# Summary:

The expected number of survivors after a shootout given as $$\lim_{n\rightarrow\infty}\operatorname{E}[n]\approx 0.284051\ n;\quad \text{(Tao/Wu - see below)}$$ is, if not correct, almost certainly very close to being correct *(see update 2)*.

However, this is disputed by Finch in **Mathematical constants** *(again, see below for details).* The results from Finch are easily replicable in *Mathematica* or similar, but I was not able to replicate even the partial results in Tao/Wu's paper *(despite leaving out the absolute values of $\alpha$ and $\beta,$ which Finch points out as being incorrect - see below for futher details)*, leaving me unsure as to whether I am missing something in my *"translation"* of the problem into Finch's more modern notation. I should be most grateful if someone could illuminate me further in this matter.

# Original answer:

Based on numerical tests, I would say the expected number of survivors for $n>3 \approx n/3.5$

Trial example `test[20]`

*(code below)*:

`anim[20,8]`

:

For $1000$ trials, $1\leq n\leq 40$ `est[40,10^3]`

:

**Note**

Using `RandomReal`

it is very unlikely that any two distances will be exactly equal, thereby fulfilling the *no isosceles triangle* requirement.

# Update 1

**History of the problem**

Robert Abilock proposed in **American Monthly** The Rifle-Problem *(R. Abilock; 1967)*,

$n$ riflemen are distributed at random points on a plane. At a signal,
each one shoots at and kills his nearest neighbor. What is the expected
number of riflemen who are left alive?

This was reposed as the Vicious neighbor problem *(R.Tao and F.Y.Wu; 1986)*, where the answer of $\approx 0.284051 n$ remaining riflemen *(or $\approx n/3.52049$)* was given as the solution in $2$ dimensions.

This agrees distinctly with tests of sample-size $10^5:$

```
ListLinePlot[{const[#, 100000] & /@ Range@40}, GridLines -> {{}, {1/0.284051}}]
```

However, in **Mathematical Constants** Nearest-neighbor graphs *(S.R.Finch; 2008)*, Finch states that

In [*Vicious neighbor problem*], the absolute value signs in the definitions of $\varphi$ and $\psi$ were mistakenly omitted.)$\dots$

Given the discrepancy between our estimate $\dots$ and their estimate $\dots$,
it seems doubtful that their approximation $\beta(2) = 0.284051\dots$ is entirely correct.

So the question (for the bounty) is then reduced to:

*Has any progress been made since 2008 on the problem? In short, is Tao and Wu's calculation incorrect, and if so, is a more precise estimate of $\beta(2)$ known?*

# Update 2

I have also tested the problem in other regular polygons (circle, triangle, pentagon, etc.) for $10^5$ trials, $1\leq n \leq 30$, and it seems that the comment by @D.Thomine below is in agreement with the data gathered, in that the constant for any bounded $2$ dimensional region appears to be the same for large enough $n,$ *ie*, independent of the global geometry of the domain:

while further simulations, using $2\cdot 10^6$ trials for $n=30$ and $n=100$ yielded the following results:

with the final averages after $2\cdot 10^6,$ compared to Tao/Wu's result, being:

\begin{align}
&n=30:&0.284090\dots\\
&n=100:&0.284066\dots\\
&\text{Tao/Wu:}&0.284051\dots\\
\end{align}

indicating that the Tao/Wu result of $\lim_{n\rightarrow\infty}\operatorname{E}[n]\approx 0.284051\ n$ is, if not correct, almost certainly very close to being correct.

# Upper and lower bounds

Following up on @mathreadler's suggestion that it may be interesting to study the spread of data, I include the following as a minor contribution to the topic:

Since arrangements like this

are possible (and their circular counterparts, however unlikely through random point selection), clearly the lower bound for odd $n$ is $1$ and for even $n$ it is $0$ (since the points can be paired).

Finding an upper bound is less obvious though. Looking at this sketch proof for upper bound $n=10$ provided by @JohnSmith in the comments, it is easy to see that the upper bound is $7:$

and by employing the same method, upper bounds for larger $n$ can be constructed:

Assuming one can repeat this process indefinitely, it is likely that an upper bound for $n\geq 6$ then is $n-\lfloor n/3\rfloor:$

which has been set against the data for $2\cdot 10^4$ trials *(red dots - see *`data`

below).

Regarding density of spread, (again with $2\cdot 10^4$ trials) produces the following plot:

```
ListPlot3D[Flatten[data, 1], ColorFunction -> "LakeColors"]
```

*(courtesy of @AlexeiBoulbitch here)*, and regarding max. density of spread along $x/z$ axes from above plot, produces the following:

```
With[{c = 0.284051},
Show[ListLinePlot[Max@#[[All, 3]] & /@ data, PlotRange -> All],
Plot[{(1 + c)/(n - (1 + c)^2)^(1/2)}, {n, 0, 100}, PlotRange -> All,
PlotStyle -> {Dashed, Red}]]]
```

It is tempting to conjecture max height of distribution to be $\approx (c+1)/\sqrt{n-(c+1)^2},$ but of course this is largely empirical.

```
test[nn_] := With[{aa = Partition[RandomReal[{0, 1}, 2 nn], 2]},
With[{cc = ({aa[[#]], First@Nearest[DeleteCases[aa, aa[[#]]], aa[[#]]]}
& /@ Range@nn)},
With[{dd = Table[Position[aa, cc[[p, 2]]][[1, 1]], {p, nn}]},
With[{ee = Complement[Range@nn, dd]},
Column[{StringJoin[ToString["Expected: "], ToString[nn/3.5]],
StringJoin[ToString["Survivors: "], ToString[Length@ee], ToString[": "],
ToString[ee]], Show[Graphics[{Gray, Line@# & /@ cc}, Frame -> True,
PlotRange -> {{0, 1}, {0, 1}}, Epilog -> {Text[Style[(Range@nn)[[#]],
30/Floor@Log@nn], aa[[#]]] & /@ Range@nn}], ImageSize -> 300]}]]]]]
est[mm_, trials_] := ListLinePlot@({Quiet@With[{nn = #},
(N@Total@(With[{aa = Partition[RandomReal[{0, 1}, 2 nn], 2]},
With[{cc = ({aa[[#]], First@Nearest[DeleteCases[aa, aa[[#]]],
aa[[#]]]} & /@ Range@nn)},
With[{dd = Table[Position[aa, cc[[p, 2]]][[1, 1]], {p, nn}]},
With[{ee = Complement[Range@nn, dd]},Length@ee]]]]
& /@ Range@trials)/trials)] & /@ Range@mm, Range@mm/3.5})
anim[nn_, range_] := ListAnimate[test@nn & /@ Range@range,
ControlPlacement -> Top, DefaultDuration -> nn]
const[mm_, trials_] := With[{ans = Quiet@With[{nn = #},
SetPrecision[(Total@(With[{aa = Partition[RandomReal[{0, 1}, 2 nn], 2]},
With[{cc = ({aa[[#]],First@Nearest[DeleteCases[aa, aa[[#]]],
aa[[#]]]} & /@ Range@nn)},
With[{dd = Table[Position[aa, cc[[p, 2]]][[1, 1]], {p, nn}]},
With[{ee = Complement[Range@nn, dd]},
Length@ee]]]] & /@ Range@trials)/trials), 20]] &@ mm}, mm/ans]
act[nn_, trials_] := With[{aa = Partition[RandomReal[{0, 1}, 2 nn], 2]},
With[{cc = ({aa[[#]], First@Nearest[DeleteCases[aa, aa[[#]]], aa[[#]]]} & /@
Range@nn)}, With[{dd = Table[Position[aa, cc[[p, 2]]][[1, 1]], {p, nn}]},
With[{ee = Complement[Range@nn, dd]}, Length@ee]]]] & /@ Range@trials
data = Quiet@ Table[With[{tt = 2*10^4},
With[{aa = act[nn, tt]}, With[{bb = Sort@DeleteDuplicates@aa},
Transpose@{ConstantArray[nn, Length@bb], bb, (Length@# & /@
Split@Sort@aa)/tt}]]], {nn, 1, 100}];
```