## asymmetric information

Posted in Kids, R with tags , , , , , on November 4, 2020 by xi'an

Take a real number θ uniformly distributed over (0,100). Among three players, the winner is whoever guessed the closest price without going over θ. In the event all guesses exceeded θ, the contestant with the lowest (and therefore closest) guess is declared the winner. The second player knows the first player’s guess and the third player knows both other guesses. What is the optimal guess for the first player, assuming all players maximise their probability of winning?

Looking at the optimal solution z for the third player leads to six possible choices, depending on the connection between the other guesses, x and y. Which translates in the R code

```topz=function(x,y){
if((2*y>=x)&(y>=1-x))  z=y-.001
if(max(4*y,1+y)<=2*x)  z=y+.001
if((2*x<=1+y)&(x<=1-y))z=x+.001
z}

third=function(x,y) ifelse(y<x,topz(x,y),topz(y,x))
```

For there, the optimal choice y for the second player follows and happens on a boundary of one of the six regions, which itself returns that the optimal choice for the first player is x=2/3, leading to equal chances of winning (although there is some uncertainty on the boundaries). It is thus feasible to beat the asymmetric information. The picture above was my attempt at representing the probabilities of gain for all three players, some of the six regions being clearly visible, with first axis being x and second being y [and z is one of x⁻,x⁺,y⁻,y⁺]. The R code is too pedestrian to be reproduced!

## sampling w/o replacement except when replacing

Posted in Books, Kids, R with tags , , , , , , , on November 3, 2020 by xi'an

Another Riddle(r), considering a box with M myrtle balls and D dandelion balls. Drawing balls without replacement while they stay of the same color as the initial draw, else put back the last ball and repeat the process until all balls are drawn. The funny thing is that, unless M=0 or D=0, the probability to draw a myrtle ball at the end is always ½..! This can be easily checked by simulation (when M=2 and D=8)

```r=function()sample(0:1,1,p=c(d,m))
for(t in 1:1e6){
m=2;d=8
i=r();m=m-!!i;d=d-!i
while(!!m*d){
j=r();i=ifelse(i==j,j,r())
m=m-!!i;d=d-!i}
F=F+(m>0)}
F/1e6
```

Now the proof that the probability is ½ is quite straightforward, for M=1 (or D=1). But I cannot find a quick fix for larger values. I thus reasoned by recursion, with the probability of emptying a given colour first is d!m!/(d+m)!, whatever the colour and whatever d>0,m>0. Hence half a chance to finish with myrtle. Any shorter sequence of a given colour reduces the value of either d or m, at which point we are using the recursion assumption that the probability is ½…

## parking riddle

Posted in Books, Kids, pictures, R, Statistics, Travel with tags , , , , , , on October 23, 2020 by xi'an

The Riddler of this week had a quick riddle: if one does want to avoid parallel parking a car over a six spot street, either the first spot is available or two consecutive spots are free. What is the probability this happens with 4 other cars already parked (at random)?

While a derivation by combinatorics easily returns 9/15 as the probability to fail to park, a straightforward R code does as well

``` l=0
for(t in 1:1e6){
k=sort(sample(0:5,4))
l=l+1*(!!k[1]|3%in%diff(k)|!k[4]%%3)}
```

since

```
> round(choose(6,2)*F/1e6)
[1] 10
```

## Fermat’s Riddle

Posted in Books, Kids, R with tags , , , , , , , , , , on October 16, 2020 by xi'an

·A Fermat-like riddle from the Riddler (with enough room to code on the margin)

An  arbitrary positive integer N is to be written as a difference of two distinct positive integers. What are the impossible cases and else can you provide a list of all distinct representations?

Since the problem amounts to finding a>b>0 such that

$N=a^2-b^2=(a-b)(a+b)$

both (a+b) and (a-b) are products of some of the prime factors in the decomposition of N and both terms must have the same parity for the average a to be an integer. This eliminates decompositions with a single prime factor 2 (and N=1). For other cases, the following R code (which I could not deposit on tio.run because of the packages R.utils!) returns a list

```library(R.utils)
library(numbers)
bitz<-function(i,m) #int2bits
c(rev(as.binary(i)),rep(0,m))[1:m]
ridl=function(n){
a=primeFactors(n)
if((n==1)|(sum(a==2)==1)){
print("impossible")}else{
m=length(a);g=NULL
for(i in 1:2^m){
b=bitz(i,m)
if(((d<-prod(a[!!b]))%%2==(e<-prod(a[!b]))%%2)&(d<e))
g=rbind(g,c(k<-(e+d)/2,l<-(e-d)/2))}
return(g[!duplicated(g[,1]-g[,2]),])}}
```

For instance,

```> ridl(1456)
[,1] [,2]
[1,]  365  363
[2,]  184  180
[3,]   95   87
[4,]   59   45
[5,]   40   12
[6,]   41   15
```

Checking for the most prolific N, up to 10⁶, I found that N=6720=2⁶·3·5·7 produces 20 different decompositions. And that N=887,040=2⁸·3²·5·7·11 leads to 84 distinct differences of squares.

## where is .5?

Posted in Statistics with tags , , , , on September 10, 2020 by xi'an

A Riddler’s riddle on breaking the unit interval into 4 random bits (by which I understand picking 3 Uniform realisations and ordering them) and finding the length of the bit containing ½ (sparing you the chore of converting inches and feet into decimals). The result can be found by direct integration since the ordered Uniform variates are Beta’s, and so are their consecutive differences, leading to an average length of 15/32. Or by raw R simulation:

```simz=t(apply(matrix(runif(3*1e5),ncol=3),1,sort))
mean((simz[,1]>.5)*simz[,1]+
(simz[,1]<.5)*(simz[,2]>.5)*(simz[,2]-simz[,1])+
(simz[,2]<.5)*(simz[,3]>.5)*(simz[,3]-simz[,2])+
(simz[,3]<.5)*(1-simz[,3]))
```

Which can be reproduced for other values than ½, showing that ½ is the value leading to the largest expected length. I wonder if there is a faster way to reach this nice 15/32.