# The Role of Visibility in Pursuit/Evasion Games

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

- We use the following notations for sets:$\phantom{\rule{4pt}{0ex}}\mathbb{N}$ denotes $\left\{1,2,\dots \right\}$; ${\mathbb{N}}_{0}$ denotes $\left\{0,1,2,\dots \right\}$; $\left[K\right]$ denotes $\left\{1,\dots ,K\right\}$; $A-B=\left\{x:x\in A,x\notin B\right\}$; $\left|A\right|$ denotes the cardinality of A (i.e., the number of its elements).
- A graph $G=(V,E)$ consists of a node set V and an edge set E, where every $e\in E$ has the form $e=\left\{x,y\right\}\subseteq V$. In other words, we are concerned with finite, undirected, simple graphs; in addition we will always assume that G is connected and that G contains n nodes: $\left|V\right|=n$. Furthermore, we will assume, without loss of generality, that the node set is $V=\left\{1,2,\dots ,n\right\}$. We let ${V}^{K}=\underset{K\phantom{\rule{4.pt}{0ex}}\text{times}}{\underbrace{V\times V\times \dots \times V}}$. We also define ${V}_{D}^{2}\subseteq {V}^{2}$ by ${V}_{D}^{2}=\{(x,x):x\in V\}$ (it is the set of “diagonal” node pairs).
- A directed graph (digraph) $G=(V,E)$ consists of a node set V and an edge set E, where every $e\in E$ has the form $e=\left(x,y\right)\in V\times V$. In other words, the edges of a digraph are ordered pairs.
- In graphs, the (open) neighborhood of some $x\in V$ is $N\left(x\right)=\left\{y:\left\{x,y\right\}\in E\right\}$; in digraphs it is $N\left(x\right)=\left\{y:\left(x,y\right)\in E\right\}$. In both cases, the closed neighborhood of $x$ is $N\left[x\right]=N\left(x\right)\cup \left\{x\right\}$.
- Given a graph $G=\left(V,E\right)$, its line graph $L\left(G\right)=\left({V}^{\prime},{E}^{\prime}\right)$ is defined as follows: the node set is ${V}^{\prime}=E$, i.e., it has one node for every edge of G; the edge set is defined by having the nodes $\left\{u,v\right\},\left\{x,y\right\}\in {V}^{\prime}$ connected by an edge $\left\{\left\{u,v\right\},\left\{x,y\right\}\right\}$ if and only if $\left|\left\{u,v\right\}\cap \left\{x,y\right\}\right|=1$ (i.e., if the original edges of G are adjacent).
- We will write $f\left(n\right)=o\left(g\left(n\right)\right)$ if and only if ${lim}_{n\to \infty}\frac{f\left(n\right)}{g\left(n\right)}=0$. Note that in this asymptotic notation n denotes the parameter with respect to which asymptotics are considered. So in later sections we will write $o\left(n\right)$, $o\left(M\right)$ etc.

#### 2.2. The CR Game Family

Adversarial Visible Robber | av-CR |

Adversarial Invisible Robber | ai-CR |

Drunk Visible Robber | dv-CR |

Drunk Invisible Robber | di-CR |

## 3. Cop Number and Capture Time

#### 3.1. The Node av-CR Game

**Theorem 1.**Let $\left({\overline{V}}_{0},{\overline{V}}_{1},\overline{E},\overline{F}\right)$ be a reachability game on the digraph $\overline{D}=\left(\overline{V},\overline{E}\right)$. Then $\overline{V}$ can be partitioned into two sets ${\overline{W}}_{0}$ and ${\overline{W}}_{1}$ such that (for $i\in \left\{0,1\right\}$) player i has a memoryless strategy ${\sigma}_{i}$ which is winning whenever the game starts in $u\in {\overline{W}}_{i}$.

- Nodes of the form $u=\left(x,y,p\right)$ correspond to positions (in the original CR game) with the cops located at $x\in {V}^{K}$, the robber at $y\in V$ and player $p\in \left\{C,R\right\}$ being next to move.
- There is single node $u=\left(\lambda ,\lambda ,C\right)$ which corresponds to the starting position of the game: neither the cops nor the robber have been placed on G; it is C’s turn to move (recall that λ denotes the empty sequence).
- Finally, there exist n nodes of the form $u=\left(x,\lambda ,R\right)$: the cops have just been placed in the graph (at positions $x\in {V}^{K}$) but the robber has not been placed yet; it is R’s turn to move.

**Definition 1.**The cop number of G is

**Theorem 2.**For every G, let $K\ge c\left(G\right)$ and consider the CR game played on G with K cops. There exists a a memoryless cop winning strategy ${\sigma}_{C}$ and a number $\overline{T}\left(K;G\right)<\infty $ such that, for every robber strategy ${s}_{R}$, C wins in no more than $\overline{T}\left(K;G\right)$ rounds.

**Definition 2.**Given a graph G, some $K\in \mathbb{N}$ and strategies ${s}_{C}\in {\mathbf{S}}_{C}^{\left(K\right)}$, ${s}_{R}\in {\mathbf{S}}_{R}^{\left(K\right)}$ the av-CR capture time is defined by

**Definition 3.**For every graph G and $K\in \mathbb{N}$, the strategies ${s}_{C}^{\left(K\right)}\in {\mathbf{S}}_{C}^{\left(K\right)}$ and ${s}_{R}^{\left(K\right)}\in {\mathbf{S}}_{R}^{\left(K\right)}$ are a pair of optimal strategies if and only if

**Theorem 3.**Given any graph G and any $K\ge c\left(G\right)$, for the av-CR game there exists a pair $\left({\sigma}_{C}^{\left(K\right)},{\sigma}_{R}^{\left(K\right)}\right)\in {\tilde{\mathbf{S}}}_{C}^{\left(K\right)}\times {\tilde{\mathbf{S}}}_{R}^{\left(K\right)}$ of memoryless time optimal strategies such that

**Definition 4.**The adversarial visible capture time of G is

#### 3.2. The Node dv-CR Game

**Theorem 4.**Given any graph G and $K\in \mathbb{N}$, for the dv-CR game played on G with K cops there exists a memoryless strategy ${\sigma}_{C}^{\left(K\right)}\in {\tilde{\mathbf{S}}}_{C}^{\left(K\right)}$ such that

**Definition 5.**The drunk visible capture time of G is

#### 3.3. The Node ai-CR Game

**Theorem 5.**On any graph G let ${\overline{s}}_{C}^{\left(K\right)}$ denote the strategy in which K cops random-walk on G. Then

**Theorem 6.**Given any graph G and $K\ge c\left(G\right)$, the ai-CR game played on G with K cops has a value ${v}^{\left(K\right)}$ which satisfies

**Definition 6.**The adversarial invisible capture time of G is

#### 3.4. The Node di-CR Game

**Theorem 7.**Given any graph G and $K\in \mathbb{N}$, for the di-CR game played on G with K cops there exists a strategy ${s}_{C}^{\left(K\right)}\in {\overline{\mathbf{S}}}_{C}^{\left(K\right)}$ such that

**Definition 7.**The drunk invisible capture time of G is

#### 3.5. The Edge CR Games

## 4. The Cost of Visibility

#### 4.1. Cost of Visibility in the Node CR Games

**Definition 8.**For every G, the adversarial cost of visibility is ${H}_{a}\left(G\right)=\frac{c{t}_{i}\left(G\right)}{ct\left(G\right)}$ and the drunk cost of visibility is ${H}_{d}\left(G\right)=\frac{dc{t}_{i}\left(G\right)}{dct\left(G\right)}$.

**Theorem 8.**For every $N\in \mathbb{N}$ we have ${H}_{a}\left({S}_{N,1}\right)=N$.

**Proof.**$\underset{\xaf}{\mathbf{(i)}\mathbf{Computing}\mathit{ct}\left({\mathit{S}}_{\mathit{N},\mathbf{1}}\right)}$. In av-CR, for every $N\in \mathbb{N}$ we have $\mathit{ct}\left({S}_{N,1}\right)=1$: the cop starts at ${X}_{0}=0$, the robber starts at some ${Y}_{0}=u\ne 0$ and, at $t=1$, he is captured by the cop moving into $u$; i.e., $\mathit{ct}\left({S}_{N,1}\right)\le 1$; on the other hand, since there are at least two vertices ($N\ge 1$), clearly $\mathit{ct}\left({S}_{N,1}\right)\ge 1$.

**Theorem 9.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have

**Proof.**$\underset{\xaf}{\mathbf{(i)}\mathbf{Computing}\mathit{dct}\left({\mathit{S}}_{\mathit{N},\mathit{M}}\right)}$. We will first show that, for any $N\in \mathbb{N}$, we have $\mathit{dct}\left({S}_{N,M}\right)=\left(1+o\left(1\right)\right)\frac{M}{2}$ (recall that the parameter $N$ is a fixed constant whereas $M\to \infty $.) Suppose that the cop starts on the $i$-th ray, at distance $(1+o(1\left)\right)\mathit{cM}$ from the center (for some constant $c\in [0,1]$). The robber starts at a random vertex. It follows that for any $j$ such that $1\le j\le N$, the robber starts on the $j$-th ray with probability $(1+o(1\left)\right)/N$. It is a straightforward application of Chernoff bounds to show that with probability $1+o\left(1\right)$ the robber will not move by more than $o\left(M\right)$ in the next $O\left(\mathit{MN}\right)=O\left(M\right)$ steps, which suffice to finish the game. This is so because, if $X$ has a binomial distribution $\mathit{Bin}(n,p)$, then $\mathit{Pr}\left(\right|X-\mathit{np}|\ge \u03f5\mathit{np})\le 2exp(-{\u03f5}^{2}\mathit{np}/3)$ for any $\u03f5\le 3/2$. Now suppose the robber starts at distance $\omega \left({M}^{2/3}\right)$ from the center. During $N=O\left(M\right)$ steps the robber makes in expectation $N/2$ steps towards the center, and $N/2$ steps towards the end of the ray. The probability to make during $N$ steps more than $N/2+{M}^{2/3}$ steps towards the center, say, is thus at most ${e}^{-{\mathit{cM}}^{1/3}}$, and the same holds also by taking a union bound over all $O\left(M\right)$ steps. Hence, with probability at least $1-{e}^{-{\mathit{cM}}^{1/3}}$ he will throughout $O\left(M\right)$ steps remain at distance $O\left({M}^{2/3}\right)$ from his initial position. In short, the expected capture time is easy to calculate.

- With probability $(1-c+o(1\left)\right)/N$, the robber starts on the same ray as the cop but farther away from the center. Conditioning on this event, the expected capture time is $M(1-c+o(1\left)\right)/2$.
- With probability $(c+o(1\left)\right)/N$, the robber starts on the same ray as the cop but closer to the center. Conditioning on this event, the expected capture time is $M(c+o(1\left)\right)/2$.
- With probability $(N-1+o(1\left)\right)/N$, the robber starts on different ray than the cop. Conditioning on this event, the expected capture time is $(c+o(1\left)\right)M+M(1/2+o(1\left)\right)$.

**(ii.1)**Suppose C starts at the end of one ray (chosen arbitrarily), goes to the center, and then successively checks the remaining rays without repetition, with probability at least $1-O\left({M}^{-1/3}\right)$, the robber will be caught. If the robber is caught (this implies that the robber did not switch rays), the capture time is calculated as follows:

- With probability $(1+o(1\left)\right)/N$, the robber starts on the same ray as the cop. Conditioning on this event, the expected capture time is $(1+o(1\left)\right)M/2$.
- With probability $(1+o(1\left)\right)/N$, the robber starts on the $j$-th ray visited by the cop. Conditioning on this event, the expected capture time is $(1+o(1\left)\right)(M+2M(j-2)+M/2)$. ($M$ steps are required to move from the end of the first ray to the center, $2M$ steps are `wasted’ to check $j-2$ rays, and $M/2$ steps are needed to catch the robber on the $j$-th ray, on expectation.)

**(ii.2)**Now suppose $C$ starts at the center of the ray, rather than the end, and checks all rays from there. By the same arguments as before, the capture time is

**(ii.3)**Similarly, suppose the cop starts at distance $\mathit{cM}$ from the center, for some $c\in [0,1]$. If he first goes to the center of the ray, and then checks all rays (suppose the one he came from is the last to be checked), then the capture time is

#### 4.2. Cost of Visibility in the Edge CR Games

**Definition 9.**For every G, the edge adversarial cost of visibility is ${\overline{H}}_{a}\left(G\right)=\frac{{\overline{ct}}_{i}\left(G\right)}{\overline{ct}\left(G\right)}$ and the edge drunk cost of visibility is defined as ${\overline{H}}_{d}\left(G\right)=\frac{{\overline{dct}}_{i}\left(G\right)}{\overline{dct}\left(G\right)}$.

**Theorem 10.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have ${\overline{H}}_{a}\left({S}_{N,1}\right)=N-1$.

**Proof.**We have ${\overline{H}}_{a}\left({S}_{N,1}\right)=\frac{{\overline{ct}}_{i}\left({S}_{N,1}\right)}{\overline{ct}\left({S}_{N,1}\right)}=\frac{c{t}_{i}\left({K}_{N}\right)}{ct\left({K}_{N}\right)}$ and, since $N\ge 2$, clearly $ct\left({K}_{N}\right)=1$. Let us now compute $c{t}_{i}\left({K}_{N}\right)$.

**Theorem 11.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have ${\overline{H}}_{d}\left({S}_{N,1}\right.)=\frac{N(N-1)}{2N-3}$.

**Proof.**This is quite similar to the adversarial case. We have ${\overline{H}}_{d}\left({S}_{N,1}\right)=\frac{{\overline{dct}}_{i}\left({S}_{N,1}\right)}{\overline{dct}\left({S}_{N,1}\right)}=\frac{dc{t}_{i}\left({K}_{N}\right)}{dct\left({K}_{N}\right)}$. Clearly we have $dct\left({K}_{N}\right)=1-1/N$ (with probability $1/N$ the robber selects the same vertex to start with as the cop and is caught before the game actually starts; otherwise is caught in the first round).

## 5. Algorithms for COV Computation

#### 5.1. Algorithms for Visible Robbers

#### 5.1.1. Algorithm for Adversarial Robber

- $C\left(x,y\right)$, the optimal game duration when the cop/robber configuration is $(x,y)$ and it is C’s turn to play;
- $R\left(x,y\right)$, the optimal game duration when the cop/robber configuration is $(x,y)$ and it is R’s turn to play.

Algorithm 1: Cops Against Adversarial Robber (CAAR) |

Input: $G=(V,E)$ |

01 For All $\left(x,y\right)\in {V}_{D}^{2}$ |

02 ${C}^{\left(0\right)}\left(x,y\right)=0$ |

03 ${R}^{\left(0\right)}\left(x,y\right)=0$ |

04 EndFor |

05 For All $\left(x,y\right)\in {V}^{2}-{V}_{D}^{2}$ |

06 ${C}^{\left(0\right)}\left(x,y\right)=\infty $ |

07 ${R}^{\left(0\right)}\left(x,y\right)=\infty $ |

08 EndFor |

09 $i=1$ |

10 While $1>0$ |

11 For All $\left(x,y\right)\in {V}^{2}-{V}_{D}^{2}$ |

12 ${C}^{\left(i\right)}\left(x,y\right)=1+{min}_{{x}^{\prime}\in N\left[x\right]}{R}^{\left(i-1\right)}\left({x}^{\prime},y\right)$ |

13 ${R}^{\left(i\right)}\left(x,y\right)=1+{max}_{{y}^{\prime}\in N\left[y\right]}{C}^{\left(i\right)}\left(x,{y}^{\prime}\right)$ |

14 EndFor |

15 If ${C}^{\left(i\right)}={C}^{\left(i-1\right)}$ And $\phantom{\rule{4pt}{0ex}}{R}^{\left(i\right)}={R}^{\left(i-1\right)}$ |

16 Break |

17 EndIf |

18 $i\leftarrow i+1$ |

19 EndWhile |

20 $C={C}^{\left(i\right)}$ |

21 $R={R}^{\left(i\right)}$ |

Output: C, R |

#### 5.1.2. Algorithm for Drunk Robber

Algorithm 2: Cops Against Drunk Robber (CADR) |

Input: $G=(V,E)$, ε |

01 For All $\left(x,y\right)\in {V}_{D}^{2}$ |

02 $\phantom{\rule{4pt}{0ex}}{C}^{\left(0\right)}\left(x,y\right)=0$ |

03 EndFor |

04 For All $\left(x,y\right)\in V-{V}_{D}^{2}$ |

05 $\phantom{\rule{4pt}{0ex}}{C}^{\left(0\right)}\left(x,y\right)=\infty $ |

06 EndFor |

07 $i=1$ |

08 While $1>0$ |

09 For All $\left(x,y\right)\in V-{V}_{D}^{2}$ |

10 ${C}^{\left(i\right)}\left(x,y\right)=1+{min}_{{x}^{\prime}\in N\left[x\right]}{\sum}_{{y}^{\prime}\in V}P\left(\left({x}^{\prime},y\right)\to \left({x}^{\prime},{y}^{\prime}\right)\right){C}^{\left(i-1\right)}\left({x}^{\prime},{y}^{\prime}\right)$ |

11 EndFor |

12 If ${max}_{\left(x,y\right)\in {V}^{2}}\left|{C}^{\left(i\right)}\left(x,y\right)-{C}^{\left(i-1\right)}\left(x,y\right)\right|<\epsilon $ |

13 Break |

14 EndIf |

15 $i\leftarrow i+1$ |

16 EndWhile |

17 $C={C}^{\left(i\right)}$ |

Output: C |

#### 5.2. Algorithms for Invisible Robbers

#### 5.2.1. Algorithms for Adversarial Robber

#### 5.2.2. Algorithm for Drunk Robber

Algorithm 3: Pruned Cop Search (PCS) |

Input: $G=(V,E)$, ${x}_{0}$, ${J}_{max}$, ε |

01 $t=0$ |

02 $S.\mathbf{x}={x}_{0}$, $S.\mathbf{p}=Pr\left({y}_{0}\right|{x}_{0})$, $S.C=0$ |

03 $\mathbf{S}=\left\{S\right\}$ |

04 ${C}_{best}^{old}=0$ |

05 While $1>0$ |

06 $\tilde{\mathbf{S}}=\varnothing $ |

07 For All $S\in \mathbf{S}$ |

08 $\mathbf{x}=S.\mathbf{x}$, $\mathbf{p}=S.\mathbf{p}$, $C=S.C$ |

09 For All $v\in N\left[{x}_{t}\right]$ |

10 ${\mathbf{x}}^{\prime}=\mathbf{x}\&v$ |

11 ${\mathbf{p}}^{\prime}=\mathbf{p}\xb7P\left(v\right)$ |

12 ${C}^{\prime}=\mathbf{Cost}({\mathbf{x}}^{\prime},{\mathbf{p}}^{\prime},C)$ |

13 ${S}^{\prime}.\mathbf{x}={\mathbf{x}}^{\prime}$, ${S}^{\prime}.\mathbf{p}={\mathbf{p}}^{\prime}$, ${S}^{\prime}.C={C}^{\prime}$ |

14 $\tilde{\mathbf{S}}=\tilde{\mathbf{S}}\cup \left\{{S}^{\prime}\right\}$ |

15 EndFor |

16 EndFor |

17 $\mathbf{S}=\mathbf{Prune}(\tilde{\mathbf{S}},{J}_{max})$ |

18 $[{\mathbf{x}}_{best},{C}_{best}]=\mathbf{Best}\left(\mathbf{S}\right)$ |

19 If $|{C}_{best}-{C}_{best}^{old}|<\epsilon $ |

20 Break |

21 Else |

22 ${C}_{best}^{old}={C}_{best}$ |

23 $t\leftarrow t+1$ |

24 EndIf |

25 EndWhile |

Output: ${\mathbf{x}}_{best}$, ${C}_{best}=C\left({\mathbf{x}}_{best}\right)$. |

**Prune**which computes “best” in terms of smallest $C\left(\mathbf{x}\right)$). Finally, the subroutine $\mathbf{Best}$ in line 18 computes the overall smallest expected capture time ${C}_{best}=C\left({\mathbf{x}}_{best}\right)$. The procedure is repeated until the termination criterion $|{C}_{best}-{C}_{best}^{old}|<\epsilon $ is satisfied. As explained above, the criterion is expected to be always eventually satisfied because of Equation (20).

## 6. Experimental Estimation of the Cost of Visibility

#### 6.1. Experiments with Node Games

**Figure 4.**$dct\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 5.**$dc{t}_{i}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 6.**${H}_{d}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

#### 6.2. Experiments with Edge Games

**Figure 8.**$\overline{dct}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 9.**${\overline{dct}}_{i}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 10.**${\overline{H}}_{d}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Kehagias, A.; Mitsche, D.; Prałat, P. The Role of Visibility in Pursuit/Evasion Games. *Robotics* **2014**, *3*, 371-399.
https://doi.org/10.3390/robotics3040371

**AMA Style**

Kehagias A, Mitsche D, Prałat P. The Role of Visibility in Pursuit/Evasion Games. *Robotics*. 2014; 3(4):371-399.
https://doi.org/10.3390/robotics3040371

**Chicago/Turabian Style**

Kehagias, Athanasios, Dieter Mitsche, and Paweł Prałat. 2014. "The Role of Visibility in Pursuit/Evasion Games" *Robotics* 3, no. 4: 371-399.
https://doi.org/10.3390/robotics3040371