4
GAME-THEORETICAL PERSPECTIVES ON THE
EMERGENCE OF SOCIAL NORMS
Thomas Voss
In: Michael Hechter und Karl-Dieter Opp (Hg.) (2001): Social Norms.
New York: Russell Sage.
ATIONAL
choice theory argues that soci al norms emerge bec ause the
norms' content yi eld s benefits for some agents. G ame theory is that
branc h of rational c hoice theo ry that d eals wi th soci al i nteractions
among rational acto rs. T he subj ec t of game theory i s therefore soci al ac tion
and soci al relatio ns among rational agent s (in M ax Weber' s sense). The
emergence o f soci al norms i s c lo sely related to the Hobbesi an p roblem of
social ord er, and i ts explanation i s among the most i mportant i ssues of
soci al theory. In thi s ch apter I di sc uss game theo retical appro ach es to soci al
norms.
A core concep t of game theory i s the N ash equi libri um, a profil e of
actio ns in w hich no ac to r has a posi tiv e inc entiv e to uni laterally devi ate
from those actions. Most game theoretic al appro aches use the basic id ea
th at a soci al norm wi ll be enforced if the ac tions that are co mpatible with
th at norm are sup ported by a N ash equi librium. H enc e, one task of game
theoretic al analysi s is to point o ut those conditions or soci al mec hani sms
that lead to those eq ui li bria.
When adopting a rational c hoic e ap pro ach, as I do here, two q uestions
are usefully disti nguished ( see, for ex amp le, Co leman 1990, chapters 10
and 11): Wh at are the struc tural f eatures of soci al si tuations that generate a
demand" for soci al norms? and Wh at mechani sms contribut e to the en­
forc ement, o r " effec tiv e reali zation," of norms? These two problems must be
addressed sep arately, bec ause not every soci al norm that po tenti ally pro­
vid es benefi ts to a c ertain class of agents will actually co me into exi stenc e.
T he effec tive reali zation of no rms d ep end s on mechani sms that render
norm confo rmity self-enforci ng. S elf- enforc ement means that rational target
actors of a norm acc ept a norm bec ause the norm ben efici ari es h ave created
certain inc entiv es that mak e no rm conformi ty consi stent wi th a N ash equi­
librium. T he i ncentiv es may v ary but often i nc lude some form of sanc tion.
The analyses in the li terature thus far rest on the assumption that the ac tors
R
U
105
4
GAME-THEORETICAL PERSPECTIVES ON THE
EMERGENCE OF SOCIAL NORMS
Thomas Voss
ATIONAL
choice theory argues that soci al norms emerge bec ause the
norms' content yi eld s benefits for some agents. G ame theory is that
branc h of rational c hoice theo ry that d eals wi th soci al i nteractions
among rational acto rs. T he subj ec t of game theory i s therefore soci al ac tion
and soci al relatio ns among rational agent s (in M ax Weber' s sense). The
emergence o f soci al norms i s c lo sely related to the Hobbesi an p roblem of
social ord er, and i ts explanation i s among the most i mportant i ssues of
soci al theory. In thi s ch apter I di sc uss game theo retical appro ach es to soci al
norms.
A core concep t of game theory i s the N ash equi libri um, a profil e of
actio ns in w hich no ac to r has a posi tiv e inc entiv e to uni laterally devi ate
from those actions. Most game theoretic al appro aches use the basic id ea
th at a soci al norm wi ll be enforced if the ac tions that are co mpatible with
th at norm are sup ported by a N ash equi librium. H enc e, one task of game
theoretic al analysi s is to point o ut those conditions or soci al mec hani sms
that lead to those eq ui li bria.
When adopting a rational c hoic e ap pro ach, as I do here, two q uestions
are usefully disti nguished ( see, for ex amp le, Co leman 1990, chapters 10
and 11): Wh at are the struc tural f eatures of soci al si tuations that generate a
demand" for soci al norms? and Wh at mechani sms contribut e to the en­
forc ement, o r " effec tiv e reali zation," of norms? These two problems must be
addressed sep arately, bec ause not every soci al norm that po tenti ally pro­
vid es benefi ts to a c ertain class of agents will actually co me into exi stenc e.
T he effec tive reali zation of no rms d ep end s on mechani sms that render
norm confo rmity self-enforci ng. S elf- enforc ement means that rational target
actors of a norm acc ept a norm bec ause the norm ben efici ari es h ave created
certain inc entiv es that mak e no rm conformi ty consi stent wi th a N ash equi­
librium. T he i ncentiv es may v ary but often i nc lude some form of sanc tion.
The analyses in the li terature thus far rest on the assumption that the ac tors
R
U
105
106
SOCIAL NORMS
employ cer tain types of endo geno usly cr eated san ctions, wh at may be
call ed " ind ir ect san ctions, " but man y real-wor ld d escr iption s of social
nor ms refer to o th er, mor e active types of sanction in this ch apter.
What Are Social Norms?
To under stand the for m and fun ction of so cial nor ms, w e migh t begin by
con sid er in g th e follow in g ex amples of h uman beh avior:
1. Every morn in g, Rober ta pr acti ces Moz ar t son atas on h er piano.
2. Dur in g the l ate w in ter, many peo ple in so me C atholic co untr ies (for
ex ampl e, south er n G er many) cel ebr ate carnival.
3. In the Leipzig con cer to house, G ew andh aus, th e aud ien ce applauds a
perfor mance of S chuber t' s Winterreise after th e final son g, r ath er than
between songs.
4. In many h igh school classes, tho se students who l earn eagerly are no t
popul ar. S imi lar ly, in workin g gro ups in lar ge or ganiz ation s, worker s
r egul arly r estr ict their output ( see, for example, Homans 1951), and
over perfor mer s ar e tar gets of infor mal san ctions .
5. Some busin ess peo pl e act in accord ance w ith the pr incipl e that commit­
men ts are to be honor ed ( M acaul ay 1963; see also Ellick son 199 1, 1899 1 ).
6. In th e community o f Inter net user s, a r ul e prohibitin g the use of e-mail
for commer cial adver tisin g is wid ely r espected ( F uk uyama 1995, 196).
7. In spor ts contests ( for ex ample, th e Tour de Fr ance), athletes ar e ex­
pected to respect the r ul e proh ibiting the use of dr ugs that enh ance
athletic perfor man ce and thus migh t improve th eir chances of w inn ing
a pr ize. Appar en tly, many athl etes systematically br eak th is r ul e.
The fir st ex ampl e d escr ibes a per sonal r ule or individual r egular ity in the
beh avior of a sin gl e actor. P er sonal r ules may be of interest, in pr in ciple, in
th e context of nor m for mation because of their r el ation to individ ual sel f­
con trol pro cesses, "ch ar acter plannin g" and the lik e ( see Lo ew enstein and
Elster 1992). For pur po ses of this expo sitio n, per sonal rules are d ismissed
as irrelevan t. The second ex ample d escr ibes a r egul ar ity in the behavior of
so me popul ation of agents, a beh avior that migh t be called a social custom.
How ever, it is a theor eticall y unin ter estin g case. Pr ima facie, this case can
be expl ain ed by th e assumptio n th at w ithin a population of actor s cer tain
pr efer en ces are shar ed. T hese agents share an in ter est in the joint produc­
tion of a consumption good. Th ere ar e- at least at a fir st gl an ce-no non­
tr ivial str ategic inter action problems involved in this social custo m: Tho se
peopl e who lik e carn ival jo in the activities, tho se who do no t stay at home.
The third case, though substantially tr ivial, d escr ibes an inter esting
GAME-THEORETICAL PERSPECTIVES
107
problem. Consider a woman who v isits a per for man ce of classical music
for the fir st time. Fro m the ver y beginn in g, she enjoys th e presen tation and
wan ts to ex press h er admir ation and approval. Should sh e break in to ap­
plause after the fir st son g of th e cycle? If sh e wer e to do so, and no o th er
per son joined her, sh e wo uld probably be embarr assed. If she does no t
know th e proper r ule, sh e w ill probably w ait un til o ther s star t to clap. Th is
scen ar io d escr ibes a coordin ation problem. Coord ination situations are str a­
tegic so cial inter action s w ith multiple eq uilibr ia-for ex ample, to clap after
each piece, no t to clap at all, to w ait until oth er s begin to clap, and so for th .
Most people would be un comfor table bein g th e only per son or among the
few per son s who applaud. Wer e th is cir cumstan ce to be r ealiz ed, even if by
accident, th e peo ple invo lved would feel th at they h ad commit ted an error.
Th e last four ex amples (4 through 7) ar e of a differ en t k ind. In e ach of these
cases, ther e ex ist social nor ms ( infor mal or for mal) th at prescr ibe or pro­
scr ibe cer tain actions. The content of each nor m is such th at the inter ests of
the par ticipan ts are met. Th e nor m encour agin g r estr icted effort in a work­
in g situation, for ex ample, ser ves th e in terests of the aver age group mem­
ber. I f th e nor m w er e enfor ced, every group member-accord in g to the
group member s' per ception of the situation -would be better off, at least
in compar ison with the outco me if no member s w er e to r espect the nor m.
Th e nor m elimin ates co mpetition. However, man y group member s may
per ceive a temptation to contr ad ict th e nor m: it m igh t be ad van tageous for
an ind ivid ual to put in effor t abo ve th e gro up stand ard because it would
improve th e work er's ch ance of r eceivin g high er r ew ard s ( in cr eased in­
co me, career oppor tun ities, and the lik e).
These last four cases illustr ate in cen tive problems. They r esemble pr is­
on er's dilemma situations. Th e pr isoner's coo per ation will ben efit him (or
her ) if all o th er s cooper ate but w ill h ar m him (or h er ) if he (or she) cooper­
ates and oth er s do no t. If an actor unilater ally cooper ates h e (or she) w ill
receive the wor st po ssible payo ff. In a coordin ation problem, lik e th at d e­
scr ibed in ex ample (3 ), r ation al in ter actin g agents co mpletely sh are th e in­
ter ests of th eir in ter action par tn er s : thus, th ey h ave an in ter est in coordinat­
in g their cho ices. If coord in ation is successful, there will be no incen tive for
individuals to deviate from a beh avior al r egular ity, commonly called a con­
ven tion (L ew is 1969). A con ven tion is self-stabiliz in g or self-enforcin g, and,
once realiz ed, confor mity ser ves the in ter ests of bo th the gro up and its
individ ual members. Thus, in th e case of a coordin ation problem, ther e is
no n eed to impo se san ctions to ach ieve un iver sal cooper ation. In a pr is­
on er's dilemma, how ever, there is an incen tive to d eviate fro m a nor m th at
pr escr ibes cooper at ion; in such a case, sanction s are gener ally n eeded to
coord in ate beh avior to ach ieve a un iver sally advan tageo us o ut come.
The conten t of a nor m may be ex pr essed as an imper ative-th at is, a
statemen t th at some k ind of action ough t to be done or ough t not to be
done under appropr iate cir cumstan ces. Some socio lo gists ( Ho mans 195 1;
108
SOCIAL NORMS
O pp 1983) use this cr it er ion as a core defin ien s of the conc ept of a nor m.
How ever, fro m a soc io lo gic al per spect ive, nor ms are more than imper atives
that ar e appro ved or no t approv ed by peo ple. To use E mile Durk heim' s
( 1895) phr ase, nor ms ar e soc ial f ac ts ( "f aits soc iaux " ), outcomes of ac tions,
and, as such, they ar e collective phenomena. Nor ms ex ist, in a sen se, inde­
pend ently from the con sc iousn ess of individ uals: they ar e objec tive and
c ausally r elevant if they are enforced within a community or group of ac­
tor s. If I d eviate from a nor m that is perfec tly r ealized and monitored, even
if I am unawar e of the norm' s content, I am lik ely to beco me the tar get of
mor e or less sever e sanct ion s.
Nor ms ar e mor e than imper atives in ano ther sense: Even if agent s are
infor med about the nor m's con tent and acc ept the norm as v alid, they do
not necessar ily confor m to the nor m. Con sider the ex ample of the no -dr ugs
nor m in sports. It may be r ealistic to assume that mo st-if not all- athletes
think that this nor m pr escr ibes "r ight," " valid," or "f air " behavior. N ev­
er theless, the same peo ple w ill systematic ally break the nor m bec ause of
the temptation-as in the pr isoner' s d ilemm a- to achieve a better po sition
in the contest. Henc e, it is impor tant to focus on ac tion s, especially regu­
lar ities in the ac tion s of a population .
Rational c ho ice theor y may ex plain many in gr ed ients of soc ial nor ms.
Fir st, it is possible to ex pr ess the d emand for c er tain nor ms in ter ms of
statements abo ut d esir able or obligator y actions in c er tain situations. T his
can be an impor tant explanator y task. Second, and per haps mor e d ifficult,
it c an poin t out tho se conditions that lead to an enforc ement of nor ms. I n
f ac t, most game theor etical explanations focus mainly on this problem of
nor m enforc ement. With the emphasis on enforcement, social nor ms c an be
d ef ined as behavior al r egular ities in a population of ac tor s. For the pur­
po ses of this ex po sition a soc ial nor m may be def ined as a regular ity R in a
po pulation P of actor s such that
1.
R ar ises in r ecurren t inter ac tions among the agents of po pulation P
2.
almo st ever y member of P prefer s to confor m to R on the cond ition
that almost ever y o ther member of P also confor ms to R
3.
almo st ever y member of P believes that almost ever y o ther member of
P co nfor ms to R
R is a N ash eq uilibr ium of the r ecurr ent inter ac tion (Lewis 1969; Bic­
chier i 1993, 232; see also, for ex ample, Sc hotter 1981; Sugd en 1986;
Leibenstein 1987 Rober t Ax elrod 1986, 1097, uses a simplif ied v er sion
of such a d ef inition)
4.
This concept may be expr essed more str ic tly by d istinguishing between
soc ial nor ms (in the proper sense of ex amples 4 thro ugh 7 at the beginn in g
GAME-THEORETICAL PERSPECTIVES
109
of the chapter) and co nventions. A ,social norm proper is a r egular ity R
such that members of P ex pect that nonconformity will (w ith positive prob­
ability) be punished w ith ( negative) sanctions, and th ese ex pectations ar e a
reason for th e f act that cond itions ( 2) and (3) hold true. A convention, on
the o ther hand, is a regular ity R such that R is a coo rd ination eq uilibrium
in a recurrent coord ination situation. H ence, social nor ms are clo sely re­
lated to, though d istinct f rom, conventions. O ne migh t even argue that so­
cial no rms accompanied by sanctions are a subset of the class of conven­
tions ( see Young 1998, 144-45).
T his concept of a so cial nor m may ser ve as an ex pl ication of some
intuitions that have been ex pr essed with in the so cio lo gical tradition.'
Norms ( and conventio ns) are tr eated as social f acts. In M ax Weber's ( 1976)
sense they are regularities in the beh av ior of agents w ho ar e involved in a
"social r elation." Theodor Geiger ( 1970,43-9 1) h as pointed o ut th at nor ms
arise from social interd epend ence (see also Po pitz 1980; Z iegler 1984; B aur­
mann 1996). T hey are regularities that are generally stabiliz ed by threats of
negative sanctions. From a game th eoretical point of view, th is "stabiliz a­
tion" results f rom a N ash eq uilibrium.
There are many classes of so cial nor ms that may be of inter est. M ax
Weber (1976) distinguishes legal norms ( law ) fro m informal no rms. The
und erlying d imension that generates this typolo gy is the k ind of sanction­
ing abuse of the nor m engend ers. L aw is enforced by a specialized "staff "
of agents who monitor th e target actors of a nor m and puni sh deviant
behavior. O ther nor ms are enfo rced by sanctio ns that are cr eated w ithin the
community of agents w ho are targets of the norm. T hese norms are, so to
speak, norms of self-help. Mor e r ecently, J ames S. Co leman (1990) h as pro ­
posed a typo lo gy based on the relation betw een the set of nor m benef ici­
aries and the set of target acto rs ( th at is, tho se agents whose actions are
r egulated by a norm). If these two sets are identical, th at is, if the targets
and the benef iciar ies of a no rm are the same ind ivid uals, the no rm is "con­
jo int"; if the targets and benef iciar ies are diff erent, o r if their inter ests do
no t conver ge, it is "d isjoint" ( Co leman 1990, 247-48) . Conjo int nor ms are
par ticularly impor tant because they are amo ng the most elementary yet
also th e mo st f und amental norms. Ex amples for tho se norms are all those
r egularities of behavior that ar ise in situations of a Hobbesian anar chy, if
any. Hobbesian anarchy is a situation in which cooperation, although prof­
itable, is difficult to achieve. T her e is no ex ter nal agency (such as the state)
that co uld enforce mutually advantageo us rules. Th erefore, ever y actor rep­
r esents a threat to every o ther actor. All actor s pref er the universal reco gni­
tion of cer tain r ules, for instance w ith r espect to proper ty rights, to th e w ar
of every man against every man. How ever, everyone perceives th at th ere is
a temptation to def ect. Concrete cases are tho se nor ms mentioned in ex am­
ples (4) through (7). The explanation of the emer gence of conjoint nor ms is
a diff icult task, in particular in prisoner's d ilemma situations, because it
110
SOCIAL NORMS
has to be demonstr ated that r ational agents cr eate sanctions that ar e sever e
eno ugh to enfor ce a so cially o ptimal eq uilibr ium.
Conditions That Generate a Demand
for Social Norms
Fo cusing on conjo int nor ms, r ational cho ice theor y offer s a gener al hypo th­
esis: Social nor ms d evelop and ar e maintained in such a w ay that their
content ser ves to improve the aggregate welfar e of the nor m beneficiar ies
( this hypo thesis is ado pted, in a slightly mod ified for m, from E llickson
1991, 167; see also U llmann-M ar galit 1977; Opp 1979; Co leman 1990, chap­
ter 10, for similar hypotheses). In other words, nor ms emer ge when they
pro mise to br ing about efficiency gains to the beneficiar ies. Efficiency gains
may be inter pr eted as P ar e to improvements ( altho ugh other gains may be
po ssible) in game theor etical terms. Nor ms help beneficiar ies avoid or es­
cape social d ilemmas.2 A so cial d ilemma is d efined as a situation of (str ate­
gic) interd ependence in which r ational agents fail to achieve an efficient
outcome. T his means that ther e ex ists an outco me (w hich is no t r ealized) in
which the situation of every agent would improve or they all would at
least r eceive the same payoff.
Figures 4.1 through 4.4 illustr ate sever al types of so cial d ilemmas that
may ar ise amo ng r ational agents (see H ar sanyi 1977; Voss 1985).3 In a clas­
sic pr isoner's d ilemma (figur e 4.1), there is no P ar eto-optimal equilibr ium.
Consequently, r ational action (d efection) yields a collectively irr ational o ut­
come that co uld (physically) be improved by univer sal coo per ation. In a
noncooper ative bar gaining game, ther e are sever al pure P areto-optimal
N ash eq uilibr ia, but there is also a "bar gaining d ead lock," owing to the
conflict of inter est among the player s. In the battle-of- the- sexes game, for
ex ample (figur e 4.2), two equilibr ia ar e in pur e str ategies ([ A, A] and [ B, Bl)
(see, for ex ample, Young 1998, 25 ). How ever, these equilibr ia ar e no t eligi­
ble, according to classic game theor y, because the agents' prefer ences ar e in
conflict. A mixed str ategy eq uilibr ium would ar ise if the row player w er e
to play A w ith a probability of two i n thr ee and B w ith a probability of one
in thr ee. T he colUllli1 player' s mixed eq uilibr ium str ategy is to play A with
a probability of one in three and B w ith a probability two in three. This
yield s a payoff of 2/3 to bo th player s, which is str ictly less than the payoff
of each player's wor st pure str ategy equilibr ium. Rational agents will
hence r ealiz e a subo ptimal outcome. Bo th player s would be better off if any
of the pur e eq uilibr ia had been selected.
Ano ther prominent bar gaining scenar io is known as the chick en game
(figur e 4.3). Consid er two agents w ho can contr ibute to a co llective good .
T he prod uction function is such that any player's contr ibution is sufficient
to produce the good. T her e ar e no economies of scale. As a concr ete ex am­
ple one may think of help in emer gency situations ( compar e the n-per son
GAME-THEORETICAL PERSPECTIVES
FIGU R E 4.1
111
Prisoner's Dilemma Game
C
D
C
3,3
-1,5
D
5,-1
0,0
Source: Author's compilation.
Note: The entries of each cell of a matrix represent the actors' payoffs as ordered pairs. The
first number is the payoff of the row player, the second is the payoff of the coluum player.
var iant of this game, called the Vo lunteer's D ilemma, in D iek mann 1985).
The two asymmetr ic N ash equilibr ia, (c, D) and (D, C), gener ate a bar gain­
ing problem. Ther e is also a mixed equilibr ium, namely, coo per ation w ith a
probability of three in f ive. This mixed eq uilibr ium gives an exp ected pay­
off of three. A ny of the two pur e equilibr ia would yield a P areto- super ior
payoff-that is, it would mak e one player better off w ithout alter ing the
payoffs of the o th er player.
A pur e coordination game ( see f igur e 4.4) has at least two efficient
Nash equilibr ia. How ever, if ther e is no preplay communicatio n among the
player s, they w ill h ave to choose the mixed eq uilibr ium of playing each
pure str ategy w ith a probability of one in two. Th is gives an exp ected pay­
off of one to each player, which is str ictly less than th e payoff of any pur e
str ategy equilibr ium.
A ll of these social d ilemmas gener ate a "d emand " for a social nor m,
or-in the case of the coordination problem-for a convention.4 This
nor m' s content includ es an imper ativ e to choose the action th at is consis­
tent w ith an efficiency gain to the actors. The "function" of the nor m or
conv ention wo uld be a P areto improvement. For example, in the pr isoner's
FIGU R E 4.2
Battle of the Sexes Game
A
B
A
1,2
0,0
B
0,0
2,1
Source: Author's compilation.
Note: The entries of each cell of a matrix represent the actors' payoffs as ordered pairs. The
first number is the payoff of the row player, the second is the payoff of the colulTU1 player.
112
SOCIAL NORMS
F I GU R E 4.3
Chicken Game
C
D
C
3,3
3,5
D
5,3
0,0
Source: Author's compilation.
Note: The entries of each cell of a matrix represent the actors' payoffs as ordered pairs. The
first number is the payoff of the row player, the second is the payoff of the column player.
dilemma, there is no efficient equilibr ium at all. A nor m prescr ibing coop­
er ation wo uld mak e ever y player better off. S imilar ly, common know led ge
of a rule that one should choo se A would mak e the player s better off in a
coordination situation.
T her e is one core problem with such an approac h that is appar ently
most sever e in the c ase of the prisoner's d ilemma. Given the f ac t that a
social nor m of cooper ation wo uld impro ve the situation, it is obvious that
the benef ic iar ies of the nor m w ill have an inter est in creating the nor m. T he
inefficient outco me gener ates a demand for a nor m ( Coleman 1990, chapter
10). If that nor m w er e to be r ealized or enforc ed, the benef ic iar ies wo uld be
better off; but how c an the nor m be r ealized? Answ er s to this q uestion c an
bro adly be c lassif ied as exo genous and endo genous so lutions. Exo genous
so lutions dr aw on the invo lvement of third par ties in the process of norm
creation and enforcement. Ano ther type of exo geno us ar gument r ef er s to
an internal sanc tioning system or inter naliz ed sentiments, such as guilt.
T he problem with exo genous so lutions is, of co ur se, their incompleteness.
T hey are probably heur istic ally fr uitf ul but need to be complemented by
ar guments that ex plain the emergenc e and stability of third -par ty inter ven­
tions or an inter nalized sanc tioning system by explic itly using the assump­
tion of r ational action. Endo genou s ex planations r eq uir e ar guments demF IGU R E 4.4
Coordination Game
A
B
A
2,2
0,0
B
0,0
2,2
Source: Author's compilation.
Note: The entries of each cell of a mah'ix represent the actors' payoffs as ordered pairs. The
first number is the payoff of the row playel� the second is the payoff of the colullm player.
GAME-THEORETICAL PERSPECTIVES
113
onstr ating that r ation al self- inter este,d agents in a soc ial d il emma situation
confor m to social nor ms of cooper ation in r esponse to inc entives created by
these actor s themselves. B ec ause in an ordinary pr ison er's d ilemma situa­
tion there is no cooper ation amon g r ation al ac tor s, the emer genc e of nor ms
may d epend on thr eats of negative sanctions that are created by the target
ac tor s of the nor ms themselves. In the c ase of coordination problems, it has
to be d emonstr ated how the member s of the population that r ecurrently
encounter s coordination problems achieve co mmon know led ge of a uniq ue
foc al point ( in Schel ling' s [ 1960] sen se) and, in this w ay, endogeno usly cre­
ate a r ule that becomes the basis of a convention.
The Effective Enforcement of Social Norms
Consid er conjoint social nor ms that may regulate gener al ized prison er's
dilemma situations. I t is w ell k now n that w ith r espec t to a nor m of univer­
sal cooper ation a free-r idin g problem w ill ar ise: Every actor w ill profit from
the collective good that is produc ed by cooper ative col lec tive ac tion (nor m
confor mity). Y et there w il l be an inc entive for an ind ividual to avoid con­
tr ibuting to the co sts of cooper ation bec ause the good c an be con sumed by
any member of the group, independen tly of that ind ivid ual' s contr ibution .
A social nor m of cooper ation would be enforceabl e if the ac tor s used ap­
propriate sanctions (n egative or positive) . How ever, the thr eat (or promise)
to provid e nor m d eviator s (or nor m conformer s) w ith sanc tions (or re­
w ard s) appar ently involves a "second-order " col lec tive good problem (see
Oliver 1980; Axelrod 1986; H eck athor n 1989; Coleman 1990, 270-73 and
passim). This problem r esults from the f act that agents w ho apply sanc­
tion s must be compensated for their co sts, assumin g that the provision of
sanctions is no t a gen er ally costl ess activity.
Although the problem situation is often fr amed in game theoretic al
ter ms, a thorough game theoretic al analysis of the " effective r ea liz ation " of
social nor ms is q uite r are. Most an alyses of the f ir st- and second-order
problems of cooper ation impl icitly use the assumption of parametr ic ( that
is, nonstr ategic) behavior. U sing a par ametr ic analy sis, Coleman, for ex am­
ple, conj ec tures that the second-order problem may be so lved if the nor m
benefic iar ies ar e " able to shar e appropr iately the co sts of sanctioning the
tar get actor s or w ill be able to gener ate second-ord er sanc tions amon g the
set of benef iciar ies that are suff icient to ind uc e effective sanc tions of
the target actor s by one or mor e of the benef ic iar ies. This condition de­
pend s on the ex istenc e of soc ial r el ationshi ps among the benefic iar ies"
(1990, 273). By and large, this proposition may be r ight, but Coleman's
ar guments for it ar e mostly uncon vinc ing (for a more d etailed d isc ussion,
see Voss 1998 a, 122-24 ) bec ause Coleman, as w ell as o ther s, d ispenses w ith
an ex plic it game theoretic al analy sis of bo th levels of individ ual d ec ision
with r egard to nor m confor mity: the f ir st-order problem of nor m confor-
114
SOCIAL NORMS
mity and the second-order problem of confor mity to a " metanor m" of
sanc tioning tar get actor s (Ax elrod 1986). These two l evels ar e treated sep­
ar ately by mak ing use of ar guments that do no t focus on the str ategic f ea­
ture of the inter ac tions.
In contr ast, a game theor etical approach has to d eal w ith a number of
desid er ata. Regul ar ities of behavior confor ming to a soc ial nor m ar e self­
enforc ing if they resul t, to speak technic ally, from a N ash equilibr ium.
H enc e, a f ir st requir ement corr espond s to answ er ing the q uestion, What
are the cond itions for an efficient equil ibr ium of univer sal cooper ation in a
social d il emma situation ( in par ticul ar, of the gener al ized pr isoner's di­
lemma type)? Providing a satisf actor y answer to this q uestion is at l east a
nec essar y cond ition of a po tential endo genous expl anatio n of conjo int
nor ms in prisoner's d ilemma situations. Cooper ation d epend s on sanc tions
created by the nor m beneficiar ies themselves. T his l ead s to a second ques­
tion: What ar e the cond itions for the emer genc e of thr eats of sanc tions that
are no t only consistent w ith N ash eq uil ibr ium behavior but also cr ed ible? I t
is w ell k nown that thr eats and promises i n str ategic inter ac tions will be
eff ec tive in changing inc entives only if the agents w ho offer them ac tually
apply them w hen the appropr iate c irc umstanc es are reached (Schell ing
1960; Sel ten 1965). Even in c ases- in which-contr af actually-a path out
of an eq uilibr ium of a game w er e to be real ized, the r el evant actor should
have no po sitive incentive to d eviate from the impl ementation of a thr eat.
Technic ally spe aking, a minimum requir ement for N ash equil ibr ia, w hich
incl ud e c er tain thr eats or promises as moves of the eq uilibr ium str ategy, is
that they be subgame perfec t (S el ten 1965).5
The Endogenous Emergence of Social Norms
with Sanctions: Some Mechanisms
One impor tant mechanism that may c hange ac tor s' incentives in a social
d il emm a is r epeated inter ac tions ( see, for ex ampl e, Taylor 1976, 1987; Ax­
elrod 1984, for treatments of iter ated pr isoner's d il emmas). R epeated inter ­
actions allow for the impl ementation o f a spec ial type o f endo geno us sanc­
tions, that is, s anc tions created by the agents themsel ves. For pur po ses of
ill ustr ation, but w ithout loss of gener ality, I c hoo se the classic two-person
pr isoner's d il emma. Figure 4.5 d ispl ays the payoff matr ix of the pr isoner's
d il emma in nor mal for m.
Assuming a standard infini tely r epeated game means roughly that (1)
the pr isoner's d il emma game is r epeated infinitely of ten amo ng the same
pl ayer s; ( 2) player s d isco unt their future payoffs w ith a constant d iscount
f ac tor a (1 > a > 0 ); and ( 3) there is ( almo st) co mpl ete infor mation abo ut
the moves of the pl ayer s in the past.6 As is well k nown ( Fr ied man 1971;
Fud enber g and M ask in 1986), there may ex ist P ar e to-optimal N ash eq ui-
GAME-THEORETICAL PERSPECTIVES
FIGUR E 4.5
115
The Classic Prisoner's<Dilemma Game
C
D
c
R,R
S,T
D
T, S
0,0
T>R >O> S
Source: Author's compilation.
Note: The entries of each cell of a matrix represent the actors' payoffs as ordered pairs. The
first number is the payoff of the row playel� the second is the payoff of the column player.
libr ia in which bo th player s cooper ate if the " shadow of the f uture" (Ax­
elrod 1984) is lar ge eno ugh. The shadow of the future corr espond s to the
actor s' discount par ameter s a and can also be inter preted as the actor s'
( cond itional subjectiv e) probabilities that the iter ation of the ga me w ill be
continued for ano ther per iod.
If the payoffs of the pr isoner's dilemma ar e r epresented by the in ­
equality (T > R > 0 > S), the payoffs of the r epeated game can be repre­
sented as w eighted sums of the payoffs in each per iod. Because it is as­
sumed that the disco unt f actor a is constant in each per iod, f uture payoff s
will be disco unted ex ponentially. For ex ample, assuming that both player s
coo per ate forev er, ev er y player w ill ex pect to r eceiv e the payoff
R
+
aR
+
2
aR
+
...
=
Rj (1 - a)
in the r epeated game.
Cooper ation in the r epeated game can be enfor ced by tr igger str ategies
(Fr iedman 1971). T hese dem and cond itional coo per ation, that is, cooper a­
tion that lasts as long as the o ther actor s cooper ate. O therwise, the str ategy
pr escr ibes d ef ection. I call this type of puni shment " indir ect punishment. "
The simplest v ar iant of a tr igger str ategy r espond s w ith " eternal d amna­
tion" ( that is, defection) to the f ir st def ection. Unilater al deviations from a
prof ile of tr igger str at egies yield a payoff of T at max imum in the iter ated
game. Conseq uently, a pair of tr igger str ategies is in eq uilibrium if and
only if,
a
2:
a*
:
=
(T - R)j(T - 0)
=
1
-
(RjT).
This eq uilibr ium d epend s on cred ible thr eats of punishment. A single d e­
fection tr igger s the parh1er' s "eternal" d efection. T his means th at the tr ig-
116
SOCIAL NORMS
ger- strategy pl ayer uses the threat to implement the uniq ue equil ibrium
action of the prisoner' s d ilemma. T here is cl early no po sitive inc entive to
d eviate from tho se thr eats out of eq uil ibrium. T he profile of trigger strate­
gies is henc e a subgame perfec t equilibrium if a 2:: a*. Note, how ever, that
no t every strategy of conditional cooperation is subgame perfect. T he c el e­
brated tit-for-tat strategy (Ax elrod 1984) is subgame perf ec t only for one
partic ul ar val ue of the d iscount fac to r. I t c an easily be d emonstrated that
conditions for tit-fo r-t at to be in sub game eq uilibrium w ith itself are
2R > T
a
2::
a*
:
=
+
5,
1 - R / T,
and
a
=
a**
:
=
(T - R) / (R - 5).7
If only the second inequality, a > a**, hold s, tit-for- tat employs threats and
pro mises that are not c redible for particul ar histo ries of the game (out of
equilibrium), namely, for histories w ith an al ternation of Cs and D s. T he
"fo rgiving" tit-for- tat is therefore-from the point of view of analytic al
game theory-a strategy in the iterated game that does not generally pass
the rationality criteria. Substanti ally, one co uld argue that tit-for- tat does
not succ essfully surmount the second -ord er d il emma.
Und er appro priate cond itions, repeated interactions generate endo ­
geno us ind irect sanctions that c an enforce conjoint social norms of cond i­
tional cooperation. One should, however, notic e that there ex ists a large
number of equilibria in a repeated game if the shadow of the f uture ( the
d iscount factor) appro aches one (see the folk theorems in F ud enberg and
M askin 1986). T here is alw ays the equil ibrium of universal d efec tion,
which is obvio usly subgame perfec t; but there are also many effic ient (P ar­
eto -optimal ) sub game perfec t equil ibria-for exampl e, d ifferent k ind s of
trigger strategies. T his means that there is an equil ibrium selection prob­
l em. In game theoretic al terms this selection problem c an be expressed as a
coord ination or (noncooperative) bargaining problem. We encounter a bar­
gaining probl em if an outcome is profitable to one of the players at the
d isavantage of his partner.
T he eq uilibrium sel ec tion problem has often been overd ramatized in
critiq ues of the game theoret ic appro ach (for ex ample, H ec hter 1992). It is
cl early a majo r difficul ty of the repeated-games pro gram. Contemporary
evolutionary game theory (see also the remark s at the end of this c hapter)
aims to provid e mod els that show how coordinatio n equil ibria are d eter­
mined endo geno usl y. T he traditional game theory appro ac h treats this se­
l ect ion probl em exogeno usly.
GAME- THEORETICAL PERSPECTIVES
117
O bviously, these coordination problems are " so lved " in groups with a
common history. Any eq uilibrium that is realiz ed at some po int in histor y
will be a fo cal po int for the pro cess of a con ver gence of recipro cal expecta­
tion s that a particular eq uilibr ium w ill be cho sen. Moreover, the con tent o f
so cial nor ms o f cooper ation may b e interpreted in such a w ay that it points
out par ticular equilibr ia. T here may be eq uilibria that are con sisten t with
egalitarian P ar eto-optimal d ivision s of the cak e and o thers w ith more un­
equal divisio ns ( see, for ex ample, B inmore and S amuelson 1994; B inmore
1994). Given the coo rd ination and bar gainin g problems in repeated games,
an important lesson emerges: d ifferent cultures ( gro ups, co mmunities) may
have enforced d ifferent so cial norms (or no efficien t social no rms at all) in
structurally similar recurrent situations. I t seems lik ely, in o ther words, that
path d ependencies ex ist (for general d iscussions of path d epend ency, see
No rth 1990, 92-104; Greif 1994; Yo ung 1996).
It could perhaps be argued that path d epend ency mak es the theory
untestable. How ev er, this impression is wrong. F irst, the game theoretical
appro ach suggests that in cases of a low d isco un t f actor ( smaller than the
cr itical thr eshold value to suppor t coo per ation), no P ar e to- improving so cial
norms can be enforced. T his statement can clear ly be falsified. S econd, in
cases of a large discount parameter, path d ependency is relevant. Even in
this con text, how ever, hypo theses w ith empirical conten t abo ut the emer­
gence of diff er ent no rms in str ucturally similar so cial systems can be d e­
r ived, and thus it is importan t to includ e information on historical d ata in
this contex t. T hird, ado ptin g an evo lutionar y appro ach (Youn g 1998), the
str ucture of r esultin g " long-run" eq uilibria of a dyn amic may be described
quite precisely. Under par ticular circumstances, there may even ex ist equi­
l ibr ia w ith no path dependen cies or efficien t "lon g-run " path-d epend en t
eq uilibr ia.
Some cr itics of the r ational cho ice appro ach to norms point out that
there are many no rms with no asso ciated efficien cy gain s. Some norms
would even stabiliz e P ar e to-inferior states ( E lster 1989). Y et this obser va­
tion is con sistent w ith the traditional as well as the evo lutionar y game
theoretical appro ach. Ow in g to histor ical accidents or fo r other reason s,
there may be instances o f norms o r conven tions that are stable but subopti­
mal.
T he analysis of repeated games suggests the gen er al id ea that so cial
norms emer ge endo genously in recurrent d ilemma situations if the popula­
tion is a close-knit community ( Ellick son 1991, 167). A clo se-knit commu­
nity is "def ined as a so cial network whose members have credible and
recipro cal pro spects for the application o f pow er against one ano ther and
good supply of infor mation on past and present intern al even ts" (E llickson
1991, 18 1).8 No tice that it is assumed that the application o f pow er against
on e ano ther" is based on particular types o f sanctions, nam ely, trigger
strategies. O ther types of sanctions will be discussed soon.
11
118
SOCIAL NORMS
M any concr ete ex amples illu str ate this mechanism of nor m emer genc e.
Amon g the mo st prominen t are nor ms of r ec iprocity in social exchange
( Ax eirod 1984; Vo ss 1985) and-in n- ac tor inter actio ns-nor ms of collusion
in oligopolistic ind ustr ies (P hlips 1995). Ano ther ex ample is nor ms of pr i­
vacy w ith r egard to po litic al ac tor s. In the United S tates, the media heavily
investigate the private lives of politic ians. In Ger many, there is an infor mal
nor m that mass media shou ld no t r epor t d elic ate or o ther pr ivate d etails of
politic ian s' lives w ithout their con sent. This may be a conjoint nor m be­
c ause enforc ement may be in the interests of most jour nalists and pub­
lisher s. T he nor m may fo ster po litic ian s' willin gness to cooper ate w ith
r egard to the tr ansf er of infor mation on po litic al issues. Also, mo st high­
q uality jo urnalists would no t w ant to investigate pr ivate affair s but would
pref er to research impor tant po litic al issues. On the other hand, there is a
clear temptation to disclo se private aff air s bec ause any newspaper (or
o ther med ium) that acted as a pr ime mo ver in this r espect wou ld sell more
copies ( and would ther eby impro ve shor t-run profits). A pparently, this
conjoint nor m is in par t enforced by tr igger str ategies among the press. A
leading Ger man jo ur nalist, commen ting on the Monic a Lew insky aff air of
1998, has r emark ed that the Ger man press r espec ts the ethic al code as lon g
as everybody confor ms to the norm.9 O therw ise, nor m conformity r apidly
collapses: one d ev iation immediately tr igger s near ly univer sal d ef ection.
T he r epeated -games approach, in pr inc iple, helps to construct empir i­
c al ( q ualitative) hypotheses abou t soc ial condition s of nor m emer gence. I t
c an b e used i n two w ays. Fir st, the core theor etic al par ameter s, like the
"shadow of the futur e" (d iscoun t f ac tor a ) and quo tients of payoff diff er­
enc es (a*) , dir ectly su ggest a number of hypotheses:
•
•
•
Coo per ation incr eases with the shadow of the f uture (a)
Cooper ation d ecreases with the "co sts of cooper ation" (T - R)
Cooper ation incr eases with the "co st of conflic t" (T - 0 )
O ther hypo theses can b e der ived b y mak ing use o f "br id ge assumptions"
( in L ind enber g' s 1985 sense) that link var iables of the game theor etic al
mod el to exo genous factor s descr ibing soc ial conditions. For ex ample, with
r egard to muItiper son d ilemma situations, such as the n-per son pr isoner's
dilemma, on e might ar gue that there w ill be a "lar ge-number s" d ilemma
(Buchanan 1965; Olso n 1965 ) in the r epeated game ( R aub 1988): T he r e­
q uirements of coo per ative equilibr ia in the r epeated game ar e less lik ely to
be met as the siz e of the gro up of nor m benef ic iar ies incr eases. Incr easing
gro up size will, in turn, decr ease the ac tor s' mo nitor in g oppor tu nities
( H ard in 1982, 40, 170-71). Ano ther aspect of the large-number s dilemma is
r elated to the agents' shadow of the f utur e. A large gro up w ill mo st lik ely
consist of a hetero geneou s group of actor s- that is, there will be so me
GAME-THEORETICAL PERSPECTIVES
119
member s w ith discount f ac tor s that �r e too low to suppor t univer sal coop­
er ation.
Moreover, the r epeated- games appro ac h c an be used to in q uir e as to
struc tur al effects on nor m emergence in a mor e dir ect way. Wer ner Ra ub
and Jeroen Weesie ( 1990) explic itl y analyze a mechanism that generally fos­
ter s coo per ation in d ilemma situations, namely, mul tilater al r eputatio n.
Consider two ac tor s, A and B, who are invo lved in an exchang e (or o ther )
soc ial r elationship that is associated with a pr iso ner 's d ilemm a. If these
agents A and B w er e embedd ed into a lar ger soc ial network consisting of
potential (structur ally simil ar ) parh1er s C, D,
, infor mation about the
behavior of A and B may spr ead to these third agents, C, D, . . . . In this
c ase, the threshold of the d iscount f ac tor suppor ting an equil ibr ium of co­
oper ation is lower than it wo uld be in an isol ated (soc ial ly ato mized) inter­
ac tion.
Another aspec t of embedd ed ness is multiplexity. Consid er two agents
A and B who are no t only in one relation, Rv but also in a second r elation,
R2, w ith each other. In this c ase, the agents may co gnitively co nnect these
relations. T hey cooper ate in any of these r el ations Ri on the condition that
the par tner l ikewise has cooper ated in all r elations RI, R 2,
It has been
show n that the cr itical value of the discount f ac tor that is nec essar y to
suppor t cooper ation is-und er c er tain cond itions-lower if ther e is multi­
plex ity (Voss 1998b, 106-7, 125-26). (These applic ations of r epe ated games
correspond to analyses of " multimark et" contac t in ind ustr ial or ganization;
see Ber nheim and Wh inston 1990; Hughes and Oughton 1993.) O ther work
combines r epeated- games theor y with social network an alysis by investi­
gating the effec ts of var ious network par ameter s (B usk ens 1999).
R esearch on struc tur al effects in this sense larg ely suppor ts the intu­
itive hypo thesis that nor ms of cooper ation ar e enforc eable in small, stable,
and cultur ally homo geneous co mmunities. I n such comm unities, multi­
later al reputation is very eff ec tive bec ause of the high d egree of network
conn ectedness. The rol e specialization is typic ally low, or, in o ther words,
social relations are "diffuse" and multiplex ( see Taylor 1982, 76, and the
refer ences therein). Much ethno gr aphic and histor ic al evidence corrobo­
r ates this intuition ( see, for ex ample, Gr eif 1994, 1998; M ac aulay 1963;
Ellick son 1991).
.
.
.
.
.
.
.
Direct (Retributive) Sanctions
The trigger str ategy w ith the implic it threat of d ef ection ( "N ash threat,"
with r espec t to the pr ison er' s d ilemma) is not the only kind of sanc tioning
mec hanism we encounter in r eal- lif e inter ac tions. N umerou s ex amples
from the sociolo gic al liter atur e (for exampl e, Homans 1951, 1 23 and pas­
sim) refer to nor ms that seem to be eff ec tive bec ause of the imposition of
mor e ac tive and possibly mor e sever e sanctions. Another mechanism is the
120
SOCIAL NORMS
applicat ion of exit t hreats: norm d ev iat ion is pun ished by o straciz in g t he
dev iators from fut ure int eract ion s. T here are many empirical descript ion s
of this san ction in g mechanism (for ex ample, Ost ro m 1990) and so mew hat
few er theoret ical an alyses (Tullock 1985; H irshleifer and R asmusen 1989;
Schtissler 1989; V anberg and Congleton 1992).
Norms suppo rt ed by exit threats can be con sid ered as cases of decen­
tral ized out com es of m arket processes. These norms may be enforced even
in cases of large n umbers of actors und er cond ition s of anonymity (Tullo ck
1985; S chti ssler 1989). T he basic po int is that d efectors can be crowd ed o ut
under cert ain struct ural condit ions ( if a large pool of cooperators is av ail­
able and exit and searching cost s are min imal). T hese pro cesses are, how­
ev er, not easily d escribed an alytically. They are best illustrat ed by intuit ive
id eas or by simulat ion studies (Schti ssler 1989; V anberg and Con gleton
1992).
In Co leman's d iscussion of norm enfo rcement, " in crement al" sanct ions
are import ant (Coleman 1990, chapt er 11). T hese san ct ion s are more act iv e
than t he thr eat t o defect. Increment al sanctions are applied b y members of
a group. Every ind ivid ual provides a small " in crement" of a sanction that
is apparent ly not very costly to the pun isher. The sum of t hese sanct ions
m ay, on t he other hand, be a severe punishment to the norm deviator. An
ex ample may be so- call ed mobbin g ( horizont al viol en ce, bullyin g) in work­
in g gro ups. T he person who d eviates from a gro up norm is pun ished by
so cial disapprov al of the other group members.
T he fo llow in g ex ample, borrowed from Fran cis F ukuyama (1995, 196),
may illumin at e t he d ist inction betw een ind irect and d irect san ction s more
clearly: The community of Internet users w as initially ( in the 1970s and
1980s) q uit e small. A mon g the net' s informal conjoint no rms w as a rule
prohibit in g the use of e- mail for co mmercial advert ising. Using el ectron ic
mail for advert isin g co uld clo g t he n et and would be a pot ential threat to
t he viability of the system as a who le. The informal norm w as broken in
1994 by a pair of lawyers who adv ertised for t heir services. Other users
react ed not by advert isin g for their own services (which would hav e been a
plausible response, at least for co mmercial users) but by sendin g hat e
e-mail t o the d ev iators and t o the service provider. T he prov ider ev entually
canceled the acco unt. The ex ample ill ust rat es increment al sanctions by
e-mail users and t he applicat ion of t he sanct ion of social ex clusion by a
third party. Both of these sanctions are clearly d ifferent from the N ash
threat of defect ion.
A simple gam e t heoret ical mod el, in spired by Robert Boyd and Pet er
Richerson' s ( 1992) evolut ionary an alysis, can be useful in an alyz ing retribu­
t iv e and other d irect san ctions. Figure 4.6 illust rates an en larged v ariant of
the classic prisoner's d ilemma present ed in figure 4.5. T his en larged game
is in fact a mod ification, rather than an ex ample, of the classic prisoner's
dil emma. I call this gam e the norm game ( act ually, it is a simpl ified version
F IG U R E 4.6
The Norm Game in Extensive Form
@
(R-k-p,R-k-p) (R-k,R-p) (R-p,R-k)
(RR) (S-k-p,T- k-p) (S-k,T-p) (S- p,T-k)
(S,T) (T-k-p,S-k-p) (T-k,S-p) (T-p,S - k)
T>R>O>S
Source: Author's compilation.
(T,S) (- k-p,-k -p) ( -k - p,-p) (-p,-k)
(0,0)
122
SOCIAL NORMS
of Ax elrod' s [1986 ] nor ms game). In the nor m game ther e ar e two stages. In
the fir st stage, both player s simultan eously choose betw een the altern ative
respon ses- coo per ation (C) and defection (D )-to the pr isoner' s dilemma.
T he second phase is a san ction ing or punishmen t stage. In this stage, both
player s can r eact on the decision s of the fir st stage (which ar e d isclosed at
the end of the pr isoner's d ilemma phase). Bo th player s can punish their
par tn er s by selectin g a negative san ction, s. A ltern ativ ely, they can select
s*- that is, they can chose not to employ a san ction. Usin g this sanction
yield s a pun ishmen t cost of -P to the tar get actor and a co st of ( actively)
san ctionin g of -le I assume that the payoH s in the nor m game are sums of
the payoH s in both stages of the game, n amely, the payoH s of the pr is­
oner' s dilemma (T, R, 0, or S) and payoffs of the sanction in g phase ( - k,
-p, or [-k -pD.
In the nor m game, an actor who defects v is-a-v is a cooper atin g par tner
r eceives a payoff of T. The actor w ho is pun ished by his or her par tn er
receives the payoff T p. The par tn er' s payoff is S - k.
As played out in figur e 4.6, there are three r esults of the norm game
that indicate cir cumstan ces in which there is no second-ord er dilemma.
-
•
Propo sition 1: In the nor m game there is a N ash eq uilibr ium of mutual
cooper ation if and only if p 2:: (T
R).
-
•
Proposition 2: A N ash eq uilibr ium of mutual cooper ation is subgame
perfect if and only if k 2:: O.
•
Propo sition 3: In an infinitely r epeated nor m game w ith disco un tin g,
there ex ist subgame perfect eq uilibr ia of un iver sal coo per ation even for
k > 0, if the shadow of the future (d iscount factor ) is l ar ge enough.
Propo sition 1 is based on the id ea that the player s use str ategies of the
fo llowing k ind: "P lay C; if your par tn er has chosen D, play s, o therw ise
select s*." A pair of tho se str ategies is an eq uilibr ium if and on ly if the co st
of cooper ation (T - R) is not lar ger than the pun ishmen t p. Intuitiv ely, the
en largemen t of the or igin al pr ison er's d ilemma by mean s of san ctions
tr an sfor ms the pr isoner's d ilemma into a game that has affinities w ith an
assur ance game ( in S en's sen se). In an assur an ce game, mutual cooper ation
is an eq uilibr ium because both player s hold the prefer en ces R > T
> 0 > S.
Propo sition 2 says that the threat to puni sh a d efection is cr edible if
and only if the cost of pun ishing the par tn er is k s; O. If k > 0, a r ational
player would not hav e an incentive to implemen t the thr eat to pun ish a
d efection because the punisher wo uld no t only san ction the par tner' s nor m
d eviation but would also pun ish himself or her self. Assumin g that a par t­
n er has defected, this defection is irrever sible. P un ishin g this defection w ill
not in cr ease the pun isher's payoffs; in fact, it w ill even decr ease the payoH s
GAME-THEORETICAL PERSPECTIVES
123
if le > O. To reit erat e, these result s d emon strat e cond ition s of rat ion al mu­
t ual condit ion al coo perat ion in a n� rm game that is not repeat ed. Und er
the condit ion s st ated in the pro po sition s, there is no second-ord er d ilemma.
How ever, the req uirement t hat le :::; 0 seems to be q uite stron g. Yet it
has been observed that the applicat ion of san ct ion s is gen erally not very
co st ly-in particular, in comparison w ith t he harm that san ction s may
cause (H ardin 1995, 52-53). How could an assumption of costle ss san ction­
in g (le :::; 0 ) be j ust ified? F irst, in t he case of int eract ion s involvin g social
approval-for in st an ce, so cial ex chan ge relat ion s in small- gro up cont exts­
the refusal to prov id e so cial approv al to a gro up member may be a n ega­
tive sanct ion that is not co st ly. Moreov er, it may even be the case that this
act ion is a po sit ive rew ard (k < 0) to the pun isher if the provision of social
approv al is co stly. S econd, emotion al dispo sit ion s may serv e as commit­
ment devices (Frank 1988). If on e has observed a parh1er' s d efect ion, san c­
t ion in g may prov id e emotional, "psychic" rew ard s that outw eigh the mate­
rial costs of pun ishin g. Strict ly speak in g, elaborat in g this idea mean s that
the payoff s of the norm game must be specified in such a way that these
rew ards are represent ed in add it ion to t he other compon ent s of ut ility.
Third, the norm game may be embedd ed into a larger game that represents
social network s. An actor in tho se network s might have an int erest in ac­
quirin g a person al reput at ion as a bully. Such an actor wo uld be w illin g to
apply sanct ion s even in cases in w hich the mat erial co sts ar e large. The
basic reason is that such a reputation will be profitable in fut ure interac­
tion s wit h t hird partners. T his id ea could be mod eled explicit ly in game
theo ret ical t erms ( see E aton and En gers 1999 for hints on such an an alysis
in t he context of intern at ion al relations). How ev er, I will not pursue t his or
other defen sive appro aches here, because there is another ent irely plausible
scen ario in which no second-order d ilemma emerges even for large (le > 0)
co st s of san ct ion in g.
P roposit ion 3 addresses the norm game that is repeat ed ind efin itely.
The actors have ex pect ation s with regard to the chan ces of another repet i­
tion, given t hat the game has already been played for a n umber of period s.
These expect ation s are measured by t he d isco unt factor (the shadow of the
fut ure). The stand ard repeat ed- games assumpt ion s apply.
The proof of propo sition 3 follow s from t he find in gs of Drew F uden­
berg and E ric M ask in (1986) on fo lk theorems and on o pt imal punishment.to
Int uitively, infin it ely repeat ed games en sure that punishment co sts to the
punisher w ill be o utw eighed by the t arget's cooperat ion in the futur e.
Therefore, ev en for posit ive co sts (le > 0 ) of san ct ion in g, the threat to pun­
ish d efection may be opt imal. To illust rate, con sider an n-person v ariant of
the norm game-for in stan ce, H ardin' s (1971, 198 2) n-prisoner' s d ilemma
game with an addition al phase of d irect pun ishment in each period of the
it erated game ( see Boyd and R icherson 1992). It is assumed th at
124
1.
SOCIAL NORMS
Ev ery actor can san ct ion every other actor by d irect pun ishment.
2. T he level of pun ishment is in depen dent of the n umber of punishers.
3. There may be positive co st s of pun ishin g, le > O. These cost s are not
shared an d are independ ent of the number of pun ishers.
The repeated no rm game makes it po ssible to use con dit ion al strate­
gies t hat are in fact a mixture of d irect and indirect pun ishments. These
st rat egies require that actors be "friend ly," that is, that they cooperate in
the first period of the game. Mo reov er, they demand that a d efector be
pun ished for a number-say, t-of period s by implement in g d irect and
ind irect sanct ion s. Furthermo re, an agent who fails to punish a norm d evia­
tor will be similarly punished for a n umber of period s by d irect and ind i­
rect sanct ion s, and so forth. S uch a patt ern of pun ishment s may be ind iv id­
ually opt imal because severe and co stly san ctionin g tod ay may en sure
perpet ual un iversal cooperat ion in the fut ure (see, fo r ex ample, Osbo rn e
and R ubinst ein 1994, chapt er 8, for a d etailed expo sit ion).
This pun ishment regime may look very complicated. Y et on e should
not e that in equilibrium these sanctions n eed not be implemented. What in
eq uilibrium w ill be observed is universal conformity. The en forcement o f
the social norm i s based o n the agent s' self-fulfillin g ex pectat ion s that
every d eviation from the norm and t he metanorm w ill be sanctioned.
In con clusion, there is no second -order d ilemma for a sufficient ly large
shadow of t he fut ure, even if the agent s implicit ly use the t hreat to imple­
ment co stly pun ishment s (prov ided the punishment co st s are not too
large). In add it ion, on e sho uld note not only that t his mechan ism of en­
forcement may sust ain P areto-opt imal ( " good " ) so cial norms but also that a
populat ion of rat ion al agent s may realize a "bad " (that is, inefficient) social
no rm by usin g t hese sanction s ( as is also argued by H irshleifer and
R asmusen 1989; Boyd and Richerson 1992).
Discussion
The result s of this ex po sit ion may be summarized as fo llows. Conjoint so ­
cial norms emerge amon g ration al actors in situation s of a prison er's d i­
lemma und er t he fo llow in g condit ion s:
1. norm d eviat ions are d et ect ed easily and immediat ely
2. t he beneficiaries use appropriat e ind irect or d irect san ction s
3. either san ct ionin g is co stless (k :s 0), or there is a sufficiently large
shadow of t he future
4.
appropriat e " cultural beliefs" are held in the po pulation of norm ben e­
ficiaries
GAME-THEORE TICAL PERSPE CTI VE S
125
The first cond ition seems t rivial, but,it is substantively very import ant. For
example, conf or mity to the n o-drugs n orm in sports probably w ould yield
efficiency gain s t o t he group of compet in g at hletes. One could argue t hat in
prin ciple m any condit ion s of a repeat ed prison er' s d ilemma are present in
this case. T he same athl et es repeat ed ly meet each other in matches ind efi­
nitely oft en. However, the inform at ion al requirem ent s of alm ost perfect in­
for mat ion on the other agents' past moves in t he game ar e obviously not
ful filled. An athlete cann ot observe whether or n ot a compet itor has t ak en
pr oscribed dr ugs. T herefore, t he beneficiar ies of the norm will n ot be abl e
t o enfor ce the n orm w ithout the assist an ce o f a third party.
Enforcement of norms also req uires the use of appropriate sanction s­
both direct and ind ir ect- on t he part of ben eficiaries ( condit ion [2]). Indi­
rect sanction s ar e simil ar t o t rigger or t it -for-t at strat egies. T hey react t o a
partner's d efection by sel ect in g a r espondin g d efection. Indire ct san ction s
can be very effective, but under many circum st an ces, they have d isad van­
tages. In large groups, tr igger strat egies pun ish n ot only those agent s who
deviat e from a n orm but al so t he members of t he group who have con­
for med to t he norm. A sin gle d efect ion tr iggers un iversal d efect ion w it hin
the whole group. D irect sanction s, on the other hand , ar e applicabl e mor e
pr ecisely tow ard those act or s who have defected . They can al so be impl e­
mented in sit uation s that are n ot repeated . H en ce, it is n ot surpr isin g that
the bulk of sociological and ethn ogr aphic d escr iption s of informal n orm
enforcement contain s statement s on this type of san ct ions.
M oreover, with regard to end ogen ously cr eat ed sanct ion in g, it is im­
port ant to n ote that the cond it ion of costl ess sanct ion in g ( con dition [3]) is
cr ucial if one accept s the ration al ity crit erion of t he cr edibility of t hreats ( or,
t echn ically, sub game perfection). T he crit ical t hr eshold value of t he shad ow
of the futur e (d iscount fact or) depend s on cert ain variables, nam ely, the
embedd edn ess of the r ecurrent interact ion in social n etw ork s . In gener al, it
can be argued that d en ser and mor e "closed" social net works are more
effective in promot in g n orms of cooperation t han m or e open" social str uc­
tur es (C oleman 1990, chapter 11). Game t heoret ical analyses m ay give a
rat ion al e for this intuition by point in g out t hat structural embedd edn ess
reduces-cet er is paribus-the cr it ical val ue of the d iscount factor, which is
necessary t o sust ain a N ash equil ibr ium of cooper at ion . Note, how evel�
that t here are except ion s t o t his r ul e of t he str en gth of cohesive groups ( see
Fl ache 1996; Fl ache and M acy 1996). An ot her aspect of the d isco unt fact or' s
cr it ical val ue i s it s (logical) d ependen ce on the payoff differen ces in the
or igin al game: The l ar ger the cost s of cooper at ion (T
R), the larger t he
cr it ical value of the shadow of t he future, a* . ( A simil ar r elat ionship hold s
for t he requir ed level of d irect pun ishment : t he l arger t he cost s of coopera­
tion, the more severe m ust be the punishm ent.)
None of the N ash equil ibria in repeat ed games ( for a sufficient ly large
shad ow of t he future) and in a n orm game are unique-that is, there is a
/I
-
126
SOCIAL NORMS
coordin ation or a bar gainin g problem of selectin g on e specific equilibrium.
T he enforcemen t of norms therefore is based on the assumption of join tly
held " cultur al beliefs." Appropriate cultural beliefs are tho se ration al be­
liefs "that capture individuals' ex pectation s with respect to actions that
o thers w ill tak e in various contin gen cies" (Greif 1994, 915 ). It is impo rt an t
that these rational cultural beliefs are restricted to those expectations that
are self- enforcin g-that is, ev ery player is expected to play his or her best
respon se, given id entical and commonly known beliefs ( see Greif 1994,
914- 17). This is a crucial property of cultur al beliefs in game theoretic
terms: T hey w ill no t be stable determin an ts of behavior if they are no t
fo cused on a N ash equilibr ium. In contrast, certain culturalist explan ation s
are based on the idea that cultural id eas d ir ectly affect behavior. T he ratio­
n al cho ice and game theory appro ach, how ev er, suggest s that cultural be­
liefs per se canno t prov id e in centiv es that chan ge behavior, altho ugh they
may help to select a mo re or less efficien t eq uilibrium profile of action s.
Cultur al beliefs are, for ex ample, for med and tr ansmitt ed in so cializ a­
tion pro cesses. On ce established, they beco me focal poin ts that d etermin e
what type of behavio ral regularity or nor m w ill b e enforced in a given
population. I t is obvio usly possible that two societies exhibit similar social
str uctures ( that is, similar incen tives to cooper ate or conform to a nor m)
but have huge d ifferen ces w ith regard to so cial in stitution s and so cial
norms. No rms are typically d epend ent on the paths tak en by history. In
this w ay, besid es the shadow of the future, a "shadow of the past" co mes
into play.
T he impor tance of cultural beliefs can hard ly be overestimated. How­
ev el� classical game theo ry canno t con tr ibute much tow ard an ex plan ation
of cultural beliefs. C ultural beliefs have to be treated as given, exo genous
variables. The main ex ception arises in so me Bayesian games w ith inco m­
plete information such that certain beliefs are r ation ally ad apted or learn ed
by rational agent s.
Learn in g pro cesses with respect to cultural beliefs are at the cent er of
some recen t d ev elopments within evolution ary game theory. These evo lu­
tionary appro aches are socio lo gically relev an t if they dispen se w ith the as­
sumption of genetically determined behavior. As one of the pioneers of the
evolution ary appro ach has noted, "Game-theoretical ideas origin ated
w ithin socio lo gy. N aturally eno ugh, the so lution concepts w hich d ev eloped
w er e based on the id ea of ration al calculation. The ideas w ere borrowed by
evolut ion ar y bio lo gists, who introduced a n ew con cept of a solution, based
on selection and heredity o per atin g in a population. I f, as seems lik ely, the
id ea of evo lution ary stability is now to be rein trod uced into sociolo gy, it is
crucial that this sho uld be don e only when a suitable mechan ism of cul­
tural hered ity ex ists" ( M ayn ard Smith 1982, 17 2). T here are many semin al
contribution s to specify socio lo gically mean in gful ideas on mechan isms of
GAME-THEORETICAL PERSPECTIVES
127
cul tural heredity (for ex ampl e, Boy� and Richerson 1985). Mo st of these
work s use the assumption of bo undedl y ration al behav ior and try to elabo­
rate an alytically tractabl e and empirical ly meanin gful assumpt ion s about
individual and social learn in g pro cesses.
T he core features of some recent appro aches ( F ud enberg and Levin e
1998; S amuel son 1997; Yo un g 1998) are summarized as follow s :
•
A popul ation of bound ed ly ration al agen ts who are recurrently en­
gaged in so cial interaction s is represented as a sto chastic dyn amical
system.
•
Bounded ration ality mean s that only l imited d ata, not perfect informa­
tion about the o thers' past behavior, is req uired.
•
A gents do not sel ect w ith certain ty best replies tow ard t heir expecta­
tion s abo ut others' behavior; t here is some small probability of an erro­
n eous respon se.
•
T here are po ssibly sto chast ic sho cks correspond in g to mutation s w ithin
biological evolution.
The basic research q uestion of t ho se mod el s con cern s t he lon g-run be­
havio r in these interactions. Con sid er a game in no rmal form w ith multipl e
strict Nash eq uilibria (fo r exampl e, a coord in ation game). It i s of in terest,
then, to inq uire w hich eq uil ibria w ill more freq uen tly persist in the lon g
run -under condition s of sto chast ic shocks. Fo r example, in a two-by-two
(matrix) coo rd in ation game w ith two eq uil ibria, on e of them being P areto
domin an t, certain k ind s of learn ing pro cesses (which are described and
illust rated in Youn g 1998) w ill real ize t he efficien t equil ibrium, provided
that t his equilibrium is con sistent w ith the risk-domin ance criterion, as
stated by John Harsanyi and Reinhard S elt en (1988). Howev er, d ependin g
on properties o f t he payoff structure, it n eed not b e the case that the " mo st
efficient" equilibria will be "sto chastically stable" long-run eq uil ibria of the
dyn amic ( Kando ri, M ail ath, and Rob 1993; Yo un g 1998).
With regard to t he repeat ed prisoner' s d il emma, con sider ation of the
lon g-run d yn amics implies an assumption that there is a nonzero proba­
bility that any part icular strat egy w ill beco me ext in ct. T his con trasts w ith
Robert Ax el rod's ( 1984) appro ach, w hich stud ies on ly the short-run dy­
n amic. In the Ax elrod appro ach, it is assumed that the con test of strategies
yield s states in which t he l ess successful strategies beco me successively
extin ct. It can be shown t hat t he evolution ary dyn amic may d ramat ically
chan ge if on e adopts the lon g-run dyn amic perspectiv e. In particular, tit­
for-tat w ill be much l ess v iabl e in the lon g run than in the short run (Yo un g
and Foster 1991).
The n ew evolution ary appro ach provid es sev eral important results
that are, in principle, empirically t est able (Youn g 199 8, chapters 4 and 5):
128
SOCIAL NORMS
For cert ain si mple games in normal form ( coordin ation and non cooperative
bargainin g games in H arsanyi's 1977 sense), learnin g pro cesses wi thin a
population of bounded ly ration al agents in fact lead to the evolution of
st able conv entions. How ever, I am not aw are of any empiri cal work that
employs t hi s appro ach. One di fficulty in the appli cation of the new con­
cept s of sto chasti cally stable equilibri a and so forth i s that it i s not easy to
d et ermin e what the lon g- run behavior perspective mean s empi ri cally.
Long-run behavior certain ly does not alw ays correspond with a lon g di s­
tan ce on a physical or hi stori cal ti mescale (Young 1998, 14-19). Most em­
piri cal studi es on norm emergence use a short-run perspective.
T he new evo lutionary appro ach is i mpo rt ant because i t shed s n ew
li ght on an i ssue that has up to now gen erally been treated as an exo g­
enous problem, the evolution of conv ent ion s. As has been argued, any so ­
ci al norm i s based on conv entions o r cultural beli ef s that there i s a st able
regularity of behavio r in a population of recurren tly interactin g agents. In
t he case of soci al norms, there are rational reci pro cal ex pectations that cer­
tain action s are forbidd en or allowed and that there wi ll be puni shment in
cases of devi ant behavior. T he new appro ach may be parti cularly fruitf ul
because it link s evo lution ary and bounded ration ality ideas to more tradi­
tional game theoreti c concepts, especi ally the N ash equi libri um. I t may also
be und erstood as a ration aliz ation of the more t radition al concepts that
have been developed fro m a hyperration ali sti c perspective of perf ect ly ra­
tional players. Some of the results of tradition al game theory, in fact, appar­
ently are robust if mo re "realistic" id eas abo ut individuals' behavior are
introd uced.
T he sk etch of a game theoretical ex plan ation o f conjoin t so ci al norms
demon strates that those no rms emerge amon g self-interest ed actors if ap­
pro pri ate soci al conditions are met. How ev er, thi s appro ach to soci al norms
is incomplete. P eo ple someti mes follow social norms in the absen ce of
credible ex tern al san ction s: for in stance, it i s diffi cult to underst and why
people confo rm to the no rm of ti ppin g in rest aurant s they are sure they
will never vi sit again. T here i s ample eviden ce of cooperat ion in si tuations
with external incentiv es to free rid e. T he lev el of cooperation observed in
soci al dilemma and publi c good s situation s i s si gnificantly hi gher than pre­
di cted by game theoretical ration ali ty st and ard s ( see, for ex ample, T haler
1992, chapt er 2; Green and Shapiro 1994, chapter 5; and Kagel and Roth
1995 for surveys of the research results). T hi s does no t mean that the v ari­
ables fostering o r inhibiting cooperation- accordin g to game theo ry- such
as group size, shadow of the f uture, and t he lik e, do not have marginal
effects.
Two po ssible mechani sms may be relev ant. Fi rst, it may be import an t
to incorporate intern al in cen tiv es into a rational choi ce an alysi s o f social
norms. ( Robert Frank [ 1992, 150 -52], for ex ample, cri tically comments on
Co leman's approach). In contrast to the tradi tional socio lo gi cal appro ach,
GAME-THEORETICAL PERSPE CTIVES
129
int ernalized norms sho uld not be !reat ed as given, exo genous factors. It
seems, how ever, d ifficult to give a consist ent ex plan at ion of t he adopt ion of
preferences in which (cond ition al) cooperat ion is v alued per se. On ly quite
recently have some id eas emerged that attempt to t ack le this problem. One
import ant observat ion in this context is that intrinsic motives may play an
instrument al ro le in an actor' s ability to achieve rew ard s that are material
or so cial ( such as social approv al) . For ex ample, con siderin g a st andard
(sin gle- shot) prison er's d ilemma situat ion, indiv id ual agent s would be bet­
ter off in t erms of t heir "n at ural" (material and social) preferences (T >
R > 0 > 5) if they w ere endowed with intern alized "mo ral" preferen ces
prescribing cond it ion al cooperation (Rill > Till > 0 > 5111) in stead of un con­
dit ional d efect ion. Moral preferen ces may result fro m an intern alized san c­
tion in g syst em that prov id es negat ive san ct ion s ( in the form of a guilty
conscien ce), vis- a.-vis a coo perat in g partner, to an actor who unilat erally
defects. For ex ample, if the co st of defectin g is In, an actor w ill receive (Til l :
= T - In) inst ead of a payoff T. I f (In > [T - R]), t hen t he strength of the
int ern al sanction s would chan ge the in centiv e struct ure of t he game, pro­
vid ed both actors had ado pt ed moral preferences and there were some
tran sparen cy ( common know led ge) wit h respect to t hese moral prefer­
ences. These moral preferences wo uld transfo rm a prisoner's d ilemma into
an assurance game. In an assuran ce game, mut ual cooperat ion is a N ash
equilibrium. Und er these circumst ances, an actor w ith moral preferen ces
will cooperat e with a partner w ho is similarly endow ed w ith moral prefer­
ences. The result in g out come wo uld be superior even in t erms of the ac­
tors' n at ural preferen ces ( see S en 1974, 80).
Another scen ario is relat ed to the norm game. Con sid er two or more
agents who are invo lved in such a prison er's d ilemma with a possibility to
use t hreat s of negat ive sanctions. T hese agent s wo uld be better off in t erms
of t heir nonmoral or "n at ural" preferen ces if intern aliz at ion meant that an
agent would feel obliged to employ san ctions even if they v iolated his or
her material int erest s. S uch an agent who is committed to punishing an
opportunist ic partn er would use a cred ible threat even if this t hreat wo uld
not be a cred ible on e in t erms of the agent's mat erial int erest s. So me contri­
butions to t he lit erature attempt to d emonstrate that rational actors would
choo se such moral preferen ces if they had an oppo rt unity to mod ify their
preferences.ll
Another body of wo rk is based on the assumption of bounded rat io­
nality. First of all, the not ion of action frames as used in "int erpretative"
sociology and co gn it ive psycho lo gy may be relev ant in t his context. T he
basic id ea is that actors use sit uation- specific d ecision rules. If a situation is
framed to be govern ed by a social norm, t he ind ividual w ill use a norma­
tive d ecision rule.12 T he main d esiderat um of such an analysis is that ad
hoc ex planations sho uld be strict ly avo id ed. Every behavior could, in prin­
ciple, be " ex plain ed" by post ulat in g specific preferen ces or a specific frame
130
SOCIAL NORMS
or d ecision rule. St ill another body of work is b ased on the id ea that
bo unded ly ration al actors are endow ed with preferences that are subj ect to
evo lution ary forces (Frank 19 88; Cuth and Kliemt 1993). An argument that
b ears some family resemb lance to this follow s from the recent evo lutionary
appro ach (Samuel son 1997, chapter 5). It can be argued that Nash equi­
lib ria that are not sub game perfect may, nev ertheless, b e stab le lon g-run
eq uilibria of evo lutionary pro cesses. A lso, certain norms of fairness-fo r
ex ample, the norm to leav e mon ey fo r the w aitperson on the tab le- may
be st ab le b ecau se " the amoun ts involv ed and the freq uen cy with which the
situation arises are too small to prov id e sufficient evo lutionary pressure to
elimin ate the pheno menon in a no isy env iron men t" (Samuelson 1997, 164).
Appendix
There are many ex cellent introd uction s and comprehen siv e textbook s on
game theory that may be con sulted, includ in g E ric R asmusen ( 1994), Av i­
n ash D ix it and Susan Skeath (1999), and, more advanced, Ken B inmo re
(1992) and M art in Osbo rne and Ariel Rub instein ( 1994). B inmore's (1994,
1998) treatise giv es an illumin atin g game theoretical analysis of moral
norms in the spirit of D av id H ume. T he core concepts of game theory are
in trod uced informally and in the con text of a d iscussion of philo sophical
topics.
T he most important concept of game theory is the Nash eq uilibrium. A
Nash eq uilibrium of a game w ith complete information in strategic o r no r­
mal form (such as the games in figures 4.1 thro ugh 4.5) is a profile of
action s such that no player has a positiv e incen tiv e to unilaterally d ev iate
from that profile. T hat there is no positiv e incen tiv e to d ev iate means that
no player wo uld be ( strictly) b etter off if he or she d eviated ( unilaterally)
from the equilib rium, provided all o ther players choose their equilib rium
actions.
For ex ample, in the b attle-of- the- sexes game charted in figure 4.2, there
are two Nash equilibria, n amely (A, A) and ( B, B ) . T hese are equilibria in
pure strategies, which means that an action of the equilib rium profile is
cho sen w ith a prob ability of one. There is also a mixed eq uilibrium in this
game. A mixed equilib rium represen ts circumstan ces under w hich the
player's b ehav ior is inf luenced by prob ab ilistic rules.
The mix ed eq uilib rium can be calculated easily usin g the theorem that
any pure strategy is a b est reply to the mixed equilib rium strategies of the
other player or players. This means that any pure strategy of an actor
again st the mixed equilib rium strategies cho sen by o ther players yields the
same payoff to this actor. To illustrate, consider the b attle-of-the-sexes
game. D enote q the row player's prob ab ility of choo sin g A. This player,
then, w ill choo se B w ith ( 1 - q) . Denot e p the co lumn player's prob ab ility
of choosin g A. T hen it must b e the case that the co lumn player is indif-
GAME-THEORETICAL PERSPECTIVES
131
fer ent between A and B if the row pJayer u ses his or her mixed equ ilibr iu m
ac tion. This means that the co lu mn player's expec ted utility ( E ll) of u sing
A equ als his or her payoff of u sing B against the row player's m ixed str at­
egy:
E UA(colu mn)
=
q
=
E UB(column)
=
2(1
-
q) .
Solving for q gives the equ ilibr iu m valu e of q, namely, q
2 / 3 ( that is, a
probability of two in thr ee). By similar reaso ning, one gets the probability p
for the co lmID1 player's mixed str ategy, p
1 / 3 ( that is, a probability of
one in thr ee) . I t c an easily be seen that the expec ted payoff of this mixed
equ ilibriu m is a 2/ 3 for bo th players. In extensive games with perf ect infor­
mation (for example, the nor m game shown in f igure 4.6) includ ing a de­
tailed descr iption of the player s' sequ ential dec isions, no t all N ash equ i­
libr ia may be sensible. Consid er two actor s, E and W. E has th e f ir st move
and pro po ses a contr act A to W, yield ing a payoff of two to E and one to
W. Alternatively, he cou ld propose a d iffer ent contr ac t B, yield ing one to E
and two to W. W c an acc ept or refu se either contr act. If W r efu ses a con­
tr ac t, bo th player s r ec eive a payoff of z ero . T here ar e two equ ilibr ia in this
game, namely (1) player E proposes contr act A, and player W is willing to
acc ept either contr ac t A or B; and (2) player E proposes contr ac t B, and
player W is willing to acc ept contr ac t B and r ejec ts contr ac t A. T he second
equ ilibr iu m contains a thr eat by player W to r efuse the und esir able con­
tr act. However, is this thr eat cr ed ible? Assu me that player E had, in f ac t,
cho sen to propose contr ac t A. W would not, then, employ his threat to
refu se bec au se this refu sal wou ld hur t riot o nly player E bu t also player W;
thu s threat is no t a cr edible one. For mally, the cr iter ion of sub game perf ec ­
tion (S elten 1965) ru les ou t tho se N ash equ ilibr ia that u se threats or pro m­
ises that are no t cred ible. Subgame perfect equilibr ia ar e stra tegy profiles
that ar e N ash equ ilibr ia for any histor y of the game, inc lud ing histor ies ou t
of the equilibr iu m path. A subgame is rou ghly d ef ined as a h istor y of the
game that star ts with a single nod e (d ecision po int) and includ es all of its
successor s. This nod e is a singleton within the player' s inform ation set,
that is, the player k nows that he or she has to choose at th is par ticu lar
node. For example, one subgame of the bar gaining game between player s
E and W or iginates with the nod e a t which player W has to c hoo se a s the
seco nd mo ver ( af ter player E has decided which contr act to propose). T he
cr iter ion of sub game perf ection r equ ires that a player' s stra tegy cho ices
mu st be optimal given any histor y of the game, that is, player W has to
choo se his or her best r eply in the sub gam e that star ts after E has cho sen
contr ac t A. In this case, W' s best r esponse will be to accept the (undesir­
able) contr ac t. Henc e, the str ategy prof ile ( If contr ac t B is propo sed, acc ept
pro po sal; if contr ac t A is propo sed, rej ec t propo sal) c anno t be subgame
=
=
132
SOCIAL NORMS
perfect because it pr escribes suboptimal behavior in a par ticular histor y of
the game.
Notes
1.
There are other well-known definitions o f social norms that prima facie do not
coincide with the definition given here. Coleman 1990, 243, states that "a norm
concerning a specific action exists when the socially defined right to control
the action is held not by the actor but by others . . . . This implies a consensus
in the social system or subsystem that the right to control the action is held by
others." Yet this concept is roughly equivalent with the game theoretical defi­
nition, if Coleman' s words " rights" and " consensus" are linked to the equilib­
rium concept. A right can be said to exist whenever an actor is free to deter­
mine an action such that there will be no punishment in equilibrium (see, for
example, Binmore
2.
1994, 129
and passim) .
Some authors in the rational choice tradition argue that a demand for a social
norm arises if a population is in a situation with externalities (Coleman 1990,
249-51;
see also Opp 1983) . Coleman' s examples make it clear that the mean­
ing of the term " externality" corresponds to social dilemmas. Coleman's usage
of the term is clearly more general than in welfare economics .
3.
S e e t h e appendix for a n explanation of some technical game theoretical con­
cepts.
4.
Note that a typology of social dilemma situations that generate a " d emand"
for norms may depend on what has been called a general " solution theory" for
noncooperative games (Harsanyi 1977; Harsanyi and Selten 1988) . This is par­
ticularly relevant for certain coordination and noncooperative bargaining
games, that is, games with multiple equilibrium points. In those games consid­
erations about the " risk dominance" of equilibria are of central importance.
Interestingly, this normative concept of Harsanyi and Selten seems to be rele­
vant even for evolutionary game theory and the " theory of learning in games"
(Fudenberg and Levine 1998, chapter 5; Young 1998), which certainly can be
understood as empirical research programs . Moreover, it is of interest to notice
that risk dominance and payoff or Pare to dominance do not necessarily coin­
cide (Fudenberg and Levine 1998, 138) . There will result a different typology of
norm-generating situations if one uses risk dominance criteria rather than
some other solution theory (for example, Harsanyi' s [1977] earlier theory) .
5.
See the appendix for an explanation of the concept of credible threats.
6.
For a more rigorous treatment, see, for example, Friedman
1986
and Myerson
1991 .
4.5.
7.
Compare the prisoner ' s dilemma w ith payoffs as in figure
8.
See also Taylor 1982 and Raub and Voss 1986 for similar arguments and for
hints on the correspondence between game theoretical conditions of coopera­
tion and social conditions of a close-knit community.
9.
Herbert Riehl-Heyse, of
(ARD) on February
10.
Siiddeu tsche Zeitung, in a German
1, 1998 ("Sabine Christiansen" ) .
television talk show
For a convenient exposition, see Osborne and Rubinstein
proposition
151 . 1 .
1 994,
in particular
GAME-THEORETICAL PERSPECTIVES
133
11.
See, i n particulm� Gauthier 1986 and, for a short game theoretical explication of
Gauthier ' s program, Bicchieri 1993, 202-13; other works in this direction are
Hegselmann, Raub, and Voss 1986 and Raub and Voss 1990.
12.
See, for example, Lindenberg 1988; for an approach that is closely related to
game theory, see Montgomery 1998.
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