Competitive target search with multi-agent teams: symmetric and asymmetric communication constraints

Abstract

We study a search game in which two multi-agent teams compete to find a stationary target at an unknown location. Each team plays a mixed strategy over the set of search sweep-patterns allowed from its respective random starting locations. Assuming that communication enables cooperation we find closed-form expressions for the probability of winning the game as a function of team sizes and the existence or absence of communication within each team. Assuming the target is distributed uniformly at random, an optimal mixed strategy equalizes the expected first-visit time to all points within the search space. The benefits of communication enabled cooperation increase with team size. Simulations and experiments agree well with analytical results.

Appendices

A Details of proofs leading up to Lemma 1

We now present technical details leading up to the proof of Lemma 1.

Given particular multipath strategies $\psi _{G}$ and $\psi _{A}$ for our team and the adversary, respectively, we can compute the ratio of space our team visits first by integrating the normalized rate new (to anybody) territory is swept by our team:

$$\begin{aligned} \frac{\mathscr {L}_{D}(X_{\mathrm {win}}(\psi _{G},\psi _{A}))}{\mathscr {L}_{D}(X)} = \int _{0}^{t_{\mathrm {final}}}\frac{d}{dt}\!\left[ \frac{\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{\mathscr {L}_{D}(X)}\right] dt,\end{aligned}$$

where ${t_{\mathrm {final}}= \min (\frac{1}{n c^*} , \frac{1}{m c^*})}$. Thus, Eq. 1 can be reformulated for the Nash equilibrium of an ideal game with cooperation within both teams as:

where integrals are Lebesgue. Using the independence of the two team’s optimal mixed strategies, i.e., $\mathcal {D}_{\mathbf {g}_{0}}(\psi _{G})$ and $\mathcal {D}_{\mathbf {a}_{0}}(\psi _{A})$ for all $\mathbf {g}_{0}$ and $\mathbf {a}_{0}$ yields:

We observe that the quantity inside the outermost integral describes the expected value of ${\frac{d}{dt}[\frac{\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{\mathscr {L}_{D}(X)}]}$ over all $\mathcal {S}_G$, $\mathcal {S}_A$, $\varPsi ^*_{G}$, and $\varPsi ^*_{A}$. Recall that the expected value of ‘$\cdot$’ over all $\mathcal {S}_G$, $\mathcal {S}_A$, $\varPsi ^*_{G}$, and $\varPsi ^*_{A}$ as ${\mathbb {E^{*}}\!\left[ \cdot \right] \equiv \mathbb {E}_{\mathcal {S}_G, \mathcal {S}_A, \varPsi ^*_{G}, \varPsi ^*_{A}}\left[ \cdot \right] }.$ Thus, formally,

and

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = \int _0^{t_{\mathrm {final}}} \mathbb {E^{*}}\!\left[ \frac{d}{dt} \frac{\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{\mathscr {L}_{D}(X)} \right] dt\end{aligned}$$

Lemma 1

Assuming optimal ideal mixed strategies exist and both teams play an optimal ideal mixed strategy,

$$\begin{aligned} \mathbb {E^{*}}\!\left[ \frac{d}{dt} \frac{\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{\mathscr {L}_{D}(X)} \right] = \left( 1 - tm c^*\right) n c^*. \end{aligned}$$

Proof

At time $t$ the adversary (operating according to its own ideal optimal strategy) has swept $tm v\frac{\mathscr {L}_{D-1}(B_r)}{\mathscr {L}_{D}(X)}$ portion of the entire search space. The interplay between the mixed ideal optimal strategies for each team forces the expected instantaneous overlap between teams to be uncorrelated. Thus, for all $t\in [0, t_{\mathrm {final}}]$, the instantaneous expected rate team $G$ sweeps $\mathscr {L}_{D}(X_{\mathrm {new}})$ is discounted by a factor of ${1 - tm v\frac{\mathscr {L}_{D-1}(B_r)}{\mathscr {L}_{D}(X)}}$ versus $\frac{d^*}{dt} \frac{\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{\mathscr {L}_{D}(X)}$.

Substitution with Proposition 2 and Corollary 1 yields the desired result.$\square$

B Details of analysis leading to theorem 1

We now present details of our analysis leading up to and including the proof of Theorem 1.

Proposition 5

Given a uniform distribution of starting locations for all $a\in A$ and target locations $q$, and assuming an ideal mixed strategy is played by $A$, then the distribution of times at which that target-sweeping adversary $a_j$ sweeps the target is uniform on the interval $[0, 1/(m c^*)]$.

Let $t_{A}$ be a realization of a uniformly random sample from $[0, 1/(m c^*)]$. Given our assumptions:

$$\begin{aligned} {\mathbb {P}\left( t_{A} < t\right) = \frac{t}{1/(m c^*)}}. \end{aligned}$$

where ${0 \le t\le 1/(m c^*)}$. Note that the probability density of $t_{A}$ on $[0, 1/(m c^*)]$ is $m c^*$, formally:

(6)

Team $G$ does not communicate and can be viewed as a confederation of n independent single-agent sub-teams ${\{g_1\}, \ldots \{g_n\}}$. Each subteam $\{g_i\}$ plays a single agent ideal mixed strategy such that $g_i$ sweeps (new to $g_i$) space at the rate $c^*$ and so $g_i$ requires $1/c^*$ time to sweep the entire space by itself. This leads to the single-agent team counterpart to Proposition 5:

Proposition 6

Given a uniform distribution of starting locations for $g_i$ and target locations $q$, and assuming a single agent ideal mixed strategy is played by $\{g_i\}$, then the distribution of times at which agent $g_i$ sweeps the target is evenly distributed on the interval $[0, 1/c^*]$.

Let $t_{g,i}$ be a realization of a uniformly random sample from $[0, 1/ c^*]$. Given our assumptions on start locations, and assuming a single-agent ideal mixed strategy is played by $\{g_i\}$, the probability $g_i$ sweeps the target before time $t$ is:

$$\begin{aligned} {\mathbb {P}\left( t_{g,i} < t\right) = \frac{t}{1/c^*}} \end{aligned}$$

where ${0 \le t\le 1/c^*}$.

The target is swept by team $G$ as soon as it is swept by any agent ${g_i \in G}$, thus team $G$ essentially gets to draw n i.i.d. uniformly random samples ${t_{g,1}, \ldots t_{g,n}}$ from $[0, 1/c^*]$ and play the best (smallest) of these versus the adversary team $A$’s single draw from $[0, 1/(m c^*)]$. Let ${t_{G} = \min _{i}(t_{g,i})}$ denote a realization of the smallest out of n values sampled uniformly at random and i.i.d. from ${[0, 1/c^*]}$.

The distribution of $t_{G}$ can be determined directly using order statistics. The probability that at least one of the n (uncommunicating) subteams ${\{g_1\}, \ldots \{g_n\}}$ sweeps the target before time $t$ is:

$$\begin{aligned} {\mathbb {P}\left( t_{G} < t\right) = 1 - \left( 1 - \frac{t}{1/c^*}\right) ^n} \end{aligned}$$

(7)

Team $G$ wins whenever ${t_{G} < t_{A}}$. We observe that $\mathbb {P}\left( \omega _{\mathrm {tie}}^*\right) = \mathbb {P}\left( t_{G} =t_{A}\right) = 0$ and given our assumptions:

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = \mathbb {P}\left( t_{G} < t_{A}\right) = 1 - \mathbb {P}\left( t_{G} > t_{A}\right) = 1 - \mathbb {P}\left( \omega _{\mathrm {lose}}^*\right) . \end{aligned}$$

Theorem 1

The probability team $G$ wins an ideal game, assuming team $G$ cannot communicate but the adversary team $A$ can communicate, and the adversary team $A$ plays optimal ideal mixed strategies, while each ${\{g_i\}\subset G}$ plays an optimal ideal single agent mixed strategy, is:

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = \frac{ \left( 1-\frac{1}{m}\right) ^n (m-1) - m + n + 1 }{n + 1} \end{aligned}$$

Proof

We can compute $\mathbb {P}\left( \omega _{\mathrm {win}}^*\right)$ using the Law of Total Probability:

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = \int _{-\infty }^{\infty } \mathbb {P}\left( t_{G} < t_{A} | t_{A} = t\right) f_{t}(t) d t. \end{aligned}$$

We note that $f_{t}(t) = 0$ for all ${t< 0}$ and for all ${t> 1/(m c^*)}$ because the game has not started yet and the adversary will have already won, respectively. Substituting Eqs. 6 and 7 we get:

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = \int _{0}^{1/(m c^*)}\left( 1 - \left( 1 - \frac{t_{A}}{1/c^*}\right) ^n \right) mc^* d t\end{aligned}$$

and performing the integration yields:

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}^*\right) = 1 + \frac{ m \left( \frac{m-1}{m}\right) ^{n+1}}{(n+1)} - \frac{m}{n + 1}. \end{aligned}$$

The final result is obtained using algebra. $\square$

C Details of proofs pertaining to non-ideal cases

In Sects. C.1, C.2 and C.3 we bound the individual non-ideal effects of type I, II, and III, respectively that were discussed in Sect. 6. The each effect is analyzed assuming the others can be ignored. The general case where I, II, and III simultaneously occur can be found by combining these results. Here we focus on the case where both teams can communicate. Bounds on cases where one or the other team cannot communicate can be found by extending these results as was done for the ideal case⁷ in Sect. 4.

Boundary effects caused by non-ideal startup locations and environmental geometry both cause the team to perform more than the ideal amount of sweeping (see Fig. 5). We use the same basic proof technique for their analysis in Sects. C.1 and C.2.

The analytical technique used in Sects. C.1 and C.2 relies on the fact that search-hindering non-ideal effects are expected to be increasingly detrimental to $G$’s probability of winning the game the earlier in the game they occur. This happens because the ratio $\frac{d \mathscr {L}_{D}(X_{\mathrm {swept}})}{d \mathscr {L}_{D}(X_{\mathrm {new}})}$ decreases versus time. It is possible to construct a scenario that is even worse than a worst-case non-ideal search, in term’s of team $G$’s probability of winning the game, by: (first) assuming that all negative ramifications of a non-ideal search happen at the beginning of the search for team $G$, instead of whenever they actually occur; (second) Assuming the adversary team $A$ sweeps at the ideal-game rate for the entire game. The relative length of the non-ideal startup phase in this modified scenario is guaranteed to be worse than the worst-case,⁸ and is bounded by a dimensionally dependent constant length of time.

A road-map of our analytical technique, including the construction of the worse than worst-case scenario we use, is now presented:

• 1. We break the space $X$ into two non-overlapping sets, $X_{ideal}$ and $X_{startup}$, depending on if the sweep search through it is “ideal” (it is swept at a time when the team is sweeping new space at the maximum rate).

  • (a) Let all search space that is not reswept be combined in set $X_{ideal}$.

  • (b) Let all search space that is ever reswept (plus, when relevant, all non-search space that is swept) be combined in set $X_{startup}$. Multiple copies of each reswept portion of space are included in $X_{startup}$—if a subset of space is swept i different times, then i different copies of that subset are included. For the following discussion each copy is considered distinct such that each contributes its own volume to $\mathscr {L}_{D}(X_{startup})$. Thus, by construction, ${\mathscr {L}_{D}(X_{startup}) + \mathscr {L}_{D}(X_{ideal}) \ge \mathscr {L}_{D}(X)}$.

• 2. We consider (a worse than worst-case scenario) where the time costs, but not the target detection benefits, of sweeping $X_{startup}$ are incurred prior to performing an ideal search through $X_{ideal}$.

  • (a) We design a multipath that is guaranteed to sweep $X_{startup}$, including all duplicate copies of reswept space, and derive an upper bound $t_{\mathrm {startup}}$ on the time required for the team to travel this multipath.

  • (b) We assume team $G$ begins the game by moving along a path sufficient to sweep $X_{startup}$—but not actually performing search as it moves along this path (e.g., with its detection sensor turned off). For each duration of non-ideal behavior that occurs in a normal scenario, this essentially shifts an equivalent duration of non-ideal behavior to the beginning of the search without providing any target detection benefits.

  • (c) However, after accounting for the penalty we receive for not searching before $t_{\mathrm {startup}}$, we assume that search through $X_{ideal}$ happens at the ideal rate. In other words, for the purposes of deriving performance bounds, we essentially pay an up-front performance penalty “with interest” to move each piece of non-ideal search such that it happens before $t_{\mathrm {startup}}$.

  • (d) Finally, we account for actually searching $X_{startup}$ (since we moved through it without searching before $t_{\mathrm {startup}}$).

    • i. Our original search would have already completed by this point; thus, we can assume that any rate of sweep for the second pass over unique elements of $X_{startup}$ and our scenario will still be worse than the original (and provide a valid performance bound).

    • ii. Thus, it is permissible to assume the ideal search rate in this phase (for convenience) without destroying the worse than worst-case bound.

We now apply this technique in Sects. C.1 and C.2.

1.1 C.1 Non-ideal starting locations

If two robots on the same team start closer than $2r$, then either some nonzero measure subset of space will be swept by both of them (see Fig. 5left), or some nonzero measure subset of space will be swept by them and during some other point in the search (see Fig. 5center). Such an event occurs with nonzero probability given an assumption of uniform random i.i.d. start locations (Fig. 12).

Fig. 12

Intuition for the worse than worst-case bound: an example in which ${|G| = 5}$ and ${|A| = 5}$. a Ideal sweep rates. b Sweep rates in a non-ideal case in which $G$ has three periods of non-ideal sweep ($\tau _{1}$, $\tau _{2}$, $\tau _{3}$) and $A$ has two ($\tau _{4}$, $\tau _{5}$). c The corresponding worse than worst-case bound, in which periods of non-ideal sweep have been moved to the beginning of search for $G$ and eliminated for $A$. d Territory capture rate for both teams in all three cases. e Total area swept in all three cases; by construction $G$ sweeps less space in the bounding case than in the non-ideal case and $A$ sweeps more (shaded regions and arrows show difference). f Total area captured in all three scenarios; by construction $G$ captures less space in the bounding case than in the non-ideal case and $A$ captures more (shaded regions and arrows show difference). Games end once all territory is captured (green) (Color figure online)

Fig. 13

An ideal strategy could be played if all robots start at ${\mathbf {g}_{0}^* = (x_{1,0}^*, \ldots , x_{n,0}^*)}$. Different colors represent area swept by different robots. In the worst-case, all robots all start at the same position, ${\mathbf {g}_{0}= (x_{1,0}, \ldots , x_{n,0}) = (x_{1,0}, \ldots , x_{1,0})}$, and movement from $\mathbf {g}_{0}$ to $\mathbf {g}_{0}^*$ requires each robot to move no further than $2 r(n-1)$, which can be accomplished in time $t_{\mathrm {startup}}= 2 r(n-1)/v$ (Color figure online)

The ill-effects of non-ideal start locations can be bounded by considering the following worse than worst-case scenario: all n team members start at exactly the same point ${\mathbf {g}_{0}= (x_{1,0}, \ldots , x_{n,0})}$, and then begin the game by moving (without actually searching) to the closest configuration $\mathbf {g}_{0}^*$ at which type-I effects would not have occurred if the robots had started at $\mathbf {g}_{0}^*$ in the first place (see Fig. 13). We assume that the entire team waits to start searching until all robots have reached their coordinate of $\mathbf {g}_{0}^*$, and that this requires $t_{\mathrm {startup}}$ time.

The probability the adversary wins before $t_{\mathrm {startup}}$ is $\frac{t_{\mathrm {startup}}m v \mathscr {L}_{D-1}(B_r) }{ \mathscr {L}_{D}(X)}$. After, $t_{\mathrm {startup}}$ our expected search rate is that of the ideal case such that $\frac{d\mathscr {L}_{D}(X_{\mathrm {new}}(t))}{dt}$ decreases versus $\frac{d\mathscr {L}_{D}(X_{\mathrm {swept}}(t))}{dt}$ by the usual factor of ${1 - \frac{tm v{\mathscr {L}_{D-1}(B_r)}}{{\mathscr {L}_{D}(X)}}}$ after $t_{\mathrm {startup}}$.

Recalling that ${c^*=v\frac{\mathscr {L}_{D-1}(B_r)}{\mathscr {L}_{D}(X)}}$, the resulting worse than worst-case bound on the probability we win the game is:

$$\begin{aligned} {\mathbb {P}\left( \omega _{\mathrm {win}}\right) \ge \left[ \int _{t_{\mathrm {startup}}}^{\hat{t}_{\mathrm {final}}} \left( 1 - t m c^*\right) n c^* \,\, dt\right] - t_{\mathrm {startup}}m c^*} \end{aligned}$$

(8)

where

$$\begin{aligned} \hat{t}_{\mathrm {final}}= \min \left( t_{\mathrm {startup}}+ \frac{1}{nc^*} , \frac{1}{mc^*}\right) . \end{aligned}$$

As discussed above, this bound accounts only for inefficiencies in search caused by non-ideal search locations.

In toroid $\mathbb {T}^D$ environments the furthest distance that any individual member of the team must move during the startup phase is $2 r(n - 1)$ (see Fig. 13) and so ${t_{\mathrm {startup}}< 2 r(n-1) / v}$.

A similar bound exists for convex subsets of $\mathbb {R}^D$ as long as other boundary effects can be ignored, and ${\mathcal {W}> 2rn}$, where $\mathcal {W}$ is the maximum distance between any two points in $X$ along a geodesic. In other words, $\mathcal {W}$ is the maximum width of the environment. We assume other boundary effects can be ignored, but address them directly in Sects. C.2 and C.3. We now limit our consideration to ${\mathcal {W}> 2rn}$, i.e., wide environments. In thin environments, such that ${\mathcal {W}\ngtr 2rn}$, moving to an ideal start location may require time on the same order as sweeping the environment.

An upper bound on the probability we win is found by swapping the roles played by team $G$ and the adversary. We let ${\tilde{t}_{\mathrm {startup}}< 2 rm / v}$ denote the startup time required for an adversary that must compete with an ideal version of team $G$ (for achieving this other bound), and define

$$\begin{aligned} \tilde{t}_{\mathrm {final}}= \min \left( \tilde{t}_{\mathrm {startup}}+ \frac{1}{mc^*} , \frac{1}{nc^*}\right) . \end{aligned}$$

The preceding discussion leads to the following theorem:

Theorem 3

Assuming that both teams can communicate, and an optimal mixed strategy exists for both teams, and that both teams play an optimal mixed strategy, and that the game is ideal in every sense except for starting locations, the probability we win is bounded as follows:

and

1.2 C.2 Turns and other non-ideal boundary effects

Re-sweeping of previously swept space occurs near the boundary of the search space due to the necessity of turning (Fig. 5right). Moreover, sweeping all of the space within the search space sometimes requires sweeping some portion of space outside the search space (Fig. 5right). Finally, a third boundary effect occurs due to the fact that, in general, the search spaces cannot be covered by an integer number of sweep passes (this also happens in $\mathbb {T}^D$, see Fig. 14). All of these effects can be analyzed simultaneously.

Fig. 14

Two robots perform a nearly ideal search over a $\mathbb {T}^2$ space. Different colors represent the different subspaces swept by either robot. The space cannot be divided into an integer number of non-overlapping wraps around the space and so some space is swept more than once (hashed). The manifold at the center of the reswept space is marked with a dot-dash line

We assume that the team starts at locations that do not suffer from the startup effects discussed in Sect. C.1. We also assume that the projection of the sensor footprint along the direction of movement permits a tiling over $D-1$ space, see Fig. 6 (we investigate the case where this is not true in Sect. C.3).

Given these assumptions, the space swept in any $\mathbb {T}^D$ search space or subset of $\mathbb {R}^D$ can be decomposed into two different non-overlapping subsets: the first ($X_{startup}$) contains all space involved in turning induced resweep as well as all non-search area that is swept, while the second ($X_{ideal}$) contains the remaining search space (that involved an ideal search). An example appears in Fig. 15.

Fig. 15

A nearly ideal two robot search strategy to cover the convex space. Red and blue represent different robots. Space boundary is black, red and blue dashed lines depict the multipath, shades of gray depict search sweeps, light blue and light red depict sweeps that re-sweep some area. The strategy can be broken into an non-ideal part over $X_{startup}$ and an ideal part over $X_{ideal}$ (Color figure online)

Similar to the analysis in Sect. C.1, we derive $t_{\mathrm {startup}}$ an upper bound on the time required to sweep $X_{startup}$ (including any necessary resweeps). And as in Sect. C.1, the worse than worst-case scenario we use for our analysis requires team $G$ to wait for $X_{startup}$ time before beginning an ideal search (during which time the adversary searches at the ideal rate).

Fig. 16

It is possible to design a strategy for sweeping $X_{startup}$, the non-ideal part of a search. a The non ideal part of a search (red, blue, gray) and the search space boundary (black). b The search space boundary is covered with grid cells that have side lengths twice the minimum sensor radius. c Adding additional grids within $r$ of the original grid cells is sufficient to cover $X_{startup}$. d We can design a strategy that sweeps the latter gridded space (there are $N_{cover}$ grid cells) in time $< c_2 6 r N_{cover}\mathscr {L}_{D- 1}(\partial X)$ where $c_2$ depends on the dimensionality of the search space. e, f Radius of the sensor decreases relative to the size of search space the portion of time spent sweeping $X_{startup}$ decreases toward 0 (Color figure online)

The first step to find $t_{\mathrm {startup}}$ is to bound the length of a path that is guaranteed to cover $X_{startup}$ (see Fig. 16). Consider the largest hypercube that is contained within the robot’s sensor footprint and that is aligned with the direction of forward movement. We call this hypercube $\hat{C}_r$. Let the boundary of the search space be denoted $\partial X$. Note, in toroid spaces the only boundary effects are related to the last pass, and so for $\mathbb {T}^D$, $\partial X$ represents the manifold located between the paths taken during the first and last sweeps (See Fig. 14).

Since $\partial X$ is essentially a lower-dimensional manifold embedded in our $D$-dimensional search space, it is possible to cover $\partial X$ with $N_{cover}$ hypercubes (each of dimension $D$), where

$$\begin{aligned} N_{cover}< c_2 r \mathscr {L}_{D-1}(\partial X) \end{aligned}$$

here $c_2$ is a constant that counts the maximum number of tiled hypercubes required to cover a non-axis aligned line segment of length $\sqrt{D}$, and where $\sqrt{D}$ is the maximum distance between any two points in the same unit grid cell. Figure 16a–c depicts such a covering.

It is important to note that $c_2$ is a dimensionally dependent constant. It is possible to construct a tour upper bounded by $\hat{\ell }$ that covers all grid cells within $c_2$ of $\partial X$ (and thus covers $X_{startup}$). We note that $\hat{\ell }$ is also, by design, longer than the cumulative length traveled along the subset of the original multipath involved in the boundary effects we are investigating in this section. We calculate $\hat{\ell }$ by considering a naive tour of the hypercube covering of the search space boundary (see Fig. 16bottom). For each hypercube, it is possible to construct such a covering by traveling a distance that is at most three times a side length (i.e., 6 times $r_c$) as follows: $2r_c$ to reach the center of the nearest face, $2r_c$ to reach the opposite face and thus sweep the entire cube, and $2r_c$ to exit through the center of any other face (in most cases it will be much less than this since multiple cubes can be swept without changing direction). Thus, ${\hat{\ell }< 6N_{cover}r_c}$ and so

$$\begin{aligned} \hat{\ell }< 6 c_2 r N_{cover}\mathscr {L}_{D}(\partial X). \end{aligned}$$

The time required to perform the startup phases is upper bounded by

$$\begin{aligned} t_{\mathrm {startup}}< \hat{\ell }/ v\end{aligned}$$

This bound implicitly assumes a worst-case situation in which all boundary effects must be dealt with by a single agent. Thus, this bound on $t_{\mathrm {startup}}$ may be up to n times too large (i.e., in the best-case boundary problems are divided evenly between agents). We can now proceed as in the previous subsection (using our new definition of $t_{\mathrm {startup}}$ in Eq. 8). This discussion leads to the following theorem:

Theorem 4

Assuming that both teams can communicate, and an optimal mixed strategy exists for both teams, and that both teams play an optimal mixed strategy, and that the game is ideal in every sense except for boundary conditions of re-sweep and sweep outside the search space, the probability we win is bounded as follows:

We note that ${\frac{N_{cover}\mathscr {L}_{D}(\hat{C}_r)}{\mathscr {L}_{D}(X)}\rightarrow 0}$, in the limit, as the size of the environment increases relative to the sensor radius.

The non-ideal effects from this section can be combined with those from the previous subsection by simply combining the startup phases used for analysis into a single startup phase of combined duration.

1.3 C.3 Sensor model when $D > 2$

A sensor sweep footprint that forms a space tiling in $D- 1$ dimensions is required for ideal search, See Fig. 6. Non-tiling footprints require neighboring sweep passes to overlap in order to sweep the full space. When ${D> 2}$ the vast majority of sensor models will not permit a sweep footprint that forms a space tiling in ${D- 1}$ dimensions. While an appropriately chosen sensor footprint, such as the $D-1$ dimensional $L_{\infty }$-ball, does permit such a covering, other common symmetrical coverings such as the $D-1$ dimensional $L_{2}$-ball do not. We now investigate the effects of what happens when a non-tiling sensor is used.

Fig. 17

Two different sweep patterns that use the same sensor footprint, projected along the direction of travel to remove depth (as in Fig. 6bottom). This particular (projected) patch could be tessellated arbitrarily many times in the horizontal and/or vertical directions. Left: Sensor footprints of the robot appear (gray, dashed red) for a two phase search strategy that covers the space using an ideal search with 12 passes per patch (6 gray, 6 red). Right: the same space can be swept using only 8 passes per patch (blue); although this requires sweeping the entire space more quickly, it requires searching below the ideal search rate after the first 1/3 of search (Color figure online)

Assume that, other than the tiling of the search sensor, the search is otherwise ideal (i.e., we are ignoring the startup and boundary effects that were addressed in Sects. C.1 and C.2). Search necessarily happens in a number of different separate phases that are characterized by different $\frac{d \mathscr {L}_{D}(X_{\mathrm {swept}})}{d t}$ (sweep rates of space we have not yet swept) and thus different $\frac{d \mathscr {L}_{D}(X_{\mathrm {new}})}{d t}$ (sweep rates of space that has not been swept by ether team).

For example, we could perform search in two meta-phases represented by the gray discs and red circles in Fig. 17. During the first phase, search happens at the ideal rate due to the fact that each pass (gray disc) covers terrain that we have never visited before. However, after some time, sweeping any new space will necessarily require re-sweeping some previously swept terrain (red circles). Thus, in the second phase, $\frac{d \mathscr {L}_{D}(X_{\mathrm {swept}})}{d t}$ and $\frac{d \mathscr {L}_{D}(X_{\mathrm {new}})}{d t}$ are substantially reduced.

We could alternatively sweep the entire space much more quickly using a multipath defined by sensor discs inscribed by the blue hexagons in Fig. 17. The price we pay for a quicker overall search turns out to be an earlier decrease in $\frac{d \mathscr {L}_{D}(X_{\mathrm {swept}})}{d t}$ and $\frac{d \mathscr {L}_{D}(X_{\mathrm {new}})}{d t}$.

The fact that we can control $\frac{d \mathscr {L}_{D}(X_{\mathrm {swept}})}{d t}$ via choosing how different sweep passes overlap adds a significant amount of complexity to our analysis. Each time either team’s search rate changes, we must use a slightly different version of Eq. 2. This can be accomplished by breaking our analysis into $K$ different intervals depending on the number of times that either team changes their sweep rate.

Let the $k$-th time interval start at time $t_k$ and end at time $t_{k+1}$. By the beginning of the $k$-th time interval the adversary has already swept $F_k$ proportion of the total search space. During the $k$-th interval the rate team $G$ sweeps new (for us) territory is determined by $nc_{G,k}$ and the rate the adversary sweeps new (for them) territory is determined by $mc_{A,k}$.

The total fraction of area that has been swept by the adversary prior to the start of the $k$-th interval is given by

$$\begin{aligned} F_k= 1 - \sum _{k= 1}^{K-1} \int _{t_k}^{t_{k+1}} tm c_{A,k} \,\, dt\end{aligned}$$

As a corollary of Lemma 1 we get:

Corollary 5

The probability we win, assuming an optimal mixed strategy exists for both teams, and both teams play an optimal mixed strategy, when both teams communicate, and the game is ideal except for sensor footprint is

$$\begin{aligned} \mathbb {P}\left( \omega _{\mathrm {win}}\right) = \sum _{k= 1}^{K-1} \int _{t_k}^{t_{k+1}} \left( F_k- tm c_{A,k}\right) nc_{G,k} \,\, dt. \end{aligned}$$

We note that these effects do not vanish as the size of the environment increases relative to the sensor footprint.

Corollary 5 works for the case that both teams communicate. In general, extending these results to cases where one team cannot communicate is more involved than how the ideal results from Sect. 4.1 were extended in Sects. 4.2 and 4.3. Each of the sub-games versus a single (i.e., non-communicating) agent must be analyzed separately to account for the fact that each agent will independently choose when and how much its own sweeps will overlap between different passes through the environment.