Monogomous wives?

Steve Landsburg has published a book called "More Sex is Safe Sex". I found out about this from econlog. Back in January 2005, I read the article that Landsburg wrote with the same title. That article no longer appears to be available. In any case, I had some questions for him, so I sent him this email:

I just your article titled "MORE SEX IS SAFER SEX". This has produced a series of questions that I have been unable to resolve on my own, and was wondering if you could provide some insite.

In your "monogamous wives" parable you mention that all the men demand two sexual partners and all the women demand one. As a consequence, prostitution starts up, and the prostitutes end up being the source of HIV for most of the men. Basically, if any one person in this country is infected, the prostitutes turn into the transmission point to ensure that everyone in the country becomes infected.

You argue that if the women were to take another partner, then prostitution will die, and most everyone will not be infected.

1. Question 1: Doesn't this assume that the prostitutes are the source of the disease?
   

   
   As example, I offer the following image, where men are represented by letters and women are represented by numbers. A & 1 are married, B & 2 are married, etc. A line indicates that they're having sex.
   

   
   A B C D E F G H
   | | | | | | | |
   1 2 3 4 5 6 7 8
   

   
   If we assume that each woman's additional partner will be the husband of her neighbor to the right, then we end up with this:
   

   
   A B C D E F G H A
   |/|/|/|/|/|/|/|/
   1 2 3 4 5 6 7 8
   

   
   Now as soon as any one of them gets infected, all of them will get infected. There's two ways that this isn't true. The first is if one of the infected people dies before they can propagate the infection. With the effectiveness of drugs at prolonging life with HIV, that seems like a bad assumption. The second way this isn't true is if the prostitutes are the original source of the infection. At which point everyone's demands are being met and there's no impetous for an external source of infection.


2. Question 2: What if the prostitutes aren't the source of the disease?
   

   
   Using my notation above, and assuming that the prostitutes aren't the source of the disease, I can construct a similar graph which optimally limits the spread of the disease and meets the assumption of 2 partners per person:
   

   
   A B C D E F G H
   |x| |x| |x| |x|
   1 2 3 4 5 6 7 8
   

   
   This additionally assumes that there's an even number of couples. In this case, the best solution is a fixed "wife-swapping" scenario. Now if any one of them gets infected, the infection will spread to only 3 other people (4 total)


3. Question 3: But if that's good, then why isn't monogamy even better?
   

   
   A B C D E F G H
   | | | | | | | |
   1 2 3 4 5 6 7 8
   

   
   If any one of the people get infected, then the spread is limited to only one other person. Yes, in order for this to work, men need to be monogamous, which violates the assumption that all men demand at least 2 partners. But doesn't this demonstrate that such a demand is part of the problem, and that monogamy is more effective at limiting the spread of the disease?


4. Question 4: What about other complications, like having children?
   

   
   Looking at my graph from Question 1 we get:
   

   
   A B C D E F G H A
   |/|/|/|/|/|/|/|/
   1 2 3 4 5 6 7 8
   

   
   Assume 16 children. The bigamist scenario is on the left and the monogamous scenario is on the right.
   

   
   A1 => boy(A1)   A1 => boy(A1)
   B1 => girl(B1)  A1 => girl(A1)
   B2 => boy(B2)   B2 => boy(B2)
   C2 => girl      B2 => girl(B2)
   C3 => boy       C3 => boy(C3)
   D3 => girl      C3 => girl(C3)
   D4 => boy       D4 => boy(D4)
   E4 => girl      D4 => girl(D4)
   E5 => boy       E5 => boy(E5)
   F5 => girl      E5 => girl(E5)
   F6 => boy       F6 => boy(F6)
   G6 => girl      F6 => girl(F6)
   G7 => boy       G7 => boy(G7)
   H7 => girl      G7 => girl(G7)
   H8 => boy       H8 => boy(H8)
   A8 => girl      H8 => girl(H8)
   

   
   Unfortunately, this bigamist scenariol creates a problem becuase the next generation has genetically close ties to more people than would be if each monogamous couple had two children, increasing the risk of birth defects in the 3rd generation. In the monogamous situation, a girl would avoid genetic problems if she avoided bearing the child of one person: her brother. But in this bigamist scenario, girl(B1) has boy(A1) as a half brother whom she has to avoid, and boy(B2) as a half brother whom she also has to avoid. Bigamy has doubled the riskiness factor for birth defects for the next generation.


5. Question 5: If we follow this through successive generations, don't we end up with a population of people who only have one safe mate in order to produce offspring? In other words, don't we end up right back at monogamy?


6. Question 6: Have I missed something fundamental that is keeping me from understanding what you're paper is saying?

I appreciate any clarity you'd be willing to provide.

Thanks,
mjh

He wrote back. He said:

Unlike some of the other arguments in the essay, the "monogamous wives" example is not meant to be terribly realistic but just to indicate, in the context of a very simple model, that certain things are logically possible.

In your question 1, the disease dies as soon as one person fails to pass it on, either through death or by missing an assignation. With the prostitutes, that's not true. And with the prostitutes, it doesn't matter where the disease starts; it still gets passed on to everyone pretty quickly.

The only point you've missed was that this particular example can yield a great variety of conclusions depending on your auxiliary assumptions (as you've shown), but the only point I was making with it was that certain conclusions are *possible*, not that they're *necessary*.

Of course other parts of the essay do make points that should hold in any model; in particular, when there are positive externalities to sexual activity, there can't be enough sexual activity without subsidies.

Perhaps I didn't understand his response, but I found it to be unsatisfactory. It seems to me that his model is unrealistic. Who cares if something is possible in a given model? If that model doesn't match reality then there's little value to the recommendations that come as a result of that model? What are the set of assumptions that are realistic? If my assumptions are closer to reality, then doesn't the recommendation that "More Sex is Safer Sex" actually hold false?