8}\left(\begin{array}[]{c|cccc|cccc|cc|cccc|c}0&2&2&2&2&0&0&0&0&0&0&0&0&0&0&0\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr 0&2&1&0&1&1&0&0&1&2&0&0&0&0&0&0\\\ 0&1&2&1&0&1&1&0&0&0&2&0&0&0&0&0\\\
0&0&1&2&1&0&1&1&0&2&0&0&0&0&0&0\\\ 0&1&0&1&2&0&0&1&1&0&2&0&0&0&0&0\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&1&1&0&0&2&0&0&0&1&1&1&0&0&1&0\\\ 0&0&1&1&0&0&2&0&0&1&1&1&1&0&0&0\\\
0&0&0&1&1&0&0&2&0&1&1&0&1&1&0&0\\\ 0&1&0&0&1&0&0&0&2&1&1&0&0&1&1&0\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&0&0&0&0&1&1&1&1&4&0&0&0&0&0&0\\\ 0&0&0&0&0&1&1&1&1&0&4&0&0&0&0&0\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&0&0&0&0&1&1&0&0&2&0&2&1&0&1&0\\\ 0&0&0&0&0&0&1&1&0&0&2&1&2&1&0&0\\\
0&0&0&0&0&0&0&1&1&2&0&0&1&2&1&0\\\ 0&0&0&0&0&1&0&0&1&0&2&1&0&1&2&0\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&0&0&0&0&0&0&0&0&0&0&2&2&2&2&0\end{array}\right).$ Figure 2: The transition
matrix $\bm{P}_{A^{\prime}}$ for the Markov chain describing game
$A^{\prime}$, with states ordered first by equivalence class and then in the
order
$\\{0\\},\\{8,4,2,1\\},\\{12,6,3,9\\},\\{10,5\\},\\{14,7,11,13\\},\\{15\\}$.

Again the condition is clearly met, and we have

$\bar{\bm{P}}_{A^{\prime}}=\frac{1}{4}\left(\begin{array}[]{cccccc}0&4&0&0&0&0\\\
0&2&1&1&0&0\\\ 0&1&1&1&1&0\\\ 0&0&2&2&0&0\\\ 0&0&1&1&2&0\\\ 0&0&0&0&4&0\\\
\end{array}\right).$

The lumpability condition (17) has been checked for $\bm{P}_{B}$ by Ethier and
Lee [5]. For $\bm{P}_{A^{\prime}}$, we can verify (17) by observing that, if
$(\sigma(1),\ldots,\sigma(N))=(2,3,\ldots,N,1)$, then, after some
calculations,
$P_{A^{\prime}}(\eta_{\sigma},\zeta_{\sigma})=P_{A^{\prime}}(\eta,\zeta)$. If
$(\sigma(1),\break\ldots,\sigma(N))=(N,N-1,\ldots,2,1)$, then the same
identity holds.

###  3.2 Means and variances

We saw in Theorems 8 and 10 that the means and variances that appear in the
SLLNs and CLTs of Sections 2.2–2.4 (namely, $\mu_{B}^{\circ}$,
$\mu_{(\gamma,1-\gamma)^{\prime}}^{\circ}$, $\mu_{[r,s]^{\prime}}^{\circ}$,
$(\sigma_{B}^{\circ})^{2}$,
$(\sigma_{(\gamma,1-\gamma)^{\prime}}^{\circ})^{2}$, and
$(\sigma_{[r,s]^{\prime}}^{\circ})^{2}$) are equal to the corresponding
quantities defined in terms of the original transition matrices (namely,
$\mu_{B}$, $\mu_{(\gamma,1-\gamma)^{\prime}}$, $\mu_{[r,s]^{\prime}}$,
$\sigma_{B}^{2}$, $\sigma_{(\gamma,1-\gamma)^{\prime}}^{2}$, and
$\sigma_{[r,s]^{\prime}}^{2}$). We claim that the corresponding quantities
defined in terms of the reduced transition matrices (namely, $\bar{\mu}_{B}$,
$\bar{\mu}_{(\gamma,1-\gamma)^{\prime}}$, $\bar{\mu}_{[r,s]^{\prime}}$,
$\bar{\sigma}_{B}^{2}$, $\bar{\sigma}_{(\gamma,1-\gamma)^{\prime}}^{2}$, and
$\bar{\sigma}_{[r,s]^{\prime}}^{2}$) are also equal. First, we define

$\displaystyle\bar{\mu}_{B}$
$\displaystyle:=\bar{\bm{\pi}}_{B}\dot{\bar{\bm{P}}}_{B}\bm{1},$
$\displaystyle\bar{\mu}_{(\gamma,1-\gamma)^{\prime}}$
$\displaystyle:=(1-\gamma)\bar{\bm{\pi}}_{C^{\prime}}\dot{\bar{\bm{P}}}_{B}\bm{1},$
$\displaystyle\bar{\mu}_{[r,s]^{\prime}}$
$\displaystyle:=\frac{1}{r+s}\sum_{v=0}^{s-1}\bar{\bm{\pi}}\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{v}\dot{\bar{\bm{P}}}_{B}\bm{1},$
$\displaystyle\bar{\sigma}_{B}^{2}$
$\displaystyle:=\bar{\bm{\pi}}_{B}\ddot{\bar{\bm{P}}}_{B}\bm{1}-(\bar{\bm{\pi}}_{B}\dot{\bar{\bm{P}}}_{B}\bm{1})^{2}+2\bar{\bm{\pi}}_{B}\dot{\bar{\bm{P}}}_{B}(\bar{\bm{Z}}_{B}-\bm{1}\bar{\bm{\pi}}_{B})\dot{\bar{\bm{P}}}_{B}\bm{1},$
$\displaystyle\bar{\sigma}_{(\gamma,1-\gamma)^{\prime}}^{2}$
$\displaystyle:=\bar{\bm{\pi}}_{C^{\prime}}\ddot{\bar{\bm{P}}}_{C^{\prime}}\bm{1}-(\bar{\bm{\pi}}_{C^{\prime}}\dot{\bar{\bm{P}}}_{C^{\prime}}\bm{1})^{2}+2\bar{\bm{\pi}}_{C^{\prime}}\dot{\bar{\bm{P}}}_{C^{\prime}}(\bar{\bm{Z}}_{C^{\prime}}-\bm{1}\bar{\bm{\pi}}_{C^{\prime}})\dot{\bar{\bm{P}}}_{C^{\prime}}\bm{1},$
$\displaystyle\bar{\sigma}_{[r,s]^{\prime}}^{2}$
$\displaystyle:=\frac{1}{r+s}\bigg{\\{}s-\sum_{v=0}^{s-1}(\bar{\bm{\pi}}\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{v}\dot{\bar{\bm{P}}}_{B}\bm{1})^{2}$
$\displaystyle\qquad\qquad{}+2\bigg{[}\sum_{0\leq u<v\leq
s-1}\bar{\bm{\pi}}\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{u}\dot{\bar{\bm{P}}}_{B}(\bar{\bm{P}}_{B}^{v-u-1}-\bm{1}\bar{\bm{\pi}}\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{v})\dot{\bar{\bm{P}}}_{B}\bm{1}$
$\displaystyle\qquad\qquad\qquad{}+\sum_{u=0}^{s-1}\sum_{v=0}^{s-1}\bar{\bm{\pi}}\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{u}\dot{\bar{\bm{P}}}_{B}\bar{\bm{P}}_{B}^{s-u-1}(\bar{\bm{Z}}-\bm{1}\bar{\bm{\pi}})\bar{\bm{P}}_{A^{\prime}}^{r}\bar{\bm{P}}_{B}^{v}\dot{\bar{\bm{P}}}_{B}\bm{1}\bigg{]}\bigg{\\}}.$

######  Theorem 13.

$\mu_{B}=\bar{\mu}_{B},\qquad\mu_{(\gamma,1-\gamma)^{\prime}}=\bar{\mu}_{(\gamma,1-\gamma)^{\prime}},\qquad\mu_{[r,s]^{\prime}}=\bar{\mu}_{[r,s]^{\prime}}$

and

$\sigma_{B}^{2}=\bar{\sigma}_{B}^{2},\qquad\sigma_{(\gamma,1-\gamma)^{\prime}}^{2}=\bar{\sigma}_{(\gamma,1-\gamma)^{\prime}}^{2},\qquad\sigma_{[r,s]^{\prime}}^{2}=\bar{\sigma}_{[r,s]^{\prime}}^{2}.$

###### Proof.

A result of Ethier and Lee [6] implies that, if $\bm{Q}$ is a $G$-invariant
square (not necessarily stochastic) matrix (i.e.,
$Q(\eta_{\sigma},\zeta_{\sigma})=Q(\eta,\zeta)$ for all
$\eta,\zeta\in\\{0,1\\}^{N}$ and all $\sigma\in G$), then

$\bm{\pi}\bm{Q}\bm{1}=\bar{\bm{\pi}}\bar{\bm{Q}}\bm{1}.$

Repeated application of this identity gives the desired conclusions. ∎

The formulas for the means with bars are computable for $3\leq N\leq 18$, at
least. We give partial results for Toral’s [2] choice of the parameter vector
$(p_{0},p_{1},p_{2},p_{3})$ in Table 1. The formulas for the variances with
bars are perhaps computable for $3\leq N\leq 12$, but we do not include them
here.

###  3.3 Computer graphics

Ethier and Lee [4] sketched, for games $A$, $B$, and
$C:=\frac{1}{2}A+\frac{1}{2}B$, the Parrondo and anti-Parrondo regions when
$3\leq N\leq 9$. They assumed that $p_{1}=p_{2}$ and relabeled $p_{3}$ as
$p_{2}$. In other words, their parameter vector was of the form
$(p_{0},p_{1},p_{1},p_{2})$. (The reason for this simplification is that a
three-dimensional figure is easier to visualize than a four-dimensional
figure.) The figures for games $A^{\prime}$, $B$, and
$C^{\prime}:=\frac{1}{2}A^{\prime}+\frac{1}{2}B$ are distinctively different
from those for games $A$, $B$, and $C$. In both cases, the general shape of
the Parrondo and anti-Parrondo regions does not change much, once $N\geq 5$.
We illustrate in the case $r=1$ and $s=2$ in Figure 4.

Table 1: Mean profit per turn at equilibrium in the games of Toral [2] and Xie
et al. [3], assuming $(p_{0},p_{1},p_{2},p_{3})=(1,4/25,4/25,7/10)$. Results
are given to six significant digits. The entries corresponding to $N=\infty$
are limits as $N\to\infty$ (see Theorem 14). mean profit per turn, Toral’s
games
---
$N$0 | $B$ | $\frac{1}{2}(A+B)$ | $AB$ | $ABB$ | $AAB$ | $AABB$
030 | $-0.0909091\hphantom{0}\hphantom{0}$ | $-0.0183774\hphantom{0}$ | $-0.00695879$ | $-0.0274821\hphantom{0}$ |  $-$0.000672486 |  $-0.0148718$0
060 | $-0.0189247\hphantom{0}\hphantom{0}$ |  $-$0.00463310 |  $-$0.00497503 |  $-$0.00590528 |  $-$0.003250990 |  $-$0.00498178
090 | $-0.00189233\hphantom{0}$ |  $-$0.00479036 |  $-$0.00493507 |  $-$0.00598135 |  $-$0.003278020 |  $-$0.00493728
120 | $-0.000676916$ |  $-$0.00479089 |  $-$0.00490464 |  $-$0.00586697 |  $-$0.003288000 |  $-$0.00490531
150 | $-0.000586184$ |  $-$0.00479089 |  $-$0.00488431 |  $-$0.00579891 |  $-$0.003292490 |  $-$0.00488449
180 | $-0.000579652$ |  $-$0.00479089 |  $-$0.00486999 |  $-$0.00575438 |  $-$0.003294830 |  $-$0.00487001
$\infty$0 |  |  $-$0.00479089 |  $-$0.00479089 |  $-$0.00554084 |  $-$0.003298530 |  $-$0.00479089
mean profit per turn, Xie at al.’s games
$N$0 | $B$ | $\frac{1}{2}(A^{\prime}+B)$ | $A^{\prime}B$ | $A^{\prime}BB$ | $A^{\prime}A^{\prime}B$ | $A^{\prime}A^{\prime}BB$
030 | $-0.0909091\hphantom{0}\hphantom{0}$ | $-0.0766158\hphantom{0}$ | $-0.105479\hphantom{0}\hphantom{0}$ | $-0.102038\hphantom{0}\hphantom{0}$ | $-0.0724638\hphantom{0}$ |  $-0.0773252$0
060 | $-0.0189247\hphantom{0}\hphantom{0}$ |  $-$0.00671656 |  $-$0.00640351 |  $-$0.00955597 |  $-$0.00363075 |  $-$0.00745377
090 | $-0.00189233\hphantom{0}$ |  $-$0.00678314 |  $-$0.00676079 |  $-$0.00887095 |  $-$0.00402382 |  $-$0.00705972
120 | $-0.000676916$ |  $-$0.00678336 |  $-$0.00682799 |  $-$0.00860524 |  $-$0.00419181 |  $-$0.00695667
150 | $-0.000586184$ |  $-$0.00678336 |  $-$0.00684381 |  $-$0.00845891 |  $-$0.00427852 |  $-$0.00691300
180 | $-0.000579652$ |  $-$0.00678336 |  $-$0.00684607 |  $-$0.00836539 |  $-$0.00433011 |  $-$0.00688859
$\infty$0 |  |  $-$0.00678336 |  $-$0.00678336 |  $-$0.00792947 |  $-$0.00451510 |  $-$0.00678336

##  4 Convergence of means

Computations suggest that $\mu_{(\gamma,1-\gamma)^{\prime}}^{N}$ and
$\mu_{[r,s]^{\prime}}^{N}$ converge as $N\to\infty$, regardless of the
parameters $p_{0},p_{1},p_{2},p_{3}\in(0,1)$. We cannot prove this, but we can
give sufficient conditions on the parameters for this convergence to hold.
These are

$\displaystyle\max\bigg{[}\bigg{|}\frac{\gamma}{2}+(1-\gamma)(p_{0}-p_{1})\bigg{|},\bigg{|}\frac{\gamma}{2}+(1-\gamma)(p_{2}-p_{3})\bigg{|}\bigg{]}$
$\displaystyle\qquad+\max\bigg{[}\bigg{|}\frac{\gamma}{2}+(1-\gamma)(p_{0}-p_{2})\bigg{|},\bigg{|}\frac{\gamma}{2}+(1-\gamma)(p_{1}-p_{3})\bigg{|}\bigg{]}<1.$
(18)

######  Theorem 14.

Fix integers $r,s\geq 1$ and put $\gamma:=r/(r+s)$. If (4) holds, then
$\lim_{N\to\infty}\mu_{(\gamma,1-\gamma)^{\prime}}^{N}$ exists, and
$\lim_{N\to\infty}\mu_{[r,s]^{\prime}}^{N}=\lim_{N\to\infty}\mu_{(\gamma,1-\gamma)^{\prime}}^{N}$.

The volume of the subset of the parameter space $[0,1]^{4}$ for which (4)
holds with $\gamma=1/2$ is, by Mathematica, 5/6. If we assume that
$p_{1}=p_{2}$, then the volume of the subset of the parameter space
$[0,1]^{3}$ for which (4) holds is, by Mathematica, 3/4. In fact, we plot the
three-dimensional volume as a function of $\gamma$ in Figure 3.

Figure 3: Assuming $p_{1}=p_{2}$, the three-dimensional volume of the subset
of the parameter space for which (4) holds is plotted as a function of
$\gamma$.

Notice that the volume is $3/4$ if and only if $\gamma\geq 1/3$.

$N=3$ $N=4$

$N=5$ $N=6$

Figure 4: For $3\leq N\leq 6$, the blue surface is the surface $\mu_{B}=0$,
and the red surface is the surface $\mu_{[1,2]^{\prime}}=0$, in the
$(p_{0},p_{2},p_{1})$ unit cube. The Parrondo region is the region on or below
the blue surface and above the red surface, while the anti-Parrondo region is
the region on or above the blue surface and below the red surface. Here
$(p_{0},p_{1},p_{1},p_{3})$ is relabeled as $(p_{0},p_{1},p_{1},p_{2})$.

## References

  * [1] A. Ajdari and J. Prost, Drift induced by a spatially periodic potential of low symmetry: Pulsed dielectrophoresis, C. R. Acad. Sci., Ser. II 315 (1992) 1635–1639.
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  * [9] Y.-F. Li, S.-Q. Ye, K.-X. Zheng, N.-G. Xie, Y. Ye and L. Wang, A new theoretical analysis approach for a multi-agent spatial Parrondo’s game, Phys. A 407 (2014) 369–379.
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  * [14] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/, 2019.


# The Rotation of Magnetic Flux Rope Formed during Solar Eruption

Zhenjun Zhou(周振军) Planetary Environmental and Astrobiological Research
Laboratory (PEARL), School of Atmospheric Sciences, Sun Yat-sen University,
Zhuhai, China  CAS Key Laboratory of Geospace Environment, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Comparative Planetology, China  Chaowei Jiang Institute of Space
Science and Applied Technology, Harbin Institute of Technology, Shenzhen
518055, People’s Republic of China  Rui Liu CAS Key Laboratory of Geospace
Environment, University of Science and Technology of China, Hefei, Anhui
230026, China  Yuming Wang CAS Key Laboratory of Geospace Environment,
University of Science and Technology of China, Hefei, Anhui 230026, China
Lijuan Liu Planetary Environmental and Astrobiological Research Laboratory
(PEARL), School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, China
Jun Cui Planetary Environmental and Astrobiological Research Laboratory
(PEARL), School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, China

###### Abstract

The eruptions of solar filaments often show rotational motion about their
rising direction, but it remains elusive what mechanism governs such rotation
and how the rotation is related to the initial morphology of the pre-eruptive
filament (and co-spatial sigmoid), filament chirality, and magnetic helicity.
The conventional view regarding the rotation as a result of a magnetic flux
rope (MFR) undergoing the ideal kink instability still has confusion in
explaining these relationships. Here we proposed an alternative explanation
for the rotation during eruptions, by analyzing a magnetohydrodynamic
simulation in which magnetic reconnection initiates an eruption from a sheared
arcade configuration and an MFR is formed during eruption through the
reconnection. The simulation reproduces a reverse S-shaped MFR with dextral
chirality, and the axis of this MFR rotates counterclockwise while rising,
which compares favorably with a typical filament eruption observed from dual
viewing angles. By calculating the twist and writhe numbers of the modeled MFR
during its eruption, we found that accompanied with the rotation, the nonlocal
writhe of the MFR’s axis decreases while the twist of its surrounding field
lines increases, and this is distinct from the kink instability, which
converts magnetic twist into writhe of the MFR axis.

Sun: flares — Sun: corona — Sun: X-rays

††journal: ApJL††journal: APJL

_Online-only material_ : animations, color figures

##  1 Introduction

Solar eruptions often show rotational motion, especially in filament
eruptions, which are commonly believed to be the manifestation of erupting
magnetic flux ropes (MFRs). For example, during some filament eruptions, the
observation of a bunch of helical threads that appear to wind around a central
axis with rotation motion is reminiscent of an MFR. The rotation motion plays
an important role in reconfiguring the erupting magnetic field. For instance,
on the one hand, it may yield a possible magnetic reconnection of the erupting
MFR with the surrounding field ruining the coherence of the erupting MFR, even
resulting in a failed eruption (Zhou et al., 2019). On the other hand, it can
persistently modulate the axis direction of the subsequent coronal mass
ejection (CME), and change the southward component of the interplanetary
magnetic field, which therefore makes the prediction of potential
geoeffectiveness more challenging (Yurchyshyn et al., 2009).

The twisting and rotating features indicate that their underlying magnetic
fields carry currents and possess magnetic helicity, and the direction of
rotation is closely related to the sign of helicity. Indeed, observations of
filament eruptions show that there is a one-to-one correlation between the
rotation direction and the filament chirality (or sign of helicity of the
corresponding magnetic field); sinistral (dextral) filaments with positive
(negative) helicity rotate clockwise (counterclockwise) when viewed from above
(Green et al., 2007). Furthermore, the pre-eruptive morphology of filaments
corroborates an moderate hemispheric preference, namely, filaments of forward
(reverse) S shape are usually located in the southern (northern) hemisphere
and have a positive (negative) helicity (Rust & Kumar, 1996; Zhou et al.,
2020). Besides, there are coronal loops presenting S shape known as sigmoids
observed in extreme ultraviolet (EUV) and soft X-ray (SXR) passbands (Cheng et
al., 2017), and in many events the sigmoid is found to be co-spatial roughly
with the pre-eruptive filament.

To explain the relationships between the rotation direction of the MFR as it
erupts, the field chirality, and the associated filament (sigmoid) morphology,
Green et al. (2007) invoked the theory based on the Titov and Démoulin (T&D)
model (Titov & Démoulin, 1999) which assumes an arched MFR pre-existing before
eruption, and argued that the observations agree well with the T&D model. In
the T&D model, the observed sigmoid is considered to be a thin current layer
formed in the bald patch separatrix layer (BPSS) or hyperbolic flux tube (HFT)
in the wake of the rising flux rope, and Green et al. (2007) suggested that
the observed relationship between the filament chirality and its rotation
direction is a manifestation of ideal MHD instability of the MFR, during which
the magnetic twist is converted into the writhe of the axis (thus rotation of
the MFR’s axis). However, Török et al. (2010) found that for the T&D’s MFRs,
the relation between writhe and the projected S shape of MFRs is not unique,
since the writhe depends largely on the height of the MFRs and on the presence
or absence of dips in the middle of the MFR, rather than the transformation of
their twist helicity into writhe helicity as is often assumed.

In this Letter, we proposed an alternative explanation for the eruption
rotation, which is more uniformly consistent with the observations, by using a
reconnection-initiated eruption model in which an MFR does not need to exist
before eruption but is formed during eruption through reconnection within a
sheared arcade configuration. Our explanation is developed based on a recent
fully 3D magnetohydrodynamic (MHD) simulation (Jiang et al., 2021) which
demonstrates for the first time that the runaway tether-cutting reconnection
alone can initiate solar eruption within a single arcade as sheared by
photospheric motion. In the simulation, the MFR is formed during the eruption
through reconnection of the sheared arcade. Here we will analyze the
morphology, chirality, and rotation direction of the erupting MFR in the
simulation, and compare them with the observations of a typical filament
eruption. Our results show that at the onset of the eruption, the MFR is built
up with a reverse S shape, and the top of the MFR shows a significant
counterclockwise rotation immediately after the initiation of the eruption,
which are entirely consistent with the observations. Furtherly, by a
quantitative measurement of writhe and twist of the MFR in the simulation, we
found that there a transfer of writhe to twist in the MFR during eruption,
which is distinct from the previous theory based on kink instability.

##  2 Observation of a typical filament eruption

We first take a typical filament eruption, which occured in NOAA active region
(AR) 11475 on May 10th, 2012, as an example to illustrate the relationship
between the orientation of the S-shaped morphology of the filament (and its
associated sigmoid), the filament chirality, and the rotation direction during
its eruption. The filament is well observed by $\mathrm{H}\upalpha$ image in
the 6563 Å wavelength from Global Oscillation Network Group (GONG; Hill et
al., 1994), showing an inverse S shape located on the solar disk center
(S15W15) as seen from the Earth view (Figure 1(a)). It has a wider appearance
in the He II 304 Å from dual-perspective imaging observations from Solar
Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA; Lemen et al.,
2012) and Solar TErrestrial RElations Observatory (STEREO)-A/ Extreme
UltraViolet Imager (EUVI; Wuelser et al., 2004). A sigmoid co-spatial with the
filament is observed with the X-ray telescope (XRT; Golub et al., 2007)
onboard Hinode (Figure 1(b)).

The chirality of the filament can be determined with the help of the radial
magnetogram provided by the Heliospheric and Magnetic Imager (HMI; Schou et
al., 2012) onboard SDO. Figure 1(c) shows that overall this AR has a bipolar
configuration where the two opposite polarities are aligned
northeast$-$southwest, and the filament observed in the SDO/AIA 304 Å (Figure
2(d)) is outlined by the green plus symbols as overplotted on the magnetogram.
The footpoints of the magnetic field supporting the filament can be located by
where the filament plasma flows down to the solar surface. The observed right-
skewed drainage sites relative to the PIL implies that the filament chirality
is dextral (Chen et al., 2014; Zhou et al., 2020). Meanwhile, during its
eruption, the apex part of the filament displays CCW rotation, which can be
seen in both SDO and STEREO-A observations (Figure 2(b),(d), and accompanied
animation). This event is a well-observed example supporting a strong one-to-
one relationship during the eruption as found by Zhou et al. (2020);
sinistral/dextral filaments rotate clockwise (CW)/counterclockwise (CCW) when
viewed from above, and the morphology of the filament and related sigmoid both
exhibit a forward (reverse) S shape.

##  3 MHD Simulation of an erupting MFR

Jiang et al. (2021) performed a high accuracy, fully 3D MHD simulation and
established a fundamental mechanism behind solar eruption initiation: a
bipolar field driven by quasi-static shearing motion at the photosphere can
form an internal current sheet, followed by fast magnetic reconnection that
triggers and drives the eruption. In this mechanism, an MFR is built up during
the eruption, and here we focus on the evolution of the erupting MFR by using
the simulation run like the one in Jiang et al. (2021), but with a lower
resolution than the original ones. The simulation solves the full set of MHD
equations with both solar gravity and plasma pressure included, and starts
from a bipolar potential magnetic field and a hydrostatic plasma stratified by
solar gravity with typical coronal temperature. Then shearing flows along the
PIL, which are implemented by rotating the two magnetic polarities at the
photosphere in the same CCW direction, are applied on the bottom boundary to
energize the coronal field until an eruption is triggered, and after then the
surface flow is stopped. During the quasi-static evolution phase as driven by
the shearing motion, a current sheet is gradually built up. Since no explicit
resistivity is used in the MHD model, magnetic reconnection is triggered when
the current sheet is sufficiently thin such that its width is close to the
grid resolution, owing to the implicit, grid-dependent numerical resistivity.
For more details of the simulation settings, the readers are referred to Jiang
et al. (2021). In that paper, the simulation is managed to be of very high
resolutions with Lundquist number achieving $\sim 10^{5}$ for a length unit
(approximately 10 Mm). Therefore, the plasmoid instability is triggered in the
current sheet and the magnetic topology becomes extremely complicated in small
scales along with formation of the large-scale MFR. Such a complexity
substantially complicates our analysis of the large-scale evolution associated
with the erupting MFR, thus in this paper we used a lower-resolution run
(corresponding to a Lundquist number of $\sim 10^{3}$). In the lower-
resolution run, the amount of shearing time before the eruption onset is a
little less than that needed in the high-resolution run, because the current
sheet required for triggering reconnection is thicker and thus needs less
shear (which has been shown in Jiang et al. (2021) with four different
resolutions). That said, the basic evolution of the MFR during the eruption is
not changed as compared to the high-resolution run, except that the small-
scale complex structure will not arise. Moreover, with the lower resolution,
we can run the simulation longer and thus follow a longer evolution of MFR.

##  4 Comparison of Simulation and Observation

Figure 2 compares the process of the filament eruption from 23:00 UT on May 9
to 00:36 UT on May 10 observed by STEREO-A/EUVI, and SDO/AIA with magnetic
field evolution seen in two different views from the MHD simulation. At the
onset of the eruption (Note that here for the simulation, $t=0$ is reset to be
the onset time of the simulated eruption), the core magnetic field of the
newly formed MFR, which is built up through reconnection of the two sets of
sheared arcades, presents a continuous reverse S-shaped sigmoid from the top
view, and subsequently exhibits a significant CCW rotation during the eruption
(Figure 2(c)). From the side (limb) view, the low-lying flux rope rises up
quickly, yielding a nearly circular shape, and further with the CCW rotation
the shape is transformed into an oval (Figure 2(a)). Therefore, the evolving
morphology of the erupting flux rope in the simulation agrees well with that
of the erupting filament in the dual-perspective observations from STEREO-A
and SDO.

To compare the thermal morphology of the observed sigmoid with that of the
simulation, we deduce the thermal evolution of this eruption based on imaging
data from six AIA EUV passbands, including 131Å (Fe XXI, $\sim$11 MK; Fe VIII,
$\sim$0.4 MK), 94Å (Fe XVIII, $\sim$7.1 MK; Fe X, $\sim$1.1 MK), 335Å (Fe XVI,
$\sim$2.5 MK), 211Å (Fe XIV, $\sim$2.0 MK), 193Å (Fe XII, $\sim$1.6 MK; Fe
XXIV, $\sim$17.8 MK), and 171Å (Fe IX, $\sim$0.6 MK) (O’Dwyer et al., 2010).
We use a sparse inversion code (Cheung et al., 2015; Su et al., 2018) to
calculate the emission measure (EM) as a function of temperature from AIA
imaging data. EM is the integral of electron density squared over the emitting
volume, it gives the amount of plasma emission at a given temperature. Due to
the limitation of HINODE/XRT observations, evolution of sigmoid can’t be
followed up in SXR passbands, we use this EM map as a substitution. The EM
maps over the temperature range from 5 -8 MK show a clear sigmoid-to-arcade
transformation during the eruption (Figure 3(a)-(d)). Initially, a coaxial,
bright feature appears in the wake of this rising filament, broadens as a
sigmoidal shape, and finally, evolves into an arcade shape. This sigmoidal
emission pattern is expected to be due to the heating in the current-carrying
magnetic fields (Kliem et al., 2004; Gibson et al., 2006). To visualize
current-carrying field lines in the simulation for comparison, a method for
the synthesis of mock coronal images is utilized. It calculates line-of-sight
integrals of the proxy emissivity from the value of $j^{2}$ (square of the
current density) averaged along magnetic field lines (Cheung & DeRosa, 2012).
From the synthetic images (Figure 3(e)-(h) and accompanied movie), an inverse
sigmoidal shape forms before the eruption and then it broadens with the
expansion of the erupting field. In the end, the two elbows of the sigmoid
fade away and become indistinguishable from the ambient as shown in Figure
3(g) and (h). Compared to the heating only along current sheets at the
interface between the helical core field (e.g., MFR) and the ambient field
(Kliem et al., 2004; Gibson et al., 2006), this result provides an alternative
scenario that the sigmoidal emission pattern is due to the heating as line-of-
sight integrals of $j^{2}$ from both the current sheet and the nearby regions
with intense currents.

##  5 Evolution of twist and writhe

To understand the variation of the MFR’s morphology during its rising, we
investigate the evolution of two parameters, namely, the writhe number
($\mathcal{W}_{r}$) and the twist number ($\mathcal{T}_{w}$), which
characterize quantitatively the helical deformation of the MFR axis and how
much the field lines winding about the MFR axis, respectively. Based on the
simulation data, these two parameters are computed using the following
methods.

For an open curve like the axis of an MFR with both endpoints on a bottom
plane (e.g., the photosphere), its temporal evolution of writhe is difficult
to be quantified (Linton et al., 1998). Berger & Prior (2006) proposed a
modified writhe expression termed the polar writhe to distinguish from the
pre-existing closed curve definition. The bottom plane has $\hat{\mathbf{z}}$
as normal. Along the $z$-direction, the open curve is split into several
pieces at turning points (extrema in z-direction). The coiled geometric
quantity of each individual piece is the local polar writhe
($\mathcal{W}_{pl}$), where the global geometric relations between the pieces
is described by the nonlocal polar writhe ($\mathcal{W}_{npl}$). The polar
writhe ($\mathcal{W}_{p}$) is the sum of local and non-local components. This
nonlocal component is useful in interpreting the presence of S shape in the
corona (Török et al., 2010). It can be calculated as (Berger & Prior, 2006):

$\mathcal{W}_{pnl}(\textbf{r})=\mathop{\sum_{i=1}^{n+1}\sum_{j=1}^{n+1}}_{i\neq
j}\frac{\sigma_{i}\sigma_{j}}{2\pi}\int_{z^{min}_{ij}}^{z^{max}_{ij}}\frac{d\Theta_{ij}}{dz}dz,$
(1)

where i,j are two different pieces, $\sigma_{i}=+1$ if this piece is moving
upward, $\sigma_{i}=-1$ if this piece is moving downward, and $n$ is the
number of the turning points. In the integration, let the relative position
vector at height z be $\mathbf{r}_{ij}(z)=x_{j}(z)-x_{i}(z)$. Note that
$\mathbf{r}_{ij}(z)$ is parallel to the xy plane. $\Theta_{ij}$ is the
orientation of this vector with respect to the x axis, $z^{min}_{ij}$ and
$z^{max}_{ij}$ are the minimum and maximum heights at which both pieces reach.
The $\mathcal{W}_{pnl}$ of the open curve then can be computed based on this
equation using the Prior & Neukirch (2016) code available
online.111https://www.maths.dur.ac.uk/~ktch24/code.html This code can also
compute the $\mathcal{W}_{pl}$ and $\mathcal{W}_{r}$, which are used to
calculate the linking number
($\mathcal{L}_{k}=\mathcal{W}_{r}+\mathcal{T}_{w}$, see the inset panel of
Figure 5(a)).

The definition of the other parameter, the twist number ($\mathcal{T}_{w}$),
is defined as follow: Let a smooth curve $\mathbf{y}(s)$ wrap around the
central axis $\mathbf{x}(s)$, where $s$ is the arc length starting from a
reference point on this central axis. $\mathbf{T}(s)$ is the unit tangent
vector to the axis curve $\mathbf{x}(s)$, and $\mathbf{V}(s)$ denotes a unit
vector normal to $\mathbf{T}(s)$ and points to $\mathbf{y}$ at the point
$\mathbf{y}(s)=\mathbf{x}(s)+\epsilon\mathbf{V}(s)$. Then $\mathcal{T}_{w}$
density can be calculated following the formula (Berger & Prior, 2006; Guo et
al., 2013),

$\dfrac{d\mathcal{T}_{w}}{ds}=\frac{1}{2\pi}\mathbf{\textit{T}}\cdot\mathbf{\textit{V}}\times\dfrac{\mathbf{\textit{dV}}}{ds},$
(2)

The total twist is the integration of Equation 2 along the axis curve
$\mathbf{x}$. If the field line is in the vicinity of the MFR’s axis and the
MFR is approximately cylindrically symmetric, it is more convenient to use the
twist number of an individual magnetic field line defined by

$\mathcal{T}_{w}^{\prime}=\int_{L}\frac{\mu_{0}J_{\parallel}}{4\pi
B}dl=\int_{L}\frac{\curl{\mathbold{B}}\cdot\mathbold{B}}{4\pi B^{2}}dl$ (3)

$\mathcal{T}_{w}^{\prime}$ measures how much two neighboring close field lines
twist about each other. It is a reliable approximation of the twist number
with respect to the axis, $\mathcal{T}_{w}$, as computed by integration of
Equation (2) (Liu et al., 2016, Appendix C). The total twist is the line
integral of the twist intensity ($\curl{\mathbold{B}}\cdot\mathbold{B}/4\pi
B^{2}$) along each individual field line.

To calculate the two parameters, the top priority is the determination of the
MFR’s axis. Due to the symmetry of modeled MFR’s geometry, the streamlines of
its transverse magnetic field forms a series of concentric rings at the
central cross-section (Figure 4(a)), and the axis of the MFR can be identified
clearly as the field line passing through the center of these concentric
rings. Furthermore, the bottom surface of the MHD model is fixed without any
motion during the eruption, thus for any field line without reconnection, its
two footpoints will not change with time owing to this line-tied boundary
condition. Therefore, once the axis is located initially, its subsequent
evolution can be followed by tracing the field line from one fixed footpoint
of the axis (red line in Figure 4(b)-(d)). To compute the twist number, we
traced eight sample field lines around the axis starting from eight points in
the neighborhood surrounding the footpoint of the axis (Figure 4(b)-(d), as an
example for one of the eight field lines). During the eruption, the other
footpoints’ location of the axis and wrapping field line moves less than 1.8
grid points, lower than 5.8% changing rate relative to the separation of their
two footponts.

Accompanied with the MFR rising and rotation motions (Figure 2(a) and (c)),
temporal evolutions of $\mathcal{W}_{pnl}$ and $\mathcal{T}_{w}$ of the
neighboring field line are shown in Figure 5. Initially, the reverse S-shaped
MFR possesses a positive $\mathcal{W}_{pnl}$ of $0.33$, and as the CCW
rotation of the apex of the MFR about its rise direction sets in, this value
decreases monotonically to $0$, indicating that the initial strong reverse
S-shaped bending is completely straightened out. Moreover, as the rotation
goes on, $\mathcal{W}_{pnl}$ even reverses its sign to a negative value of
$-0.07$. On the other hand, the neighboring field lines winding around the
axis have initially a left-handed twist on average of ($\mathcal{T}_{w}$)
$\approx-1.70$, and as the eruption goes on, the twist is enhanced with its
absolute value growing gradually to $2.09$, which indicates that the CCW
rotation motion twists up these spiral field lines. Based on the extension for
the open field lines of Călugăreanu theorem (Berger & Prior, 2006), the total
linking number ($\mathcal{L}_{k}=\mathcal{T}_{w}+\mathcal{W}_{r}$) is proved
mathematically invariant to all motions (as long as no reconnection happens
between these field lines). This is consistent with our result (the changing
rate of $\mathcal{L}_{k}$ is less than 8%, see the inset panel of Figure 5(a))
which shows a negative correlation between $\mathcal{W}_{r}$ and
$\mathcal{T}_{w}$ and indicates that writhe transfers to twist during the
eruption. We have also checked the energy and helicity evolution during the
simulated eruption. As can be seen in the bottom panel of Figure 5, the free
magnetic energy in the volume is rapidly released by 50% through the eruption,
while the magnetic helicity is preserved pretty well in the volume with only a
small variation of less than 2%.

##  6 Discussion and Conclusion

In this Letter, the relationship between the direction of filament rotation
during eruption, the orientation of S-shaped morphology of filament (and co-
spatial sigmoid), and the chirality of the filament is studied using a fully
3D MHD simulation of solar eruption initiation. The simulated flux rope
eruption resembles the initial morphology and rotation motion of an erupted
filament. The emission image as synthesized from the electric current density
in the model shows an inverse sigmoidal pattern in the wake of the eruption
and the sigmoid-to-arcade transformation. Furtherly, $\mathcal{W}_{pnl}$ and
$\mathcal{T}_{w}$ of the simulated MFR, the quantitative parameters describing
the deformation of the axis and its wrapping field lines, are calculated,
which shows clearly an accumulation of the $\mathcal{T}_{w}$ and a reduction
of the $\mathcal{W}_{pnl}$ during the eruption. Such a transfer from writhe to
twist is at variance with the existing explanation for MFR rotation invoking
the helical kink instability in which the twist of MFR is converted to writhe.

Many attempts have been made before to determine this observed
rotation–chirality relationship. For example, Green et al. (2007) has
comprehensively reviewed various models of sigmoid formation and considered
this observed property as a consequence of the conversion of twist into writhe
under the ideal MHD constraint of helicity conservation. But this leaves a
mystery: through the rotation the original inverse S-shaped filament spine is
straightened and even over rotated to become forward S-shaped, which
contradicts the expectation of the kink instability. Some observation and
simulation suggest that eruption of a low-lying MFR with downward-bent axis
also accommodates this scenario (e.g., Török et al., 2010; Zhou et al., 2017).
But for the studied filament eruption, no obvious dip (i.e., with a concave-
upward motion) is present in the middle, and the initial reverse S-shape of
the filament is formed largely by the two curved ends rather than downward-
bent or flat portions in its mainbody. This mystery is solved in our analysis
here, that a forward (reverse) S-shaped filament eruption shows CW (CCW)
rotation is consistent with the eruption scenario as demonstrated by Jiang et
al. (2021)’s simulation, namely, the erupting MFR is formed during the
eruption by tether-cutting reconnection. The physics behind the key different
behavior of MFR formed during eruption from that formed prior to eruption will
be investigated in future works.

##  7 acknowledgements

The authors wish to express their special thanks to the referee for
suggestions and comments which led to the improvement of the paper. The
authors appreciate discussions with Guo Yang, Xin Cheng, and Xudong Sun. We
acknowledge the _SECCHI_ , _AIA_ , _GONG_ , _XRT_ , and _HMI_ consortia for
providing excellent observations.

This work is supported by the B-type Strategic Priority Program XDB41000000
funded by the Chinese Academy of Sciences. The authors also acknowledge
support from the National Natural Science Foundation of China (NSFC 42004142,
41822404, 41731067, 11925302, 42188101, 41822404, 41731067, 41574170, and
41531073), Open Research Program of CAS Key Laboratory of Geospace
Environment, the Fundamental Research Funds for the Central Universities
(grant No. HIT.BRETIV.201901), and Shenzhen Technology Project
JCYJ20190806142609035.

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Figure 1: Panel (a) shows a reverse S-shaped filament observed by
GONG/H$\alpha$ on 2012 May 10 at 00:00 UT. The corresponding sigmoid observed
by Hinode/XRT is shown in (b). In Panel (c), the green plus symbols outlining
the EUV filament spine are projected onto an HMI magnetogram of local $B_{r}$.

Figure 2:  Comparison of the filament eruption on 10 May 2012 with the
numerical simulation of Jiang et al. (2021). Panels (a) and (c) Side and top
views of the MFR eruption sequentially in the simulation. The colored thick
lines represent the MFR’s core magnetic field lines, and the colors denote the
value of the nonlinear force-free factor defined as $\alpha$=$\bf J\cdot\bf B$
/$B^{2}$, where $\bf J$ is the current density and $\bf B$ is the magnetic
field. Panels (b) and (d) provide sequential snapshots of the filament
eruption from the limb view in STEREO-A/EUVI 304 Å and disk view in SDO/AIA
304 Å, respectively, showing a CCW rotation of the filament, all the images
here from STEREO and SDO has been rotated to accommodate with the simulation
for comparison purpose. The video duration is 24s. (An animation of this
figure is available.)

Figure 3: Thermal evolution of the solar eruption and simulation. EM maps
ranging from 5 - 8 MK (a)-(d) for the filament eruption and synthetic EUV
images (e)-(h) for the numerical simulation. The video duration is 18s. (An
animation of this figure is available.)

Figure 4: (a) Streamlines of the magnetic field at central plane at t = 28
mins, showing the cross-section of the MFR; Panels (b)-(d) Evolutions of the
MFR’s axis (red) and one twisted field line (green) wrapping around.

Figure 5: Evolution of the parameters in the simulation. Top panel (a):
temporal evolution of twist (black line) and nonlocal writhe (red line) of the
flux rope during its eruption. In the inset panel, the green line shows the
temporal changing rate of linking number ($\mathcal{L}_{k}$); Bottom panel
(b): the temporal change ratios of free magnetic energy E (black line) and
relative magnetic helicity H (blue line) compared with that of the eruption
onset. For reference, at onset of the eruption the magnetic free energy is
$0.91$ of the corresponding potential magnetic energy, and the relative
helicity is $-0.06$ as normalized by the square of the total unsigned magnetic
flux.