A Family Tree Voter Model

Election Theory
#1

Let me begin by saying that, though I oppose ideological voter models in general, the two-dimensional model is least unrealistic. Why? Because the number of points a given distance away from a given point is proportional to the distance to the power of one less than the number of dimensions. Thus, only in two dimensions is the number of points a given distance away from a given point proportional to the distance. That’s consistent with kinship, in which the average individual’s number of relatives of a given coefficient of relationship is proportional to the reciprocal of the coefficient of relationship.

But the problem with ideological voter models is that they allow voters to inhabit an objective center, whereas in reality we’re all extremists, each the center of his own universe, with concern for others that diminishes as rapidly as kinship. Our parents are marginally more centrist, having twice our kinship with everyone but ourselves, but only Adam and Eve are truly centrist.

What I propose is a family tree voter model, in which a candidate’s utility is equal to the product of his fitness and his coefficient of relationship to the voter. It is unnecessary that a candidate actually have his coefficient of relationship, only that he acts as if he does (though it’s worth noting that we evidently prefer elderly candidates, verging on senility, which is hard to explain without a literal family tree model).

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#2

I think the idea of this is interesting, it would probably make for more versatile simulations if we produced random “ideology graphs.” It sort of generalizes the ideology space by allowing it to be an approximate manifold rather than a flat plane.

Still, I don’t think it’s necessary or actually that useful of a model. The ideology space should I think be a convex space. Philosophically speaking maybe it should be infinite-dimensional, but practically speaking there are probably only a few dimensions that are the most important for decision-making. You’re right that it is probably more than two, but the advantage of two is that we are able to visualize the situation. I would imagine probably stepping up to three dimensions would be the most natural approach for improvement, since we can still sort of visualize it and it is a more versatile model. I’m sure people have tried it and found diminishing returns.

#3

I agree, the effect of adding dimensions diminishes rapidly, but I never proposed adding dimensions or going non-convex or anything like that. I’m not proposing a Euclidean space of any kind. I’m proposing a family tree, which is easily visualizable and which there are strong a priori reasons to expect is more predictive: that the utility of other-investment is the product of responsiveness to investment and the coefficient of relationship is basic sociobiology; that utility loss is the Euclidean distance between points in issue-space is an insane hypothesis whose only support is that the impartial culture model is even worse.

#4

Some of the properties of the situation project into the model and do give useful information, at least about ranked choice systems. I agree that Yee Diagrams are abused, especially in the analysis of range systems.

However, a family tree model or a graph model could be simply embedded into a Euclidean space of high-enough dimension. Would you be able to produce a small example so that I can better understand what you are proposing?

#5

Picture a family tree. The voters are the nodes or a subset thereof (for instance, the last 2 generations). In the simplest model, each candidate would inhabit a single node and be defined by that node and his “fitness” (perhaps visualizable by the size of whatever mark is used to indicate that that candidate is at that node). The candidate’s utility, to a given voter, is the product of his fitness and his coefficient of relationship (CoR) to the voter (in the simplest model, CoR would simply be 1/2^x, where x is the length of the path from the voter to the candidate).

Polyphyletic candidates (i.e. candidates with polyphyletic coalitions) could be accommodated by allowing the candidate to inhabit multiple nodes, each of his “identities” having its own “fitness”, which represents how effectively he’s expected to serve the part of his coalition centered on that node. Paraphyly could be accommodated the same way or more simply, via identities with negative fitness (perhaps visualizable by the orientation of the mark) at the nodes central to the groups of voters the candidate neglects.

In the most complex version, each voter would have an exclusive utility for each candidate, representing how well the candidate is expected to serve that voter. But the voter would ultimately vote on the basis of the inclusive utility of the candidate, i.e. the sum of the products of the voters’ exclusive utilities of the candidate and their CoRs to the voter in question.

#6

I like the concept. I think it could give very interesting results to analyze. I personally think one could expand on the notion of a tree and work with a random graph over a large number of nodes, and evaluate candidate preference as a function of the minimal path between a voter and that candidate. One could then assign random weights to particular nodes corresponding to the fraction of voters of that state in the graph, and calculate very standard measures of disappointment, etc.

That would I think work very well for ordinal systems, but cardinal systems would still demand a model for assigning scores.

#7

I think I’m taking the model too literally, but how would you explain people whose political views diverge significantly from those of their parents?

#8

There are a couple of ways to explain that.

The first is that our genes diverge significantly from those of our parents. The difference between 1 (my CoR with myself) and 1/2 (my CoR with my mother) is very significant, especially when it comes to what I call polyphyletic candidates.

For example, historically first-born sons of the nobility would inherit everything, including, naturally, the ideology that apologized for their privilege. Their brothers would join the Church and adopt the ideology of and for the Church. A candidate representing the nobility would thus be a polyphyletic candidate, representing the left-most (in the genealogical sense) noble lines; a candidate representing the Church would be a polyphyletic candidate representing the remainders of the ancestrally noble families. A second-born son would back the Church candidate quite simply because he cares more about himself than his father or his eldest brother.

The second way to explain it is ideological division of labor. My divergent ideology is only a betrayal of my mother if we assume my mother’s ideology is the one that serves her inclusive interest. But what if what serves her inclusive interest is the composite family ideology, in which her feminism cancels out my anti-feminism, leaving only our shared white supremacism? Why don’t we just both be neither feminist nor anti-feminist? Well, because when she dies a neutral position is no longer in her interest. Her interest becomes the interests of her children, males who are better off under an anti-feminist regime. It’s just easier to let nature take its course than to have to change one’s ideology whenever a family member dies.