Reciprocity models revisited: intention factors and reference values

Hinz, Janna; Nicklisch, And eas; Somme , Mey-Ling
A icle — Published Ve sion
Recip oci y models e isi ed: in en ion ac o s and
e e ence alues
In e na ional Jou nal o Game Theo y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Hinz, Janna; Nicklisch, And eas; Somme , Mey-Ling (2024) : Recip oci y models
e isi ed: in en ion ac o s and e e ence alues, In e na ional Jou nal o Game Theo y, ISSN
1432-1270, Sp inge , Be lin, Heidelbe g, Vol. 53, Iss. 2, pp. 299-324,
h ps://doi.o g/10.1007/s00182-024-00898-z
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/314965
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h p://c ea i ecommons.o g/licenses/by/4.0/
Vol.:(0123456789)
In e na ional Jou nal o Game Theo y (2024) 53:299–324
h ps://doi.o g/10.1007/s00182-024-00898-z
1 3
ORIGINAL PAPER
Recip oci y models e isi ed: in en ion ac o s
and e e ence alues
JannaHinz1· And easNicklisch2 · Mey‑LingSomme 3
Accep ed: 14 Ap il 2024 / Published online: 21 May 2024
© The Au ho (s) 2024
Abs ac
We p esen a es o he wo mos es ablished ecip oci y models, an in en ion ac o
model and a e e ence alue model. We es cha ac e is ic elemen s o each model
in a se ies o wel e mini-ul ima um games. Resul s om online expe imen s show
majo di e ences be ween ac ual beha io and p edic ions o bo h models: he dis-
ance o ac ual o e s o he p oposed e e ence alue p o ides a poo measu e o
he kindness o o e s, while a compa ison o o e s wi h ex eme o e s as sugges ed
by he in en ion ac o model makes o e s indisc iminable in iche se ings. We dis-
cuss possible combina ions o bo h models be e desc ibing ou obse a ions.
Keywo ds Expe imen s· In en ions· Mini-ul ima um game· Re e ence alue·
Recip oci y models
JEL Classi ica ion C52· C72· C91
* And eas Nicklisch
and eas.nicklisch@ hg .ch
1 Depa men o Economics, Uni e si y o Hambu g, on-Melle-Pa k 5, 20146Hambu g,
Ge many
2 Cen e o Economic Policy Resea ch, Uni e si y o Applied Sciences o  heG isons,
andResea ch Uni “Needs based Jus ice andDis ibu ion P ocedu es”, Come cials . 20,
7000Chu , Swi ze land
3 Facul y o Economics und Social Sciences, Helmu Schmid Uni e si y, Hols enho weg 85,
22043Hambu g, Ge many
300
J.Hinz e al.
1 3
1 In oduc ion
Recip oci y is one undamen al co ne s one o human beha io , and an in eg al ele-
men o o he - ega ding p e e ences. The impo ance o ecip ocal beha io o
human in e ac ions has been s essed by a la ge body o economic li e a u e (e.g.,
Cox and Deck 2003, And eoni e al. (2003); Falk e al. (2003); Gäch e and Thöni
(2007)).1 Consequen ly, beha io al economis s ha e been s i ing o explain how
people dig ess om sel -in e es ed beha io o ewa d kind ac ions and punish
unkind ac ions o hei opponen s. Modeling ecip oci y, howe e , has u ned ou
o be a e y complex endea o . The speci ic o mula ion o ecip ocal p e e ences
ollows p edominan ly wo dis inc ways: Rabin (1993) and Du wenbe g and Ki ch-
s eige (2004) ocus on an compa ison be ween a “sugges ed” payo and a e e ence
alue. In u n, Falk and Fischbache (2006) ely on a combina ion o he inequal-
i y o he “sugges ed” payo s and an in en ion ac o measu ing how delibe a e he
“sugges ion” is o cap u e unde lying mo i a ions. Despi e he ambigui y o bo h
app oaches, hei scien i ic impac is eno mous2 leading subsequen s udies o ely
on one o he o he heo y (e.g., S anca e al. (2009), Amb us and Pa hak 2012).
In his cu en s udy we es key cha ac e is ic ea u es o bo h app oaches in a
numbe o mini-ul ima um games simila o he one used in he adi ion o Bol on
and Zwick (1995) and Falk e al. (2003). Pa icula ly, we ocus on wo key di e -
ences be ween he wo app oaches: i s ly, he e e ence alue app oach measu es he
ex en o (un)kindness acco ding o he dis ance o he speci ic o e o he e e ence
alue, while he in en ion ac o app oach measu es he unkindness by he inequi y
o he speci ic o e . Secondly, al hough bo h models ake ex eme ou comes in o
conside a ions o assessing (un)kindness, he e e ence alue appea s mo e obus
agains ou lie s: ou lie s along o he al e na i e ou comes o he game a e con e ed
in o a e e ence alue, whe eas he in en ion ac o app oach measu es he in en-
ion by a pai wise compa ison be ween an ex eme ou come and he speci ic o e .
We show ha bo h app oaches ha e ad an ages and disad an ages when explaining
ac ual decisions so ha a combina ion o bo h app oaches seems o p o ide a good
desc ip ion o beha io .
The gene al idea o ecip ocal p e e ences is pe haps bes summa ized by he
La in p inciple ‘quid p o quo.’ The o e a ching non-pa ame ic model by Cox e al.
(2008) o malizes hese wo ds in he ollowing way: suppose a p opose (playe P)
has a numbe o al e na i es om which she can choose one. He choice has conse-
quences in e ms o payo s no only o he sel , bu also o he esponde (playe
R). Among he al e na i es playe P can choose om, playe R conside s P’s choice
blue o be mo e gene ous han ed i blue yields a highe payo o R han he choice
o ed, while P’s gain om choosing blue and no ed is a mos as la ge as R’s gain
om choosing blue and no ed. R may o may no accep he p oposed al e na i e.
1 Al e na i e app oaches such as dis ibu ional conce ns (e.g., Feh and Schmid (1999)) o guil -a e -
sion (e.g., Ba igalli and Du wenbe g (2007)) ha e been shown o explain p o-social beha io pa ly, bu
no comp ehensi ely (e.g., And eoni e al. (2003); Nicklisch and Wol (2012)).
2 In Feb ua y 2024, he pape by Falk & Fischbache is ci ed mo e han 3920 imes, while Du wenbe g
& Ki chs eige is ci ed mo e han 2690 imes acco ding o Google Schola .
301
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
Wi h R holding ecip ocal p e e ences, he likelihood o R’s app o al o a sugges ed
al e na i e inc eases, he mo e gene ous P’s choice is: R gains imma e ial u ili y
om P’s ma e ial payo s i P beha es gene ously, whe eas R’s imma e ial u ili y
can e en be nega i e, i P chooses a mean al e na i e. Consequen ly, R may wan
o punish P i P beha ed unkindly, o ins ance by des oying all payo s al oge he .
Con empo a y ecip oci y concep s ansla e he gene al idea o quid p o quo in o
well-de ined closed p e e ence models. We e e o he i s app oach as he “ e e -
ence alue model,” i s o malized by Rabin (1993).3 In his belie -dependen model,
ecip oci y is analyzed o wo-playe , no mal- o m games. Du wenbe g and Ki ch-
s eige (2004; he a e D &K) ex ended Rabin’s model o belie -dependen p e e -
ences o ex ensi e n-playe games. In bo h models, R anks P’s al e na i es om he
one yielding he lowes payo o R o he highes payo o R. Hal way be ween
R’s lowes and highes payo lies R’s equi able payo di iding P’s al e na i es in o
unkind ones below he equi able payo and kind ones abo e (he ea e , we deno e
he equi able payo as he e e ence alue). P’s (un)kindness owa ds R inc eases
in he di e ence be ween he payo co esponding o P’s choice and he e e ence
alue. We would like o s ess ha he e e ence alue model measu es he ac ion’s
kindness by a “global assessmen .” Tha is, he midpoin o he en i e se o al e na-
i es in he game de e mines he e e ence alue which, in u n, de e mines he kind-
ness o a speci ic o e .
We es he p edic i e success o he e e ence alue model acco ding o wo
cha ac e is ics. Fi s ly, we check o he con inui y o he e e ence alue: we ana-
lyze whe he he likelihood ha R accep s an al e na i e inc eases in he dis ance
be ween he equi able payo and he payo “no mally esul ing” om P’s chosen
ac ion.4 Secondly, we es o he p edic i e success o he e e ence alue: a ying
he game, bu keeping he e e ence alue and he dis ance om he e e ence alue
cons an , we analyze whe he he likelihood ha R esponds unkindly (i.e., she does
no accep ) emains cons an .5
The second class o ecip oci y models, “in en ion ac o models,” con as s he
e e ence alue models in wo ways: hey decompose P’s (un)kindness owa ds R
in o he p oduc o an in en ion e m and an ou come e m (e.g., Falk and Fisch-
bache (2006), he ea e F &F). The i s e m de e mines whe he R pe cei es P’s
ac ion as in ended o no , he second e m de e mines he se e eness o P’s pe cei ed
(un)kindness. F &F place in R’s imma e ial pa ial u ili y om ecip oci y p io
impo ance on he di e ence o payo s be ween P and R wi hin an op ion be o e he
di e ence o he o he possible payo s o R is conside ed. As men ioned ea lie , he
in en ion ac o model assesses he ac ion’s kindness in a pai wise compa ison wi h
ex eme ou comes o he game. This may yield a p oblem o his app oach: one
kind al e na i e can u n all o he al e na i es ine i able in o ully in en ional unkind
3 O he models inco po a ing ecip oci y ollow he same logic, bu apply sligh ly di e en echniques
(e.g., Cox e al. (2007)).
4 We cla i y he meaning o “no mally” below.
5 No ice ha we do no es he comple e app oach by D &K. We dismiss some equilib ia based on
mu ual meanness as no plausible in ou se ing. Fo u he de ails see Sec ion2.
302
J.Hinz e al.
1 3
al e na i es. We will show he consequences o his ea u e o p edic ions in games
wi h se e al al e na i es in Sec ion3.
The c ucial impo ance o in en ions o ecip ocal beha io has been shown else-
whe e (e.g. Falk e al. (2008)). We es he p edic i e success o in en ions by check-
ing o he con inui y o he in en ion ac o : we analyze whe he he likelihood ha
R esponds unkindly inc eases, i he in en ion ac o inc eases, keeping he inequi y
o P and R’s payo s cons an . Secondly, we es o consis ency o he in en ion ac-
o : a ying he game, bu keeping he in en ion ac o and he inequi y o payo s
cons an , we analyze whe he he likelihood ha R esponds unkindly emains con-
s an . In o he wo ds, we es whe he an assessmen o kindness elying on he pai -
wise compa ison desc ibes P’s pe cei ed kindness p ope ly.
Fo ou pu pose, we p opose a se ies o wel e mini-ul ima um games. Some o
hem o e P wo al e na i es o choose om, some o hem o e ou al e na i es.
All o hem allow R o ejec a p oposed al e na i e and o go he own income o
he sake o punishing P. The games a e designed such ha hey allow us o assess
he p edic i e success o e e ence alue models and in en ion ac o models. We
e ie e ou da a in online expe imen s wi h almos 500 pa icipan s. As such, ou
analysis ollows Sobel (2005) c i icism ha exis ing ecip oci y models seem o
be i ing o speci ic si ua ions, bu lack a clea cha ac e iza ion o his e y si ua-
ion. Along he same line o a gumen s, he e a e some o he s udies discussing and
es ing he p edic i e success o ecip oci y models. Fi s ly, hey p o ide e idence
on he impo ance o in en ions o ecip oca ion: i he e is no al e na i e bu o
beha e unkindly, subjec s ecip oca e less se e ely (Falk e al. 2003); he same holds
ue i an ac ion is aken ha is no unambiguously kind, bu sel ish o some deg ee
(S anca e al. 2009). Secondly, Dhaene and Bouckae (2010) elici i s and second
o de belie s o pa icipan s in a sequen ial p isone s’ dilemma and a mini-ul ima-
um game. They show ha belie s and beha io , pa icula ly o second mo e s, a e
e y consis en wi h D &K’s ecip oci y model. Fu he mo e, Pellig a (2011) a ies
sys ema ically he ou side op ions in a us game, whe e he i s mo e ’s us ing
is ei he kind o unkind o he second mo e . Con as ing he heo e ical p edic-
ions o he D &K model, he us wo hiness o he second mo e emains cons an
ac oss ea men condi ions sugges ing ha o he mo i es domina e beha io in his
se ing (c ., Pellig a (2011)). Finally, Nicklisch and Wol (2012) es an o e all cha -
ac e is ic o e e ence alue models and in en ion ac o models: i punishmen is
su icien ly cheap, ecip oca ion is modeled as an “all-o -no hing” decision. Tha is,
i P beha es kindly (unkindly), R maximizes he u ili y by choosing he mos kind
(unkind) esponse possible. By means o a modi ied ul ima um game, he au ho s
show ha decisions o a majo i y o pa icipan s in a labo a o y expe imen do no
ollow his assump ion.
Along his pa ly pessimis ic assessmen o con empo a y ecip oci y models,
ou da a shows impo an sho comings o bo h models when p edic ing beha io .
Pa icula ly, he con inui y o he e e ence alue model ails o cha ac e ize ac ual
beha io : inc easing he dis ance be ween he e e ence alue and he payo o he
ac ual o e does no necessa ily co espond wi h inc easing ejec ion a es. Mo eo-
e , a ia ion o he game yields di e ences in he ejec ion a es al hough he e e -
ence alue and he dis ance om he e e ence alue emain cons an . We conclude

303
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
om hose indings ha he dis ance o he e e ence alue se es as a poo desc ip-
o o he ex en o (un)kindness. On he o he hand, expe imen al esul s o sim-
ple games wi h wo al e na i es a e nicely p edic ed by he in en ion ac o model.
The likelihood o ejec ion inc eases o inc easing in en ion ac o s. Howe e , he e
is li le consis ency be ween p edic ions and decisions in he iche games wi h
ou al e na i es. We conclude om his ha he pai wise compa ison o al e na-
i es does no cha ac e ize beha io adequa ely, and sugges a combina ion o bo h
app oaches. This combina ion includes a global assessmen o he in en ion o a
choice and he inequi y o an al e na i e o he ex en o (un)kindness.
The emainde o his a icle is o ganized as ollows: The ollowing sec ion ( e)
acquain s wi h bo h ecip oci y models o be es ed wi h an emphasis on he elemen
we sc u inize. In Sec ion 3 we in oduce ou expe imen al design and p ocedu e.
Sec ion4 p esen s esul s. In Sec ion5 we discuss ou indings and sugges po en ial
de elopmen s o ecip oci y models e lec ing ou esul s. Sec ion6 concludes.
2 Recip oci y based u ili y
Le us conside a wo-playe ex ensi e o m game wi h wo s ages and common
knowledge abou he game and decisions in p e ious s ages.6 Fo mally, suppose
playe P and playe R be p opose and esponde in an ul ima um-like game. P
chooses an ac ion
ap
om he se o al e na i es
Ap
. Suppose
ap
a ec s P’s and R’s
payo s (
𝜋p
and
𝜋
, espec i ely). R obse es P’s ac ion. Then i’s ecip oci y based
u ili y unc ion wi h
i,j∈{P,R}
has he ollowing s uc u e:
U ili y consis s o a ma e ial payo
𝜋i
and an imma e ial u ili y componen . The
ma e ial pa o
Ui
e e s o he payo assigned o he ou come o a speci ic choice.
The imma e ial pa o u ili y is ini ia ed wi h an indi idual sensi i i y pa ame e o
ecip ocal conce ns,
𝜓i
. I
𝜓i=0
, hen u ili y will equal ma e ial payo as sugges ed
by na ow sel -in e es .
𝜅ji(.)
deno es i’s pe cei ed (un)kindness o j’s ac ion, and
𝜆ij(.)
is he (un)kindness o i’s esponse. The la e wo depend on i’s ( i s -o de )
expec a ions conce ning j’s beha io in he consecu i e game (
b′
ij
), and i’s (second-
o de ) expec a ions conce ning j’s belie s o i’s belie s o he consecu i e game (
b′′
iji
).
Depending on he i s - and second-o de belie s o playe s, ecip oci y models
can accommoda e gene ally a e y la ge ange o beha io . Fo ins ance, a e y
ecip ocal esponde may expec o ecei e a e y low o e , pe cei ing a la ge o e
o be mean. While she assumes ha he p opose expec s he o accep he o e , she
may decide o ejec he la ge o e yielding subs an ial imma e ial u ili y esul ing
om he mean ejec ion o a mean o e . In u n, a e y ecip ocal p opose 7 may
(1)
U
i=𝜋i(ai,aj)+𝜓i𝜅ji(aj,b
�
ij
,b
��
iji
)𝜆ij(ai,b
�
ij
,b
��
iji)
6 O cou se, bo h app oaches co e much iche se ings. He e, we es ic ou delibe a ions o he expe -
imen al se ing o he no a ional simplici y.
7 Fo his a gumen a ion, “ e y” e e s o cases in which he imma e ial u ili y componen ou weighs he
ma e ial u ili y componen o bo h playe s.
304
J.Hinz e al.
1 3
conside he ejec ion o o e s o be mean. An icipa ing ha esponde s conside
la ge o e s as mean, he p opose cas s a la ge (mean) o e acing he (an icipa ed)
mean ejec ion. Consequen ly, bo h p opose and esponde yield highe u ili y by a
ejec ion o a la ge o e han h ough he accep ance o a small o e .8
Whe eas ecip oci y allows gene ally o “masochis ic” equilib ia ha maximize
he imma e ial u ili y componen o mu ual mean ac ions (see ou las oo no e), we
ule hose equilib ia ou as non-plausible in ou dic a o games. Tha is, we assume
ha p opose s do no submi o e s o which hey seek a ejec ion in o de o maxi-
mize hei imma e ial u ili y.9 No ice ha D&K’s app oach allows in p inciple o
hose equilib ia based on mu ual meanness. Hence, we do no es D &K’s ull
model bu dismiss some o i s equilib ia on plausibili y g ounds.
Thus, we assume in he ollowing ha i a p opose cas s an o e , she expec s i s
accep ance, and she expec s he esponde o belie e ha she aims a he accep ance
o he o e . Fo mally,
b�
PR =
“accep ance” and
b��
PRP =
“
b�
RP
=
accep ance
”, while
b�
RP =
“accep ance” and
b��
RPR =
“
b�
PR
=
accep ance
”. As a consequence, he e ms o
i’s pe cei ed (un)kindness o j’s ac ion and he (un)kindness o i’s esponse simpli y
o
𝜅ji(aj)
and
𝜆ij(ai)
, since p opose ’s and esponde ’s i s o de and second o de
belie s assume he accep ance o he o e . Tha is, “ he no mally esul ing way” o
he ul ima um game assumes he accep ance o he p oposed o e . Whe he ac ual
beha io
aR
de ia es om he expec ed way, is in bo h models a ma e o he imma-
e ial pa ial u ili y componen s om ecip oci y.
In he nex sec ions, we b ie ly p esen he key componen s o bo h e e ence
alue models and in en ion ac o models. Bo h models p o ide a solu ion concep
ha equi e an upda ing o playe s’ belie s as he play un olds.10 We will pay special
a en ion o he di e ences in he upda ing be ween bo h app oaches, and o mula e
hypo heses based on he heo e ical analysis.
2.1 Sequen ial ecip oci y acco ding oDu wenbe g & Ki chs eige
Wi h espec o he e e ence alue model by Du wenbe g & Ki chs eige , R’s imma-
e ial pa ial u ili y om ecip oci y equals he p oduc o P’s (un)kindness owa ds
R,
𝜅PR(aP)
, and R’s (un)kindness owa ds P by de ia ing om “ he no mally esul -
ing way”,
𝜆RP(aR)
.
8 No ice ha he desc ibed beha io c ea es a logical wis o ecip ocal p e e ences. I appea s ques-
ionable whe he a mu ual mean ac ion allowing playe s o maximize he ecip ocal u ili y is no in ac
nice: i c ea es a highe u ili y han he one esul ing om he accep ance o a small o e . I so, i u ns
he s a ing poin o his hough expe imen – ejec ing o e s is mean – absu d.
9 I he p opose seeks ejec ions, his esponse may no be conside ed o be mean any longe , while
accep ing a mean o e c ea es an equally bad op ion o he esponde . We guess ha he coo dina ion
on such an unusual equilib ium appea s e y di icul in an anonymous expe imen al se ing and may be
desi able only o a e y small numbe o playe s. Simila di icul ies wi h dis ibu ional ai ness p e e -
ences (e.g., inequali y a e sion) a e excluded by he assump ion ha playe s do no h ow away money
o he sake o a educ ion o income equali ies (see, e.g., Feh and Schmid (1999),p. 824).
10 In hei e e ence alue model, Du wenbe g and Ki chs eige (2004) e e o i as a “sequen ial eci-
p oci y equilib ium.”
305
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
R’s e e ence alue sepa a ing P’s ac ions in o kind and unkind ac ions is he
alue hal way be ween he lowes and highes payo a ailable a he ime when P
makes he decision. Fo mally, R’s equi able payo
𝜋
eP
R
(i.e., R’s a e age payo ol-
lowing P’s ac ion) is:
whe e
𝜋′
R
is he se o payo s induced by P’s e icien s a egies. In u n, no con-
side ed a e P’s ine icien s a egies, ha is, s a egies o which one inds a Pa e o
imp o emen – in e ms o R’s and P’s payo s – among P’s s a egies o any s a -
egy choice o R (compa e D &K, pp. 275–276).
P’s (un)kindness owa ds R in D &K’s app oach is e alua ed acco ding o he
di e ence be ween he payo esul ing om accep ing P’s o e and he e e ence
alue:
A e R obse es P’s mo e, i is on he o espond, again in luencing he mone a y
ou comes o bo h playe s. Tha is, anking R’s e icien al e na i es om he one
yielding he lowes o he highes payo o P, he midpoin o he anking de e -
mines R’s e e ence alue o he (un)kindness o he esponse. In o he wo ds, R’s
(un)kindness owa ds P is measu ed acco ding o he dis ance be ween P’s ac ual
payo o P’s equi able payo . I is impo an o s ess ha P’s equi able payo
𝜋
eR
P
in he ul ima um game is he payo esul ing om accep ing he o e : ecall ha
P’s equi able payo a e ages he maximum and he minimum in he se o payo s
induced by R’s e icien s a egies. Since ejec ions a e ine icien , R’s only e icien
s a egy equals he accep ance o he o e . The e o e, i ollows:
D &K’s explici quan i ica ion o (un)kindness wi h he equi able payo as a e e -
ence alue allows us o es he p edic i e success o hei model:
Hyp
D&K
: Keeping he o e cons an , bu dec easing he dis ance o he equi able
payo ac oss he p opose ’s ac ions implies non-inc easing ejec ion a es o hose
ac ions, while keeping he dis ance cons an implies a cons an ejec ion a e.
2.2 In en ion‑based ecip oci y acco ding oFalk & Fischbache
Wi hin he con ex o ou simple mini-ul ima um games, one can show o Falk &
Fischbache ’s app oach ha
𝜆F&F
RP
(a
R
)=𝜆
D&K
RP
(a
R)
: acco ding o F &F, he kindness
o R’s ecip oca ion is measu ed wi h espec o he deg ee by which R al e s P’s
ac ual om his expec ed payo . Since R expec s P o p opose he an o e o which
P seeks R’s accep ance in ou mini-ul ima um games, R al e s P’s payo by ejec -
ing he o e implying a nega i e ecip oca ion.
In u n, he di e ence be ween he wo app oaches is se led in he speci ic o m
o
𝜅F&F
PR
(a
P)
. F &F sepa a e he pe cei ed (un)kindness in o wo e ms, he ou come
(2)
𝜋
e
P
R
=0.5 max(𝜋
�
R
)+0.5 min(𝜋
�
R
)
,
(3)
𝜅D&K
PR
(a
P
)=𝜋
R
(a
P
)−𝜋
eP
R
(4)
𝜆D&K
RP
(a
R
)=𝜋
P
(a
R
)−𝜋
P
(accep ance
)
306
J.Hinz e al.
1 3
e m
ΔP(aP)
and he in en ion ac o
𝜗F&F
P
(a
P)
. The ou come e m is o malized
such ha he e alua ion o kindness is based on he inequi y be ween p opose ’s and
esponde ’s payo s a a speci ic end node assuming ha he esponde chooses he
e icien s a egy:
Again, wi hin he con ex o ou mini-ul ima um games, we can simply inse he
payo s ollowing accep ed o e s in o Eq. (5).
To de i e pe cei ed kindness, he ou come e m is mul iplied wi h he in en ion
e m, whe e ecip ocal conce ns come in o play. He e, F &F dis inguish be ween
i e payo cons ella ions om which di e en in en ions a e de i ed. Mo e speci i-
cally, he in en ion ac o accoun s o he p opose ’s in en ional and unin en ional
choices depending on how she could ha e al e ed payo cons ella ion wi h ega d o
he own in combina ion wi h he esponde ’s payo :
whe e
𝜋0
R
,
𝜋0
P
a e payo s esul ing om accep ing he speci ic o e in ou design.
Le

ΠR
be he se o payo s esul ing om he accep ance o an al e na i e o e (bu
no he speci ic o e ),
𝜋 R
be one elemen in

ΠR
, and
𝜖R
be an indi idual pa ame e
wi h
0
≤
𝜖R
≤
1
. This pa ame e is deno ed as he pu e ou come conce n pa ame e .
Tha is,
𝜖R
measu es he esponde ’s unease wi h he inequi y be ween p opose ’s
and esponde ’s payo , al hough he p opose has no op ion o a oid he kind o
mean o e .
The i s wo cases o (6) e e o in en ions o he p opose ’s ac ions a o ing
he esponde money-wise: in he i s one, he p opose o e s he esponde no he
smalles payo possible al hough i is highe han his own one. This case is con-
side ed as ully in en ional. In he second case, he p opose o e s he esponde a
highe payo han he own one, bu has no chance o a oid his. In his case, he ou -
come is nice, bu he p opose does no ac in en ionally so ha he in en ion ac o
is educed. The las h ee cases e e o nega i e in en ions: he inal case mi o s he
second case in o he nega i e domain; he p opose o e s he esponde a smalle
payo han he own one, bu has no chance o a oid his. In his case, he ou come
is mean, bu he p opose does no ac in en ionally so ha he in en ion ac o is
educed, whe eas he hi d and ou h case e e o in en ionally mean choices. In he
ou h one, he p opose o e s he esponde a smalle payo han a possible al e -
na i e, bu he o e is “somehow unde s andable” in he sense ha he al e na i e
yields less o he p opose han o esponde . The e o e, he p opose ’s in en ion is
discoun ed acco ding o he p opose ’s ela i e disad an age unde he al e na i e.
In con as , in he hi d case, he p opose ’s unkindness is ully in en ional, since
(5)
ΔP(aP)=𝜋R−𝜋P
(6)
𝜗
F&F
P(aP)=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1 i 𝜋0
R≥𝜋0
Pand ∃𝜋 R∈
ΠR∶𝜋 R<𝜋
0
R,
𝜖Ri 𝜋0
R≥𝜋0
Pand ∀𝜋 R∈
ΠR∶𝜋 R≥𝜋0
R,
1 i 𝜋0
R<𝜋
0
Pand ∃𝜋 R∈
ΠR∶𝜋 R>𝜋
0
Rand 𝜋 R≤𝜋
P
max �1−𝜋 R−𝜋 P
𝜋0
P−𝜋0
R
,𝜖R�i 𝜋0
R<𝜋
0
Pand ∃𝜋 R∈
ΠR∶𝜋 R>𝜋
0
Rand 𝜋 R> 𝜋
P
𝜖Ri 𝜋0
R<𝜋
0
Pand ∀𝜋 R∈
ΠR∶𝜋 R≤𝜋0
R,
313
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
Simila ly, p edic ions acco ding o Hyp
F&F
a e misaligned wi h esul s due o he
lowe ejec ions a e o (8,2) in
Γ8
and
Γ11
compa ed o
Γ9
,
Γ10
and
Γ12
. In con-
as o ou ea lie esul s, which co obo a e wi h F &F’s model, we p o ide he e
Fig. 1 Fi s mo e decisions (in pe cen )
Fig. 2 The y-axis displays he ejec ion a es o (8,2) in
Γ1
o
Γ12
. On he x-axis, he espec i e ank is
plo ed acco ding o
D&K
and
F&F
wi h 0 e lec ing he lowes and 4 he highes likelihood o ejec ion

314
J.Hinz e al.
1 3
e idence o a lack o gene ali y (o a limi ing speci ici y) o F &F’s model. Mo e
p ecisely, by changing he s uc u e o he games in a way ha we add a numbe o
al e na i es, ecip ocal beha io canno be sa is ac o ily desc ibed by F &F’s in en-
ion-based model anymo e. In e es ingly, he compa ison ac oss simple and iche
games also cas s some doub s on o F &F’s model (e.g.,
Γ4
and
Γ12
:
p=0.04
; o
Γ2
and
Γ8
:
p<0.001
), al hough in hose cases he al e na i es o he simple games a e
subse s o he al e na i es in he iche games. We discuss his poin in g ea e de ail
in he Sec ion5.
P o ided ou obse a ions, i seems ha nei he D &K’s model no F &F’s model
cha ac e ize ejec ion beha io in he iche games accu a ely. Pa icula ly, bo h
models ail o p edic he occu ence o he wo blocks. The esul s sugges ha bo h
models miss o inco po a e an impo an cha ac e is ic o ecip oci y.
4.2 Indi idual ecip oci y
In he ollowing, we es o he consis ency o ou esul s on an indi idual le el. Fo
his, we es he pe sonal implica ions o bo h models ac oss games. In a i s s ep,
we es whe he pai wise compa ison o accep ances and ejec ions o wo games
leads o consis en esul s. Tha is, p o ided he decision in one game, he same sub-
jec has o make he same decision in ano he game associa ed o he same ank.
Mo eo e , i he subjec ejec s (accep s) an o e , o he o e s associa ed wi h highe
(lowe ) anks ough o be ejec ed (accep ed) by he same subjec as well. The e o e,
we will es he con inui y o he wo ankings ac oss all games in a second s ep.
Tha is, we will es whe he he ejec ion o an o e implies he ejec ion o all
Table 3 The i s column epo s obse ed ejec ion a es o (8,2) in ascending o de o
Γ1
o
Γ12
The second and and hi d column display he ank o ejec ion acco ding o
D&K
and
F&F
wi h 0
e lec ing he lowes and 4 he highes likelihood o ejec ion along he game’s alues and anks acco d-
ing o he modi ica ion o
𝜗CA
P
(8, 2
)
( o his see below)
Game
{Al e na i es}
Rejec ion a es
o 8,2
ΨD&K(
8, 2
)
ΨF&F(
8, 2
)
𝜗CA
P
(8, 2
)
ΨCA (
8, 2
)
Γ3{(9, 1)}
0.32 0 1
𝜖R
1
Γ1{(8, 2)}
0.35 0 1
𝜖R
1
Γ11{(9, 1),(10, 0),(6, 4)}
0.48 0 4
𝜖R
1
Γ6{(3, 7)}
0.48 4 2
𝜎R
2
Γ7{(3, 4)}
0.49 2 3
𝜎R
2
Γ8{(10, 0),(9, 1),(5, 5)}
0.50 1 4
𝜖R
1
Γ5{(4, 3)}
0.54 1 4 1 3
Γ4{(7, 3)}
0.55 1 4 1 3
Γ2{(5, 5)}
0.57 3 4 1 3
Γ12{(9, 1),(7, 3),(6, 4)}
0.59 1 4 1 3
Γ9{(9, 1),(7, 3),(5, 5)}
0.61 2 4 1 3
Γ10{(7, 3),(6, 4),(5, 5)}
0.62 3 4 1 3
315
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
games associa ed o a highe ank. Likewise, he accep ance o a game implies he
accep ance o all games wi h a lowe ank.
Le us s a wi h he i s s ep and he pai wise compa ison o simple games
acco ding o D &K. In line wi h Hyp
D&K
, we canno ejec he hypo hesis ha he
same subjec s ejec
Γ4
and
Γ5
(
p=.61
).18 A he same ime, we ha e o ejec he
hypo hesis ha a leas all esponde s ejec ing (8,2) in
Γ4
and
Γ5
(203 espond-
e s do so) ejec (8,2) in
Γ2
,
Γ6
and
Γ7
as well (
p<0.001
): only 151 esponde s
ejec (8,2) in all i e games. Likewise, he p edic ion ha esponde s ejec ing
(8,2) in
Γ7
ejec (8,2) in
Γ2
and
Γ6
is no suppo ed by he da a (
p<0.001
):
only 160 esponde s ejec (8,2) in all h ee games. Finally, he p edic ion ha
esponde s who ejec (8,2) in
Γ2
do so in
Γ6
as well is also no suppo ed by he
da a (
p<0.001
): 186 esponde s ejec (8,2) in bo h games.
Tu ning o F &F, we ind ha 107 esponde s ejec (8,2) in
Γ1
and
Γ3
(ou
o 152/137 ejec ing
Γ1∕Γ3
), so ha he e is weakly signi ican e idence ha no
he same subjec s ejec his o e in bo h games (
p=0.08
). Simila ly, Hyp
F&F
is
no suppo ed in he sense ha om 204 (209) esponde s ejec ing (8,2) in
Γ6
(
Γ7
), 153 (180) ejec he same o e in
Γ4
,
Γ5
and
Γ7
(
Γ4
and
Γ5
). He e, we ha e
o ejec he hypo hesis ha he same subjec s ejec his o e in all ou games
(
p<0.001
, and
p=0.002
, espec i ely). Howe e , we canno disca d he hypo h-
esis ha he same subjec s ejec (8,2) in
Γ4
and
Γ5
(
p=0.61
): om 235 (231)
ejec ing (8,2) in
Γ4
(
Γ5
), 203 ejec he o e in bo h games. In o he wo ds, F
&F o ganizes he da a well, i o e s a e ully in en ional e e ing o hei model.
Fo he iche games, D &K p edic ha esponde s ejec ing (8,2) in
Γ8
(
Γ9
)
a e expec ed o ejec he same o e in
Γ9
and
Γ10
(
Γ10
). The e is li le e idence
o he claims: om 214 (260) esponde s ejec ing (8,2) in
Γ8
(
Γ9
), 194 (236)
esponde s do so in
Γ9
and
Γ10
(
Γ10
) as well (
p<0.001
o bo h compa isons).
Likewise, no he same esponde s ejec (8,2) in
Γ4
,
Γ5
,
Γ8
and
Γ12
(
p<0.001
),
Γ7
and
Γ9
(
p<0.001
), and
Γ2
and
Γ10
(
p=0.01
). Finally, om 167 esponde s
ejec ing (8,2) in
Γ4
,
Γ5
,
Γ8
and
Γ12
, 137 ejec (8,2) in
Γ7
,
Γ9
,
Γ2
,
Γ10
and
Γ6
as well. Again, he claim ha he same esponde s ejec (8,2) ac oss all hose
games is no suppo ed (
p<0.001
).
We con inue ou analysis by es ing F &F in he con ex o iche games:
acco ding o Hyp
F&F
, he same esponde s ejec (8,2) in games
Γ2
,
Γ4
,
Γ5
and
Γ8
o
Γ12
. This claim is no suppo ed by he da a (
p<0.001
). E en i we es ic ou
analysis o games
Γ8
o
Γ12
, he e is li le e idence ha he same esponde s ejec
(8,2) in hose games (
p<0.001
). Howe e , we canno ejec he claim ha he
same esponde s ejec (8,2) in
Γ9
,
Γ10
and
Γ12
(
p=0.19
). Thus he e seems o be
an impo an di e ence be ween
Γ9
,
Γ10
and
Γ12
on he one hand, and
Γ8
and
Γ11
on he o he ha in luences ecip oci y subs an ially in ou expe imen .
Now, we u n o he second s ep o he analysis: we check he con inui y o he
wo ankings ac oss all games. Tha is, p o ided ha a subjec ejec s a game, she
ejec s all games wi h he same and lowe anks as well. Likewise, p o ided ha
18 Th oughou his subsec ion, we use a wo-sided F iedman es o he assessmen o s a is ical signi i-
cance.
316
J.Hinz e al.
1 3
a subjec accep s a game, she accep s all games wi h he same and highe anks
as well. Fo ins ance, suppose a subjec accep s he o e (8,2) in
Γ7
. Since
Γ7
has
he ank 2 acco ding o D &K, he subjec ough o accep all o e s in games wi h
highe o equal anks, oo (i.e.,
Γ1,Γ3,Γ4,Γ5,Γ8,Γ9,Γ11
, and
Γ12
). In u n, ejec -
ing (8,2) in
Γ7
( ank 3 e e ing o F &F), a subjec ough o ejec all o e s in
games wi h lowe o equal ank 3 acco ding o F &F (i.e.,
Γ2,Γ4,Γ5,Γ8,Γ9,Γ10,Γ11
,
and
Γ12
).
Al oge he , he p edic ion o bo h D &K and F &F leads o i e pa e ns o
accep ances and ejec ions each.19 We check o each subjec whe he he subjec ’s
accep ances and ejec ions ma ch one o he pa e ns and, i no , he minimum mis-
ma ches om one o he pa e ns bo h o D &K and F &F (and CA, see below).
Figu e3 illus a es he equency o subjec s whose accep ances and ejec ions o
(8,2) ma ch exac ly one p edic ed pa e n (deno ed as ‘0’), ma ch a p edic ed pa -
e n excep one o wo choices a mos (‘1’ and ‘2’), espec i ely ( wo ou o wel e
choices is he maximum numbe o indi idual misma ches). O e all, only 114 sub-
jec s (ou o 427) show no misma ch acco ding o he p edic ions om D &K, while
220 acco ding o he p edic ions om F &F. Thus, he o me model yields signi i-
can ly ewe no-misma ches han he la e (
p<0.001
, wo-sided p opo ion es
compa ing bo h numbe s). In line wi h his obse a ion, we ind an a e age numbe
o 1.47 misma ches o D &K and 0.48 misma ches o F &F. No ice ha a la ge
p opo ion o indi idual misma ches wi h D &K’s p edic ions esul s om ejec ions
in
Γ1,Γ3
, and
Γ11
(i.e., D &K p edic ank ze o o hose games). Thus, an ad an age
o he F &F model and a disad an age o he D &K model is ha he o me (la e )
model can (canno ) a ionalize ejec ions based on inequi y conside a ions.
5 Discussion
Ou esul s indica e o bo h models weaknesses when p edic ing pe cei ed kind-
ness. D &K p o ide wi h hei
𝜅D&K
PR
(a
P)
a pa i ion in he deg ee o kindness which
is oo de ailed. F &F’s kindness measu emen yields a pa i ion o kindness which is
oo gene al in iche se ings due o he pai wise compa ison o o e s o he ex eme
al e na i es. Consequen ly, nei he model can desc ibe beha io ully. The e o e,
we wan o discuss a po en ial combina ion o bo h ecip oci y models (‘Combined
app oach’, abb e ia ed ‘CA’). Howe e , we do no a emp o p esen a ully elabo-
a ed model, bu we wan o ske ch a possible a enue o he u he de elopmen
o esea ch on ecip oci y. On he one hand, we wan o adjus F &F’s model so
ha i yields mo e p edic i e powe in iche en i onmen s while keeping he num-
be o misma ches as low as possible ( hough addi ional p edic i e powe will lead
almos ine i able o mo e misma ches). On he o he hand, we wan o inco po a e
19 Fo example, accep ing (8,2) in
Γ1,Γ3,Γ4,Γ5,Γ7,Γ8,Γ9,Γ11
, and
Γ12
, while ejec ing (8,2) in
Γ2,Γ6
,
and
Γ10
is one pa e n p edic ed by D &K. T i ially, accep ance o (8,2) in all games is ano he pa e n
p edic ed bo h by D &K and F &F, whe eas ejec ing (8,2) in
Γ1,Γ3
, and
Γ11
canno be a ionalized by D
&K. All p edic ed pa e ns a e desc ibed in he Appendix.
317
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
he assessmen o al e na i es’ kindness acco ding o he e e ence alue app oach o
p o ide a su icien ly simple pa i ion o kindness.
As ou da a co obo a e in simple games F &F’s dis inc ion be ween i e gene ic
cases o in en ion (and he dis inc ion be ween he in en ion and he ex en o which
his al e na i e is conside ed o be kind o unkind), we do no seek o modi y his
ea u e.20 Hence, we p opose a
𝜅CA
PR
(a
P)
which consis s o he p oduc o an in en ion
ac o and he ou come e m
ΔP(aP)
acco ding o equa ion (6). Howe e , he majo
ex ension is an assessmen o P’s al e na i es acco ding o a e e ence alue. No
only do ou expe imen al esul s suppo his app oach, bu we conside i as un e-
alis ic ha in mo e complex si ua ions – and we s udy in ou expe imen complexi y
only o he ex en ha we es games wi h ou ins ead o wo al e na i es – he exis -
ence o one clea ly (un)kind ac ion de e mines he choice o ano he al e na i e as a
ully in en ional ac o (un)kindness.
Thus, simila o D &K’s app oach, we model in en ion ela i e o some e e ence
alue. Speci ically, he e e ence alue we p opose, he median o e (i.e., he “mid-
dle” payo wi hin he se o co esponding end no es esul ing om he consecu i e
choice o e icien s a egies) p o ides he addi ional bene i o being obus agains
ou lie s in he se o al e na i es (D &K discuss he p oblem o ou lying payo s
ex ensi ely in hei pape ). Fo mally, le us deno e wi h
𝜋M
i
i’s median payo among
he se o payo s esul ing om j’s choice.21 Then, we de ine he in en ion ac o o
P’s mo e
𝜗CA
P
(a
P)
o o e ing a speci ic payo combina ion
𝜋0
P
,𝜋
0
R
as ollows:
012
D&K
F&F
CA
Numbe o misma ches pe pe son
F equency
0.00.2 0.40.6 0.
8
Fig. 3 F equency o indi idual de ia ions om he p edic ion pa e ns acco ding o he heo ies ( o ‘CA’
see below)
20 On a side-no e, we conside he ac ha he e is no signi ican di e ence o ejec ion a es wi hin bo h
blocks – al hough one game pe block yields in sum less han 10 Tale s – as e idence ha e iciency con-
ce ns a e less impo an in his se ing.
21 We de ine
𝜋M
i
implici ly wi h espec o he cumula i e dis ibu ion unc ion F(x) on i’s se o payo s
esul ing om i’s and j’s choice o any combina ion o e icien s a egies in a game:
𝜋M
i
sa is ies bo h
inequali ies
∫
(−∞,𝜋M
i
]dF(x)
≥0.5
and
∫
[𝜋M
i
,∞) dF(x)
≥0.5
.
318
J.Hinz e al.
1 3
whe e

ΠP
be he se o p opose ’s payo s esul ing om he accep ance o an al e -
na i e o e ,
𝜋 P
be one elemen in

ΠP
, and
𝜎R
be ano he indi idual pa ame e wi h
0
≤
𝜖R
≤
𝜎R
≤
1
.
Tha is, like F &F’s app oach, ou in en ion ac o di e en ia es be ween i e ca -
ego ies o R’s ou comes esul ing om P’s ac ion: payo s implying smalle payo s
o P han o R a e conside ed as ully in en ionally kind i hey a e la ge han he
e e ence alue, whe eas hey a e acciden ally kind i hey a e smalle o equal o
he e e ence alue. In u n, he e a e acciden ally unkind o e s which imply la ge
payo s o P han o R i hey a e la ge o equal o he e e ence alue. Finally, P’s
ac ion leading o R’s payo being smalle han R’s e e ence alue and P’s payo is
conside ed o be ully in en ional unkind only i P could choose be e al e na i es
o he sel . Tha is, i he unkind o e is “somehow unde s andable” in he sense
ha all o he al e na i es yield less o P han R’s e e ence alue, he o e is s ill
pe cei ed as in en ionally unkind bu no ha much. Only, i he e is a leas one
o he al e na i e which yields o P mo e han R’s e e ence alue, choosing he spe-
ci ic al e na i e is ully in en ional (and unkind).
No ice ha he la e wo cases ansla e F &F’s obse a ion ha “ he pe cep ion
o he un ai o e depends on how much j has o sac i ice in o de o make he mo e
iendly o e ” (F &F, 2006, p. 297) in o he con ex o a global assessmen o i’s
payo s.22 Tha is, i making a mo e iendly o e han (8,2) implies ha P ea ns
less han R’s e e ence alue, his o e is s ill unkind, bu wi h limi ed in en ion.
Based on ou e o mula ion o
𝜗CA
P
(a
P)
, we ob ain a new ank o de o he likeli-
hood o a ejec ion o (8,2) which is epo ed in he las column o Table3: he
p edic ions based on
𝜗CA
P
(a
P)
ollow quali a i ely he one based on
𝜗F&F
P
(a
P)
o he
simple games. Howe e , using
𝜗CA
P
(a
P)
one can p edic he wo blocks o ejec ion
a es in he iche games. In addi ion, he ejec ion a es o (7,3) in
Γ9
,
Γ10
and
Γ12
ollow he p edic ed pa e n: in
Γ9
and
Γ12
whe e (7,3) is un a o able bu mildly
unkind, 101 and 106 esponde s ejec he o e , while 151 do so in
Γ10
whe e his
o e is ully in en ionally unkind.23
Compa ing he numbe o indi idual misma ches be ween decisions and p e-
dic ions be ween bo h models, we ob ain a simila pe o mance o he combined
app oach and he F &F model (see ou Fig.3): o 48% o all subjec s we compu e
no misma ch based on he p edic ion o he combined app oach, whe eas 52% based
(8)
𝜗CA
P(aP)=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1 i
𝜋
0
R>𝜋
M
R
and
𝜋
0
R>𝜋
0
P
,
𝜖Ri 𝜋0
R≤𝜋M
Rand 𝜋0
R≥𝜋0
P,
𝜖Ri 𝜋0
R≥𝜋M
Rand 𝜋0
R≤𝜋0
P,
1 i 𝜋0
R<𝜋
M
R,𝜋0
R<𝜋
0
P, and ∃𝜋 P∈
ΠP:𝜋 P>𝜋
M
R
𝜎Ri 𝜋
0
R<𝜋
M
R,𝜋
0
R<𝜋
0
P, and
∀
𝜋 P
∈
Π
P:𝜋 P≤𝜋
M
R,
22 In his quo e, j e e s o he p opose and i o he esponde in he con ex o ou se ing.
23
p
<
0.001
o he hypo hesis ha he ejec ion a e in
Γ10
equals one in he o he wo games, whe eas
p=0.466
o he hypo hesis ha he ejec ion a es in
Γ9
and
Γ12
a e he same.

319
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
on he p edic ion o he F &F model. The di e ence be ween bo h equencies is no
signi ican (
p=0.338
, wo-sided p opo ion es compa ing bo h numbe s). Hence
i seems ha he combined app oach is able o sha pen he p edic i e powe while
keeping he good pe o mance wi h ega ds o he low numbe o misma ches.
Finally, we un a se ies o linea p obabili y eg essions on he indi idual deci-
sions whe he o accep o ejec he o e (8,2) ( epo ed in Table4). We con ol o
indi idual cha ac e is ics by including indi idual dummy a iables o all subjec s.
Independen a iables a e he anks esul ing om ou h ee models F &F, D &K
and CA. We ob ain ema kably high
R2
alues in all eg essions along cons an ly
signi ican coe icien s o all h ee anks. Mos impo an ly, we assess whe he
including he anks o he Combined app oach adds o he p edic i e powe o he
eg essions by unning likelihood- a io es s. Resul s indica e ha in all cases he
explana o y powe o he eg essions inc ease signi ican ly by adding he anks o
he Combined app oach. Tha is, adding he new anks o he anks o he F &F
model (i.e., (1) s. (5)) and o he anks o he D &K model (i.e., (2) s. (6)), as well
as o a combina ion o bo h models (i.e., (4) s. (7)) con ibu es signi ican ly o he
explana ion o a iance in he ejec ions o (8,2).
O e all, i seems ha
𝜅CA
PR
(a
P)
p o ides a good combina ion be ween F &F’s idea
o a kindness e m which di e en ia es be ween he in en ion o an ac ion and he
ex end o kindness wi h D &K’s app oach o assess an en i e game by means o a
e e ence alue. This esul s in mo e di e se p edic ions pa icula ly in he con ex
o iche decision en i onmen s. Fo ins ance, in an ul ima um- ype game wi h se -
e al al e na i es, he assump ion ha playe s e alua e he kindness o a speci ic o e
by pai wise compa isons be ween al e na i es seems un ealis ic o us a leas due
o he shi e compu a ional e o i akes o compa e al e na i es agains each o he .
Ra he , we ollow D &K’s idea ha playe s condense al e na i es by means o e e -
ence alues. As such, ou app oach is loca ed in some sense hal way be ween he
models by D &K and F &F: i p ocesses mo e in o ma ion han he model by D &K.
On he o he hand, ou app oach gene alizes o e al e na i es by a la ge ex en han
F &F by o ming e e ence alues. Whe he ou modi ica ions op imize he adeo
be ween he gene alisabili y o he model o a ious si ua ions and he accu a e p e-
dic ion o speci ic beha io is an open ques ion and equi es u u e esea ch.
6 Conclusion
Al hough ecip oci y is one undamen al co ne s one o human beha io , modeling
ecip oci y s ill is a challenge o social scien is s. The cu en s udy analyzes wo
o he mos es ablished app oaches, he e e ence alue model by D &K and he
in en ion ac o model by F &F. We poin ou ha he e a e wo majo di e ences
be ween he wo app oaches: he i s model measu es pe cei ed kindness o an
ac ion in ela ion o a e e ence alue while he second model dis inguishes be ween
he in en ion o an ac ion and he ex en o which he ac ion is pe cei ed as being
unkind o kind. The la e elemen o F &F’s model elies on he inequi y o he
p oposed payo s whe eas he o me elemen esul s om a pai wise compa ison
be ween al e na i es.
320
J.Hinz e al.
1 3
We es elemen s o bo h models wi hin he con ex o mini-ul ima um games wi h
wo and ou al e na i es. Resul s show ha F &F’s app oach wo ks ine in he games
wi h wo al e na i es, bu has impo an d awbacks in he games wi h ou al e na-
i es, bo h wi h espec o he a e age numbe s bu also once we un a wi hin-subjec
analysis. On he o he hand, D &K’s model ails o cha ac e ize beha io wi hin bo h
con ex s. Howe e , we ha e o admi ha we do no es D &K’s comple e model
excluding some equilib ia based on mu ual meanness. This may impai i s p edic i e
success in ou se ing. Despi e he sho coming, we conclude ha D &K’s idea o meas-
u e pe cei ed kindness in one a iable, he dis ance o he equi able payo , does no
su icien ly cap u e he na u e o pe cei ed (un)kindness. Pa icula ly o games wi h
ou al e na i es, i seems ha he pai wise compa ison o al e na i es does no p e-
dic beha io accu a ely. The e o e, we p esen and discuss a po en ial modi ica ion o
F &F’s ecip oci y model which includes elemen s o D &K’s app oach. Tes ing he
Combined app oach’s p edic ions wi h ou expe imen al da a yields an app op ia ely
low numbe o misma ches be ween subjec s’ decisions and he p edic ions. Also he
esul s o he eg ession analyses indica e a signi ican imp o emen o p edic i e
powe added by he Combined app oach o he exis ing models.
To conclude, mo e esea ch is needed o model ecip oci y in a su icien way.
Pe haps, he ques ion is no whe he he e is a ue model mapping ecip oci y,
bu whe he he e is a model ha adequa ely balances he need o gene alisabili y
ac oss di e en games wi h a sa is ac o y good p edic abili y o beha io wi hin a
speci ic en i onmen . Elsewhe e, i has been shown ha ecip oci y i sel encom-
passes a numbe o di e en sub ypes o social u ili y (e.g., Nicklisch and Wol
(2012)). The e o e, we ha e o ask ou sel es whe he we wan o model he beha io
Table 4 Linea p obabili y eg essions
Signi icance le els *p<0.1, **p<0.05 and ***p<0.01
Dependen a iable Rejec ing he o e (8,2)
(1) (2) (3) (4) (5) (6) (7)
ΨF&F
0.068
∗∗∗
0.062
∗∗∗
0.046
∗∗∗
0.049
∗∗∗
(0.004) (0.004) (0.005) (0.005)
ΨD&K
0.037
∗∗∗
0.025
∗∗∗
0.009
∗∗
0.015
∗∗∗
(0.003) (0.003) (0.004) (0.004)
ΨCA
0.083
∗∗∗
0.047
∗∗∗
0.077
∗∗∗
0.034
∗∗∗
(0.005) (0.006) (0.006) (0.007)
Cons an
−
0.220
∗∗
−
0.055
−
0.180
∗∗
−
0.241
∗∗∗
−
0.252
∗∗∗
−
0.179
∗∗
−
0.255
∗∗∗
(0.088) (0.090) (0.089) (0.088) (0.088) (0.089) (0.088)
Obse a ions 5124 5124 5124 5124 5124 5124 5124
Indi idual Dummies Yes Yes Yes Yes Yes Yes Yes
R
2
0.662 0.645 0.659 0.666 0.666 0.659 0.667
F-s a is ic 21.50*** 20.01*** 21.24*** 21.82*** 21.89*** 21.22*** 21.94***
LR- es :
𝜒2(
1
)
70.78*** 204.90*** 27.22***
(1) s.(5) (2) s.(6) (4) s.(7)
321
1 3
Recip oci y models e isi ed: in en ion ac o s and e e ence…
in one speci ic game which may igge one speci ic o m o ecip oci y, o whe he
we wan o ely on a gene al model, which, howe e , has less p edic ing powe in
special si ua ions. The answe o his ques ion we canno p o ide he e. The e o e,
we would like o in i e u u e esea ch o ollow his a enue, o , maybe, p o e i
w ong.
Appendix: P edic ions acco ding oou come conce ned social u ili y
Inequi y a e sion
P o ided inequi y a e se p e e ences, we ha e o claim ha esponde s ei he accep
o ejec (8,2) in all games
Γ1
o
Γ12
. The eason o his is a he ob ious: as he
same o e (8,2) is conside ed h oughou all games and inequi y a e sion p e e -
ences ake only he ou come o a p oposal in o conside a ion, he e is no di e ence
in he u ili y esul ing om accep ance ac oss games, no om ejec ion ac oss
games.
Le ine’s (1998) model
In he ollowing we wan o de i e a p edic ion o beha io in ou games acco ding
o Le ine’s (1998) model. We choose his model, since i can be cha ac e ized as
some in e media e s ep be ween models based on ou come conce ns and ecip oci y
models. The eason o his is ha he p opose pa ly e eals he as e o al uism
h ough he choice among he al e na i es o he game. This in o ma ion upda es he
weigh o al uism in he esponde ’s u ili y unc ion and may lead o ejec ions i
al uism is nega i e. Fo mally, we can de ine he u ili y unc ion o playe i (pai ed
wi h j) acco ding o Le ine (1998) as:
whe e
𝜋i(ai,aj)
is i’s mone a y payo o an o e , while
𝛼i
is i’s as e o al uism
(
−1<𝛼
i<1
); inally,
𝜚i
measu es he impo ance o j’s al uism o i’s u ili y
(
0≤𝜚i≤1
).
Suppose i is a p opose , while j is a esponde . O cou se,
𝛼i
and
𝜚i
a e i’s p i a e
in o ma ion. Howe e , by choosing a speci ic o e in he game, he p opose i pa ly
e eals he as e o al uism o he esponde j who upda es he u ili y
UL
j
acco d-
ingly. The e o e, j’s u ili y changes wi h i’s choice o al e na i es in he game. Ye , i
u ns ou ha i’s choice o (8,2) is unin o ma i e in
Γ1,Γ2
,
Γ4
,
Γ5
,
Γ6
,
Γ7
, and
Γ10
in
he sense ha i only e eals
ai<1
which is known om he beginning. As an exam-
ple, conside
Γ2
. He e, choosing (8,2) implies
8
+
𝛼
i
+𝜚
i
𝛼
j
1
+
𝜚
i
2>5+
𝛼
i
+𝜚
i
𝛼
j
1
+
𝜚
i
5
which is
equi alen o
1+𝜚i(1−𝛼j)>𝛼
i
. I ollows ha he maximum o
𝛼i
is 1.
Thus he choice o a speci ic o e in ou mini-ul ima um games does no esul in
an upda e o j’s belie conce ning
ai
. I ollows ha in all o he p e iously
U
L
i=𝜋i(ai,aj)+
𝛼
i
+𝜚
i
𝛼
j
1
+𝜚i
𝜋j(ai,aj
)
322
J.Hinz e al.
1 3
men ioned games j ejec s he o e o (8, 2) i
0
>2+
𝛼
j
+𝜚
j
𝛼
i
1+𝜚
j
8
. I ollows ha j
ejec s i
𝛼j< 𝛼
wi h
𝛼 ∈ (−1, …, 0.5]
depending on j’s speci ic
𝜚j
. Tha is o say, i
a esponde ejec s (8,2) in one o he games
Γ1,Γ2
,
Γ4
,
Γ5
,
Γ6
,
Γ7
, o
Γ10
, she should
ejec (8,2) in all se en games. Following he same a ional, Le ine’s model p e-
dic s ha i does no o e 8,2 in
Γ3
,
Γ8
,
Γ9
,
Γ11
, and
Γ12
. The e o e, we a e ha dly
able o o m p edic ions wi h espec o esponde ’s beha io .
No ice ha ou delibe a ions ely on he linea speci ica ion o Le ine’s model.
One may hink o a mo e gene al in e p e a ion o he model. Le us assume ha
we can use he equency by which p opose s choose (8,2) as an indica o o he
deg ee o unkindness o his o e . This leads hen o an upda e o R’s belie ega d-
ing
𝛼P
and, consequen ly, R’s ejec ion a e: ha is, he mo e equen (8,2) is cho-
sen by he p opose s, he less mean i is conside ed by he esponde s, and, he e o e,
ejec ed less equen ly (e.g., U ikal and Fischbache (2014) ollow his idea). This
leads, o ins ance, o he iche se ings
Γ8
o
Γ12
o he p edic ion ha he ejec-
ion a e o (8,2) in
Γ8
is lowe han in
Γ9
o
Γ12
(since (8,2) is o e ed signi ican ly
mo e equen ly in
Γ8
han in
Γ9
,
Γ10
,
Γ11
, and
Γ12
, see ou p opose analysis a he
beginning o Sec ion4.1). Al hough no explaining ou da a ully – he p edic ion
is iola ed by compa ing
Γ8
o
Γ11
(
p=0.584
, wo-sided p opo ion es compa ing
bo h ejec ion a es) – we would like o in i e esea che s o conside his o en o e -
looked model o hei u u e s udies.
P edic ed accep ance and ejec ion pa e ns acco ding o he h ee app oaches
See Table5.
Table 5 Choice pa e ns o (8,2) in
Γ1
o
Γ12
acco ding o he h ee app oaches; ‘a’ indica es p edic ed
accep ances, ‘ ’ p edic ed ejec ions
Game D &K pa e n F &F pa e n CA pa e n
(I) (II) (III) (IV) (V) (I) (II) (III) (IV) (V) (I) (II) (III) (IV)
Γ1
a a a a a a a a a a a a
Γ2
a a a a
Γ3
a a a a a a a a a a a a
Γ4
a a a a a a
Γ5
a a a a a a
Γ6
a a a a a a
Γ7
a a a a a a a
Γ8
a a a a a a a a
Γ9
a a a a a
Γ10
a a a a
Γ11
a a a a a a a a a
Γ12
a a a a a a