Theoretical Economics Letters, 2011, 1, 134-136
doi:10.4236/tel.[phone] Published Online November 2011 (http://www.SciRP.org/journal/tel)
Elective Affinities
Daniela Di Cagno1, Emanuela Sciubba2, Marco Spallone3
1LUISS Guido Carli, Rome, Italy
2Birkbeck College, London, UK
3University of Chieti and Pescara, Pescara, Italy
Received September 14, 2011; revised October 22, 2011; accepted October 30, 2011
Abstract
We propose a marriage model where assortative matching results in equilibrium for reasons other than those
driving similar results in the search and matching literature. A marriage is a joint venture where husband and
wife contribute to the couple’s welfare by allocating their time to portfolios of risky activities. Men and
women are characterised by different preferences over risk and the optimal match is between partners with
the same level of risk aversion. In our model no two men (women) rank the same woman (men) as most de-
sirable. Given that there is no unanimous ranking of candidates, everyone marries in equilibrium their most
preferred partner.
Keywords: Marriage Model, Risk Aversion
1. Introduction
The view of marriage we have in economics since
Becker’s contribution [1] is rather grim.1 Becker ex-
plained the common observation that like marries with
like on the basis of a matching model where assortative
pairing results as an equilibrium outcome. The model
builds on a heterogeneous population of males and fe-
males whose distinctive characteristic, say beauty, can be
ranked in terms of desirability to the partn er in an objec-
tive way.2 A common ordering of types guarantees that
all the boys agree that females are ranked in desirability
from the most plain of women all the way to, say, Ange-
lina Jolie. Similarly all girls agree that males can be or-
dered from the world’s worst looking man to, obviously,
Brad Pitt. Pairing occurs through search. Boys and girls
look for their best match. All boys would love to marry
Angelina Jolie and all girls would lov e to marry Brad Pitt.
However matches have to be agreed by both parties in-
volved and Angelina Jolie certainly would not settle for
anything less than Brad Pitt. Now, if Brad takes Angelina
(as it happens), the best match for the second boy down
the line is the girl that falls one rank short to Angelina.
And so forth. The result is that we are all matched with
the best partner we are able to reach, and this is someone
who is equally ranked to us in beauty. Like marries with
like, as in the common wisdom.
Now, for those of us who are not Angelina Jolie this
provides a rather grim view of marriage. When we ex-
press our vows we like to think that we are marrying the
only one we would ever consider to marry, the best man
in the world. By contrast, Becker’s view is that we are
actually marrying the only one we could get!
In this note we propose a happier view of marriage,
where matching results out of elective affinities. We
think of marriage as a joint venture: in their married life,
husband and wife commit to share an y outcome resulting
from the risky choices they make. Hence what may or
may not make a potential partner desirable is his or her
attitude towards risk. In a population of males and fe-
males who are heterogeneous in risk preferences, we
derive assortative matching as an equilibrium outcome
where like marries with like. Unlike in Becker’s story,
here all men and women manage to marry the best part-
ner they could ever dream for themselves because there
is an Angelina Jolie and a Brad Pitt for everyone.
Although these two alternative views of marriage end
up with the same matching outcome, we believe that we
are proposing a much happier view of marriage, with no
regrets.
1For more recent reviews, see Becker [2], Bergstrom [3], Weiss [4],
Browning Chiappori and Weiss [5].
2Of course beauty is only one of the possible desirable characteristics o
f
a man or a woman. We take beauty here just for the sake of an example.

135 D. D. CAGNO ET AL.
2. The Model
Men and Women are endowed with one unit of time
which they must allocate to a portfolio of risky activities.
To make the analysis simple, we assume that there are
only two types o f activities: a risky activity which returns
a payoff r > 1 with prob ability p and zero with probab il-
ity (1 - p); and a safe activity which returns 1 for each
unit of time invested.3 The risky activity has a (strictly)
higher expected return than the safe. Payoffs generated
by life activities are here expressed in terms of income,
but could easily be interpreted as lifestyle quality, chil-
dren education, health, or any other variable that matters
for the w elfar e of th e coup le. Pre feren ces o ver r isk are of
the expected utility form, with Bernoulli u tility functions
over income
 
,wuw

where θ is a risk aversion
parameter. We assume that utility is monotonically in-
creasing and strictly concave in income. We normalise
the Bernoulli utility function so that

,0,uw



.
Formally:
Assumption 1. The risky activity has a higher ex-
pected return than the safe: pr > 1.
Assumptio n 2. Individual utilities over income


,uw

are such that.
(,) 0 uw
w


,2
2
(,) 0
()
uw
w
 

and (0, )0u


.
Men and women are heterogeneous in their risk aver-
sion parameter. In particular, both women’s and men’s
risk aversion parameter ranges between a lower bound

and an upper bound

, and it is uniformly distrib-
uted. The double continuum assumption guarantees that,
for each risk aversion parameter θ, there are one man and
one woman (and no more than one) with preferences
represented by

,uw

.
Marriage is a joint venture. Man and wife both decide
simultaneously and indep endently on their preferred time
allocation over the two available activities. Any win-
nings are equally shared by the couple.4 We assume that
the realisations of the risky activity in which man and
wife invest are independent random draws. Under this
assumption, expected utility of man i when wife j’s time
share in the risky activity is
j
x
is eq ual to:
 

 

 

 
i
2
2
11
,
2
11
1,
2
11
1,
2
11
1,
2
ij j
i
ii j
i
ijj
i
ij
i
xrxx rx
pu
xr xx
ppu
xxr x
ppu
xx
pu





 




 

 


 

 




 

The marriage game is in two stages: a proposal stage
and a married life. In the first stage, couples are formed.
In married life, partners independently decide how to
allocate their time across the two activities. Uncertainty
is revealed and payoffs are distributed at the end of mar-
ried life.
We show that the marriage game admits a unique
Subgame Perfect Nash Equilibrium (SPNE) where in the
proposal stage agents match with partners of equal pref-
erences over risky outcomes. In married life, man and
wife go on to invest more in the risky activity than they
would if they were investing on their own, due to mutual
insurance.5
Proposition Optimal partner choice is such that ij



.
Proof. W e solve for Subgame Perfect Nash Equilibria
(SPNE). Hence we proceed by backward induction. In
the second stage agent i chooses his optimal investment
in the risky activity



,
ij
xx

 , given his own risk
aversion parameter i

and given the choice of his wife


j
x

. For notational convenience denote by i
x
and
j
x
man i’s and wife j’s choices respectively. For any
given choice of wife j,
j
x
, the optimal investment in the
risky activity by i solves:
 

 

 
 

2
2i
i
111
,
22
11 1
1,
22
11
1
1,
22
11 1
1,
22
iij j
i
ii j
i
ij j
i
j
xr xxr x
ur
pw
xr xx
ur
pp
w
xxr x
u
pp
w
xx
u
pw





 






 



 


 

 




 

3Lotteries are here assumed to be binary only for mathematical conven-
ience. We believe that our qualitative results would go through even i
f
we modelled uncertainty through non-binary variables.
4As long as we assume homotetic preferences, the results of our model
carry through when one considers an exogenous sharing rule (α,1-α).
5We do not report second stage results here, but the increased riskiness
in the individual portfolios of activities chosen by the couple follows
from mutual insurance (see Di Cagno, Sciubba and Spallone [6]).
Similarly, wife j takes man i’s allocation choice as
given and solves:

D. D. CAGNO ET AL.
136
 






 

2
2i
111
,
2
11 1
1
2
11
1
1
22
11 1
1,
22
iij j
j
ii j
j
ij j
j
j
j
xr xxrx
u
pw
xr xx
u
pp
w
xxr x
u
pp
w
xx
u
pw





 





 


 


 

 




 

2
,
2
,
r
r












Notice now that, for ij

 , the first order conditions
of agents i and j coincide and correspond to a global op-
timum for player i (and j). In fact, wife j with the same
preferences as man i chooses for herself the same portfo-
lio of activities that man i would have chosen for his wife,
had he been free to optimise both with respect to his own
portfolio and with respect to his wife’s. Given that sec-
ond period expected payoffs correspond to a global
maximum for agent i whenever ij

 , first stage
choice falls on a partner with equal risk aversion pa-
rameter.
3. Conclusions
We propose a marriage model where assortative match-
ing results from elective affinities. The risky choices th at
partners make in life affect the couple’s welfare. Hence a
good match is a match between man and wife with simi-
lar risk preferences. There is no given level of risk aver-
sion that is desirable per se and if we could order poten-
tial partners by their desirability, our orderings would no t
coincide, so that there is no need for us to fight for the
same man or the same woman. Every man has his Ange-
lina Jolie. Every woman has her Brad Pitt. Surely a
happy world.
4. Acknowledgements
We thank an anonymous referee for his/her comments.
5. References
[1] G. Becker, “A Theory of Marriage: Part I,” Journal of
Political Economy, Vol. 81, No. 4, 1973, pp. 813-846.
doi:10.1086/260084
[2] G. Becker, “Treatise on the Family,” Harvard University
Press, Boston, 1993.
[3] T. Bergstrom, “A Survey of Theories of the Family,” In:
M. Rosenzweig and O. Stark, Eds., Handbook of Popula-
tion Economics, Elsevier, Amsterdam, 1997.
[4] Y. Weiss, “Economic Theory of Marriage and Divorce,”
In: N. Smelser and B. Baltes, Eds., International Ency-
clopedia of Social and Behavioral Sciences, Elsevier,
Amsterdam, 2000.
[5] N. Browning, P. A. Chiappori and Y. Weiss, “Family
Economics,” Cambridge University Press, Cambridge, in
press.
[6] D. Di Cagno, E. Sciubba and M. Spallone, “Choosing a
Gambling Partner: Testing a Model of Mutual Insurance
in the Lab,” Theory and Decision, in press.
doi:10.1007/s11238-011-9267-2