The magic of the d’Alembert

Simulations of the d’Alembert on a faIr roulette wheel with 36 paying outcomes and one “0”. Even odds bets (e.g., red versus black). Each line is one game. Each picture is 200 games. Parameters: initial capital of 25 units, maximum number of rounds is 21, emergency stop if capital falls below 15.

Source:


Harry Crane and Glenn Shafer (2020), Risk is random: The magic of the d’Alembert. https://researchers.one/articles/20.08.00007

Stewart N. Ethier (2010), The Doctrine of Chances Probabilistic Aspects of Gambling. Springer-Verlag: Berlin, Heidelberg.

set.seed(12345)
startKapitaal <- 25
eersteInzet <- 1
noodstopKapitaal <- 15
aantalBeurten <- 21
K <- 100
J <- 200
winsten <- rep(0, K)

for (k in (1:K)){

	plot(x = -2, y = -1, ylim = c(-5, 45), xlim = c(0, 22), xlab = "Beurt", ylab = "Kapitaal")
	abline(h = 25)
	abline(h = 0, col = "red")

	aantalKeerWinst <- 0
	totaleWinst <- 0

	for (j in (1:J)) {

		huidigeKapitaal <- startKapitaal
		huidigeInzet <- eersteInzet
		resultaten <- sample(x = c(-1, +1), prob = c(19, 18), size = aantalBeurten, replace = TRUE)
		verloop <- rep(0, aantalBeurten)
		stappen <- rep(0, aantalBeurten)
		for(i in 1:aantalBeurten) {
			 huidigeResultaat <- resultaten[i]
			 if(huidigeInzet > 0){
				  stap <- huidigeResultaat * huidigeInzet
				  stappen [i] <- stap
				  huidigeKapitaal <- huidigeKapitaal + stap
				  huidigeInzet <- max(1, huidigeInzet - stap)
				  if(huidigeKapitaal < noodstopKapitaal) {huidigeInzet <- 0}
				  verloop[i] <- huidigeKapitaal
			 } else {
				  stappen[i] <- 0
				  verloop[i] <- huidigeKapitaal
			 }
		} 
	aantalKeerWinst <- aantalKeerWinst + (verloop[aantalBeurten] > startKapitaal)
	totaleWinst <- totaleWinst + (huidigeKapitaal - startKapitaal)
	lines(0:aantalBeurten, c(startKapitaal, verloop) + runif(1, -0.15, +0.15 ), add = TRUE)
	}
print(c(k, aantalKeerWinst, totaleWinst))
winsten[k] <- totaleWinst
}

The program repeatedly runs and plots 200 games of each maximally 21 rounds. Below are the total number of times that the player made a profit, and the final net gain, for 100 sets of 200 games. The sets are numbered 1 to 100.

[1]    1  100 -483
[1]    2  108 -336
[1]    3  103 -517
[1]    4  110 -275
[1]   5 123 -40
[1]   6 125 148
[1]    7  115 -209
[1]    8  104 -427
[1]    9  108 -356
[1]   10  110 -225
[1]   11  101 -440
[1]  12 120  80
[1]   13  108 -334
[1]   14  110 -279
[1]   15   99 -538
[1]   16  114 -101
[1]  17 113 -92
[1]  18 117 -87
[1]   19  104 -363
[1]   20  103 -320
[1]  21 114 -52
[1]   22  107 -422
[1]   23  108 -226
[1]   24  115 -173
[1]   25  110 -209
[1]   26  109 -261
[1]   27  114 -186
[1]  28 120 -62
[1]  29 123  35
[1]   30  101 -442
[1]   31  111 -215
[1]   32  104 -378
[1]  33 120  49
[1]  34 117 -49
[1]   35  119 -102
[1]   36  104 -488
[1]   37  107 -402
[1]  38 122  38
[1]   39  100 -549
[1]  40 116 -31
[1]  41 127 220
[1]   42  105 -427
[1]   43  114 -153
[1]   44  109 -256
[1]   45  119 -166
[1]  46 121  47
[1]   47  105 -417
[1]   48  113 -134
[1]  49 121 111
[1]   50  112 -307
[1]  51 114 -92
[1]  52 123 123
[1]  53 118  24
[1]   54  113 -188
[1]  55 124 127
[1]   56  110 -229
[1]   57  113 -255
[1]   58  101 -554
[1]   59  114 -345
[1]  60 124 236
[1]   61   97 -599
[1]   62  115 -220
[1]  63 120  55
[1]   64  102 -512
[1]  65 121 109
[1]   66  112 -219
[1]   67  112 -181
[1]  68 115 -45
[1]   69  107 -474
[1]   70  109 -272
[1]   71  116 -134
[1]   72  107 -440
[1]   73  108 -470
[1]  74 119 -85
[1]  75 115   1
[1]  76 115 -88
[1]   77  113 -219
[1]  78 118 -55
[1]   79  115 -150
[1]  80 124  70
[1]   81  115 -203
[1]   82  115 -153
[1]   83  109 -219
[1]   84   97 -675
[1]   85  108 -396
[1]   86  112 -220
[1]   87  115 -187
[1]   88  108 -290
[1]   89  114 -182
[1]   90  105 -439
[1]   91  113 -183
[1]   92  115 -216
[1]  93 124 110
[1]   94  115 -173
[1]  95 125 177
[1]   96  110 -203
[1]  97 128 160
[1]  98 114 -83
[1]  99 118 -90
[1] 100 123 106

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