A366996 -id:A366996 - cp's OEIS frontend

A366566 a(n) is the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each. The end time, rounded to the nearest integer, is given for games in which one of the two poor players wins.

3, 6, 9, 13, 17, 22, 28, 34, 41, 49, 58, 67, 76, 87, 98, 109, 122, 135, 149, 163, 178, 194, 210, 227, 245, 263, 282, 302, 322, 343, 365, 387, 410, 434, 458, 483, 509, 535, 562, 590, 619, 648, 677, 708, 739, 770, 803, 836, 869, 904, 939, 974, 1011, 1048, 1085

Offset: 1

Hugo Pfoertner, Oct 13 2023

nonn

For details see the Diaconis link.
Initially, terms up to a(25) were calculated using Monte Carlo simulation of 10^9 games at each value of n.
The expected end times without rounding to nearest integer are: 3.00, 5.57, 8.76, 12.57, 17.03, 22.14, 27.91, 34.33, 41.41, 49.15, 57.55, 66.61, 76.33, 86.72, ... .
The expected shorter end time also allowing the rich player to win would be 2*n+1 (Bachelier, 1912, page 149).

Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
Persi Diaconis and Stewart N. Ethier, Gambler’s Ruin and the ICM, Statist. Sci. 37 (3) 289 - 305, August 2022.
Persi Diaconis, Gambler's ruin with k gamblers (slide 3), talk in the Rutgers Experimental Mathematics Seminar, Fall 2023 Semester, Oct. 12, 2023.
Experimental Mathematics, GAMBLER’S RUIN WITH K GAMBLERS, recording of talk, Vimeo video (time after 11:55), Oct 22, 2023.
Hugo Pfoertner, Example of the time history of a game with n=3, i.e., the "rich" player starts with 3 chips.
Hugo Pfoertner, Distribution of the number of games won, n=3, plotted vs end time.
Hugo Pfoertner, Distribution of the number of games won, n=5, plotted vs end time.
Hugo Pfoertner, Distribution of the number of games won, n=6, plotted vs end time.
Hugo Pfoertner, Distribution of the number of games won, n=10, plotted vs end time.
Hugo Pfoertner, Distribution of the number of games won, n=20, plotted vs end time.

Cf. A366166, A366567 (mode of corresponding probability distributions), A366995, A366996.

a(n) equals A366995(n)/A366996(n) rounded to the nearest integer. - Pontus von Brömssen, Oct 31 2023

a(26)-a(55) from Pontus von Brömssen, Oct 31 2023

A366995 a(n) is the numerator of the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each, conditional on the event that one of the two poor players wins.

Original entry on oeis.org

3, 39, 359, 2477, 119667, 1522705, 46419629, 6143100517, 5472109127035, 790136773603303, 278129286200597661, 16684426086791338103, 503067648850136040148699, 2626565018569118643191009, 10920130209346850287269887104735, 236686188450953790757840351941895

Offset: 1

Pontus von Brömssen, Oct 31 2023

In each round of the game, 1 unit is transferred from one randomly chosen player to another. Players play until they are out of money, so when the first player is out the other two continue to play. The winner is the player who ends up with all n+2 units of money.

Pontus von Brömssen, Table of n, a(n) for n = 1..53
Persi Diaconis and Stewart N. Ethier, Gambler’s ruin and the ICM, Statist. Sci. 37 (3) 289-305, August 2022.
Persi Diaconis, Gambler's ruin with k gamblers, slides from talk in the Rutgers Experimental Mathematics Seminar, Fall 2023 Semester, Oct 12, 2023.
Experimental Mathematics, Gambler’s ruin with k gamblers, recording of talk, Vimeo video, Oct 22, 2023.

Cf. A366566 (a(n)/A366996(n) rounded to nearest integer), A366996 (denominators).

from itertools import permutations
def T(n):
    nodes = [(i,j) for i in range(n+2) for j in range((n+2-i)//2+1)]
    m = len(nodes)
    Q0 = {x:{y:0 for y in nodes} for x in nodes}
    for x in nodes:
        c1 = x+(n+2-sum(x),)
        for i,j in permutations(range(3),int(2)):
            if c1[i] and c1[j]:
                c2 = list(c1)
                c2[i] -= 1
                c2[j] += 1
                y = (c2[0],min(c2[1:]))
                if c2[0] != n+2:
                    Q0[x][y] += n+2-c2[0]
    Q0 = matrix(QQ,[list(R.values()) for R in Q0.values()])
    s = sum(Q0.columns())
    Q = identity_matrix(QQ,m-1)
    for i in range(1,m):
        for j in range(1,m):
            if s[i] != 0: Q[i-1,j-1] -= Q0[i,j]/s[i]
    return (Q**(-1)*ones_matrix(QQ,m-1))[-2,0]
def A366995(n):
    return T(n).numerator()
def A366996(n):
    return T(n).denominator()

cp's OEIS Frontend

A366566 a(n) is the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each. The end time, rounded to the nearest integer, is given for games in which one of the two poor players wins.

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