Marriage & Divorce Simulation The goal of this exercise to show the simulation of a “society”. In the larger context, it’s an example of how students might perform a simulation. Given a body of data, how do we arrange that data in order to represent how the society is behaving. This is essentially a “model” using the data. There are three ways we go about putting numerical values on this model.: 1.Given a series of equations, can we simply solve the equations? 2.If the equations don’t have a closed form solution, can we solve them recursively. There are no statistics involved here, but all we do is solve each equation over and over again and hope that it converges. This method gives us no details about the population since we’re simply solving equations. 3.We can try for a “real” simulation. In this case, we use the probabilities and a random generator to try to simulate good years and bad years. This allows us to answer much more complex situations. We could now track characteristics for each individual in our society. We could, possibly, see how long a person in our society stays married for instance.

Marriage & Divorce Statistics There’s lots of stuff on the web, confusing and maybe contradictory: All data is for the US (unless otherwise noted) In 2007, there were 2,200,000 marriages. This represents a rate of 7.5 per 1000 total population. Note this is 2.2M / 296M = 7.5. (Total US population is higher but some states don’t report.) Another metric which may be saying the same thing is that there are 39.9 marriages per 1000 single women. We’re going to use the first number here. In 2007, there were 856,000 divorces. This is 3.6 per 1000 total population. Interesting numbers, but not used here: 41% of 1 st marriages end in divorce. 60% of 2 nd marriages end in divorce. 74% of 3 rd marriages end in divorce. The average remarriage occurs 3.3 years after a divorce. In 2007 there were 2.400,000 deaths representing a rate of 8.2 per Details of this on next page. 60% of all marriages last until 1 partner dies Birth rate is 13.8 per 1,000 population Recent statistics say that 51% of the adult population is married. This is important because we don’t use it directly as one of our equations – we use it to test if our model gives approximately this answer.

Marriage & Divorce Statistics In 2007 there were 2.400,000 deaths representing a rate of 8.2 per thousand. Details on this mortality data are for men and women 65+ : Death rate for married man is defined as 1.00 Death rate for a widowed man is 1.06 times that of a married man. Death rate for a divorced or separated man is 1.14 times that of a married man. Death rate for a never-married man is 1.05 times that of a married man. Death rate for married woman is defined as 1.00 Death rate for widowed woman is defined as 1.15 Death rate for divorced or separated woman is defined as 1.26 Death rate for a never-married woman is 1.18 times that of a married woman. This information is from “US Mortality by Economic, Demographic, and Social Characteristics: The National Longitudinal Mortality Study”, Sorlie, Backlund, and Keller, 1995 We use a rate that’s above and below the 8.2 per 1000 for the national average to take into account single and married rates. DeathMarriedRate = 7.6 per 1000 DeathSingleRate = 8.7 per 1000

Zombie Single Married Reincarnation = 100% Death while Married Death while Single Birth Rate Marriage Rate Divorce Rate Widowed

Marriage & Divorce Equations Leaving Zombie:  Z = - R birth * ( S + M ) Entering Zombie:  Z = + R death-single * S + R death-married * M Leaving Single:  S = -2 * R marriage * ( S + M ) - R death-single * S Entering Single:  S = + R birth * ( S + M ) + 2 * R divorce * ( S + M ) + R death-married * M Leaving Married:  M= -2 * R divorce * ( S + M ) - R death-married * M Entering Married:  M= + 2 * R marriage * ( S + M ) In Steady State – leaving equals entering + R death-single * S + R death-married * M - R birth * ( S + M ) = 0 + R birth * ( S + M ) + 2 * R divorce * ( S + M ) + R death-married * M -2 * R marriage * ( S + M ) - R death-single * S = * R marriage * ( S + M ) - 2 * R divorce * ( S + M ) - R death-married * M = 0

Marriage & Divorce Equations In Steady State – leaving equals entering + R death-single * S + R death-married * M - R birth * ( S + M ) = 0 + R birth * ( S + M ) + 2 * R divorce * ( S + M ) + R death-married * M -2 * R marriage * ( S + M ) - R death-single * S = * R marriage * ( S + M ) - 2 * R divorce * ( S + M ) - R death-married * M = 0 Rearranging these equations gives: - R birth * ( S + M ) + R death-single * S + R death-married * M = 0 + R birth * ( S + M ) - 2 * R marriage * ( S + M ) + 2 * R divorce * ( S + M ) - R death-single * S + R death-married * M = * R marriage * ( S + M ) - 2 * R divorce * ( S + M ) - R death-married * M = 0 Maybe there’s a solution, but they seem redundant to me.