A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R

Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning. This paper derives important formulas for the martingale strategy, such as the distribution, the expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented. The results of doubling to gambling with a constant sized bet on simple chances (red or black numbers, even or odd numbers, and low (1 – 18) or high (19 – 36) numbers) and on single numbers (straight bets) are compared. In the long run a loss is inevitable because of the negative expected value. The martingale strategy and the constant bet strategy are more risky than the constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a positive profit in the short term. But on the other hand higher risk means that the losses suffered by doublers and by single number bettors are much greater than that suffered by constant bettors. In the long run a martingale strategy cannot beat an unbiased roulette because of the negative expected value. After a large number of coups it is very likely that the gambler suffers a loss. The martingale strategy and the single number bet strategy are more risky than the constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a positive profit in the short term. But on the other hand higher risk means that the losses suffered by doublers and by single number bettors were much greater than that suffered by constant bettors. Gamblers should be aware that the martingale play is a very risky strategy which could produce extremely high losses. Often casual gamblers underestimate the risk of the martingale play or versions of it because they used to leave the roulette table with a profit. The reason is that the probability of winning a few martingale rounds is high if the number of rounds is low. E.g., a player stops after 20 martingales rounds, then the probability of winning is nearly 98 percent, assuming the assumptions, which are made in this paper. But the comparatively low profit is offset by the possibility of an extremely high loss which is inevitable in the long run

Leonhard Euler’s Research on the Multiplication of the Human Race with Models of Population Growth

The renowned Swiss mathematician Leonhard Euler created three variations of a simple population projection model, including one stable model and two non-stable models, that consider a couple with different fertility behaviors and life-spans. While one of the models was published by a German demographer, Johann Peter Süßmilch, in his book “The Divine Order”, the other two are not widely known in contemporary literature. This paper compares and reanalyzes the three variants of Euler's population projections using matrix algebra, providing diagrams and tables of the population time series and their growth rates, as well as age structures of selected years. It is demonstrated that the non-stable projection models can be explained in the long run by their geometric trend component, which is a special case of strong ergodicity in demography as described by Euler. Additionally, a continuous variant of Euler's stable model is introduced, allowing for the calculation of the age structure, intrinsic growth rate, and population momentum in a straightforward manner. The effect of im¬mortality on population size and age structure at high growth rates is also examined

A Demometric Analysis of Ulpian’s Table

Ulpian’s table is a famous ancient text that is preserved in edited form in Justinian’s Digest, a compendium of Roman law compiled by order of the emperor Justinian I in the sixth century AD. This passage probably provides a rough estimation of Roman life expectancy in the early third century AD. The paper begins with a discussion of the demographic properties and peculiarities of Ulpian´s table. Then the Gompertz distribution and some of its extensions are used to fit life expectation functions to Ulpian´s data. The model can be used to estimate important demographic functions and parameters of the Roman life table. Inter alia, the average and median remaining life expectancies are calculated, and compared with the results of other investigations, e.g., Frier’s life table for the Roman Empire. It turns out that Ulpian´s life table is characterized by a steep decline of the life expectancy function in the advanced age classes, which is much steeper than in life expectancy functions of other life tables based on data. The modal or normal age at death, which is between 55 and 60 years, is comparatively high

Risk and Return of the Tontine: A Brief Discussion

This article analyzes the stochastic aspects of a tontine using a Gompertz distribution. In particular, the probabilistic and demographic risks of a tontine investment are examined. The expected value and variance of tontine payouts are calculated. Both parameters increase with age. The stochastic present value of a tontine payout is compared with the present value of a fixed annuity. It is shown that only at very high ages the tontine is more profitable than an annuity. Finally, the demographic risks associated with a tontine are discussed. Elasticities are used to calculate the impact of changes in modal age on the tontine payout. It is shown that the tontine payout is very sensitive to changes in modal age

Life Table Forecasting with the Gompertz Distribution

First, this paper investigates the properties of the Gompertz distribution and the relationships of their constants. Then the use of Gompertz´s law to describe mortality is discussed with male and female period life table data of the United States between 1900 and 2000. For this purpose a model incorporating time trends has been formulated with age, time and the product of age and time as independent variables and the force of mortality as the dependent variable. The parameters of the model are estimated using the least squares method. Since the mortality of modern developed population is largely the mortality of old age this generalized Gompertz model provides a good approximation of life tables in these populations, and can be used to estimate and forecast many parameters of the life table and the stationary population like expectation of life, modal age, Keyfitz´entropy or old age dependency ratios. These and other parameters are forecast up to the year 2100 and compared with recent mortality forecasts of the Social Security Administration. While similar results for the male population can be observed, a greater difference between male and female mortality are forecast. Although the time dependent Gompertz model reveals systematic underestimation of mortality at young ages and overestimation at the oldest ages it is a very useful, an easy, and a quick tool for obtaining forecasts of important parameters of life tables with low mortality

How Migration can Contribute to Achieving a Stationary Population

Methods from mathematics of finance and demography are presented in order to investigate the influence of migration on the long-term population development. Methods from mathematics of finance do not take the age structure of a population into consideration and can therefore only be used as an approximation. The less the age structures in question deviate from those of stable populations, the more exact the approximation will be. In the empirical section quantitative measures for population policy are described and analyzed using the population of Germany and of the world as examples. The long-term goal of quantitative population policy is zero growth. Whereas in less developed countries, this goal can be achieved for the most part only by a reduction of fertility, it is possible in more developed countries with below-replacement fertility to achieve stationarity by means of immigration. Under the assumptions made here, Germany would have to take in between 350.000 and 500.000 immigrants each year for the population to remain at the present level. Immigration has demographic consequences for the age structure and the composition of the population which will be described at the end

Forecasting the U.S. Population with the Gompertz Growth Curve

Population forecasts have received a great deal of attention during the past few years. They are widely used for planning and policy purposes. In this paper, the Gompertz growth curve is proposed to forecast the U.S. population. In order to evaluate its forecast error, population estimates from 1890 to 2010 are compared with the corresponding predictions for a variety of launch years, estimation periods, and forecast horizons. Various descriptive measures of these forecast errors are presented and compared with the accuracy of forecasts made with the cohort component method (e.g., the U.S. Census Bureau) and other traditional time series models. These models include quadratic and cubic trends, which were used by statisticians at the end of the 19th century (Pritchett and Stevens). The measures of errors considered are based on the differences between the projected and the actual annual growth rate. It turns out that the forecast accuracies of the models differ greatly. The accuracy of some simple time series models is better than the accuracy of more complex models

Tackling the Challenge of Aging Populations: The Impact of Increasing Life Expectancy and Low Fertility on the Old-Age Dependency Ratio

The old-age dependency ratios are indicators of the number of elderly people who are generally economically inactive compared to the number of people of working age. They significantly affect the financial burden of social public pension schemes, making it essential to analyze the influence of mortality on this ratio. In this paper, the Gompertz model is used to investigate the effect of mortality and fertility on the old-age dependency ratio, with a focus on the impact of changes in life expectancy. Elasticity formulas are derived to analyze this effect, and the results indicate that an increase in life expectancy leads to a considerable rise in the old-age dependency ratio

Refining Mortality Projections at Advanced Ages: Evaluating the Significance of Wittstein's Mortality Law

Age-specific mortality rates for semi-supercentenarians and supercentenarians play a pivotal role in comprehending longevity and population dynamics at advanced ages. In this study, we introduce a modified Wittstein Model, offering an alternative to the conventional S-shaped curve models used in mortality forecasting. The Wittstein Model, originally formulated by Theodor Wittstein, has been adapted to suit contemporary contexts. Utilizing life table data for German women from 2019/2021, we project age-specific mortality rates, construct life tables commencing from age 100, and conduct a sensitivity analysis to assess the impact of model parameters on mortality patterns. The sensitivity analysis unveils the influence of parameter values on the shape of age-specific mortality rates. This study contributes to research in mortality forecasting, with a specific focus on semi-supercentenarians and supercentenarians, shedding light on an understudied population segment. Accurate projections carry profound implications for public health, healthcare planning, and social policy. Further research should explore the model's applicability in different contexts, providing a deeper understanding of mortality patterns at advanced ages. As the empirical database of centenarians expands, the model is expected to enhance its precision and reliability in forecasting age-specific mortality rates at advanced ages

Measuring the Rectangularization of Life Tables Using the Gompertz Distribution

The rectangularization of life tables is defined as a trend towards a more rectangular shape of the survival curve due to increased survival and concentration of deaths around the mean age at death. Since the mortality of modern developed population is largely the mortality of old age, the Gompertz model provides a good approximation of life tables in these populations and can be used to estimate and forecast many parameters of the life table and the stationary population, such as expectation of life, modal age, age dependency ratios, and indices of the rectangularization of life tables. Formulas of known rectangularization indices are developed assuming the Gompertz distribution, whereas some new indices are proposed, too. The mathematical relationships between the single indices are shown. It is demonstrated that some mentioned indices are a function of the coefficient of variation