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Friday, January 6, 2017

Our better halves




This post is a first draft of a paper that I'm working on. Comments and criticism are both welcome.

The secular rise in IQ scores was noticed by James R. Flynn (1984, 1987) and “popularized” in The Bell Curve (1994) as the “Flynn Effect.”  Offered explanations include an increase in the use of abstract problem-solving ability, improved nutrition, a trend towards smaller families, better education, environmental complexity, and heterosis. They Flynn effect was first discovered by Richard Lynn (1982) while comparing Japanese and US demographic data.  It was later rediscovered by James Flynn. (1984)  A strong summary of the Lynn Flynn effect and the remaining unexplained variance was offered by Williams (2011).  After close to 100 articles have attempted to explain the effect and have succeeded in only describing less than half of the total impact, it was dubbed “officially mysterious” by Deary (2001).
Jensen demonstrated that heritable intelligence changes load on “g” which was then demonstrated to be necessarily the basis for the Flynn effect by J te Nijenhuis & der Flier (2013).  This isolation demonstrates that social effects are themselves insufficient to explain the effect.  Jensen turned towards environmental factors, claiming that in order for breeding to explain the produced magnitudes observed in the Flynn Effect that breeding would have to be restricted to people in the upper half of the IQ distribution.  Though he did not name it, this “straight line rule” was referenced by Bob Williams (2011) as the reason to reject preferential mating as explaining the Flynn Effect.  Jensen does not detail his method to arrive at this number, yet the properties of the normal curve are clear: to see a 15 point increase in the average score of a normal distribution, everyone within a population beneath 106 points would need to be removed.  Over three generations, the same effect could be accomplished much more modestly by removing all members from a population whose intelligence is beneath 88 with re-normalization each generation.  This is equivalent to eliminating 21% of the population in each of three rounds and suggests that the problem may be more tractable than Jensen had figured.
In their 1998 review of algorithmic mate selection, Miller and Todd identify that a “major challenge for mate choice research is to develop [...] the search strategies that people follow in trying to form mutually attracted pairs.”  This algorithmic limitation is at odds with social observation: Darwin’s theory of natural selection (1871) has “selection” in the title, yet models of assortative mating with ongoing competition generally use a predation mechanism in preference to a sexual-selection process.  Though single sided selection models do occur with ducks, squid, and dolphins, (Brownmiller 1975, Thornhill & Thornhill 1992) rape is the exception and not the rule in human societies.  
Assortative mating models and agent-based computation demography (ABCD) were reviewed by Evert van Imhoff and Wendy Post (1998) who lay out an ontology of agent-based mating models.  Our approach aligns most closely to microsimulation, sharing the components gender (m and f), age (a), aging (a+1), mortality, and fertility.  Within this class of mating models they discuss sorting methods, monte Carlo draws until a certain number of pairs have been created (a forcing method), and collision models.  Looking solely to mate choice, Simao and Todd (2002) consider four kinds of mating strategies in which agents consider multiple partners then seek to court the potential partner of the highest quality which are primarily concerned with how long partners ought to date before seeking marriage based on quality perceptions and aspiration-setting strategies.   Their most performant algorithm is one where when a partner is left, they set their aspiration to be lower than the partner who left them which forms the basis for the relaxation rule I will propose.
Many of the mate selection models seek to model observed behaviors where “like marries like,” known as Assortative Mating.  A representative example, Todd and Billari (2003) consider satisficing heuristics.  This was the basis for an agent-based model known as MADAM (Marriage and Divorce Annealing Model) (Hills & Todd, 2008) which combines a desire for homophily and gradual relaxation of standards.  A potentially exhaustive consideration of these was provided by Walker and Davis (2013).  Additional approaches include “market clearing” models such as those proposed by Kalick & Hamilton (1986) and mating via costly signaling (Hoppe, Moldovanu & Sela, 2005). These models are in wide use in considering population dynamics (Frisell et al., 2012) as they are tractable to linear regression in that they explain the influence of one’s parents without considering those who did not become parents.  Assortative Mating models do not lead to changes in population means since their objective function implies a preference for homophily which lead less objectively desirable agents to prefer similarly undesirable agents.  This approach has some support in early work that saw homophily a predicting marriage “success” (Burgess & Cottrell 1939) yet is unsuitable for any work that seeks to understand the attributes of those who were not married.
Brozovsky & Petricek’s (2007) work sought to predict whether or not a searcher would prefer a profile versus others.  Though they did not position their work as seeking to understand whether or not there was a preference for assortative mating, they identified that the mean rating of viewers outperforms known algorithms 35.62% of the time.  This finding is supportive of the claim that quality is universally agreed upon - as is prerequisite for any beauty contest to be possible.  Since their work also correlates with user behavior, it also extends to indicating that desirability is a meaningful input to dating behaviors.  Though their data included interactions, they concluded by indicating that there is a “need for reciprocal matching algorithms.”  One such algorithm is considered here.
Model one: The Optimal Stopping Rule:
In “Algorithms to Live By: The Computer Science of Human Decisions”, Tom Griffiths states that the right approach to dating is a “solved problem”, saying that each of us should date 37% of the available pool and then marry the first person who is the best that we’ve seen.  He brags that you “don’t need a therapist, [you] need an algorithm [to] find the right, comfortable balance between impulsivity and overthinking.  The algorithm tells [you] the balance is thirty-seven percent.”
This is a port from Merrill Flood’s solution to the Secretary Problem, presented in Martin Gardner’s Mathematical Games column in 1960.  It imagines a single position available, a known number of applicants, random screening order, relative rank revealed on consideration, and a rule that once someone is rejected they cannot be recalled.  This problem makes a number of assumptions:
  1. Your sole objective function is to marry the sole best candidate
  2. Search is costless and time spent married has no value
  3. All potential candidates will accept your entreaties; this is a one-sided market
  4. You are not in competition with any other suitors
The combination of these rules explain the 37% rule - and also why the advice that Griffiths gives is so inappropriate: if any of those assumptions is wrong, then the rule also is wrong.  Since each of these assumptions is incorrect, the advice itself is popular yet wildly incorrect. And yet, it was used by Kepler who called it the “Marriage Problem” after the death of his wife, has been presented as optimal by Robert Krulwich on NPR (2014), and is the basis of a class of optimal stopping rules.  It was used by Rapoport and Tversky (1970) alongside a complete distribution of item quality to understand human behaviors.  Kogut (1990) proceeded in the same way.  
Dating, however, is somewhat famous for having a property known to mathematicians “negative drift,” known to readers of Teen Vogue as “all the good ones are taken.”  In their consideration of stopping rules, Dvoretzky, Kiefer and Wolfowitz (1956) show that this requires each round face a discrete maximization rule where choosers adopt an optimal stopping rule equal to the minimum of their current round offer and their future expectation.  In solving this, their consideration reduces using “s”, a discount which is equal to the expectation of the expected return E(M-EM)+.  This discount could contain decreasing expectations of being accepted as well as decreasing quality of the available populace.   
However, in extending this to markets of unknown size, Seale and Rapoport (1997) found the problem “mathematically intractable” and instead simulated different decision rules.   In their simulation, they found that decision makers in laboratory environments make decisions as though they are following a cut-off rule but that they are biased towards early stopping, where they claim in their discussion that people are too impatient because they assign a cost to search.  Since mathematical intractability is thus likely inevitable, we proceed by modeling decision rules.
In our initial experiment (contained in Appendix One), we modeled a population of 20,000 eligible persons, evenly split between those choosing and those being chosen.  In these populations we model a high type and a low type, each occurring with 50% probability.  We specify the decision rule as a chooser selecting any partner that is at least as good as the best that they’ve seen after “n” draws.  When n=0, all choosers choose the first candidate that they consider, all candidates are chosen, and the chance of getting a high type = n/2.  When n=1, nobody gets matched in the first round, 1/4 of the low types are matched in the second round, ⅜ of the candidates (all low types) are not matched, and the chance of getting a high type is ⅘.  When n=2, ⅛ of the low types are matched in the third round.  These results are simply explained by recognizing that the chance that someone hasn’t seen a high type is equal to (1/p(low))^n.  As n approaches infinity, the fraction of low types selected approaches zero and all high types are immediate selected.  Since this decision rule is analogous to a “better than average” rule, this exactly replicates the Jensen rule whereby half of the population does not breed.  In contrast to Jensen’s hypothetical, this is accomplished via an emergent cut-off at the median, not a draconian eugenics program.
When the market is extended to a two-sided market where both sides must affirmatively select the other, low types being selected is reduced significantly.  It still remains the case that all high types immediately marry, though (1 / PlowN) do marry a low type.
When “high types” and “low types” are replaced by a draw from a normal distribution, the model becomes mathematically intractable.  However, it is the case that roughly half as many people pair relative to the single-sided approach.  The average quality of those pairing off is significantly higher than the mean: the average paired-off quality in the first round is 108.3 across a total population of 20,000 with n=1.  In following rounds, the average quality of those paired increases to 113.6.  
Perhaps obviously, when the decision rule is that people should see 37% of the field and then only marry the best, there is a 13.7% chance that a single marriage of “the bests” takes place in a population of 10,000 within one million periods: in essence, the 37% chance success rate of a single individual seeking to marry the best translates into a 37% chance of a 37% chance of that marriage taking place.  While there is a single normative rule that suggests this, this demonstrates the unsuitability for the traditionally constrained optimal stopping rule for mating markets.
Taken together, these experiments demonstrate that even without a clearly identified average, individual agents seeking to choose the “best they’ve seen” can result in dramatic changes in population mean quality.  Utilizing the optimal stopping rule translates into stochastic filtering with mean of 108 within a population with N=1.  With regular renormalization, mean skew is additive across generations, meaning that the three generations elapsed in the time in which the Flynn effect has been observed could result in a net population intelligence of 124, assuming that regression to the mean is likewise adjusted.  These rules, however are likely unrealistic in that they describe a population in which roughly two-thirds of those eligible to mate do not do so.  This suggests that while assortative mating models fail to capture individual preferences for universal quality the optimal stopping model also fails to capture some important dynamics in mate selection.  Despite these failings, I utilize comparative stopping rules as the basis for modeling non-assortative choice.


Model two: Adaptive rules and population dynamics
In the second model (Complete code contained in Appendix 2), additional complexity is added by a second quality factor, multi-period mating, and breeding to renew the pool and to model genetic evolution.  An individual is comprised of a “Brains” score and a “Brawn” score.  “Brains” and “Brawn” are both normally distributed scores with mean=100 and s.dev=15.  Further, each agent has a preference for “Brains” uniformly distributed between zero and one and a reciprocal interest for “Brawn.”  Each agent assesses the quality of a potential partner using a simple mixture model where they weight their brains with their preference for brains plus their preference for brawn times the “Brawn” score.  Finally, each agent is initiated with an age, in months, between 0 and 924.  Agents older than 192 months begin dating.  Dating proceeds each month by randomly matching all male & unmatched agents with all female & unmatched agents.  In the basic model, each agent utilizes a stopping rule whereby they date at least 3 others and then seek to marry the first person whom they date whose assessed.  After “marriage”, partners will breed with Monte Carlo likelihood, “l.”  Likelihood will be experimentally varied.  Children directly inherit the average of their parents preferences for brains or brawn while their own brains and brawn are comprised of 40% of the father’s attribute, 40% of their mother’s attribute, and 20% an additional draw from the standard distribution, representing regression towards the mean and roughly in-line with the inheritability of intelligence.  
As demonstrated by model one, this results in a brittle population where selectivity leads populations inexorably downwards.  When considering only a single attribute (Brains in the graph below), starting with a population of 10,000, no trial survived a generation of burn-in time and ninety years.  Across 10 trials, no population survived 3,000 months (166 years) with the majority (~89%) of the initial population never breeding.
Groups which have a preference for both brains and brawn benefit from additional heterodoxy, leading to multiple sub-populations which occasionally inter-breed but whose lower conditional quality leads them to be less interesting to other populations.  This is equivalent in practice to lowering “N” and reducing learning.  The addition of a second characteristic enabled some populations of 10,000 to survive as long as 2800 months (235 years)
Since the two-sided version of the stopping rule tends to limit populations, a relaxation term is added.  This term is the percentage within the best that a person has seen that will lead them to positively choose a potential partner.  Population dynamics are extremely sensitive to this term; with a term as high as 2.5%, populations become homogenous, losing the spread that is characteristic of intelligence.  With scores until 1%, populations remain at risk. This leads to stable population characteristics.  This relaxation term suggests that in most marriages at least one partner is settling somewhat.  With this relaxation term, populations are highly stable and evidence regression towards the mean with stable preferences.
This bounding creates a “Goldilocks zone” between 1.0% and 1.5% relaxation where populations are stable.  Thus, I assume a 1.25% relaxation bound.
Scenario: environmental benefit to intelligence
None of these models have indicated what environmental or social change would indicate that these effects would take place recently or since intelligence scores were recorded.  Indeed, it is beyond possibility that we might have seen the mean intelligence of mankind increasing one standard deviation each century since Socrates.  Lead abatement in the Port Pirie Cohort Study showed increases in the cognitive performance of children equivalent to an increase of 5.3 points when lead was decreased from 30 micrograms to 10 micrograms (Baghurst et al., 1992).  Though the mean decrease in lead from lead-based paint hazard remediation and soil abatement was only 4.2 micrograms, it may be that this abatement passed a critical threshold or that changes in nutrition may have made up the difference.  For the benefit of demonstration, we model a 5 point increase in the intelligence of all children born after period 1200 in a population of 10,000 with a 1.25% relaxation term.  
This exogenous shock results in a significant change in both the mean behavior and the tendency of the population.  By conditionally increasing the quality of children based on the preference for intelligence of the general population, the addition of these new and smarter children has the effect of also increasing the preference for intelligence within the population.  This tendency offsets the usual regression to the mean.  The further benefit to future generations leads to an outsized growth in the underlying characteristic as the benefit from lead abatement is compounded within the population.  
To isolate the impact of the change in preference, the model is adjusted such that the increase in intelligence only takes place once within the population.  Since the regression towards the mean regresses towards a mean which is no longer applicable, this tendency is removed in this data in order to isolate the one-time effect of an environmental benefit to intelligence.  The net effect is that small but significant increases in preferences for intelligence which were caused by the one-time increase in selectability from increased intelligence “breed true” and create more stable (and smarter) populations over time.  The combination of slight relaxation of the secretary rule, a one-time 5-point increase in intelligence, and multi-attribute selection lead to a simplified model that demonstrates an 18-point increase in population intelligence, greater than that of the Flynn effect.


Discussion:
Though Darwin’s theory of natural selection was based more on selection than predation, popular conceptions of evolution are based on fitness, predation, and extinction.  With the world becoming increasingly safe, these factors are becoming increasingly unimportant for the consideration of human society.  As a consequence, social scientists have increasing need for preference-based mating models.  Current models specify models that either seek to “clear the market” through assortative choice or assume a single decision-maker for mathematical tractability.  By modeling different decision strategies and docking to real-world data we should be able to understand how preferential mating might lead to changes within populations.
In pursuing this work, there have been several instances where discussions of this work returned to unproven biases.  The most frequent was the belief that since people with lower socioeconomic status are more likely to breed, it is claimed that those with less intelligence are breeding as well.  This model would consider socioeconomic status as “brawn”, which would tend to suggest that in contrast to the naive assumption that “stupid people are breeding”, the converse is true: those who breed tend to have some quality that leads to them being selected by their mates.  If correct, this naive assumption can lead to misinformed policy, social unrest, and significant misunderstanding.
Decision rules such as the optimal stopping rule have been suggested and defended as optimal heuristics by Griffiths and Giggerenzer making frequent appearances in popular press.  Indeed, my initial pursuit of this project was misinformed by their guidance.  Recognizing that dating markets are competitive and two-sided may show what appears to be a biased impatience as a performative heuristic and may have normative consequences.
These decision rules require the formation of a scoring model.  Models that imagine men as benefitting most from having the largest number of children and women requiring a quality threshold are easily formulated but are more appropriate in modeling species which don’t have social safety nets.  Determining what optimality is from the perspective of an individual, society, or gene will each create different scoring models and different normative recommendations.

References:


Baghurst, Peter A., et al. "Environmental exposure to lead and children's intelligence at the age of seven years: the Port Pirie Cohort Study." New England Journal of Medicine 327.18 (1992): 1279-1284.


Brozovsky, Lukas, and Vaclav Petricek. "Recommender system for online dating service." arXiv preprint cs/0703042 (2007).


Burgess, Ernest Watson, and Leonard S. Cottrell Jr. "Predicting success or failure in marriage." (1939).7


Christian, Brian and Griffiths, Tom. Algorithms to Live By: The Computer Science of Human Decisions (2016)


Dvoretzky, Aryeh, Jack Kiefer, and Jacob Wolfowitz. "Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator." The Annals of Mathematical Statistics (1956): 642-669.


Frisell, Thomas, et al. "Heritability, assortative mating and gender differences in violent crime: results from a total population sample using twin, adoption, and sibling models." Behavior genetics 42.1 (2012): 3-18.


Krulwich, Robert. “How to Marry The Right Girl: A Mathematical Solution.”  Robert Krulwich on Science May 15, 2104  


Miller, Geoffrey F., and Peter M. Todd. "Mate choice turns cognitive." Trends in cognitive sciences 2.5 (1998): 190-198.


Seale, Darryl A., and Amnon Rapoport. "Optimal stopping behavior with relative ranks: The secretary problem with unknown population size." Journal of Behavioral Decision Making 13.4 (2000): 391.
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te Nijenhuis, Jan, and Henk van der Flier. "Is the Flynn effect on g?: A meta-analysis." Intelligence 41.6 (2013): 802-807.


Todd, Peter M., and Francesco C. Billari. "Population-wide marriage patterns produced by individual mate-search heuristics." Agent-based computational demography. Physica-Verlag HD, 2003. 117-137.


Van Imhoff, Evert, and Wendy Post. "Microsimulation methods for population projection." Population: An English Selection (1998): 97-138.


Walker, Lyndon, and Peter Davis. "Modelling" marriage markets": A population-scale implementation and parameter test." Journal of Artificial Societies and Social Simulation 16.1 (2013): 6.
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Williams, Bob. "Understanding the Flynn Effect." (2011).