2500+ words Essay

Partner search

7th October 2019

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Theory: summary  As long as men and women can rank each other,

there exists a stable matching (Gale & Shapley, 1962)

 If M or F all rank the other sex exactly the same way, there will be stable & unique matching (Eeckhout, 2000)

 If partners can bargain over the gains from relationship (transferable utility), the matching will maximize total utility across all couples – (Shapley & Shubik, 1972; Becker, 1973)

 This means it is globally optimal  If the utility fn is supermodular, the unique stable

matching will also be positively assortative (i.e. “likes attract”) as partners seek complements to each other (Becker, 1973)

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Evidence on homogamy

Table 3 – Marriage Homogamy (in %) of Recently Married Young Couples by:

Census year   race nativity literacy all three age all four

1900 actual 99.9 94.2 91.1 85.3 80.7 69.4

random cf. 78.1 74.3 83.8 51.1 44.4 22.7

1910 actual 99.9 94.8 92.1 86.9 81.2 71.0

random cf. 79.5 70.0 86.8 50.3 44.5 22.4

1930 actual 99.9 95.1 96.9 92.0 82.9 76.6

random cf. 80.1 82.3 94.7 62.6 43.2 27.0

Note: “Recently married young couples” are couples married for 5 years or less and under 30 years of age. The rows denoted “actual” show the percentage of couples matched on each characteristic in each census year. Rows denoted “random cf” show the rate of matching in each year and column that would result if the same populations of men and women were matched randomly.

Source: Cvrcek (2012)

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High-school dating PAM

Source: McElroy et al (2010)

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Other correlations

 5000+ online daters

 PAM on all (if weak for some)

 Source: Hitsch et al. (2010)

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So: should you date out of your league?

 Strong evidence that on many chars, people look for someone like them

 Heterogamous marriages are less stable  Race difference, age difference, even

religious difference are strong predictors of divorce

 Sudden changes in characteristics (fame, wealth – or unemployment) also predict divorce

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And yet: HotOrNot.com strikes back

Source: Lee, Leuwenstein et al. (2008)

And yet: Dating out of one’s league is certainly attempted by some

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And yet: Hitsch et al. (2010)

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Why do men try their luck more often?

 Men are more likely to try to start a relationship with women out of their league than women do with men

 Type I error: accepting false hypothesis  “She is so obviously into me!” => makes a

move  Type II error: rejecting true hypothesis

 “I’m probably reading too much into her behavior.” => does not make a move

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HotOrNot.com – date requests

Source: Lee, Leuwenstein et al. (2008)

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Why do men try their luck more often?

 Type I error (believing she’s interested) has low costs and large potential benefits for him

 Type II error = opportunity forgone  Logical conclusion:

 Type II error is not very likely among men  If he’s not calling you, don’t kid yourself he’s

committing Type II error (“he did not pick up on my signals”)

 He did pick up on them, he just doesn’t like you

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The search problem

 To some extent, all first dates are blind dates

 We date/court each other precisely to learn about our partner’s quality / trait / joint U

 Meeting new people and getting to know them is therefore costly

 At least we know what we’re looking for: if PAM applies, we want someone similar

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Finding Mr Right (m*)

m*

U(marriage) = Bliss – (m – f)2

U(single )

range of men who’d be acceptable relative to staying single

partner trait

U

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Finding Mr Right (m*)

m*

U(single )

range of men who’d be acceptable relative to staying single

partner trait

U/f(m)

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Let the search begin  Assume the woman undertakes some costly

action to meet men (e.g. she goes out)  Here: the costs of partner search are not zero

 As a result she meets some given number (X) of men

 They are drawn from the blue distribution  Only the most extreme-value men are not

acceptable  The range of acceptables is very wide

 Ergo, chances of meeting an acceptable man are high

 She will likely come home with a date

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Finding Mr Right (m*)

m*

U(single )

m1

U(m1)

Increase in U

Her new boyfriend, m1 range of men who’d be acceptable relative to

dating m1

partner trait

U/f(m)

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She goes out again, why not!

m* m1

U(m1)

Increase in U with m2

m2

U(m2)

range of men who’d be acceptable

relative to dating m2

U(single )

Increase in U with m1

Her EX- boyfriend, m1

Her new boyfriend, m2

partner trait

U/f(m)

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How’s the search going?

 m2 is a definite improvement on that weirdo, m1

 Her utility is increasing as she moves closer to m*  But the increments are getting smaller

 The range of acceptable men is shrinking  Therefore the chances of bumping into an

“upgrade” are getting slimmer  This is because her threshold (“standards”) are

rising

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Third time a charm!

m*

U(m1)

Increase in U with m2

m2

U(m2)

U(single )

Increase in U with m1

Her EX- boyfriend, m2

Her new boyfriend, m3

U(m3)

m3

range of men

who’d be acceptabl

e relative to dating m3

Increase in U with m3

partner trait

U/f(m)

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The emerging patterns

 Extra gain from search is falling  (each successive man is an improvement

but an ever smaller improvement)  Marginal cost of search is rising

 The acceptable range is getting narrower  This means having to check out more men

before she bumps into an improvement

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The optimal search effort

Distance to m*

MC of search are increasing

m*

MC = MB

mMC=MB quit searching, settle down

MB of search are declining

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Mr Right vs Mr Right-now

 There is optimal amount of search to be undertaken

 MC = MB determines the optimal moment to stop and marry your mcurrent

 m* is obviously unattainable (MC ≠ 0)  Some amount of mismatch is inevitable  (unless you bump into Mr Right by dumb

luck)

Big-picture conclusion: Whomever you marry,

you could always find someone who would make you happier

than your spouse. It’s just that it’s not worth looking.

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If my trait is not typical…

m*m**

U(single )

Orange’s optimal trait is less common than purple’s: higher search costs.

partner trait

U/f(m)

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If I am more picky…

m*m**

U(single )

Orange’s utility function is also more compressed. This means higher MB from any given change in m.

M B

M B

partner trait

U/f(m)

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The optimal search effort

Distance to m*

0

MC = MB

mMC=MB mMC=MB

Orange is farther from her ideal man.

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Determinants of the MB/MC

 What shifts the MB curve:  Gains from marriage  Shape of the utility fn (how important the

trait is)  What shifts the MC curve:

 Technical constraints (e.g. geographic distance)

 Social constraints (e.g. gender segregation)  Changes in population characteristics

 Implications for likelihood of divorce

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Extensive vs Intensive search

 Extensive search = meeting new people  a.k.a. hanging out, bar-hopping, going to a

Startrek convention  Search over traits that are readily observable

(physical characteristics, basic social skills)  Intensive search = getting to know them in

depth  a.k.a. dating, spending some quality time  Estimating more precisely the value of a trait

(potential parental skill, earning potential, love- making skills, cooking skills, views on politics)

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Learning-by-dating

 We date to discover info about the other person  If we obtain unfavorable info => break-up  If we obtain favorable info => good

 But:  It takes time (which is costly: clock is

ticking)  Info, too, is subject to diminishing returns

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Model: Discovering his type

 Assume men come in two varieties:  Cads: nice 60% of time, mean 40% of time  Lads: nice 90% of time, mean 10% of time  Overall, 80% of men are Lads (λ=0.8), 20% are

Cads  Problem: who is who?  Solution: observe their behavior

 Each date, they either play nice or mean  Nice dates are more likely with a Lad…  True type will show eventually (more or less)

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Bayesian updating  For Cads (20% of

men):  P(Nice|Cad) = 0.6  P(Mean|Cad) = 0.4

 For Lads (80% of men):  P(Nice|Lad) = 0.9  P(Mean|Lad) = 0.1

 A woman’s problem is to infer the opposite:  P(Lad|Nice) = ?

Mean Nice

Cad 0.2*0.4 = 0.08 0.2*0.6 = 0.12

Lad 0.8*0.1 = 0.08 0.8*0.9 = 0.72

Σ 0.16 0.84 857.0

7

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12.072.0

72.0

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CNPLNP

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NiceLadP

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Bayesian updating

 How does the learning work?  She goes out with a random guy  Initially, she assumes he’s a lad with 0.8 chance  When the date goes well (he’s nice), she re-

evaluates chances of his type (Lad) upwards  Here:

 0.8 was her prior probability  0.857 is her posterior probability

 If he acted mean, she’d reevaluate P(L|M) = 0.5 < 0.8

Mean Nice

Cad 0.08 0.12

Lad 0.08 0.72

Σ 0.16 0.84

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How the learning/dating progresses

0.80

0.86

0.50

0.90

0.60

0.20

0.93

0.69

0.27

0.06

0.95

0.77

0.36

0.09

0.02

First date

First sleepove

r

First family dinner

First vacation

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How will she respond to new info?

0.80

0.86

0.50

0.90

0.60

0.20

0.93

0.69

0.27

0.06

0.95

0.77

0.36

0.09

0.02

0.97

0.84

1. Dump him if P(L|…)<0.8 (Why? Even a random guy is better.)

2. Marry him if P(L|…) is high (But how high?)

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Three-way decision  At any point, she can

 Break-up with him – if P(L|…) < 0.8  Keep dating him – if P(L|…) > 0.8 but not by much  Seal the deal/Tie the knot – if P(L|..) is high

 Marry vs Date will depend on expected utility  EU(marriage) = P(L)UL +P(C)UC  EU(break-up) = EU(first date with random guy)  EU(keep dating) = EU(how our next date pans out)  Utility of being/remaining single U0 = 0

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Decisions about the first date

 Assume:  UL = 20  UC = -100  δ = 0.95

 Before first date:  U0 = 0  EU(M) = P(L)UL +P(C)UC =

0.8*20 + 0.2*(-100) = - 4  EU(keep dating) = EUD0 >

0  The best option is to try

him out

0.80 Mean Nice

Cad 0.08 0.12

Lad 0.08 0.72

Σ 0.16 0.84

P N = 0

.84

P M = 0.16

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After the first date…

-4

2.9

-40

8

-28

11.7

-16.9

14.4

-7.4

16.2

0.2

EUM1 = P(L)UL + P(C)UC =

= 0.857*20 + 0.133*(-100) = 2.9

EU(B) = EUD0 > 0

EUD1 = δ[PN2EUD2 + (1-PN2)EUD0]

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As dating progresses…  The higher the P(L) the higher the expected

utility from marriage  This makes sense: it looks ever more likely that he

is a lad and the marriage will be happy  But the increments to EUMt get ever smaller

 That’s the diminishing returns to new info  At any point t, the questions is:

 EUMt = P(Lt)UL + P(Ct)UC > EUDt = δ[PNt+1max(EUMt+1;EUDt+1)+ (1- PNt+1)max(EUDt+1;EUD0)]

 In short: is the expected utility from extra dating higher than from getting married today?  If not, get married now. If yes, get another round.

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Solving the model

0.80 0.86 0.90 0.93 0.95 0.97

0.88 0.920.50 0.60 0.69 0.77 0.84

0.56 0.66

0.95

0.74 0.81

0.96

0.87

0.97

0.91

Solving for cutoff: μ = 0.965; EUMt = 15.8

This is moment when she decides that the extra waiting is not worth it and agrees to marriage.

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EUMt = P(Lt)UL + P(Ct)UC > EUDt+1 = δ[PNt+1EUDt+1 + (1- PNt+1)EUD0]

 So what does the decision depend on?  Utilities from marriage: UL and UC  Impatience: δ  Cad/Lad mix in the population: λ  How different the two types are: P(N|C) vs P(N|L)

 Predicted effects:  Higher UL will speed up marriage  Low UC will slow down marriage (caution)  Greater impatience (lower δ) will speed up

marriage  Closer types will make the search more difficult

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Solving the model with δ = 0.9

0.80 0.86 0.90 0.93 0.95 0.97

0.88 0.920.50 0.60 0.69 0.77 0.84

0.56 0.66

0.95

0.74 0.81

0.96

0.87

0.97

0.91

Solving for cutoff: μ = 0.941; EUMt = 12.9

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Solving the model – changing P(N|C)

0.80 0.818 0.835 0.851 0.865 0.878

0.800.67 0.69 0.72 0.74 0.76 0.82

0.70

0.837

0.72

0.851

0.74

Solving for cutoff: μ = 0.91; EUMt = 9.2

0.81

0.89 0.91

0.67

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Process of intensive search

Amount of intensive search (e.g. time)

trai t

m*

Similar logic for intensive search: • MB decline with search • MC increase (it is getting harder to learn new

stuff)

What shifts the MB and MC curves? • Importance of a trait (increases MB across the

board) • Number of traits to zero in on (increases costs) • Social constraints (e.g. is premarital sex taboo?) • Ease of exit from relationship (divorce)

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Theory: a summary  PAM/NAM may not obtain to the same degree

as predicted by frictionless models  Multiple equilibria may obtain (depending on

luck)  Matches will have some measure of

instability  Even couples who put high likelihood on their

eventual break-up may currently date  Age at marriage will depend on search costs  Marriage rate will depend on search costs  Divorce/break-up rate will depend on search

costs  Timing of divorce/break-up: clustered early

on

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The record of the 20th century Whites, aged 25 - 34, by birth decade

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

1866-75 1876-85 1886-95 1896-1905 1906-15 1916-25 1926-35 1936-45 1946-55 1956-65 1966-75

Never married men Never married women Never in union men Never in union women

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Can search costs account for any of the 20th century trend?

Whites, aged 25 - 34, by birth decade

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

1866-75 1876-85 1886-95 1896-1905 1906-15 1916-25 1926-35 1936-45 1946-55 1956-65 1966-75

Never married men Never married women Never in union men Never in union women

 Extensive search has become easier  Easier communication, higher mobility, co-

ed college  It is less time-consuming  Works to speed up marriage

 Intensive search…? Note: Harder intensive search tends to speed up marriage (why keep dating if I cannot learn anything new anyway?) as long as the benefits from marriage are high

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A closer look at early 20th century

 A time of:  Rising employment of young women  Rising urbanization  High and then low immigration  Falling age at first marriage  Emergence of dating as a search strategy  (Slowly) rising divorce rate

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Defining the right trait/match

 Assume people look for someone of similar:  Age (assume 0 < M – W < 10)  Race (assume perfect PAM)  Education level (assume PAM on literacy)  Nativity (Assume PAM with some overlapping)  Location (within one’s county)

 Example: SWF 19 literate Irish immigrant in Pickens co. seeks a SWM 19-29 literate immigrant or Irish American from Pickens co.

 Estimate the search cost as the (inverse of) the proportion of potential matches in the general population

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Rare crossing of ethnic lines

Table 4 - Marriage Homogamy (in%) Among Recently Married Couples by Nativity and Census Year

Census year 1880 1900 1910 1920 1930

    White men

US-born with at least one US-born parent 97.1 96.8 96.6 96.8 97.1

2nd generation immigrant 93.4 94.0 94.1 94.2 94.6

Immigrant 84.7 83.4 84.0 82.9 82.2        

White women

US-born with at least one US-born parent 94.9 95.0 95.4 95.5 96.0

2nd generation immigrant 88.9 91.2 91.7 91.1 92.0

Immigrant 90.6 88.2 87.5 87.1 86.0 Note: “Recently married couples” are couples married for 5 years or less. Source: Cvrcek (2010)

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Figure 6 - Mean trait by age and year for men

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

age

white 1880 white 1900

white 1910 white 1920 white 1930 black 1800 black 1900 black 1910 black 1920 black 1930

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Figure 7 - Mean trait by age and year for women

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

16 17 18 19 20 21 22 23 24 25

age

white 1880 white 1900 white 1910 white 1920 white 1930

black 1800 black 1900 black 1910 black 1920 black 1930

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As immigration dried up  Immigrants had children who were second-

generation children  Second-generation parents had American

children  The fragmentation of the marriage market

declined  The proportion of M/W with the “right” trait

increased  Finding a mate became easier

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T h

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o f y

e llo

w a

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Table 11 - Effect of Explanatory Variables on Predicted Probability of Marriage

  White Black

  Men Women Men Women

Sex ratio 5.4 -0.9 12.6 2.6

Trait 8.5 6.4 7.5 6.9

Variance in number of breadwinners 1.2 0.9 -1.3 -1.0

Variance in number of children -0.5 -0.3 0.9 -1.6

All partner search variables 14.5 3.1 17.2 2.5

Single women's LFP -6.0 -10.4 0.9 -7.0

Married women's LFP 0.9 0.3 0.6 -2.1

Men's LFP 1.2 3.7 1.6 4.2

Own OCCSCORE 20.8   25.7  

Average men's job quality 11.0 9.3

All labor market variables 16.9 2.2 26.1 1.6

All marriage market variables 30.9 5.5 41.4 4.1 Note: The reported values are percentage point changes in the probability of being ever married by each race-sex groups median age (25 for white men, 22 for white women, 23 for black men, 20 for black women) as each variable is varied between its 10th and 90th percentile.

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Legal vs Real ends of marriage, period rates per 1000 marriages

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

1860 1870 1880 1890 1900 1910 1920 1930 1940

Divorce rate (Jacobson, 1959)

Disruption rate - baseline estimate

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BLM (1977) on divorce

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BLM (1977) on divorce

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Implications of BLM (1977)

Age at marriage

P(divorce)

28. 9

Estimation results: -1.561AM + 0.027AM 2

Derivative: -1.561 + 0.054AM = 0

AM = 28.9

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Conclusions

 Search matters a great deal  There is an extensive and an intensive

margin  Many relationship characteristics

depend on it  Duration, chances of break-up,

quality/happiness  Marriage market characteristics depend

on it  Stability of matching  Time to convergence