3.1. A brief discussion on PCAs in general

A d-dimensional deterministic cellular automaton (CA) is a discrete dynamical system with the following components:

• • the universe $\mathbb{Z}^{d}$ consisting of cells;

• • a finite set of states $\mathcal{A}$ referred to as the alphabet;

• • a finite set of indices $\mathcal{N} = \left\{\mathbf{y}_{1}, \mathbf{y}_{2}, \ldots, \mathbf{y}_{m}\right\} \subset \mathbb{Z}^{d}$ , referred to as the neighborhood-marking set; and

• • a local update rule $f\;:\; \mathcal{A}^{\mathcal{N}} \rightarrow \mathcal{A}$ .

A PCA F is, as the name suggests, a CA with a random or stochastic update rule, represented by a stochastic matrix $\varphi\;:\; \mathcal{A}^{m} \times \mathcal{A} \rightarrow [0,1]$ . A PCA can be viewed as a discrete-time Markov chain on the state space $\Omega = \mathcal{A}^{\mathbb{Z}^{d}}$ . Given any element $\eta = \left(\eta(\mathbf{x})\;:\; \mathbf{x} \in \mathbb{Z}^{d}\right)$ of the state space $\Omega$ , referred to as a configuration, the state $\eta(\mathbf{x})$ of the cell $\mathbf{x}$ , for each $\mathbf{x} \in \mathbb{Z}^{d}$ , is updated, independent of the updates happening at all other cells, to the random state $F\eta(\mathbf{x})$ whose distribution is given by

(9) \begin{align}\textrm{P}[F\eta(\mathbf{x}) = b|\eta(\mathbf{x}+\mathbf{y}_{i}) = a_{i} \text{ for all } 1 \leqslant i \leqslant m] = \varphi(a_{1}, a_{2}, \ldots, a_{m}, b) \text{ for all } b \in \mathcal{A},\end{align}

for any $a_{1}, a_{2}, \ldots, a_{m} \in \mathcal{A}$ . We may imagine that the updates happen at discrete time steps, starting with an initial configuration $\eta_0$ at time step $t=0$ , and recursively defining, for each $t \in \mathbb{N}$ , $\eta_{t} = \left(\eta_{t}(\mathbf{x})\;:\; \mathbf{x} \in \mathbb{Z}^{d}\right) = F \eta_{t-1} = \left(F\eta_{t-1}(\mathbf{x})\;:\; \mathbf{x} \in \mathbb{Z}^{d}\right)$ . This yields $\eta_{t} = F^{t}\eta_{0}$ , where $F^{t}$ refers to the update (i.e. the application of F) having happened t times.

Given a random configuration $\boldsymbol{\eta}$ which follows a probability distribution $\mu$ (measurable with respect to the $\sigma$ -field generated by all cylinder sets of $\Omega$ ), we let $F^t\mu$ denote the probability distribution of $F^{t}\boldsymbol{\eta}$ (obtained by starting from $\boldsymbol{\eta}$ and performing t updates sequentially via the stochastic update rules of F, as shown in (9)). We call a probability distribution $\mu$ stationary for F if $F\mu = \mu$ . A PCA F is said to be ergodic if:

1. (i) it has a unique stationary measure, say $\mu$ ;

2. (ii) $F^t\nu \rightarrow \mu$ weakly, as $t\rightarrow \infty$ , for each probability distribution $\nu$ on $\Omega$ .

Figure 4.

Stochastic update rules for the PCA $\widehat{G}_{p,q,r}$ .

3.2. The specific PCAs we work with in this paper

We begin by describing the PCA that represents the game rules arising from the generalized percolation games described in Section 2. This PCA, denoted $\widehat{G}_{p,q,r}$ (in order to emphasize its dependence on the parameter triple (p, q, r)), is a one-dimensional PCA (i.e. its universe is $\mathbb{Z}$ ), with:

1. (i) the alphabet $\hat{\mathcal{A}}=\{W,L,D\}$ (already, this choice of notation hints at the intimate connection between the game and this PCA: if $(x,y) \in \mathbb{Z}^{2}$ serves as the initial vertex for a generalized percolation game, we say that $(x,y) \in W$ if this game is won by the player who plays the first round, $(x,y) \in L$ if this game is lost by the player who plays the first round, and $(x,y) \in D$ if this game results in a draw);

2. (ii) the neighborhood-marking set $\mathcal{N}=\{0,1\}$ ; and

3. (iii) stochastic update rules that can be described via the stochastic matrix $\widehat{\rm \varphi}_{p,q,r}\;:\; \hat{\mathcal{A}}^{2} \times \hat{\mathcal{A}} \rightarrow [0,1]$ , defined as (see Figure 4 for a pictorial illustration)

   (10) \begin{equation} \widehat{\rm \varphi}_{p,q,r}(W,W,b) = \begin{cases} p & \text{if } b = W, \\ (1-p) & \text{if } b = L, \end{cases}\end{equation}
   (11) \begin{equation} \widehat{\rm \varphi}_{p,q,r}(a_{0},a_{1},b) = \begin{cases} p+(1-p-q)(1-r) & \text{if } b = W, \\ q+(1-p-q)r & \text{if } b = L, \end{cases} \quad\!\! \text{for } (a_{0}, a_{1}) \in \{(W,L), (L,W)\},\end{equation}
   (12) \begin{equation} \varphi_{p,q,r}(L,L,b) = \begin{cases} p+(1-p-q)(1-r^{2}) & \text{if } b = W, \\ q+(1-p-q)r^{2} & \text{if } b = L, \end{cases}\end{equation}
   (13) \begin{equation} \widehat{\rm \varphi}_{p,q,r}(a_{0}, a_{1}, b) = \begin{cases} p & \text{if } b = W, \\ (1-p-q)(1-r) & \text{if } b = D, \\ (1-p-q)r+q & \text{if } b = L, \end{cases} \quad \text{for } (a_{0}, a_{1}) \in \{(W,D), (D,W)\},\end{equation}
   (14) \begin{equation} \widehat{\rm \varphi}_{p,q,r}(a_{0}, a_{1}, b) = \begin{cases} p+(1-p-q)(1-r) & \text{if } b = W, \\ (1-p-q)r(1-r) & \text{if } b = D,\\ q+(1-p-q)r^{2} & \text{if } b = L, \end{cases} \quad\!\! \text{for } (a_{0}, a_{1}) \in \{(L,D), (D,L)\},\end{equation}
   (15) \begin{equation} \widehat{\rm \varphi}_{p,q,r}(D,D,b) = \begin{cases} p & \text{if } b = W,\\ (1-p-q)(1-r^{2}) & \text{if } b = D, \\ q+(1-p-q)r^{2} & \text{if } b = L. \end{cases}\end{equation}

A detailed explanation as to why these stochastic update rules are equivalent to the game rules governing the generalized percolation game is provided in Section 6. When we restrict $\hat{\mathcal{A}}$ to the smaller alphabet, $\mathcal{A}=\{W,L\}$ , we obtain yet another PCA, denoted $G_{p,q,r}$ , with stochastic update rules described via the stochastic matrix $\varphi_{p,q,r}\;:\; \mathcal{A}^{2} \times \mathcal{A} \rightarrow [0,1]$ , where $\varphi_{p,q,r}(a_{0},a_{1},b)=\widehat{\rm \varphi}_{p,q,r}(a_{0},a_{1},b)$ for all $a_{0}, a_{1}, b \in \mathcal{A}$ (in other words, $\varphi_{p,q,r}$ can be defined using (10), (11), and (12)). We refer to $\widehat{G}_{p,q,r}$ as the envelope to $G_{p,q,r}$ (see Section 7, where we include a discussion on the importance of invoking the notion of envelope PCAs).

Figure 5.

Stochastic update rules for the PCA $\widehat{E}_{r',s'}$ .

We now describe the PCA, denoted $\widehat{E}_{r',s'}$ , that represents the game rules arising from our bond percolation game, with the underlying parameters, r ^′ and s ^′, described in Section 2. Endowed with $\mathbb{Z}$ as its universe, $\widehat{E}_{r',s'}$ is, just like $\widehat{G}_{p,q,r}$ , a one-dimensional PCA with alphabet $\hat{\mathcal{A}}$ and neighborhood-marking set $\mathcal{N}$ , and its stochastic update rules are captured by the stochastic matrix $\widehat{\rm \varphi}_{r',s'}\;:\; \hat{\mathcal{A}}^{2} \times \hat{\mathcal{A}} \rightarrow [0,1]$ defined as follows (see Figure 5 for a pictorial illustration):

(16) \begin{equation} \widehat{\rm \varphi}_{r',s'}(W,W,b) = \begin{cases} 2s'-{s'}^2 & \text{if } b = W, \\ (1-s')^2 & \text{if } b = L, \end{cases}\end{equation}

(17) \begin{equation} \widehat{\rm \varphi}_{r',s'}(a_{0},a_{1},b) = \begin{cases} 1-r'+r's' & \text{if } b = W, \\ r'-r's' & \text{if } b = L, \end{cases} \quad \text{where } (a_{0}, a_{1}) \in \{(W,L), (L,W)\},\end{equation}

(18) \begin{equation} \varphi_{r',s'}(L,L,b) = \begin{cases} 1-{r'}^{2} & \text{if } b = W, \\ {r'}^{2} & \text{if } b = L, \end{cases}\end{equation}

(19) \begin{equation} \widehat{\rm \varphi}_{r',s'}(a_{0}, a_{1}, b) = \begin{cases} (1-s')(1-r'-s') & \text{if } b = D, \\ r'(1-s') & \text{if } b = L, \\ 2s'-{s'}^2 & \text{if } b = W, \end{cases} \quad \text{where } (a_{0}, a_{1}) \in \{(W,D), (D,W)\},\end{equation}

(20) \begin{equation} \widehat{\rm \varphi}_{r',s'}(a_{0}, a_{1}, b) = \begin{cases} 1-r'+r's' & \text{if } b = W, \\ r'(1-r'-s') & \text{if } b = D,\\ {r'}^{2} & \text{if } b = L, \end{cases} \quad \text{where } (a_{0}, a_{1}) \in \{(L,D), (D,L)\},\end{equation}

(21) \begin{equation} \widehat{\rm \varphi}_{r',s'}(D,D,b) = \begin{cases} (1-r'-s')(1+r'-s') & \text{if } b = D,\\ {r'}^{2} & \text{if } b = L, \\ 2s'-{s'}^2 & \text{if } b = W. \end{cases}\end{equation}

Once again, we refer the reader to Section 6 for a detailed explanation as to how the game rules associated with the bond percolation game give rise to the above-mentioned stochastic update rules. If we restrict $\hat{\mathcal{A}}$ to $\mathcal{A}$ , we obtain the PCA $E_{r',s'}$ with the same neighborhood-marking set $\mathcal{N}$ , and with stochastic update rules captured by the stochastic matrix $\varphi_{r',s'}\;:\; \mathcal{A}^{2} \times \mathcal{A} \rightarrow [0,1]$ such that $\varphi_{r',s'}(a_{0},a_{1},b)=\widehat{\rm \varphi}_{r',s'}(a_{0},a_{1},b)$ for all $a_{0}, a_{1}, b \in \mathcal{A}$ (in other words, the equations defining $\varphi_{r',s'}$ are the same as (16), (17), and (18)). As is the case with $G_{p,q,r}$ and $\widehat{G}_{p,q,r}$ , we refer to $\widehat{E}_{r',s'}$ as the envelope for the PCA $E_{r',s'}$ .

We end Section 3 with the main results of this paper pertaining to the PCAs described in Section 3.2.

3.3. How the PCA $\widehat{\boldsymbol{E}}_{\boldsymbol{r}',\boldsymbol{s}'}$ is obtained as a special case of the PCA $\widehat{\boldsymbol{G}}_{\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}}$

Let us consider a bond percolation game with the underlying parameter pair (r ^′, s ^′), i.e. where r ^′ indicates the probability of an edge being labeled as a trap and s ^′ indicates that of it being labeled as a target. Letting $\eta$ denote this random assignment of labels (i.e. $\eta$ is the infinite tuple made up of $\eta((x,y),(x+1,y))$ and $\eta((x,y),(x,y+1))$ for all $(x,y)\in\mathbb{Z}^{2}$ ), we tweak $\eta$ as follows:

1. (i) if $\eta((x,y),(x+1,y))$ (respectively, $\eta((x,y),(x,y+1))$ ) equals a target, we relabel the edge $((x,y),(x+1,y))$ (respectively, the edge $((x,y),(x,y+1))$ ), independent of all else, to $\eta'((x,y),(x+1,y))$ (respectively, to $\eta'((x,y),(x,y+1))$ ), where

   \[ \eta'((x,y),(x+1,y)) = \left\{\begin{array}{l} \text{trap with probability } r'(1-s')^{-1} ,\\ \text{open with probability } (1-r'-s')(1-s')^{-1}, \end{array}\right.\]
   (the same distribution is adopted for $\eta'((x,y),(x,y+1))$ if $\eta((x,y),(x,y+1))$ equals a target), and we label the vertex (x, y) a trap, i.e. we set $\sigma'(x,y)$ to be equal to a trap;

2. (ii) if $\eta((x,y),(x+1,y))$ (respectively, $\eta((x,y),(x,y+1))$ ) is either open or a trap, we keep its label unchanged, i.e. we set $\eta'((x,y),(x+1,y))=\eta((x,y),(x+1,y))$ (respectively, $\eta'((x,y),(x,y+1))=\eta((x,y),(x,y+1))$ );

3. (iii) for any $(x',y') \in \mathbb{Z}^{2}$ for which neither $\eta((x,y),(x+1,y))$ nor $\eta((x,y),(x,y+1))$ equals a target, we label (x, y) open, i.e. we set $\sigma'(x,y)$ to be open.

The modified labeling is now $(\sigma',\eta')$ , where $\sigma'$ is the tuple comprising $\sigma'(x,y)$ for all $(x,y) \in \mathbb{Z}^{2}$ , and $\eta'$ is the tuple comprising $\eta'((x,y),(x+1,y))$ and $\eta'((x,y),(x,y+1))$ for all $(x,y)\in\mathbb{Z}^{2}$ . From the definition of $(\sigma',\eta')$ , it is evident that the labels $\sigma'(x,y)$ are i.i.d. over all $(x,y) \in \mathbb{Z}^{2}$ , with

(22) \begin{align}p={}&\textrm{P}[\sigma'(x,y)=\text{trap}]\nonumber\\={}&\textrm{P}[\text{at least one of } \eta((x,y),(x+1,y)) \text{ and } \eta((x,y),(x,y+1)) \text{ equals a target}]\nonumber\\={}& 1-(1-s')^{2}=2s'-{s'}^{2},\end{align}

and likewise, the labels $\eta'(e)$ are independent and identically distributed (i.i.d.) over all directed edges e of our graph, with

(23) \begin{align}r={}&\textrm{P}[\eta'(e)=\text{trap}]\nonumber\\={}&\textrm{P}[\eta(e)=\text{trap}]+\textrm{P}[\eta'(e)=\text{trap}|\eta(e)=\text{target}]\textrm{P}[\eta(e)=\text{target}]\nonumber\\={}&r'+\frac{r'}{1-s'} \cdot s' = \frac{r'}{1-s'}.\end{align}

Moreover, it is easily verified that the tuples $\sigma'$ and $\eta'$ are independent of each other as well: for instance, we have

\begin{align}{}&\textrm{P}[\sigma'(x,y)=\text{trap}, \eta'((x,y),(x+1,y))=\text{trap}]\nonumber\\={}&\textrm{P}[\eta((x,y),(x+1,y))=\text{target},\eta'((x,y),(x+1,y))=\text{trap}]\nonumber\\&+\textrm{P}[\eta((x,y),(x+1,y))=\text{trap},\eta((x,y),(x,y+1))=\text{target}]\nonumber\\={}&s' \cdot \frac{r'}{1-s'} + r' \cdot s' = \frac{r' s' (2-s')}{1-s'} = \big(2s'-{s'}^{2}\big) \frac{r'}{1-s'} = pr,\nonumber\end{align}

as desired. It is now straightforward to see that by choosing $q=0$ , p as in (22), and r as in (23), we can reduce (10)–(16), (11)–(17), and so on.

This transformation, from a generalized percolation game with the underlying parameter-triple (p, 0, r) to a bond percolation game with the underlying parameter pair (r ^′, s ^′), where p is as given by (22) and r is as given by (23), as established previously, lies at the heart of the proof that in the regime covered by (B1) of Theorem 2, the probability of draw equals 0.

3.4. A major contribution of this paper in establishing ergodicity of elementary PCAs

A one-dimensional PCA is said to be elementary if the cardinality of its alphabet as well as its neighborhood-marking set equals 2. The definitions of $G_{p,q,r}$ and $E_{r',s'}$ in Section 3.2 immediately reveal that each of them is an elementary PCA. In Chapter 7 of [Reference Dobrušin, Kriukov and Toom13], two fundamental results pertaining to ergodicity of elementary PCAs have been proposed. We first state these two results in terms of the notation used in this paper. We let $\mathcal{A}=\{W,L\}$ (as has been defined in Section 3.2) denote the alphabet of any elementary PCA F, and we let

\[\theta_{i,j}=\textrm{P}[F\eta(x)=L|\eta(x+y_{1})=i,\eta(x+y_{2})=j], \quad \text{for } i,j \in \mathcal{A},\]

where $\{y_{1},y_{2}\}$ forms the neighborhood-marking set of F. Thus, the PCA F is completely specified by the parameters $\theta_{W,W}$ , $\theta_{W,L}$ , $\theta_{L,W}$ , and $\theta_{L,L}$ . We call F symmetric when $\theta_{W,L}=\theta_{L,W}$ , in which case F is specified by only three parameters, namely, $\theta_{W,W}$ , $\theta_{W,L}$ , and $\theta_{L,L}$ . The following are two well-known results when it comes to ergodicity properties of symmetric elementary PCAs, stated in Chapter 7 of [Reference Dobrušin, Kriukov and Toom13]:

1. (a) the PCA F is ergodic when $\theta_{W,W}$ , $\theta_{W,L}$ , and $\theta_{L,L}$ satisfy the inequalities

   (24) \begin{align}{}& 0 < \theta_{W,W}, \theta_{W,L}, \theta_{L,L} < 1,\end{align}
   (25) \begin{align}{}& \theta_{L,L} > \theta_{W,W} - 2\theta_{W,L},\end{align}
   (26) \begin{align}{}& \theta_{L,L} > \theta_{W,W} - 2(1-\theta_{W,L});\end{align}

2. (b) the PCA F is ergodic when $\theta_{W,W}$ , $\theta_{W,L}$ , and $\theta_{L,L}$ satisfy the inequality

   (27) \begin{align}\max\{|\theta_{i,j}-\theta_{k,\ell}|\;:\;i,j,k,\ell \in \{W,L\}\}+2\max\{|\theta_{L,L}-\theta_{W,L}|,|\theta_{W,W}-\theta_{W,L}|\} < 2.\end{align}

It has been mentioned in Chapter 7 of [Reference Dobrušin, Kriukov and Toom13] that these two regimes together cover more than 90% of the unit cube $[0,1]^{3}$ defined by the parameter triple $(\theta_{W,W},\theta_{W,L},\theta_{L,L})$ , and that the only region where no method of proving or disproving ergodicity for symmetric elementary PCAs, in general, is known is the union of the neighborhoods of the points $(\theta_{W,W},\theta_{W,L},\theta_{L,L})=(1,0,0)$ and $(\theta_{W,W},\theta_{W,L},\theta_{L,L})=(1,1,0)$ .

We now come to the two symmetric elementary PCAs that are of interest to us in this paper, namely $G_{p,q,r}$ and $E_{r',s'}$ . We see, from (10), (11), and (12), that $\theta_{W,W}=(1-p)$ , $\theta_{W,L}=q+(1-p-q)r$ , and $\theta_{L,L}=q+(1-p-q)r^{2}$ for $G_{p,q,r}$ , so that we have $(\theta_{W,W},\theta_{W,L},\theta_{L,L})=(1,0,0)$ if and only if $p=q=r=0$ . Thus, the neighborhood of (1, 0, 0) is obtained when we consider as small a value of each of p, q, and r as we desire. We then show, in Theorem 1, that if, in addition to $(\theta_{W,W},\theta_{W,L},\theta_{L,L})$ being in the neighborhood of (1, 0, 0) (equivalently, (p, q, r) being arbitrarily close to (0, 0, 0)), at least one of p, q, and r is strictly positive and the constraints specified in Theorem 1 are satisfied, the PCA $G_{p,q,r}$ is ergodic, thereby rigorously proving the ergodicity of $G_{p,q,r}$ in a considerable chunk of the neighborhood of (1, 0, 0). Likewise, from (16), (17), and (18), we see that $\theta_{W,W}=(1-s')^{2}$ , $\theta_{W,L}=r'(1-s')$ , and $\theta_{L,L}={r'}^{2}$ for the PCA $E_{r',s'}$ , so that we have $(\theta_{W,W},\theta_{W,L},\theta_{L,L})=(1,0,0)$ if and only if $r'=s'=0$ . Thus, the neighborhood of (1, 0, 0) is obtained when we consider as small a value of each of r ^′ and s ^′ as we desire. Part (B1) of Theorem 2 then shows that, in addition to $(\theta_{W,W},\theta_{W,L},\theta_{L,L})$ being in the neighborhood of (1, 0, 0) (equivalently, (r ^′, s ^′) being arbitrarily close to (0, 0)), if at least one of r ^′ and s ^′ is strictly positive and (8) holds, the PCA $E_{r',s'}$ is ergodic, thereby rigorously proving the ergodicity of $E_{r',s'}$ in a considerable chunk of the neighborhood of (1, 0, 0).

In fact, even the regimes described by (B2) and (B3) of Theorem 2 go significantly beyond those that are obtained via (a) and (b) for the PCA $E_{r',s'}$ . For instance, when $s'=0$ , the criterion in (24) fails to hold for $E_{r',s'}$ since $\theta_{W,W}=1$ , so that (B2) falls outside of the regime described in (a). On the other hand, $\max\{|\theta_{i,j}-\theta_{k,\ell}|\;:\;i,j,k,\ell \in \mathcal{A}\} = 1-{r'}^{2}$ and $\max\{|\theta_{L,L}-\theta_{W,L}|,|\theta_{W,W}-\theta_{W,L}|\} = 1-r'$ when $s'=0$ , so that for (27) to hold, we must have $1-{r'}^{2}+2(1-r') = 3-2r'-{r'}^{2} < 2 \Longleftrightarrow 2r'+{r'}^{2} > 1 \Longleftrightarrow r' > 0.414214$ . This tells us that the regime described in (B2), i.e. $s'=0$ and $r' > 0.157175$ , goes far beyond the regime obtained by implementing (b).

When $r'=s'$ (obviously, the common value for r ^′ and s ^′ in this case must be bounded above by $0.5$ , since $r'+s' \leqslant 1$ ), we have $\theta_{W, W} = (1-r')^{2}$ , $\theta_{W,L} = r'-{r'}^{2}$ , and $\theta_{L,L} = {r'}^{2}$ . It is immediate that these values of $\theta_{W,W}$ , $\theta_{L,W}$ , and $\theta_{L,L}$ satisfy (24) when $r' \in (0,1)$ . When examining if the inequality in (25) is satisfied or not, we observe that

(28) \begin{align} \theta_{L,L} > \theta_{W,W} - 2\theta_{W,L} \Longleftrightarrow {r'}^2 > (1-r')^2 - 2(r'-{r'}^2) &\Longleftrightarrow 2{r'}^2 - 4r' + 1 < 0\nonumber \\ &\Longleftrightarrow r' > 1 - \frac{1}{\sqrt{2}} \approx 0.293.\end{align}

When checking whether the inequality in (26) is satisfied or not, we obtain

\begin{equation*} \theta_{L,L} > \theta_{W,W} - 2(1 - \theta_{W,L}) \Longleftrightarrow {r'}^2 > (1-r')^2 - 2(1-r'+{r'}^2) \Longleftrightarrow 2{r'}^2 + 1 >0,\end{equation*}

which is indeed true for all (r ^′, s ^′) belonging to the regime in (B3). We thus conclude that when $r'=s'$ , we require $r' > 0.293$ for the inequalities in (a) to hold simultaneously, whereas our result in this paper is able to establish ergodicity for $E_{r',s'}$ whenever $r' = s' > 0.10883$ . We now focus on the regime covered by (b) when $r'=s'$ . We observe that

\begin{multline} \max\{|\theta_{W, W} - \theta_{W, L}|, |\theta_{W, W} - \theta_{L, L}, |\theta_{W, L} - \theta_{L, L}|\} \\= \max\{|r'(2r'-1)|, |2r'-1|, |(1-r')(2r'-1)|\} = 1-2r' \nonumber \end{multline}

and

\begin{equation} \max\{|\theta_{W, W} - \theta_{W, L}|, |\theta_{W, L} - \theta_{L, L}|\} = \max\{|r'(2r'-1)|, |(1-r')(2r'-1)|\} = (1-r')(1-2r'), \nonumber \end{equation}

so that 27 reduces to

\[ 1-2r' + 2(1-r')(1-2r') < 2 \Longleftrightarrow 4{r'}^2-8r'+1<0 \Longleftrightarrow r'>\frac{2-\sqrt{3}}{2} \approx 0.13397.\]

Once again, our result, in (B3), establishes ergodicity for $E_{r',s'}$ whenever $r' = s' > 0.10883$ , a significant improvement on the lower bound that is provided by (b).

4.1. In oligopolistic competitions

The bond percolation game can be interpreted as a stylized model of oligopolistic competition between two firms. At each stage, an action labeled as a trap represents a significant loss that forces a firm to exit the market, whereas an action labeled as a target represents a sufficient gain that secures monopoly. The environment (nature) randomly and independently selects the set of available actions at each stage, and these are revealed to the firms at the outset, modeled via a labeled directed graph. Each firm aims to strategically navigate the graph to reach a winning situation. A natural extension of this model would be one in which the environment selects available actions based on the history of previous moves, introducing a dependency structure and dynamic information revelation.

Analogous dynamics arise in courtroom settings, where the prosecution and defense take sequential actions. A target corresponds to decisive evidence in favor of a party, whereas a trap denotes conclusive evidence leading to defeat. The sequential nature of legal argumentation and information disclosure closely mirrors the game-theoretic framework described previously.

4.2. As an adversarial avatar of percolation

Another usefulness of our generalized/bond percolation game lies in incorporating an adversarial element into the notion of ordinary site and/or bond percolation on directed infinite graphs. Percolation, by itself, is a vast and ever-expanding area of research that spans a multitude of disciplines, including statistical physics, chemistry, computer science, biology, and sociology (for instance, in the understanding of phenomena such as polymeric gelation, the spread of a forest fire or the propagation of an epidemic, or the transportation of a fluid through a porous material). Percolation games (more specifically, site percolation games) were first introduced in [Reference Holroyd, Marcovici and Martin23], where each vertex of the infinite two-dimensional square lattice $\mathbb{Z}^{2}$ was assigned, independently, a label that read trap with probability p, target with probability q, and open with probability $(1-p-q)$ . The two players were allowed the same moves as those permitted in our game in Section 2, and a player won if she could move the token to a vertex labeled as a target or force her opponent to move the token to a vertex labeled as a trap. It was shown that the probability of draw in this game is 0 whenever $(p+q) > 0$ . The set-up described for our generalized percolation game in Section 2 is a natural and important generalization of this model, as it additionally considers a random assignment of labels to the edges of $\mathbb{Z}^{2}$ . The result presented in [Reference Holroyd, Marcovici and Martin23] was extended to the set-up considered in [Reference Bhasin, Karmakar, Podder and Roy4], where the token was permitted to be moved from where it was currently located, say the vertex (x, y), to any one of $(x,y+1)$ , $(x+1,y+1)$ and $(x+2,y+1)$ . As in [Reference Holroyd, Marcovici and Martin23], it was shown in [Reference Bhasin, Karmakar, Podder and Roy4] that the probability of draw is 0 whenever at least one of p and q is strictly positive. A different direction of generalization of the above-mentioned site percolation game on $\mathbb{Z}^{2}$ was pursued in [Reference Bhasin, Karmakar, Podder and Roy3], by allowing the token to be moved from (x, y) to one of $(x,y+1)$ and $(x+1,y+1)$ if x is even, and from (x, y) to one of $(x+1,y+1)$ and $(x+2,y+1)$ if x is odd.

It is worthwhile to note here that when $s'=0$ and $r' > 0$ in the set-up described for our bond percolation game in Section 2, the game is essentially a normal game that is being played on the infinite oriented two-dimensional square lattice, $\mathbb{Z}^{2}$ , once the process of bond percolation, with edge-deletion probability r ^′, has been performed on it. This can be justified as follows: each edge of $\mathbb{Z}^{2}$ is, independently, retained with probability $(1-r')$ and deleted with probability r ^′. A move in the normal game played on this premise involves relocating the token from where it is currently located, say the site (x, y), to either $(x+1,y)$ (if the directed edge $((x,y),(x+1,y))$ has not been deleted) or to $(x,y+1)$ (if the directed edge $((x,y),(x,y+1))$ has not been deleted). A player loses if she is unable to make a move (i.e. there are no outgoing edges from the site where the vertex is currently located). The game continues indefinitely if neither player ever encounters a site from which both outgoing edges have been deleted, and this is possible only if there exists an infinite path starting from the initial vertex, i.e. if percolation happens. Thus, in such a scenario, the probability of draw is bounded above by the probability of occurrence of percolation, i.e. the probability of the existence of an infinite path that begins from the initial vertex. In this context, we refer the reader to [Reference Holroyd and Martin24], where the normal game, as well as the misère and the escape games have been studied on rooted Galton–Watson trees.

4.3. A brief discussion on maker–breaker percolation games

A different class of two-player combinatorial games pertaining to percolation, broadly referred to as the maker–breaker percolation games, was introduced in [Reference Day and Falgas-Ravry11], and further explored in [Reference Day and Falgas-Ravry12, Reference Dvocřák and Mond16, Reference Dvocřák and Mond17, Reference Wallwork30]. Each such game is played on a graph referred to as a board, and two players, titled maker and breaker, take turns, with maker claiming m (as yet unclaimed) edges of the board during each of her turns, and breaker deleting b (as yet unclaimed) edges of the board during each of hers. In [Reference Day and Falgas-Ravry11], the board considered is an $m \times n$ rectangular grid, and Maker wins the (m,b)-crossing game if she manages to claim all edges constituting a crossing path joining the left boundary of the grid to its right boundary. In [Reference Day and Falgas-Ravry12], the board considered is an infinite connected graph $\Lambda$ (such as the two-dimensional infinite square lattice $\mathbb{Z}^{2}$ and the d-regular tree in which each vertex has degree d), with a vertex $v_{0} \in \Lambda$ specified, and breaker wins if at any point of time during the game, the connected component of $\Lambda$ containing $v_{0}$ becomes finite. In [Reference Dvocřák and Mond16], a critical lower threshold for b in terms of m was found such that if b were to exceed this value, Breaker would win the maker–breaker percolation game on $\mathbb{Z}^{2}$ ; it was also shown that when the board is $\mathbb{Z}^{2}$ after the usual bond percolation process with parameter p has been performed on it, with p not too large compared with $1/2$ , breaker almost surely wins the unbiased maker–breaker percolation game (i.e. where $m=b=1$ ). In [Reference Dvocřák and Mond17], the same random board as in [Reference Dvocřák and Mond16] was considered, and it was shown that:

• • when $p < 1$ , breaker almost surely wins the unbiased maker–breaker percolation game; and

• • when $m=2$ and $b=1$ , maker almost surely wins whenever $p > 0.9402$ , whereas breaker almost surely wins whenever $p < 0.5278$ .

In [Reference Wallwork30], the (m, b)-crossing game was studied on a triangular grid that was m vertices across and n vertices high.

6.1. Game rules for the generalized percolation game

Fix any $k \in \mathbb{Z}$ , any $(x,y) \in S_{k}$ , and set $Out(x,y)=\{(x+1,y),(x,y+1)\}$ . We assume that (x, y) serves as the initial vertex for a generalized percolation game, and we let $P_{1}$ denote the player who plays the first round, and $P_{2}$ the player who plays the second round of this game. Recalling the rules of the generalized percolation game described in Section 2, we are able to make the following observations.

1. (i) When both vertices of Out(x, y) are in W, $P_{2}$ wins no matter how $P_{1}$ moves in the first round, unless (x, y) has been labeled as a trap. Therefore, in this case, $(x,y) \in L$ with probability $(1-p)$ and $(x,y) \in W$ with probability p.

2. (ii) When $(x+1,y) \in L$ and $(x,y+1) \in W$ (an analogous situation arises when $(x+1,y) \in W$ and $(x,y+1) \in L$ ), $P_{1}$ wins (by moving the token from (x, y) to $(x+1,y)$ in the first round) if either (x, y) has been labeled as a trap or (x, y) has been labeled open and the directed edge $((x,y),(x+1,y))$ is also open, whereas $P_{2}$ is the winner otherwise. Therefore, in this case, $(x,y) \in W$ with probability $\{p+(1-p-q)(1-r)\}$ , and $(x,y) \in L$ with probability $\{q+(1-p-q)r\}$ .

3. (iii) When both vertices of Out(x, y) are in L, $P_{1}$ wins if (x, y) has been labeled as a trap or (x, y) is open and at least one of the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ is also open, whereas $P_{2}$ wins otherwise. Therefore, in this case, $(x,y) \in W$ with probability $\{p+(1-p-q)(1-r^{2})\}$ , and $(x,y) \in L$ with probability $\{q+(1-p-q)r^{2}\}$ .

4. (iv) When $(x+1,y) \in W$ and $(x,y+1) \in D$ (an analogous situation arises when $(x+1,y) \in D$ and $(x,y+1) \in W$ ), $P_{1}$ wins if (x, y) has been labeled as a trap, the game results in a draw if (x, y) is open and the edge $((x,y),(x,y+1))$ has not been labeled as a trap, and $P_{2}$ wins in all other cases. Therefore, we have $(x,y) \in W$ with probability p, $(x,y) \in D$ with probability $(1-p-q)(1-r)$ , and $(x,y) \in L$ with probability $\{q+r(1-p-q)\}$ .

5. (v) When $(x+1,y) \in L$ and $(x,y+1) \in D$ (an analogous situation arises when $(x+1,y) \in D$ and $(x,y+1) \in L$ ), $P_{1}$ wins if (x, y) has been labeled as a trap, or if both the vertex (x, y) and the edge $((x,y),(x+1,y))$ are open. The game results in a draw if (x, y) is open, the edge $((x,y),(x+1,y))$ has been labeled as a trap and the edge $((x,y),(x,y+1))$ is open, whereas $P_{2}$ wins in all other cases. Therefore, we have $(x,y) \in W$ with probability $\{p+(1-p-q)(1-r)\}$ , $(x,y) \in D$ with probability $(1-p-q)r(1-r)$ , and $(x,y) \in L$ with probability $\{q+r^{2}(1-p-q)\}$ .

6. (vi) Finally, when both vertices of Out(x, y) are in D, $P_{1}$ wins if (x, y) is a trap, the game results in a draw if (x, y) is open and at least one of the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ is open, and in all other cases, $P_{2}$ wins. Therefore, $(x,y) \in W$ with probability p, $(x,y) \in D$ with probability $(1-p-q)(1-r^{2})$ and $(x,y) \in L$ with probability $\{q+r^{2}(1-p-q)\}$ .

For each $k \in \mathbb{Z}$ , we identify $S_{k}$ with a copy of the integer line $\mathbb{Z}$ , by identifying the vertex $(x,k-x)$ on $S_{k}$ with x on $\mathbb{Z}$ . For an arbitrary but fixed $k \in \mathbb{Z}$ , let $\omega(x) \in \hat{\mathcal{A}}=\{W,L,D\}$ , referred to as the state of the vertex $(x,k+1-x)$ , denote the subset (out of W, L, and D) to which $(x,k+1-x)$ belongs, for each $x \in \mathbb{Z}$ . Conditioned on the infinite tuple $\omega = (\omega(x)\;:\; x \in \mathbb{Z})$ that specifies the state of each vertex on $S_{k+1}$ , the random state of the vertex $(x,k-x)$ , lying on $S_{k}$ , equals $\widehat{G}_{p,q,r}\omega(x)$ for each $x \in \mathbb{Z}$ , as is evident from the recurrence relations described previously, and from (9), (10), (11), (12), (13), (14), and (15)). This is how the game rules governing the generalized percolation game give rise to the PCA $\widehat{G}_{p,q,r}$ .

6.2. Game rules for the bond percolation game

The recurrence relations arising from the bond percolation game described in Section 2 can be deduced in much the same manner as those derived in Section 6.1 for the generalized percolation game. We let an arbitrarily chosen $(x,y) \in S_{k}$ , for some $k \in \mathbb{Z}$ , be the initial vertex, and assume, as before, that $P_{1}$ plays the first round of the game whereas $P_{2}$ plays the second. We then deduce the state of the vertex (x, y), i.e. which of the subsets W, L, and D it belongs to, as follows.

1. (i) When both vertices of Out(x, y) are in W, $P_{1}$ loses unless at least one of the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ has been labeled as a target, so that $(x,y) \in L$ with probability $(1-s')^{2}$ and $(x,y) \in W$ with probability $(2s'-{s'}^{2})$ .

2. (ii) When $(x+1,y) \in L$ and $(x,y+1) \in W$ (analogously, when $(x+1,y) \in W$ and $(x,y+1) \in L$ ), $P_{1}$ wins unless the edge $((x,y),(x+1,y))$ has been labeled as a trap and the edge $((x,y),(x,y+1))$ is not a target. Hence, $(x,y) \in L$ with probability $r'(1-s')$ , and $(x,y) \in W$ with probability $\{1-r'(1-s')\}$ .

3. (iii) When both vertices of Out(x, y) are in L, $P_{1}$ wins unless both the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ are traps. Therefore, $(x,y) \in L$ with probability ${r'}^{2}$ and $(x,y) \in W$ with probability $(1-{r'}^{2})$ .

4. (iv) When $(x+1,y) \in W$ and $(x,y+1) \in D$ (analogously, when $(x+1,y) \in D$ and $(x,y+1) \in W$ ), $P_{1}$ wins if at least one of the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ has been labeled as a target, the game results in a draw if $((x,y),(x+1,y))$ is not a target and $((x,y),(x,y+1))$ is open, and in every other case, $P_{2}$ wins. Thus, $(x,y) \in W$ with probability $\{1-(1-s')^{2}\}$ , $(x,y) \in D$ with probability $(1-s')(1-r'-s')$ , and $(x,y) \in L$ with probability $r'(1-s')$ .

5. (v) When $(x+1,y) \in L$ and $(x,y+1) \in D$ (analogously, when $(x+1,y) \in D$ and $(x,y+1) \in L$ ), $P_{1}$ wins as long as the edge $((x,y),(x+1,y))$ is not a trap or at least one of the edges $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ is a target, the game results in a draw if $((x,y),(x+1,y))$ is a trap and $((x,y),(x,y+1))$ is open, and in all other cases, $P_{2}$ wins. Thus, $(x,y) \in W$ with probability $\{1-r'(1-s')\}$ , $(x,y) \in D$ with probability $r'(1-r'-s')$ , and $(x,y) \in L$ with probability ${r'}^{2}$ .

6. (vi) When both vertices of Out(x, y) are in D, $P_{1}$ wins if at least one of $((x,y),(x+1,y))$ and $((x,y),(x,y+1))$ is a target, the game results in a draw if neither $((x,y),(x+1,y))$ nor $((x,y),(x,y+1))$ is a target but not both are traps, and $P_{2}$ wins otherwise. Thus, $(x,y) \in W$ with probability $\{1-(1-s')^{2}\}$ , $(x,y) \in D$ with probability $(1-r'-s')(1+r'-s')$ , and $(x,y) \in L$ with probability ${r'}^{2}$ .

As argued in Section 6.1, the above-mentioned recurrence relations or game rules governing the bond percolation game are captured precisely by the stochastic update rules of the PCA $\widehat{E}_{r',s'}$ in (16), (17), (18), (19), (20), and (21).

8.1. The principal ideas behind the construction of our weight functions

We present to the reader a very broad, but hopefully comprehensive, outline of how we aim to accomplish the proof of Theorem 4. We denote by $\mathcal{R}$ the subset of the parameter-space $\Theta$ (defined in Section 2) that consists of all those triples (p, q, r) that satisfy the constraints stated in Theorem 1; in other words, p, q, and r are sufficiently small, the inequality in (3) holds, and precisely one of (4), (5), (6), and (7) is true. Although, in the rest of Section 8.1, we focus on $(p,q,r) \in \mathcal{R}$ and work with the envelope PCA $\widehat{G}_{p,q,r}$ corresponding to generalized percolation games, this overview is equally pertinent to the construction of the weight functions corresponding to the three different regimes addressed in Theorem 5 for bond percolation games.

Recall, from Section 3.2, that the state space corresponding to our envelope PCA, $\widehat{G}_{p,q,r}$ , is $\Omega = \hat{\mathcal{A}}^{\mathbb{Z}}$ , where $\hat{\mathcal{A}}=\{W,L,D\}$ is its alphabet. Let $\mathcal{M}$ denote the space of all reflection-invariant and translation-invariant (recall these definitions from Section 7) probability measures on $\Omega$ , defined with respect to the $\sigma$ -field $\mathcal{F}$ generated by all cylinder sets of $\Omega$ . The goal is to come up with a ‘suitable’ real-valued function w, referred to as a weight function or potential function, defined on $\mathcal{M}$ . As such, a weight function (that serves the purpose outlined in the following) need not be unique, but as far as this paper is concerned, we construct w as a linear combination of the form

(29) \begin{equation}w(\mu) = \sum_{i=1}^{t}c_{i}\mu(\mathcal{C}_{i}),\end{equation}

where $t \in \mathbb{N}$ , each of $\mathcal{C}_{1}$ , $\ldots$ , $\mathcal{C}_{t}$ is a cylinder set (thus belonging to $\mathcal{F}$ ), and each of $c_{1}$ , $\ldots$ , $c_{t}$ is a real-valued function (in fact, a polynomial) in the parameters p, q, and r. Our goal is to construct a weight function, of the form (29), that satisfies an inequality, henceforth referred to as a weight function inequality, of the form

(30) \begin{equation}w(\widehat{G}_{p,q,r}\mu) \leqslant w(\mu) - \sum_{i=1}^{t'}c'_{i}\mu(\mathcal{C}'_{i}),\end{equation}

where $t' \in \mathbb{N}$ , and $\mathcal{C}'_{i}$ is a cylinder set and $c'_{i}$ is a real-valued function (once again, a polynomial) in p, q, and r, for each $i \in \{1,2,\ldots,t'\}$ . It must be ensured that

(31) \begin{equation}c'_{i} \geqslant 0 \text{ for each } i \in \{1,2,\ldots,t'\} \text{ for each } (p,q,r) \in \mathcal{R}.\end{equation}

Let $\mathcal{I}$ denote the set of all i, with $i \in \{1,2,\ldots,t'\}$ , such that $c'_{i} > 0$ whenever $(p,q,r) \in \mathcal{R}$ . If $\mu$ is stationary for $\widehat{G}_{p,q,r}$ (so that $\widehat{G}_{p,q,r}\mu = \mu$ ) and $\mathcal{I}$ is nonempty, by (31) we conclude that $\mu\left(\mathcal{C}'_{i}\right) = 0$ for each $i \in \mathcal{I}$ whenever $(p,q,r) \in \mathcal{R}$ . If the cylinder sets $\mathcal{C}'_{1}, \ldots, \mathcal{C}'_{t'}$ have been chosen judiciously enough, suitable set-theoretic operations performed on $\mathcal{C}'_{i}$ , for $i \in \mathcal{I}$ , allow us to conclude that $\mu\left((D)_{0}\right)=0$ whenever $(p,q,r) \in \mathcal{R}$ (here, $(D)_{0}$ indicates the cylinder set that consists of all configurations $\eta = (\eta(x)\;:\; x \in \mathbb{Z})$ with $\eta(0) = D$ , i.e. in which the 0th coordinate is occupied by the symbol D). This concludes the primary idea propelling the proof of Theorem 4.

From here onward, we make use of some simplified notation as follows. In its most general form, a cylinder set is indicated by $\left(a_{1},a_{2},\ldots,a_{\ell}\right)_{x_{1},x_{2},\ldots,x_{\ell}}$ , where $a_{1}, a_{2}, \ldots, a_{\ell} \in \hat{\mathcal{A}}$ and $x_{1}, x_{2}, \ldots, x_{\ell} \in \mathbb{Z}$ , signifying that this cylinder set comprises all tuples $\eta = (\eta(x)\;:\; x \in \mathbb{Z})$ in which $\eta(x_{i}) = a_{i}$ for each $i \in \{1,2,\ldots,\ell\}$ . When $x_{1}, x_{2}, \ldots, x_{\ell}$ are consecutive integers, i.e. $x_{i+1}=x_{i}+1$ for each $i \in \{1,2,\ldots,\ell-1\}$ , we have

\[\mu((a_{1},a_{2},\ldots,a_{\ell})_{0,1,\ldots,\ell-1}) = \mu((a_{1},a_{2},\ldots,a_{\ell})_{x_{1},x_{1}+1,\ldots,x_{1}+\ell-1})\]

as $\mu$ is translation-invariant. This allows us to abbreviate $\mu((a_{1},a_{2},\ldots,a_{\ell})_{x_{1},x_{1}+1,\ldots,x_{1}+\ell-1})$ as simply $\mu\left(\left(a_{1},a_{2},\ldots,a_{\ell}\right)\right)$ for any $x_{1} \in \mathbb{Z}$ . We further remove the commas and the outer parentheses and shorten $\mu\left(\left(a_{1},a_{2},\ldots,a_{\ell}\right)\right)$ to $\mu\left(a_{1}a_{2}\ldots a_{\ell}\right)$ . We call a cylinder set, of the form $\left(a_{1},a_{2},\ldots,a_{\ell}\right)_{x_{1},x_{2},\ldots,x_{\ell}}$ , D-inclusive if $a_{i}=D$ for at least one $i \in \{1,2,\ldots,\ell\}$ .

We emphasize here that even proceeding via the approach outlined previously, one may be able to come up with different weight functions all of which serve the same purpose (i.e. satisfy an inequality of the form (30), along with the criterion stated in (31)). Since the ultimate goal is to draw a conclusion about $\mu(D)$ , it makes sense:

• • to consider each cylinder set $\mathcal{C}_{i}$ in (29) to be D-inclusive; and

• • to begin our step-by-step construction of the weight function by setting $\mathcal{C}_{1} = (D)_{0}$ in (29).

The subsequent choices for $\mathcal{C}_{2}$ , $\mathcal{C}_{3}$ etc. will be motivated in Section 8.3. As shown in Section 8.3, in every step, up to and including the penultimate step, of constructing our weight function, we obtain a weight function inequality that, although of the same form as (30), fails to satisfy (31). This is what leads us to carry out an ‘adjustment’ of the weight function obtained until that step. In Section 8.1.2, we present to the reader the heuristics of how such an adjustment is implemented, and how it contributes to satisfying a weight function inequality of the form (30). However, the implementation of this idea becomes much clearer when the reader goes through the specific details laid out in Section 8.3.

8.1.1. How computations of the pushforward measures of various cylinder sets have been carried out in the sequel.

Letting $\mathcal{C}=(a_{1},a_{2},\ldots,a_{\ell})_{x_{1},x_{1}+1,\ldots,x_{1}+\ell-1}$ be a cylinder set, we denote by $\mu(a_{0}\mathcal{C})$ the probability of the cylinder set $\mathcal{C}'=(a_{0},a_{1},a_{2},\ldots,$ $a_{\ell})_{x_{1}-1,x_{1},x_{1}+1,\ldots,x_{1}+\ell-1}$ , and by $\mu(\mathcal{C}a_{\ell+1})$ the probability of the cylinder set $\mathcal{C}''=(a_{1},a_{2},\ldots,a_{\ell},a_{\ell+1})_{x_{1},x_{1}+1,\ldots,x_{1}+\ell-1,x_{1}+\ell}$ . We now state the following lemma (whose proof is immediate and has been, therefore, omitted).

Lemma 2. For any cylinder set $\mathcal{C}=(a_{1},a_{2},\ldots,a_{\ell})_{x_{1},x_{1}+1,\ldots,x_{1}+\ell-1}$ , and any probability measure $\mu\in\mathcal{M}$ (i.e. $\mu$ is translation-invariant, reflection-invariant, and measurable with respect to the $\sigma$ -field $\mathcal{F}$ generated by all cylinder sets of $\Omega$ ), we can write

\[\mu(\mathcal{C})=\mu(W\mathcal{C})+\mu(D\mathcal{C})+\mu(L\mathcal{C})=\mu(\mathcal{C}W)+\mu(\mathcal{C}D)+\mu(\mathcal{C}L).\]

Lemma 2 has been used repeatedly in the computations of Section 8.3. In addition, another general result has been used, which is captured by Lemma 3. Before stating Lemma 3, we introduce the notation $\mathcal{C}\in\{W,L,D\}^{i}$ , for any $i\in\mathbb{N}$ , to mean that $\mathcal{C}$ is a cylinder set of the form $(a_{0},a_{1},\ldots,a_{i-1})_{0,1,\ldots,i-1}$ . Given cylinder sets $\mathcal{C}=(a_{0},a_{1},\ldots,a_{i-1})_{0,1,\ldots,i-1}$ and $\mathcal{D}=(b_{0},b_{1},\ldots,b_{i})_{0,1,\ldots,i}$ , we denote by $\textrm{P}[\mathcal{C}\big|\mathcal{D}]$ the probability of the event that $\widehat{G}_{p,q,r}\eta(j)=a_{j}$ for each $j\in\{0,1,\ldots,i-1\}$ , conditioned on the event that $\eta(j)=b_{j}$ for each $j\in\{0,1,\ldots,i\}$ .

Lemma 3. For each $i\in\mathbb{N}$ and each $\mathcal{C}\in\{W,L,D\}^{i}$ ,

\[\widehat{G}_{p,q,r}\mu(\mathcal{C})=\sum_{\mathcal{D}\in\{W,L,D\}^{i+1}}\textrm{P}[\mathcal{C}\big|\mathcal{D}]\mu(\mathcal{D}).\]

When applying Lemma 3, such as in the proofs of Lemmas 4 and 5, we often simply write $\textrm{P}[a_{0}a_{1}\ldots a_{i-1}\big|b_{0}b_{1}\ldots b_{i}]$ in place of $\textrm{P}[\mathcal{C}\big|\mathcal{D}]$ .

8.1.2. How each adjustment to the weight function has been carried out in the sequel.

Let us denote by $w_{i-1}$ the weight function we have constructed up until the start of the ith adjustment (thus, the very first weight function we begin with is denoted by $w_{0}$ , the weight function obtained after the first adjustment is denoted by $w_{1}$ , and so on). The weight function inequality at the beginning of the ith adjustment will then be of the form

(32) \begin{equation}w_{i-1}(\widehat{G}_{p,q,r}\mu) \leqslant w_{i-1}(\mu) - \sum_{t=1}^{k_{i-1}}\alpha_{i-1,t}\mu(\mathcal{C}_{i-1,t}),\end{equation}

for some $k_{i-1} \in \mathbb{N}$ , coefficients $\alpha_{i-1,1}, \ldots, \alpha_{i-1,k_{i-1}}$ that are polynomials in p, q, and r, and cylinder sets $\mathcal{C}_{i-1,1}, \ldots, \mathcal{C}_{i-1,k_{i-1}}$ . However, if there exists at least one coefficient among $\alpha_{i-1,1}, \ldots, \alpha_{i-1,k_{i-1}}$ that is negative when $(p,q,r) \in \mathcal{R}$ , the inequality (32) is unable to fulfill (31) when $(p,q,r) \in \mathcal{R}$ , thereby necessitating an adjustment to the weight function $w_{i-1}$ derived so far. We now define, for suitably chosen cylinder sets $\mathcal{D}_{i,1}$ , $\ldots$ , $\mathcal{D}_{i,\ell_{i}}$ , and suitably chosen coefficients $\beta_{i,1}$ , $\ldots$ , $\beta_{i,\ell_{i}}$ that are polynomials in p, q, and r, the updated/adjusted weight function

(33) \begin{equation}w_{i}(\mu) = w_{i-1}(\mu) - \sum_{t=1}^{\ell_{i}}\beta_{i,t}\mu(\mathcal{D}_{i,t}).\end{equation}

When incorporated into (32), $w_{i}$ yields the updated weight function inequality (obtained at the end of the ith adjustment):

(34) \begin{align}{}&w_{i}(\widehat{G}_{p,q,r}\mu) + \sum_{t=1}^{\ell_{i}}\beta_{i,t}\widehat{G}_{p,q,r}\mu(\mathcal{D}_{i,t}) \leqslant w_{i}(\mu) + \sum_{t=1}^{\ell_{i}}\beta_{i,t}\mu(\mathcal{D}_{i,t}) - \sum_{t=1}^{k_{i-1}}\alpha_{i-1,t}\mu(\mathcal{C}_{i-1,t})\nonumber\\\Longleftrightarrow{}& w_{i}(\widehat{G}_{p,q,r}\mu) \leqslant w_{i}(\mu) + \sum_{t=1}^{\ell_{i}}\beta_{i,t}\mu(\mathcal{D}_{i,t}) - \sum_{t=1}^{k_{i-1}}\alpha_{i-1,t}\mu(\mathcal{C}_{i-1,t}) - \sum_{t=1}^{\ell_{i}}\beta_{i,t}\widehat{G}_{p,q,r}\mu(\mathcal{D}_{i,t}).\end{align}

Attempt is usually made to select $\mathcal{D}_{i,1}, \ldots, \mathcal{D}_{i,\ell_{i}}$ as subsets of elements from the set

(35) \begin{equation}\left\{\mathcal{C}_{i-1,t} \;:\; 1 \leqslant t \leqslant k_{i-1} \text{ and } \alpha_{i-1,t} > 0\right\}\!,\end{equation}

with the added restriction that if $\mathcal{D}_{i,t} \subseteq \mathcal{C}_{i-1,t'}$ for some $\mathcal{C}_{i-1,t'}$ belonging to the set in (35), then $\beta_{i,t} \leqslant \alpha_{i-1,t'}$ , so that the coefficient of $\mu(\mathcal{C}_{i-1,t'})$ in the right-hand side of (34) remains nonpositive. Attempt is also made to select $\mathcal{D}_{i,1}, \ldots, \mathcal{D}_{i,\ell_{i}}$ in such a manner that terms from the sum $-\sum_{t=1}^{\ell_{i}}\beta_{i,t}\widehat{G}_{p,q,r}\mu(\mathcal{D}_{i,t})$ are able to aid in negating terms of the form $-\alpha_{i-1,t}\mu(\mathcal{C}_{i-1,t})$ in which $\alpha_{i-1,t} < 0$ when $(p,q,r) \in \mathcal{R}$ .

This gives a broad overview of:

• • why adjustments to the weight function, in several rounds, are required; and

• • how they are usually accomplished.

The idea is reiterated until the final weight function inequality is of the form given in (30) and satisfies the criterion in (31).

8.2. The final weight function obtained for the generalized percolation game when $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}) \in \mathcal{R}$

We state here, for the reader’s convenience, the final weight functions obtained, and the corresponding weight function inequalities, of the form (30) (and obeying the criterion stated in (31)), that these weight functions satisfy when $(p,q,r) \in \mathcal{R}$ . The final weight function that aids us in proving Theorem 4, and, in turn, Theorem 1, is

(36) \begin{equation}w(\mu)=\mu(D)+\mu(WD)+\mu(LWD)\end{equation}

when (4) is true, and

(37) \begin{align}w(\mu)={}&\mu(D)+\mu(WD)+\{4r-2(p+q)+(p+q)^{2}-6r^{2}-3r(p+2q)\}\mu(LD)+(1-p-q)\nonumber\\&\times (1-r)(\!-\!2r+r^{2}+3pr+3qr+r^{3})\mu(LDW)+(1-p-q)(1-r)\nonumber \\&\times (\!-\!1+p+q+r^{2}+2qr)\nonumber\\&\times \mu(LDL)-[2-(3+p+q+r-2q^{2}-2r^{2}-5qr+2pq+pr+3pqr+4q^{2}r\nonumber\\&+4qr^{2} +5pr^{2}-2p^{2}r-5pqr^{2}-3p^{2}r^{2}-2q^{2}r^{2})(1-p-q)(1-r)]\mu(LWD)\end{align}

when any one of (5), (6), and (7) is true. The weight function inequality that each of the two functions in (36) and (37) satisfies (and which is of the form (30)) is given by

(38) \begin{align}w(\widehat{G}_{p,q,r}\mu)&\leqslant{}w(\mu)-[2-(1-p-q)(1-r)\{2+2p+q+r-q^{2}-r^{2}-3qr+pq\nonumber \\[5pt]&\quad +2q^{2}r+2qr^{2}\nonumber\\[5pt]&\quad +pqr+2pr^{2}-p^{2}r-2pqr^{2}-p^{2}r^{2}-q^{2}r^{2}\}]\{\mu(WD)-\mu(WWWD)-\mu(DWDD)\nonumber\\[5pt]&\quad -\mu(LWD)\}-\{1-(1+p+q+pq+pr-q^{2}-qr-p^{2}r+q^{2}r)(1-p-q)(1-r)\nonumber \\[5pt]&\quad \times (1+r)\}\mu(DD)\nonumber\\[5pt]&\quad -r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD).\end{align}

Let us now deduce, from (38), our desired conclusion, i.e. that $\mu(D)=0$ whenever $\mu$ is a translation-invariant and reflection-invariant stationary distribution for $\widehat{G}_{p,q,r}$ . To this end, we focus on the term involving $\mu(DD)$ in (38). Expanding its coefficient, we obtain

(39) \begin{align}{}&-1+(1+p+q+pq+pr-q^{2}-qr-p^{2}r+q^{2}r)(1-p-q)(1-r)(1+r)\nonumber\\[5pt]& \quad ={}-p^{3}r^{3}-p^{2}q\{1-r-r^{2}+r^{3}\}-p^{2}r\{2-r-2r^{2}-p\}-pq^{2}r(1-r^{2})-pr^{3}-2q^{2}r^{3}\nonumber\\[5pt]&\quad\quad -q^{3}r\{1+r-r^{2}\}-qr\{1-pr-2q-r^{2}-2qr\}-q^{2}(1-q)-p(p+q-r)-q^{2}-r^{2}.\end{align}

Since (3) is assumed to hold no matter which of (4), (5), (6), and (7) (p, q, r) satisfies, we have

\begin{align}{}&2(p+q)-(p+q)^{2}+3r(p+2q) \geqslant 4r-6r^{2}\nonumber\\[5pt]\Longleftrightarrow{}& p+q-r \geqslant r\left(1-3r-\frac{3p}{2}-3q\right)+\frac{1}{2}(p+q)^{2},\nonumber\end{align}

and this lower bound on the right-hand side of the inequality above is strictly positive as long as p, q, and r are sufficiently small but at least one of them is strictly positive. Note, furthermore, that if $p=0$ , the term $-p(p+q-r)$ disappears from (39), but $-q^{2}-r^{2}$ remains, and this is strictly negative whenever at least one of q and r is nonzero. All these observations show us that when p, q, and r are sufficiently small and (3) holds, the expression in (39) is strictly negative as long as at least one of p, q, and r is strictly positive (recall, from Section 2, that the condition $p+q+r > 0$ is included in the definition of the parameter-space $\Theta$ ). Consequently, when $\mu$ is stationary for $\widehat{G}_{p,q,r}$ , we deduce, from (38) and the identity $\widehat{G}_{p,q,r}\mu=\mu$ , that $\mu(DD)=0$ .

Since $\widehat{G}_{p,q,r}\mu=\mu$ for any stationary $\mu$ , hence $\mu(DD)=0 \implies \widehat{G}_{p,q,r}\mu(DD)=0$ . Setting $\mathcal{C}=(D,D)_{0,1}$ , and considering $\mathcal{D}\in\{(W,D,W)_{0,1,2},(W,D,L)_{0,1,2},(L,D,L)_{0,1,2}\}$ , an application of Lemma 3 yields the inequality

\begin{equation}(1-p-q)^{2}(1-r)^{2}\{\mu(WDW)+2r\mu(WDL)+r^{2}\mu(LDL)\} \leqslant \widehat{G}_{p,q,r}\mu(DD),\nonumber\end{equation}

so that when $r > 0$ , we have

\begin{equation}\widehat{G}_{p,q,r}\mu(DD)=0 \implies \mu(WDW)=\mu(WDL)=\mu(LDL)=0.\nonumber\end{equation}

Moreover, we have $\mu(WDD)=\mu(LDD)=\mu(DDD)=0$ since each of them is bounded above by $\mu(DD)$ (follows from Lemma 2), and $\mu(DD)=0$ . Since $\mu(D)=\mu(WDW)+\mu(LDL)+\mu(DDD)+2\mu(WDL)+2\mu(WDD)+2\mu(LDD)$ (by Lemma 2 and the reflection invariance of $\mu$ ), we have $\mu(D)=0$ , as desired.

Note that, when $r=0$ , we are back to the set-up considered in [Reference Holroyd, Marcovici and Martin23], and it has already been established there that the associated PCA, $\widehat{G}_{p,q,0}$ , is ergodic whenever $p+q > 0$ .

8.3. Detailed construction of the weight function for generalized percolation games when $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}) \boldsymbol\in \mathcal{R}$

Recall that $\mathcal{R} \subset \Theta$ comprises all those values of (p, q, r) that satisfy the constraints stated in Theorem 1, i.e. p, q, and r are sufficiently small, the inequality in (3) holds, and precisely one of (4), (5), (6), and (7) is true. Throughout Section 8.3, it is assumed that the probability measure $\mu$ that we deal with is an element of $\mathcal{M}$ , i.e. it is defined on the state space $\Omega=\hat{\mathcal{A}}^{\mathbb{Z}}=\{W,L,D\}^{\mathbb{Z}}$ of $\widehat{G}_{p,q,r}$ , with respect to the $\sigma$ -field $\mathcal{F}$ generated by all cylinder sets of $\Omega$ , and that $\mu$ is both reflection-invariant and translation-invariant. Recall, from the discussion in the paragraph preceding Section 8.1.2, that beginning the construction of our weight function with the cylinder set $\mathcal{C}_{1}=(D)_{0}$ seems to be a sensible and natural choice. Since, at each step, we must obtain an inequality of the form given by (30), it is crucial that we write down the pushforward measure $\widehat{G}_{p,q,r}\mu(\mathcal{C}_{i})$ for each cylinder set $\mathcal{C}_{i}$ that we incorporate into our gradually unfolding weight function. Let us, therefore, begin with the following lemma, which captures $\widehat{G}_{p,q,r}\mu(\mathcal{C}_{1})$ for $\mathcal{C}_{1}=(D)_{0}$ .

Lemma 4. For each $\mu\in\mathcal{M}$ , we have

(40) \begin{align}\widehat{G}_{p,q,r}\mu(D)&=2(1-p-q)(1-r)\mu(WD)+2(1-p-q)r(1-r)\mu(LD)\nonumber \\&\quad +(1-p-q)(1-r^{2})\mu(DD).\end{align}

Proof. We apply Lemma 3 to the cylinder set $(D)_{1}$ , obtain the conditional probabilities $\textrm{P}[D\big|WD], \ldots, \textrm{P}[D\big|DD]$ from (13), (14), and (15), and use of the assumption that $\mu$ is reflection-invariant, to obtain

\begin{align*}\widehat{G}_{p,q,r}\mu(D)&=(1-p-q)(1-r)\left\{\mu(WD)+\mu(DW)\right\}+(1-p-q)r(1-r)\left\{\mu(LD)+\mu(DL)\right\}\nonumber\\&\quad +(1-p-q)(1-r^{2})\mu(DD)\nonumber\\&=2(1-p-q)(1-r)\mu(WD)+2(1-p-q)r(1-r)\mu(LD)\nonumber\\&\quad +(1-p-q)(1-r^{2})\mu(DD),\nonumber\end{align*}

as claimed.

We now explain the intuition behind our choice of the second cylinder set, $\mathcal{C}_{2}$ , as we attempt to shape up our weight function in the form given by (29). The coefficient $2(1-p-q)r(1-r)$ of $\mu(LD)$ in (40) is always bounded above by 1 (in fact, by $1/2$ ), since $r(1-r) \leqslant 1/4$ and $(1-p-q) \leqslant 1$ . The coefficient $(1-p-q)(1-r^{2})$ of $\mu(DD)$ in (40) is obviously bounded above by 1. On the other hand, when p, q, and r are all sufficiently small, the coefficient $2(1-p-q)(1-r)$ of $\mu(WD)$ in (40) is very close to 2. We can thus write, using (40) and the fact that $\mu(D)=\mu(WD)+\mu(LD)+\mu(DD)$ , the following inequality: $\widehat{G}_{p,q,r}\mu(D) \leqslant \mu(D)+\mu(WD)$ . Comparing this with (30), it becomes evident that if $c_{1}=1$ and $\mathcal{C}_{1}=(D)_{0}$ in (29), a sensible choice for our next cylinder set and the corresponding coefficient would be $\mathcal{C}_{2}=(W,D)_{0,1}$ and $c_{2}=1$ . This now requires us to find the pushforward measure $\widehat{G}_{p,q,r}\mu(WD)$ .

Lemma 5. For any $\mu\in\mathcal{M}$ , we have

\begin{align*}\widehat{G}_{p,q,r}\mu(WD)={}&2p(1-p-q)(1-r)\mu(WD)+p(1-p-q)(1-r)(1+r)\mu(DD)\nonumber\\&+(1-p-q+r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)\nonumber\\&\times(1-r)\mu(LD)-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)\nonumber \end{align*}

(41) \begin{align}\qquad\qquad &-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&+(1-p-q)^{2}(1-r)^{2}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD).\end{align}

Proof. We apply Lemma 3, keeping in mind that the only way the event $\{\widehat{G}_{p,q,r}\eta(1)=D\}$ happens with positive probability is if at least one of $\eta(1)$ and $\eta(2)$ equals D. For $\mathcal{D}=(b_{0},b_{1},b_{2})_{0,1,2}\in\{W,L,D\}^{3}$ , computing the conditional probabilities

\begin{equation}\textrm{P}[WD\big|\mathcal{D}]=\textrm{P}\left[\widehat{G}_{p,q,r}\eta(0)=W\big|\eta(0)=b_{0},\eta(1)=b_{1}\right]\textrm{P}\left[\widehat{G}_{p,q,r}\eta(1)=D\big|\eta(1)=b_{1},\eta(2)=b_{2}\right]\nonumber\end{equation}

by making use of (10) through (15), we obtain

\begin{align*}\widehat{G}_{p,q,r}\mu(WD)={}&p(1-p-q)(1-r)\mu(WDW)+p(1-p-q)(1-r)\mu(DDW)\nonumber\\&+\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(LDW)+p(1-p-q)(1-r^{2})\nonumber\\&\times \mu(WDD)+p(1-p-q)(1-r^{2})\mu(DDD)\nonumber \\&+\{p+(1-p-q)(1-r)\}(1-p-q)\nonumber\\&\times(1-r^{2})\mu(LDD)+p(1-p-q)r(1-r)\mu(WDL)\nonumber \\&+p(1-p-q)r(1-r)\mu(DDL)\nonumber\\&+\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LDL)+p(1-p-q)(1-r)\nonumber\\&\times\mu(WWD)+p(1-p-q)(1-r)\mu(DWD)\nonumber \\&+\{p+(1-p-q)(1-r)\}(1-p-q)\nonumber\\&\times(1-r)\mu(LWD)+\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(WLD)\nonumber\\&+\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(DLD)\nonumber\\&+\{p+(1-p-q)(1-r^{2})\}(1-p-q)r(1-r)\mu(LLD)\nonumber\\={}&p(1-p-q)(1-r)\mu(WDW)+p(1-p-q)(1-r)(2+r)\mu(DDW)\nonumber\\&+\{1-q-r+qr+2pr\}(1-p-q)(1-r)\mu(LDW)+p(1-p-q)(1-r^{2})\nonumber\\&\times\mu(DDD)+\{1-q-r^{2}+2pr+qr^{2}+pr^{2}\}(1-p-q)(1-r)\mu(DDL)\nonumber\\&+(1-q-r+pr+qr)(1-p-q)r(1-r)\mu(LDL)+p(1-p-q)(1-r)\nonumber\\&\times\mu(WWD)+p(1-p-q)(1-r)\mu(DWD)\nonumber\\&+(1-q-r+pr+qr)(1-p-q)\nonumber\\&\times(1-r)\mu(LWD)+(1-q-r+pr+qr)(1-p-q)r(1-r)\mu(WLD)\nonumber\\&+(1-q-r+pr+qr)(1-p-q)r(1-r)\mu(DLD)\nonumber\\&+(1-q-r^{2}+pr^{2}+qr^{2})(1-p-q)r(1-r)\mu(LLD)\nonumber\\&(\text{using reflection-invariance})\\={}&\underbrace{p(1-p-q)(1-r)\mu(WDW)+p(1-p-q)(1-r)\mu(DDW)}_{(1)}\nonumber\\&\underbrace{+p(1-p-q)(1-r)(1+r)\mu(WDD)}_{(2)}\underbrace{+p(1-p-q)(1-r)\mu(LDW)}_{(1)}\nonumber\\&\underbrace{+(1-p-q-r+qr+2pr)(1-p-q)(1-r)\mu(LDW)}_{(3)}\nonumber\end{align*}

\begin{align}\qquad\qquad\qquad &\underbrace{+p(1-p-q)(1-r^{2})\mu(DDD)+p(1-p-q)(1-r)(1+r)\mu(LDD)}_{(2)}\nonumber\\&\underbrace{+(1-p-q-r^{2}+pr+pr^{2}+qr^{2})(1-p-q)(1-r)\mu(LDD)}_{(3)}\nonumber\\&\underbrace{+(1-q-r+pr+qr)(1-p-q)r(1-r)\mu(LDL)}_{(3)}\nonumber\\&\underbrace{+p(1-p-q)(1-r)\mu(WWD)}_{(4)}\nonumber\\ &\underbrace{+p(1-p-q)(1-r)\mu(DWD)+p(1-p-q)(1-r)\mu(LWD)}_{(4)}\nonumber\\&+(1-p-q)^{2}(1-r)^{2}\mu(LWD)\nonumber\\&\underbrace{+(1-q-r^{2}+pr^{2}+qr^{2})(1-p-q)r(1-r)\mu(WLD)}_{(5)}\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&\underbrace{+(1-q-r^{2}+pr^{2}+qr^{2})(1-p-q)r(1-r)\mu(DLD)}_{(5)}\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\nonumber\\&\underbrace{+(1-q-r^{2}+pr^{2}+qr^{2})(1-p-q)r(1-r)\mu(LLD)}_{(5)}\nonumber\\={}&\underbrace{p(1-p-q)(1-r)\mu(DW)}_{\text{combining terms underbraced by }(1)}\underbrace{+p(1-p-q)(1-r)(1+r)\mu(DD)}_{\text{combining terms underbraced by }(2)}\nonumber\\&\underbrace{+(1-p-q-r^{2}+pr+pr^{2}+qr^{2})(1-p-q)(1-r)\mu(LD)}_{\text{combining terms underbraced by }(3)}\nonumber\\&-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&\underbrace{+p(1-p-q)(1-r)\mu(WD)}_{\text{combining terms underbraced by }(4)}+(1-p-q)^{2}(1-r)^{2}\mu(LWD)\nonumber\\&\underbrace{+(1-q-r^{2}+pr^{2}+qr^{2})(1-p-q)r(1-r)\mu(LD)}_{\text{combining terms underbraced by }(5)}\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\nonumber\\={}&2p(1-p-q)(1-r)\mu(WD)+p(1-p-q)(1-r)(1+r)\mu(DD)\nonumber\\&+(1-p-q+r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)\nonumber\\&\times(1-r)\mu(LD)-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&+(1-p-q)^{2}(1-r)^{2}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD),\nonumber \end{align}

where, in the last step, we combine terms using the reflection-invariance property of $\mu$ .

Recall that, so far, we have set $c_{1}=c_{2}=1$ , $\mathcal{C}_{1}=(D)_{0}$ , and $\mathcal{C}_{2}=(W,D)_{0,1}$ . The weight function, so far, equals $\mu(D)+\mu(WD)$ , and the right-hand side of the weight function inequality (of the form (30)) is given by

(42) \begin{align}{}&\widehat{G}_{p,q,r}\mu(D)+\widehat{G}_{p,q,r}\mu(WD)\nonumber\\={}&2(1-p-q)(1-r)\mu(WD)+2(1-p-q)r(1-r)\mu(LD)+(1-p-q)(1-r^{2})\mu(DD)\nonumber\\&+2p(1-p-q)(1-r)\mu(WD)+p(1-p-q)(1-r)(1+r)\mu(DD)\nonumber\\&+(1-p-q+r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)(1-r)\mu(LD)\nonumber\\&-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&+(1-p-q)^{2}(1-r)^{2}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\nonumber\\&\times\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\nonumber\\={}&2(1+p)(1-p-q)(1-r)\mu(WD)+(1+p)(1-p-q)(1-r)(1+r)\mu(DD)\nonumber\\&+(1-p-q+3r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)(1-r)\mu(LD)\nonumber \\&-r(1-p-q)^{2}(1-r)^{2}\times\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)+(1-p-q)^{2}(1-r)^{2}\nonumber \\&\times \mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\times\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD).\end{align}

We now pause and focus on the coefficient of $\mu(LD)$ in (42) (we need not focus on the coefficients of $\mu(WD)$ and $\mu(DD)$ since we can readily see that the former is bounded above by 2 and the latter by 1). Writing

\begin{align}pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3}=pr-qr-r^{2}(1-p-q)(1+r),\nonumber\end{align}

we see that the coefficient of $\mu(LD)$ in (42) is given by

(43) \begin{align}1+2r-2(p+q)&+(p+q)^{2}-3r^{2}-(p+q)r+3(p+q)r^{2}-(p+q)^{2}r\nonumber\\&+pr(1-p-q)(1-r)-qr(1-p-q)(1-r)-r^{2}(1-p-q)^{2}(1-r)^{2}\nonumber\\={}&1+2r-2(p+q)+(p+q)^{2}-3r^{2}-2qr+pr\{-1+(1-p-q)(1-r)\}\nonumber\\&+r^{2}\{-(1-p-q)^{2}(1-r)^{2}+3p+4q-pq-q^{2}\}-p^{2}r-pqr\nonumber\\ \!\leqslant{}&1+2r-2(p+q)+(p+q)^{2}-3r^{2}-2qr \; \text{when } p, q, r \text{ are sufficiently small,}\end{align}

and when (3) holds, we have

\begin{align}2(p+q)-(p+q)^{2}-2r+3r^{2}+2qr&\geqslant 4r-6r^{2}-3pr-6qr-2r+3r^{2}+2qr\nonumber \\&=r(2-3p-4q-3r) \geqslant 0\nonumber\end{align}

for all p and r sufficiently small, thus proving that the expression in (43), and, consequently, the coefficient of $\mu(LD)$ in (42), is bounded above by 1.

We now continue with (42), and write (decomposing $\mu(D)$ as in Lemma 2):

\begin{align*}{}&\widehat{G}_{p,q,r}\mu(D)+\widehat{G}_{p,q,r}\mu(WD)\nonumber\\={}&\mu(D)+\mu(WD)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)-\{1-(1+p)(1-p-q)(1-r)\nonumber\\&\times (1+r)\}\mu(DD)-\{1-(1-p-q+3r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3}) \end{align*}

(44) \begin{align}&\times(1-p-q)(1-r)\}\mu(LD)-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\nonumber\\&\times \mu(LDL)+(1-p-q)^{2}(1-r)^{2}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber \\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD),\end{align}

and given the observations made above regarding the coefficient of $\mu(LD)$ in (42), we conclude that, when p, q, and r are all sufficiently small and (3) holds, the only term on the right-hand side of (44), other than $\mu(D)$ and $\mu(WD)$ , that has a non-negative coefficient is $(1-p-q)^{2}(1-r)^{2}\mu(LWD)$ . Moreover, when p, q, and r are all sufficiently small, we have $(1-p-q)^{2}(1-r)^{2}=\Theta(1)$ (i.e. its order of magnitude is the same as the constant 1 when p, q, and r approach 0). Hence, it seems sensible to set $\mathcal{C}_{3}=(L,W,D)_{0,1,2}$ and $c_{3}=1$ , and the weight function obtained thus far becomes

(45) \begin{equation}w_{0}(\mu)=\mu(D)+\mu(WD)+\mu(LWD).\end{equation}

This necessitates the computation of $\widehat{G}_{p,q,r}\mu(LWD)$ , which has been accomplished via Lemma 6.

Lemma 6. For any $\mu\in\mathcal{M}$ , we can write $\widehat{G}_{p,q,r}\mu(LWD)=\sum_{i=1}^{9}A_{i}$ , where

(46) \begin{equation}A_{1}\leqslant{} (1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LD),\end{equation}

(47) \begin{align}A_{2}\leqslant{} \{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)\mu(DD),\end{align}

(48) \begin{align}A_{3}\leqslant{}&\{q+(1-p-q)r\}(1-p-q)(1-r)\{1+p-q-r+2pr+qr\}\mu(LWD),\end{align}

(49) \begin{align}A_{4}\leqslant{}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LDL),\end{align}

(50) \begin{align}A_{5}={}&(1-p)p(1-p-q)r(1-r)\mu(WDL),\end{align}

(51) \begin{align}&A_{6}\leqslant{}\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(DW)\nonumber \\&\quad +\{p(1-p)-(q+r-pr-qr)(1-q-r+pr+qr)\}(1-p-q)(1-r)\mu(WWDW),\end{align}

(52) \begin{align}A_{7}\leqslant{}&p(1-p-q)^{2}(1-r)^{2}\mu(WWWD)+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(WD),\end{align}

(53) \begin{align}A_{8}\leqslant{}&(q+r-pr-qr)(1-q-r+pr+qr)(1-p-q)(1-r)(1+r)\mu(LDD),\end{align}

(54) \begin{align}&A_{9}\leqslant{}q(1-q-r+pr+qr)(1-p-q)(1-r)(1+r)\mu(WDD)\nonumber\\&\quad +\{p(1-p)-q(1-q-r+pr+qr)\}(1-p-q)(1-r)(1+r)\mu(WWDD)\nonumber\\&\quad +(pr+qr-q)(1-p-q)^{2}(1-r)(1+r)\mu(DWDD).\end{align}

Proof. Extending the notation introduced just before stating Lemma 3, we now let $R_{0}\times R_{1} \times \cdots \times R_{i-1}$ , for $i\in\mathbb{N}$ and $R_{j}\subset\{W,L,D\}$ for each $j\in\{0,1,\ldots,i-1\}$ , denote the set of all cylinder sets $(a_{0},a_{1},\ldots,a_{i-1})_{0,1,\ldots,i-1}$ such that $a_{j}\in R_{j}$ for each $j\in\{0,1,\ldots,i-1\}$ . Applying Lemma 3 to the cylinder set $\mathcal{C}=(L,W,D)_{0,1,2}$ , and keeping in mind that in order for the event $\{\widehat{G}_{p,q,r}\eta(2)=D\}$ to take place with positive probability, at least one of the events $\{\eta(2)=D\}$ and $\{\eta(3)=D\}$ must occur, we can write

\begin{align}\widehat{G}_{p,q,r}\mu(LWD)={}&\sum_{\mathcal{D}\in\{W,L,D\}^{4}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D})=\sum_{i=1}^{9}A_{i},\nonumber\end{align}

where we set

\begin{align}A_{1}={}&\sum_{\mathcal{D}\in\{W,L,D\}^{2}\times\{L\}\times\{D\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{2}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{D\}^{2}\times\{W,L,D\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{3}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{L\}\times\{W\}\times\{D\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D})+\sum_{\mathcal{D}\in\{L\}\times\{W\}\times\{D\}\times\{W,L,D\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{4}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{L\}\times\{D\}\times\{L\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{5}={}&\sum_{\mathcal{D}\in\{W,D\}\times\{W\}\times\{D\}\times\{L\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{6}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{L\}\times\{D\}\times\{W\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D})+\sum_{\mathcal{D}\in\{W,D\}\times\{W\}\times\{D\}\times\{W\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{7}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{W,D\}\times\{W\}\times\{D\}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{8}={}&\sum_{\mathcal{D}\in\{W,L,D\}\times\{L\}\times\{D\}^{2}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}),\nonumber\\A_{9}={}&\sum_{\mathcal{D}\in\{W,D\}\times\{W\}\times\{D\}^{2}}\textrm{P}[LWD\big|\mathcal{D}]\mu(\mathcal{D}).\nonumber\end{align}

In the computation of $A_{i}$ for each $i\in\{1,2,\ldots,9\}$ , we make use of the relation

\begin{align}\textrm{P}[LWD|\mathcal{D}]={}&\textrm{P}[\widehat{G}_{p,q,r}\eta(0)=L|\eta(0)=b_{0},\eta(1)=b_{1}]\textrm{P}[\widehat{G}_{p,q,r}\eta(1)=W|\eta(1)=b_{1},\eta(2)=b_{2}]\nonumber\\&\times\textrm{P}[\widehat{G}_{p,q,r}\eta(2)=D|\eta(2)=b_{2},\eta(3)=b_{3}],\nonumber\end{align}

where $\mathcal{D}=(b_{0},b_{1},b_{2},b_{3})_{0,1,2,3}$ , and we apply the stochastic update rules of (10) through (15).

We begin with

(55) \begin{align*}A_{1}={}&(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(WWLD)+\{q+(1-p-q)r\}\nonumber\\&\times\{p+(1-p-q)(1-r^{2})\}(1-p-q)r(1-r)\mu(WLLD)+\{q+(1-p-q)r\}\nonumber\\&\times\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LWLD)+\{q+(1-p-q)r^{2}\}\nonumber\\&\times\{p+(1-p-q)(1-r^{2})\}(1-p-q)r(1-r)\mu(LLLD)+\{q+(1-p-q)r\}\nonumber\\&\times\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(WDLD)+\{q+(1-p-q)r\}\nonumber\\&\times\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(DWLD)+\{q+(1-p-q)r^{2}\}\nonumber\\ \qquad\qquad &\times\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LDLD)+\{q+(1-p-q)r^{2}\}\nonumber\\&\times\{p+(1-p-q)(1-r^{2})\}(1-p-q)r(1-r)\mu(DLLD)+\{q+(1-p-q)r^{2}\}\nonumber\\&\times\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(DDLD).\end{align*}

A quick comparison of the coefficients in the various terms of (55) reveals that the coefficient of each of $\mu(LWLD)$ , $\mu(LLLD)$ , $\ldots$ , $\mu(DDLD)$ is bounded above by the coefficient of $\mu(WLLD)$ , and

\begin{align}(1-p)\{p+(1-p-q)&(1-r)\}-\{q+(1-p-q)r\}\{p+(1-p-q)(1-r^{2})\}\nonumber\\={}&(1-p)\{p+(1-p-q)(1-r)\}-\{q+(1-p-q)r\}\nonumber\\&\times \{p+(1-p-q)(1-r)+r(1-p-q)(1-r)\}\nonumber\\={}&(1-p-q)(1-r)\{p+(1-p-q)(1-r)\}\nonumber \\&-\{q+(1-p-q)r\}r(1-p-q)(1-r)\nonumber\\={}&(1-p-q)(1-r)\{p+(1-p-q)(1-r-r^{2})-qr\}\nonumber\end{align}

is non-negative whenever p, q, and r are all sufficiently small, so that the coefficient of $\mu(WWLD)$ exceeds that of $\mu(WLLD)$ in (55). For each cylinder set $\mathcal{D}\in\{W,L,D\}^{2}\times\{L\}\times\{D\}$ , let us denote the corresponding coefficient by $\alpha_{\mathcal{D}}$ (for instance, the coefficient of $\mu(DLLD)$ is $\alpha_{DLLD}=\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r^{2})\}(1-p-q)r(1-r)$ ), and let us define

\begin{equation}\alpha'_{\mathcal{D}}=\alpha_{\mathcal{D}}-\alpha_{WWLD}=\alpha_{\mathcal{D}}-(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\nonumber\end{equation}

for each cylinder set $\mathcal{D}\in\{W,L,D\}^{2}\times\{L\}\times\{D\}\setminus\left\{(W,W,L,D)_{0,1,2,3}\right\}$ . From our discussion above, it is evident that when p, q, and r are all small enough, each $\alpha'_{\mathcal{D}} \leqslant 0$ , so that we may rewrite (55) as

\begin{align}A_{1}={}&(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LD)+\alpha'_{WLLD}\mu(WLLD)\nonumber\\&+\alpha'_{LWLD}\mu(LWLD)+\alpha'_{LLLD}\mu(LLLD)+\alpha'_{WDLD}\mu(WDLD)+\alpha'_{DWLD}\mu(DWLD)\nonumber\\&+\alpha'_{LDLD}\mu(LDLD)+\alpha'_{DLLD}\mu(DLLD)+\alpha'_{DDLD}\mu(DDLD)\nonumber\\\leqslant{}&(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LD),\nonumber \end{align}

which proves the inequality in (46). Next, we have

(56) \begin{align}A_{2}={}&\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(WDDW)\nonumber\\&+\{q+(1-p-q)r^{2}\}p(1-p-q)(1-r)\mu(DDDW)\nonumber\\&+\{q+(1-p-q)r^{2}\}p(1-p-q)(1-r)\mu(LDDW)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)(1-r^{2})\mu(WDDD)+\{q+(1-p-q)r^{2}\}p(1-p-q)\nonumber\\&\times(1-r^{2})\mu(DDDD)+\{q+(1-p-q)r^{2}\}p(1-p-q)(1-r^{2})\mu(LDDD)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)r(1-r)\mu(WDDL)+\{q+(1-p-q)r^{2}\}p(1-p-q)\nonumber\\&\times r(1-r)\mu(DDDL)+\{q+(1-p-q)r^{2}\}p(1-p-q)r(1-r)\mu(LDDL).\end{align}

A quick comparison of the coefficients of the various terms in (56) reveals that the coefficient of $\mu(WDDD)$ , namely, $\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)$ , is the largest. As in the analysis of $A_{1}$ , letting $\alpha_{\mathcal{D}}$ denote the coefficient of $\mu(\mathcal{D})$ in (56) for every $\mathcal{D}\in\{W,L,D\}\times\{D\}^{2}\times\{W,L,D\}$ , and setting

\begin{equation}\alpha'_{\mathcal{D}}=\alpha_{\mathcal{D}}-\alpha_{WDDD}=\alpha_{\mathcal{D}}-\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)\nonumber\end{equation}

for each $\mathcal{D}\in\{W,L,D\}\times\{D\}^{2}\times\{W,L,D\}\setminus\left\{(W,D,D,D)_{0,1,2,3}\right\}$ , we know, from our previous observation, that each $\alpha'_{\mathcal{D}} \leqslant 0$ , and we can rewrite (56) as

\begin{align}A_{2}={}&\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)\mu(DD)+\alpha'_{WDDW}\mu(WDDW)\nonumber\\&+\alpha'_{DDDW}\mu(DDDW)+\alpha'_{LDDW}\mu(LDDW)+\alpha'_{DDDD}\mu(DDDD)+\alpha'_{LDDD}\mu(LDDD)\nonumber\\&+\alpha'_{WDDL}\mu(WDDL)+\alpha'_{DDDL}\mu(DDDL)+\alpha'_{LDDL}\mu(LDDL)\nonumber\\\leqslant{}&\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)\mu(DD),\nonumber \end{align}

which proves the bound in (47). From the definition of $A_{3}$ , we have

(57) \begin{align}A_{3}={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(WLWD)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(DLWD)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(LLWD)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(LWDW)+\{q+(1-p-q)r\}p(1-p-q)\nonumber\\&\times (1-r^{2})\mu(LWDD)+\{q+(1-p-q)r\}p(1-p-q)r(1-r)\mu(LWDL)\nonumber\\={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(LWD)\nonumber\\&+\alpha'_{DLWD}\mu(DLWD)+\alpha'_{LLWD}\mu(LLWD)+\{q+(1-p-q)r\}p(1-p-q)(1-r^{2})\mu(LWD)\nonumber\\&+\alpha'_{LWDW}\mu(LWDW)+\alpha'_{LWDL}\mu(LWDL)\nonumber\\={}&\{q+(1-p-q)r\}(1-p-q)(1-r)\{1+p-q-r+2pr+qr\}\mu(LWD)\nonumber\\&+\alpha'_{DLWD}\mu(DLWD)+\alpha'_{LLWD}\mu(LLWD)+\alpha'_{LWDW}\mu(LWDW)+\alpha'_{LWDL}\mu(LWDL),\end{align}

where each of the coefficients

\begin{align}{}&\alpha'_{DLWD}=\alpha'_{LLWD}=-r(1-p-q)^{2}(1-r)^{2}\{p+(1-p-q)(1-r)\},\nonumber\\{}&\alpha'_{LWDW}=-r\{q+(1-p-q)r\}p(1-p-q)(1-r),\nonumber\\{}&\alpha'_{LWDL}=-\{q+(1-p-q)r\}p(1-p-q)(1-r)\nonumber\end{align}

is non-positive, letting (57) imply the inequality in (48). Next, we have

(58) \begin{align}A_{4}={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(WLDL)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(DLDL)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LLDL)\nonumber\\={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LDL)\nonumber\\&+\alpha'_{DLDL}\mu(DLDL)+\alpha'_{LLDL}\mu(LLDL),\end{align}

where $\alpha'_{DLDL}=\alpha'_{LLDL}=-\{p+(1-p-q)(1-r)\}(1-p-q)^{2}r^{2}(1-r)^{2}$ are both nonpositive coefficients, thus letting (58) imply that the inequality in (49) is true. Keeping in mind that we have already taken into account $\mathcal{D}=(L,W,D,L)_{0,1,2,3}$ in $A_{3}$ , we obtain, from the definition of $A_{5}$ ,

\begin{align}A_{5}={}&(1-p)p(1-p-q)r(1-r)\mu(WWDL)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)r(1-r)\mu(DWDL)\nonumber\\={}&(1-p)p(1-p-q)r(1-r)\mu(WDL)-pr(1-p-q)^{2}(1-r)^{2}\mu(DWDL)\nonumber\\&-(1-p)p(1-p-q)r(1-r)\mu(LWDL)\leqslant (1-p)p(1-p-q)r(1-r)\mu(WDL),\nonumber \end{align}

proving (50). Keeping in mind that $\mathcal{D}=(L,W,D,W)_{0,1,2,3}$ has already been taken into account in $A_{3}$ , we obtain, from the definition of $A_{6}$ :

(59) \begin{align}A_{6}={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(WLDW)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(DLDW)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(LLDW)+(1-p)p\nonumber\\&\times (1-p-q)(1-r)\mu(WWDW)+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(DWDW)\nonumber\\\leqslant{}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)\mu(DW)+\alpha'_{DLDW}\mu(DLDW)\nonumber\\&+\alpha'_{LLDW}\mu(LLDW)+\{p(1-p)-(q+r-pr-qr)(1-q-r+pr+qr)\}\nonumber\\&\times(1-p-q)(1-r)\mu(WWDW)+\alpha'_{DWDW}\mu(DWDW),\end{align}

(the sum of $\mu(WLDW)$ , $\mu(DLDW)$ , $\mu(LLDW)$ , $\mu(WWDW)$ , and $\mu(DWDW)$ is bounded above by $\mu(DW)$ , which follows from Lemma 2) where each of

\begin{align}{}&\alpha'_{DLDW}=\alpha'_{LLDW}=-r\{p+(1-p-q)(1-r)\}(1-p-q)^{2}(1-r)^{2},\nonumber\\{}&\alpha'_{DWDW}=-\{q+(1-p-q)r\}(1-p-q)^{2}(1-r)^{2}\nonumber\end{align}

is nonpositive, allowing us to conclude, from (59), that (51) is true. Next, we observe that

(60) \begin{align}A_{7}={}&(1-p)p(1-p-q)(1-r)\mu(WWWD)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(DWWD)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(LWWD)+\{q+(1-p-q)r\}\nonumber\\&\times p(1-p-q)(1-r)\mu(WDWD)+\{q+(1-p-q)r^{2}\}p(1-p-q)(1-r)\mu(DDWD)\nonumber\\&+\{q+(1-p-q)r^{2}\}p(1-p-q)(1-r)\mu(LDWD)\nonumber\\\leqslant{}&p(1-p-q)^{2}(1-r)^{2}\mu(WWWD)+\{q+(1-p-q)r\}p(1-p-q)(1-r)\mu(WD)\nonumber\\&+\alpha'_{DDWD}\mu(DDWD)+\alpha'_{LDWD}\mu(LDWD),\end{align}

where $\alpha'_{DDWD}=\alpha'_{LDWD}=-pr(1-p-q)^{2}(1-r)^{2}$ (the last inequality follows from the fact that the sum of $\mu(WWWD)$ , $\mu(DWWD)$ , $\mu(LWWD)$ , $\mu(WDWD)$ , $\mu(DDWD)$ , and $\mu(LDWD)$ is bounded above by $\mu(WD)$ , which, in turn, follows from Lemma 2), thus allowing (60) to imply that (52) holds. The second last term in the expansion of $\widehat{G}_{p,q,r}\mu(LWD)$ equals

\begin{align}A_{8}={}&\{q+(1-p-q)r\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)(1+r)\mu(WLDD)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)(1+r)\mu(DLDD)\nonumber\\&+\{q+(1-p-q)r^{2}\}\{p+(1-p-q)(1-r)\}(1-p-q)(1-r)(1+r)\mu(LLDD)\nonumber\\={}&(q+r-pr-qr)(1-q-r+pr+qr)(1-p-q)(1-r)(1+r)\mu(LDD)\nonumber\\&-r\{p+(1-p-q)(1-r)\}(1-p-q)^{2}(1-r)^{2}(1+r)\{\mu(DLDD)+\mu(LLDD)\},\nonumber \end{align}

corroborating the inequality in (53), whereas the final term in the expansion of $\widehat{G}_{p,q,r}\mu(LWD)$ equals

(61) \begin{align}A_{9}={}&(1-p)p(1-p-q)(1-r)(1+r)\mu(WWDD)\nonumber\\&+\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)\mu(DWDD)\\\leqslant{}&q(1-q-r+pr+qr)(1-p-q)(1-r)(1+r)\mu(WDD)\nonumber\\&+\{p(1-p)-q(1-q-r+pr+qr)\}(1-p-q)(1-r)(1+r)\mu(WWDD)\nonumber\\&+(pr+qr-q)(1-p-q)^{2}(1-r)(1+r)\mu(DWDD),\nonumber\end{align}

(the second step is obtained by first applying Lemma 2, then ignoring the term involving $\mu(LWDD)$ since its coefficient is nonpositive), corroborating (44). This concludes the proof of Lemma 6.

The weight function inequality of the form given by (30), corresponding to the weight function in (45), can now be derived as follows, making use of (44) and Lemma 6:

(62) \begin{align}w_{0}(\widehat{G}_{p,q,r}\mu)={}&\widehat{G}_{p,q,r}\mu(D)+\widehat{G}_{p,q,r}\mu(WD)+\widehat{G}_{p,q,r}\mu(LWD)\nonumber\\={}&\mu(D)+\mu(WD)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)-\{1-(1+p)\nonumber\\&\times (1-p-q)(1-r)(1+r)\}\mu(DD)-\{1-(1-p-q+3r+pr-qr-r^{2}+pr^{2}\nonumber\\&+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)(1-r)\}\mu(LD)-r(1-p-q)^{2}(1-r)^{2}\nonumber\\&\times \mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)+(1-p-q)^{2}(1-r)^{2}\mu(LWD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\nonumber\\&+\widehat{G}_{p,q,r}\mu(LWD)\nonumber\\={}&\mu(D)+\mu(WD)+\mu(LWD)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)\nonumber\\&-\{1-(1+p)(1-p-q)(1-r)(1+r)\}\mu(DD)-\{1-(1-p-q+3r+pr\nonumber\\&-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)(1-r)\}\mu(LD)\nonumber\\&-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&-\{1-(1-p-q)^{2}(1-r)^{2}\}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)+\sum_{i=1}^{9}A_{i}\nonumber\\\leqslant{}&w_{0}(\mu)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)\nonumber\\&-\{1-(1+p)(1-p-q)(1-r)(1+r)\}\mu(DD)\nonumber\\&\underbrace{-\{1-(1-p-q+3r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})}\nonumber\\&\underbrace{(1-p-q)(1-r)\}\mu(LD)}-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)\nonumber\\&-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&-\{1-(1-p-q)^{2}(1-r)^{2}\}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\underbrace{+(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r}\nonumber\\&\times \underbrace{(1-r)\mu(LD)}+\sum_{i=2}^{9}A_{i},\end{align}

where the last step is obtained by implementing the inequality of (46). Combining the terms involving $\mu(LD)$ (indicated by underbraces) in (62), we obtain

(63) \begin{align} {}&-\{1-(1-p-q+3r+pr-qr-r^{2}+pr^{2}+qr^{2}-r^{3}+pr^{3}+qr^{3})(1-p-q)(1-r)\}\nonumber\\&\times \mu(LD)+(1-p)\{p+(1-p-q)(1-r)\}(1-p-q)r(1-r)\mu(LD)\nonumber\\={}&[\!-\!1+(1-p-q+4r-2qr-2r^{2}+pqr+3pr^{2}+2qr^{2}-r^{3}+pr^{3}+qr^{3}-p^{2}r^{2}-pqr^{2})\nonumber\\&\times (1-p-q)(1-r)]\mu(LD)\nonumber\\={}&[\!-\!1+\{1-p-q+4r-qr(2-p-2r)-r^{2}(2-3p)-r^{3}(1-p-q)-p^{2}r^{2}-pqr^{2}\}\nonumber\\&\times (1-p-q)(1-r)]\mu(LD)\end{align}

(64) \begin{align}\leqslant{}&[\!-\!1+(1-p-q+4r)(1-p-q)(1-r)]\mu(LD) \quad \text{when } p, r \text{ are sufficiently small;}\nonumber\\ ={}&[\!-\!1+\{1+3r-4r^{2}-2(p+q)+(p+q)^{2}-(p+q)r-(p+q)r(1+p+q-4r)\}]\mu(LD)\nonumber\\ \leqslant{}&\{3r-4r^{2}-2(p+q)+(p+q)^{2}-(p+q)r\}]\mu(LD) \quad \text{when } r \text{ sufficiently small}.\end{align}

When p, q, and r are sufficiently small and the inequality in (3) holds, we have

\begin{align}{}&2(p+q)-(p+q)^{2}-3r+4r^{2}+(p+q)r\nonumber\\\geqslant{}& 4r-6r^{2}-3pr-6qr-3r+4r^{2}+(p+q)r=r(1-2p-5q-2r) \geqslant 0,\nonumber\end{align}

so that the coefficient of $\mu(LD)$ in (64) is nonpositive and, consequently, the coefficient of $\mu(LD)$ in (63), is nonpositive. Incorporating (63) into (62), we obtain

(65) \begin{align}w_{0}(\widehat{G}_{p,q,r}\mu)\leqslant{}&w_{0}(\mu)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)-\{1-(1+p)(1-p-q)\nonumber\\&\times (1-r)(1+r)\}\mu(DD)-[1-\{1-p-q+4r-qr(2-p-2r)-r^{2}(2-3p)\nonumber\\&-r^{3}(1-p-q)-p^{2}r^{2}-pqr^{2}\}(1-p-q)(1-r)]\mu(LD)-r(1-p-q)^{2}(1-r)^{2}\nonumber\\&\times \mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)-\{1-(1-p-q)^{2}(1-r)^{2}\}\mu(LWD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)+\sum_{i=2}^{9}A_{i}\nonumber\\\leqslant{}&w_{0}(\mu)-2\{1-(1+p)(1-p-q)(1-r)\}\mu(WD)\nonumber\\&\underbrace{-\{1-(1+p)(1-p-q)(1-r)(1+r)\}\mu(DD)}-[1-\{1-p-q+4r\nonumber\\&-qr(2-p-2r)-r^{2}(2-3p)-r^{3}(1-p-q)-p^{2}r^{2}-pqr^{2}\}\nonumber\\&\times (1-p-q)(1-r)]\mu(LD)\nonumber\\&-r(1-p-q)^{2}(1-r)^{2}\mu(LDW)-(1-r)^{2}(1-p-q)^{2}\mu(LDL)\nonumber\\&-\{1-(1-p-q)^{2}(1-r)^{2}\}\mu(LWD)-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(WLD)\nonumber\\&-r^{2}(1-r)^{2}(1-p-q)^{2}\mu(DLD)\nonumber\\&\underbrace{+\{q+(1-p-q)r\}p(1-p-q)(1-r)(1+r)}\nonumber\\&\times\underbrace{\mu(DD)}+\sum_{i=3}^{9}A_{i},\end{align}

where the last step follows from the inequality in (47). Before proceeding further, we observe that

(66) \begin{align}{}&\{(1-p-q)(1-r)(1+r)\}^{-1}=\{1-p-q-r^{2}+pr^{2}+qr^{2}\}^{-1}\nonumber\\={}&\sum_{j=0}^{\infty}(p+q+r^{2}-pr^{2}-qr^{2})^{\,j}\nonumber\\\geqslant{}&1+p+q+r^{2}-pr^{2}-qr^{2}+(p+q+r^{2}-pr^{2}-qr^{2})^{2}\nonumber\\={}&1+p+q+r^{2}+p^{2}+q^{2}+2pq+p^{2}r^{4}-2p^{2}r^{2}+2pqr^{4}-4pqr^{2}-2pr^{4}+pr^{2}+q^{2}r^{4}\nonumber\\&-2q^{2}r^{2}-2qr^{4}+qr^{2}+r^{4}\end{align}