An analysis of the classical gambler's ruin problem through multiple devices variation

Figures & data

Table 1. Absorption probabilities, the ruin time, conditional ruin times and variance of the game for c = 2.

p	k	a	${\mu }_k$	$E(T_k)$	$E_k$	$F_k$	$Var(k)$
	1	20	0.2899	44.00	3.33	60.61	879.74
	5	20	0.0020	49.87	16.67	49.93	252.07
0.65	10	20	0.0000	33.33	33.33	33.33	168.50
	15	20	0.0000	16.67	49.93	16.67	84.26
	19	20	0.0000	3.34	60.61	3.33	16.86
	1	20	0.5244	53.21	6.25	104.97	3270.89
	5	20	0.0396	88.80	31.24	91.17	1665.42
0.58	10	20	0.0016	62.31	62.31	62.31	1161.24
	15	20	0.0001	31.25	91.17	31.24	592.75
	19	20	0.0000	6.25	104.97	6.25	118.85
	1	20	0.7846	55.12	15.51	199.45	9573.60
	5	20	0.2950	151.67	74.85	183.82	12988.86
0.53	10	20	0.0830	139.02	139.02	139.02	10785.82
	15	20	0.0192	76.94	183.82	74.85	7190.80
	19	20	0.0022	15.92	199.95	15.51	1680.53
	1	20	0.9500	38.00	26.00	266.00	9158
	5	20	0.7500	150.00	116.67	250.00	24950
0.50	10	20	0.5000	200.00	200.00	200.00	26,600
	15	20	0.2500	150.00	250.00	116.67	24,950
	19	20	0.0500	38.00	266.00	26.00	9158

Figure 1. Some simulated durations for c = 2 with various values of k and probabilities p.

Table 2. The results based on “20, 000” simulations.

p	k	a	${\mu }_k$	$E(T_k)$	$E_k$	$F_k$	$Var(k)$
	1	20	0.2906	43.81	3.31	60.40	885.2094
	5	20	0.0028	49.86	17.54	49.95	246.9837
0.65	10	20	0.0000	33.41	N.O*	33.41	166.3313
	15	20	0.0000	16.53	N.O*	16.53	81.8198
	19	20	0.0000	3.27	N.O*	3.27	15.4067
	1	20	0.5259	53.47	6.20	105.91	3306.4400
	5	20	0.0393	89.15	28.93	91.61	1676.7260
0.58	10	20	0.0017	62.51	59.59	62.51	1180.8172
	15	20	0.0000	31.06	N.O*	31.06	577.7697
	19	20	0.0000	6.38	N.O*	6.38	119.8919
	1	20	0.7821	55.30	15.63	197.70	9513.1150
	5	20	0.2981	152.12	74.31	185.17	13053.9303
0.53	10	20	0.0814	140.84	145.92	140.38	10872.1225
	15	20	0.0201	78.56	188.71	76.30	7229.9460
	19	20	0.0024	15.60	190.17	15.18	1614.5110
	1	20	0.9504	37.70	26.16	258.93	9173.8870
	5	20	0.7452	150.66	117.70	247.06	24868.4300
0.50	10	20	0.4966	200.79	199.75	201.82	26902.3500
	15	20	0.2514	150.43	248.86	117.38	25030.3100
	19	20	0.0513	38.71	271.58	26.12	9289.7061

*Not observed

Table 3. Comparison results for classic game (C), involvement of draw (tie) (D), successive trials (S) and proposed game (P).

			${\mu }_k$				$E(T_k)$
p	k	C	D	S	P	C	D	S	P
	1	0.5385	0.3846	0.2899	0.2899	27.43	28.27	88.00	44.00
	5	0.0453	0.0084	0.0020	0.0020	46.98	37.08	99.74	49.87
0.65	10	0.0020	0.0001	0.0000	0.0000	33.20	25.00	66.66	33.33
	15	0.0001	0.0000	0.0000	0.0000	16.66	12.50	33.34	16.67
	19	0.0000	0.0000	0.0000	0.0000	03.33	02.50	06.66	03.33
	1	0.7237	0.5517	0.5244	0.5244	28.29	30.64	106.42	53.21
	5	0.1979	0.0511	0.0396	0.0396	69.02	53.76	177.60	88.80
0.58	10	0.0381	0.0026	0.0016	0.0016	57.73	38.263	124.62	62.31
	15	0.0063	0.0001	0.0001	0.0001	30.46	19.22	62.50	31.25
	19	0.0006	0.0000	0.0000	0.0000	06.18	03.85	12.50	06.25
	1	0.8755	0.6979	0.7846	0.7846	24.82	31.51	110.24	55.12
	5	0.5035	0.1652	0.2950	0.2950	82.17	73.10	303.34	151.67
0.53	10	0.2312	0.0268	0.0830	0.0830	89.59	59.16	278.04	139.02
	15	0.0819	0.0038	0.0192	0.0192	56.04	30.77	153.88	76.94
	19	0.0127	0.0003	0.0022	0.0022	12.43	06.21	31.88	15.92
	1	0.9500	0.7977	0.9500	0.9500	19.00	30.47	76.00	38.00
	5	0.7500	0.3198	0.7500	0.7500	75.00	86.03	300.00	150.00
0.50	10	0.5000	0.0970	0.5000	0.5000	100.00	80.61	400.00	200.00
	15	0.2500	0.0239	0.2500	0.2500	75.00	45.21	300.00	150.00
	19	0.0500	0.0029	0.0500	0.0500	19.00	09.42	76.00	38.00