Share And Share Alike

I am gladdened to hear it Sir! Gladdened to hear it indeed!

This day's sweltering heat has put me in mind of the time that I found myself temporarily misplaced in the great Caloris rainforest on Mercury. I had been escorting the Velikovsky expedition, which had secured the patronage of the Russian Imperial court for its mission to locate the source of the Amazon, and on one particularly close evening our encampment was attacked by a band of Salamanders which, unlike their diminutive Earthly cousins, stood some eight feet tall and wielded vicious looking barbed spears.

Naturally I leapt into action, dispatching two of their number with my trusty rapier ere they realised that I was there. The vigour with which I prosecuted my attack sufficiently startled them that they took to their heels and fled. I pursued them some distance in their rout and, when I eventually returned to the camp, found it quite abandoned.

In my long search for my fellow explorers I chanced upon the very spring that was the object of their quest, close to which was a small village in which I determined that I should await their arrival. The inhabitants were of a most egalitarian disposition, happily dividing the fruits of their labours between their neighbours; the hunters their catch, the gatherers their pickings and so on and so forth.

To instil this instinct to share and share alike in the minds of their children they would play a curious dice game with them, which I propose that we employ for our wager.

Here I shall set two coins from my purse upon the table to begin. You shall then cast this four sided die and I shall add to those coins the number that you throw. I shall then divide up the pile of coins into as many piles of equal numbers of coins as I may; if there were six I should make of them three piles of two coins apiece, if there were five then I could make just one pile of five. Of these piles you may keep one for the table and I shall have back the rest for myself. We shall then start again with the pile of coins left upon the table.

The game shall last sixteen such turns and cost you a mere three coins and fifty cents to play.

When I told that loathsome student, whose presence it seems that I am incapable of entirely escaping, of the rules of this wager he paid them no mind whatsoever but instead set to lamenting that the path to his local market had been chained off; why he should not think to take another is quite beyond me!

But enough of his petty grumblings! Come refill your tankard whilst you decide whether or not this wager is to your taste!

October 12, 2017 6:17 PM

I think it's just worth it.

I drew a directed graph where the nodes where amounts and the arcs were labelled probabilities of moving from one amount to another.

From this and a little multiplication I concluded:

- The probability of being on 2 is always 1/2 after the first turn.

- The probability of being on 3 or 5 is always 3/16 after the first turn.

- The prob... of 7 converges to 3/32, the prob of 11 converges to 3/128,

and so on through the primes up to 23 (each 1/4 as likely as the last).

This gives an expectation of 3.52

October 19, 2017 9:41 PM

That you should think of representing the game as a directed graph makes it plain to me that you are well studied in the theory of wager, but I must ask whether you are certain that the sixteen rounds of the game are sufficient to bring its outcome to convergence.