Day 35 in math modeling

The crazy:

Developing nations to study ways to dim sunshine, slow warming

• You have a homework for Friday (evaluating our current model).

• The Togolese have replied to our questions, meaning you have a homework for Wednesday: evaluate their responses, and comment on their responses and propose follow up questions (one page, submitted electronically to both me and to Maria McMahon).

• men proceed through five "grades", each eight years long (p. 314-315):

  1. Daballe ("infancy" -- Olinick spells it "Dabella"; Hoffmann "Daballe")

  2. Folle (party time! Seems like there could be some incidental fathers at this stage....)

  3. Kondalla (marriage and war)

  4. Luba (administration)

  5. Yuba (wise counsel)
  Their sons all enter the grade system at the moment the father exits -- so they will be of varying ages when they enter, forty years after their father did.

• The anthropological issue here is one of cultural stability: if the age variation within classes (especially the younger ones) is too great, there may be friction -- because older men will be forced to act like "infants", and younger boys may be too young to fulfill the roles expected of them (they may not be able to party as hearty as they should, for example).

  Under what conditions will the age-grade system be stable?

  So Hoffmann sought to address this issue of stability with a definition, an axiom, and two different models.

• Let's go through the model on pages 317 and 318 (or from Hoffmann's original paper (1965, p. 112 and following)).

• Upshot: the system is stable if dads have their first son at 40, on average.

  • A few comments

    • Hoffmann was responding to Prins's (perhaps imaginary!:) argument that a necessary condition for stability was that procreation be confined to the last two grades (p. 115 of the paper).

    • Is having children at the age of 40 realistic? Reflect on the society's grades and responsibilities....

      Ask someone who's had a child at the age of 40....

    • From a review of Prins's 1953 thesis:

      As the author [Prins] remarks, some social recognition of growth and aging is universal. In East Africa, however, this recognition of the aging process has developed into a complex social institution that should be called a social age-class system. To a degree, actual age is ignored in favor of other criteria for entering a given age class. Thus, all the sons of a Galla man, and all the sons of his brothers, enter the same age class which is forty years behind that of their fathers. The author has coined the terms "infra-puerilization" for cases where the actual years are less than their social years and "ultra-senectation" for those individuals who are too old for their age classes. The prohibition against fatherhood for the warrior class is seen as a device that reduces ultra-senectation where, as with the Galla, the social age separating father and son is so great.

      ....Each Galla male is supposed to go through five age classes, each of eight years' duration. The first two classes are for children, the third and half of the fourth are for warriors, while the second half of the fourth and the fifth are for elders. The forty-year interval rule discussed in the previous paragraph keeps all members of a given patrilineal line of the same generation in the same age class. [my emphasis]

• One problem: what about all those other sons?

  • "This model deals only with one-dimensional father-son links and ignores the branching of descent lines representing siblings. In other words, the model assumes that a man has only one son when, in fact, many men have several sons. Of course, some men have no sons, and this shows another weakness of the model. The model transforms every given family into points of future time when it may, in fact, no longer exist." (p. 319)

  • So how might address Olinick's concern?

  • Hoffmann gives a hint: "... it is not necessary that every age of parenthood equal 40, but merely that they fluctuate about a mean of 40."

    1. How do we make ages "fluctuate"?

    2. How many sons?

    3. What happens long term?
  • A Galla Simulation

• Hoffmann's model is predicated on this (fictitious) data for three states (13-19, 20-29, over 29). The data represents transitions from father to son's age at entry: \[ \left [ \begin{array}{c|ccc} {}&{S_1}&{S_2}&{S_3}\cr \hline {S_1}&{10}&{25}&{30}\cr {S_2}&{55}&{60}&{35}\cr {S_3}&{5}&{15}&{5} \end{array} \right ] \] Normalizing using the number of fathers in each row, we obtain the Markov matrix \[ \left [ \begin{array}{c|ccc} {}&{S_1}&{S_2}&{S_3}\cr \hline {S_1}&{10/65}&{25/65}&{30/65}\cr {S_2}&{55/150}&{60/150}&{35/150}\cr {S_3}&{5/25}&{15/25}&{5/25} \end{array} \right ] \] with starting vector \[ \left [ \begin{array}{c} {.25}\cr {.55}\cr {.20} \end{array} \right ] \]

  "If the Galla system is to survive, the set of males about to enter the grade system should consist largely of younger men. The set S₁ should contain a relatively large proportion of the set while S₃ should include a relatively smaller fraction. As new generations enter the system, there should not be a significant drift from S₁ to S₃." (p. 319)

• We know what to do with that: let's go to InsightMaker and see how it call comes out. We can edit our birthmother Markov models to carry out Hoffmann's example. It's regular, as were they: that is, every state can lead to every other state. (If you don't have one already made, you can clone mine.)

• "Criticisms of the Model"....

  • "It should be pointed out that the particular data he used were not obtained by an actual observation or census of the Gallas, but were chosen arbitrarily." (p. 321, bottom)

  • "For completeness, he could have included a state S₀, corresponding to those who were initiated before the age of 13." (p. 322)

  • I'll add one of my own. If this society's system is really stable, and if the system is Markovian, then the distribution as it stands should remain constant, because their society has been going for many generations. The distribution should already have reached equilibrium.

    Even supposing that that were the case, that the system had evolved to an equilibrium (fixed) vector of state population probabilities, it turns out that we can't assume that the Markov matrix is constant.

    We can't deduce the transition probabilities from the equilibrium:

    See this Markov chains non-uniqueness of transitions demo.

Eventually everyone ended up in the recovered class. This would therefore be the absorbing state.

• Absorbing states ($m$ of these), and

• Transient states ($n$ of these).

• The state graph

• The transition matrix, and

• The steady state solution.