Biella, Ma co; Hennig, Max; Oswald, Lau a
A icle
In es iga ing he social bounda ies o ai ness by
modeling Ul ima um Game esponde s' decisions wi h
mul inomial p ocessing ee models
Games
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Biella, Ma co; Hennig, Max; Oswald, Lau a (2025) : In es iga ing he social
bounda ies o ai ness by modeling Ul ima um Game esponde s' decisions wi h mul inomial
p ocessing ee models, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 1, pp. 1-21,
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Published: 3 Janua y 2025
Ci a ion: Biella, M., Hennig, M., &
Oswald, L. (2025). In es iga ing he
Social Bounda ies o Fai ness by
Modeling Ul ima um Game
Responde s’ Decisions wi h
Mul inomial P ocessing T ee Models.
Games,16(1), 2. h ps://doi.o g/
10.3390/g16010002
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A icle
In es iga ing he Social Bounda ies o Fai ness by Modeling
Ul ima um Game Responde s’ Decisions wi h Mul inomial
P ocessing T ee Models
Ma co Biella 1,2,* , Max Hennig 3and Lau a Oswald 4
1Facul y o Business and Economics, Uni e si y o Basel, 4052 Basel, Swi ze land
2Facul y o Psychology, Ebe ha d Ka ls Uni e si ä Tuebingen, 72076 Tübingen, Ge many
3Depa men o Psychology, Julius-Maximilians-Uni e si ä Wue zbu g, 97070 Wü zbu g, Ge many;
[email p o ec ed]
4Depa men o Psychology, Albe -Ludwigs-Uni e si ä F eibu g, 79104 F eibu g im B eisgau, Ge many;
[email p o ec ed]g.de
*Co espondence: [email p o ec ed] o [email p o ec ed]
Abs ac : Fai ness in compe i i e games such as he Ul ima um Game is o en de ined
heo e ically. Acco ding o some o he li e a u e, in which ai ness is de e mined only based
on esou ce alloca ion, a p oposal spli ing esou ces e enly (i.e., 5:5) is gene ally assumed
as ai , and minimal de ia ion (i.e., 4:6) is conside ed enough o classi y he p oposal as
un ai . Relying on mul inomial p ocessing ee models (MPTs), we in es iga ed whe e he
bounda ies o ai ness a e loca ed in he eye o esponde s, and pi ai ness agains ela i e
and absolu e gain maximiza ion p inciples. The MPT models we de eloped and alida ed
allowed us o sepa a e h ee indi idual p ocesses d i ing esponses in he s anda d and
Thi d-Pa y Ul ima um Game. The esul s show ha , om he esponde ’s pe spec i e,
he bounda ies o ai ness encompass p oposals spli ing esou ces in a pe ec ly e en way
and include une en p oposals wi h minimal de iance (4:6 and 6:4). Mo eo e , he esul s
show ha , in he con ex o Thi d-Pa y Ul ima um Games, he esponde mus no be
indi e en be ween a o ing he p opose and he ecei e , demons a ing a bounda y
condi ion o he de eloped model. I he esponde is pe ec ly indi e en , absolu e and
ela i e gain maximiza ion a e heo e ically uniden i iable. This heo e ical and p ac ical
cons ain limi s he scope o ou heo y, which does no apply in he case o a pe ec ly
indi e en decision-make .
Keywo ds: ai ness; compe i i e games; Ul ima um Game; mul inomial p ocessing ee;
ela i e gain maximiza ion; u ili y heo y
1. In oduc ion
When engaging in economic ansac ions, people o en ace a ade-o be ween maxi-
mizing hei own p o i and ollowing he social no m o ai ness (Messick & Schell,1992).
This easoning can be ex ended o a b oade se o social in e ac ions, such as helping
beha io s o a o exchanges ha do no in ol e economic aspec s in s ic e ms.
Social in e ac ions o his kind ha e been in es iga ed by esea che s in an expe imen-
al pa adigm called he Ul ima um Game (Gü h e al.,1982). This simple game, and i s la e
de elopmen s (Biella & Sacchi,2018;Ci ai e al.,2013), o e an oppo uni y o obse e
how people engage wi h s a egic esou ces spli ing and how hey deal wi h (po en ially)
un ai si ua ions. Speci ically, he ul ima um has he po en ial o show people’s beha io
when con on ed wi h he decision be ween main aining an equi able ou come a he cos
Games 2025,16, 2 h ps://doi.o g/10.3390/g16010002
Games 2025,16, 2 2 o 21
o losing pe sonal mone a y u ili y on one hand, o gaining some mone a y u ili y a he
cos o iola ing he no m o ai ness on he o he hand. Apa om he deba e abou
when one o he wo op ions is p e e ed o e he o he , he in es iga ion o he cogni i e
p ocesses unde lying obse able decisions is a om o e . In his pape , we p opose an
inno a i e way o aming he p ocesses d i ing decision-making in he Ul ima um Game,
which lends i sel o expe imen al es ing and o e comes some o he open con o e sies
on he opic.
1.1. Ul ima um Game and he Fai ness/U ili y T ade-O
The Ul ima um Game is a simple sequen ial game in which wo playe s s a egically
in e ac o di ide some esou ces (Gü h e al.,1982). These wo playe s a e he i s -
mo e , gene ally e e ed o as he “p opose ”, and he second-mo e , gene ally e e ed
o as he “ ecei e ” o “ esponde ”. In ou esea ch, we will ocus on he esponde ’s
pe spec i e. In addi ion o acili a ing he discussion o ou easoning, his ocus is necessa y,
as mul inomial p ocessing ee models (see nex pa ag aph) equi e ca ego ical esponses.
In he s anda d Ul ima um Game, he p opose is in o med abou he o al amoun o
esou ces a ailable (i.e., 10 Eu o) and is asked o come up wi h a spli p oposal (i.e., 6 Eu o/4
Eu o). This p oposal will be e alua ed by he second-mo e , he esponde / ecei e . The
second-mo e has wo op ions o de e mine he inal payo o bo h playe s. I he second-
mo e accep s he o e , each playe ecei es he designa ed amoun (i.e., 6 Eu o o he
p opose and 4 Eu o o he ecei e ), bu i hey ejec he o e , bo h playe s ecei e no hing.
I mone a y gain is he only d i e , he p opose should c a an o e ha assigns he
g ea e amoun o him/he sel , lea ing only a minimal bu non-ze o quan i y o esou ces
o he ecei e . The esponde should accep such an o e , as ejec ing i would imply
ecei ing no hing, which is less ha he non-ze o mone a y u ili y ecei ed i hey accep .
Fo he same eason, he esponde should accep any o e ha g an s him/he any non-
ze o mone a y gain, ega dless o how much he p opose is ea ning. Howe e , hese
p edic ions hold only i one assumes ha mone a y esou ces a e he only componen o he
u ili y unc ions o bo h he esponde and he p opose . These p edic ions a e a ely me
in he li e a u e and iola ions o mone a y u ili y maximiza ion a e o en epo ed (Gü h
e al.,1982;Biella & Sacchi,2018;Aina e al.,2020;Ci ai,2013;Ruessmann & Topolinski,
2020). Indeed, a easonable p opose migh maximize hei own payo by c a ing ai
o e s, which a e mo e likely o be accep ed. Indeed, any o e ha is likely o be ejec ed
will p oduce no gain o he p opose . The e o e, ad ancing such o e s is no a easonable
beha io . The iola ion o p edic ions assuming ha mone a y gain maximiza ion is he
only componen o bo h playe s’ u ili y unc ions should no be seen as a ailu e o economic
heo y bu a he as a signal ha u ili y does no en ail only mone a y gain maximiza ion.
Fo example, ai ness can be embedded in o he u ili y unc ions o bo h he p opose and
he ecei e .
Se e al accoun s ha e been p oposed o explain de ia ions om he p edic ions
de i ed using mone a y gain maximiza ion as he only componen o he u ili y unc ion.
Some au ho s endo se an emo ional explana ion. Speci ically, hey pos ula e ha ecei ing
an un ai , disad an ageous o e , an o e in which he amoun des ined o he ecei e is
lowe han he amoun des ined o he p opose , igge s nega i e emo ions esponsible
o he ejec ion and he iola ion o economic p inciples (Aina e al.,2020;Ci ai e al.,
2010;Pillu la & Mu nighan,1996). Indeed, he “wounded p ide/spi e model” endo ses
such an explana ion (Pillu la & Mu nighan,1996), which is suppo ed by neu oscien i ic
e idence epo ing g ea e ac i a ion o he an e io insula and do sola e al p e on al
co ex, wo a eas ela ed o ange and disgus , when he ecei e is exposed o un ai ,
disad an ageous o e s (San ey e al.,2003). Simila ly, ejec ions may be explained in e ms
Games 2025,16, 2 3 o 21
o ecip oci y (Feh & Schmid ,1999). Thus, he nega i e beha io o ejec ing an o e
can be unde s ood as ecip oca ing he nega i e beha io o p oposing an un ai o e in
he i s place. O he accoun s sugges ha he ejec ion o Ul ima um Game o e s, and
he consequen iola ion o economic assump ions, is based on cogni i e heu is ics and
he no m o ai ness (Messick & Schell,1992;Messick,1995). Such a no m pos ula es ha ,
when he e is no eason o do o he wise, bo h playe s should ecei e he same sha e o
esou ces. This de ini ion o ai ness is e y s ingen , and some adjus men s a e equi ed
o games ha a e inhe en ly asymme ic, such as he Ul ima um Game (Kamas & P es on,
2012;K a i z & Gun o,1992;Suleiman,2017;Suleiman,2022). Howe e , his decision ule
has he bene i o le eling he ela i e mone a y u ili y ecei ed by bo h playe s (Messick
& Tho nga e,1967) and is in line wi h o he economic accoun s sugges ing ha a ecei e
ejec ing he p opose ’s o e is engaging in nega i e ecip oci y owa d he p opose
(Rabin,1993). The la e is e en in line wi h e olu iona y heo ies and he li e a u e on
jus ice wi hin social economic games, which sugges ha i he no m o ai ness is b oken,
a punishmen should be imposed (Ha din,1968;Sch oede e al.,2003) and ha such a
social con ol de ice p omo es su i al in he long un (Alexande ,1987;Boehm,1999;
Da win,1871). Indeed, he e is an ex ensi e li e a u e on he e olu ion o ai ness in
esou ce-sha ing games (Kahneman e al.,1986) in gene al, and ela ed o he Ul ima um
Game, speci ically (Came e & Thale ,1995;Debo e e al.,2016;Thale ,1988). Howe e ,
he s anda d pa adigm con ounds ai ness and mone a y u ili y, as any disad an ageous
o e is bo h un ai and o lowe mone a y u ili y o he ecei e . The e o e, an e olu ion o
he Ul ima um Game has been p oposed. In he Thi d-Pa y Ul ima um Game (Ci ai e al.,
2013;Ha u y & Ro h,2022), an addi ional playe called he decision-make is in oduced.
In oducing he decision-make e ec i ely sepa a es he ole o he ecei e and o he
esponde . The decision-make has he ole o e alua ing he p opose ’s o e on he behal
o he ecei e . I he decision-make accep s he o e , bo h he p opose and he ecei e
ecei e pa o he esou ces, and i hey ejec , p opose and ecei e ecei e no hing. In
bo h cases, he decision-make does no ecei e any money and, he e o e, should no be
a ec ed o guided by mone a y u ili y. This e sion emo es he u ili y/ ai ness con ound.
Mo eo e , his pa adigm allowed esea che s o demons a e ha people show inequi y
a e sion (Bol on,1991;Bol on & Ocken els,2000;Feh & Schmid ,1999) when mone a y
u ili y is no a decision d i e , bu ole a e i when such a u ili y is a s ake and in hei
a o , as in he case o an un ai ad an ageous o e in he s anda d Ul ima um Game,
in which he decision-make / ecei e ecei es pa o he esou ces (Ci ai e al.,2013).
In his pa adigm, esea che s claim ha inequi y a e sion is ole a ed, as hey documen
pa icipan s accep ing une en o e s. Addi ionally, i has been shown ha , i he decision-
make and he ecei e a e socially close (Biella e al.,2023) o belong o he same minimal
social g oup (Biella & Sacchi,2018), he mone a y u ili y di ec ed owa d he ecei e a ec s
he decision made by he decision-make such ha inequi y a e sion is mo e ole a ed i
he mone a y u ili y is in a o o he ecei e . Again, esea che s used decision-make s’
accep ance o une en o e s as key e idence o inequi y a e sion ole ance.
A e e iewing he accoun s aimed a explaining decision-making in he Ul ima um
Game, we can iden i y h ee main d i e s ha play a c ucial ole. These d i e s a e he
inna e sense o ai ness ha is cap u ed by inequi y a e sion (Bol on,1991;Bol on & Ocken-
els,2000;Feh & Schmid ,1999), he economic p inciple o absolu e gain maximiza ion, and
a si ua ionally g ounded e o o maximize ela i e gain (Rabin,1993). Beyond he deba e
on whe he such p ocesses a e cogni i e (i.e., heu is ics) o emo ional (i.e., ange /disgus ,
wounded p ide/spi e model) in na u e (Ci ai e al.,2013), he ac ha hese ee p ocesses
a e conside ed he main d i e s o decision-making in he Ul ima um Game s ands, and i
is he subjec o ou p esen analysis.
Games 2025,16, 2 4 o 21
I is wo h no ing ha ou in es iga ion does no a emp o e u e any o he heo e ical
accoun s ad anced o explain decisions in he Ul ima um Game. On he con a y, we build
upon such heo ies, and we aim a de eloping a model ha complies wi h such heo ies’
p edic ions. Ou con ibu ion mos ly comes om he heo e ical wo k equi ed o alida e
he mul inomial p ocessing ee model we a e p oposing (see nex sec ion). Such a alida-
ion equi es o malizing a su icien se o p ocesses esponsible o he decisions made by
he esponde s, de i ing heo e ically sound p edic ions (coming om he accoun s we jus
e iewed), and compa ing he model’s pe o mance agains such p edic ions.
1.2. Mul inomial P ocessing T ee Models
Mul inomial p ocessing ee models (MPT) a e powe ul analy ical ools ha o e
he capabili y o disen angling co-occu en p ocesses guiding esponses wi hin a speci ic
ask (Hü e & Klaue ,2016). Relying on ca ego ical esponses, hese models quan i y he
p obabili y o each p ocess d i ing he ou come, condi ioned upon he con ibu ion o all
o he p ocesses. Adop ing he esponde / ecei e ’s pe spec i e allows us o le e age he
ca ego ical na u e o esponses o model he unde lying p ocesses behind decisions. In
his amewo k, esea che s can ob ain p ocess-pu e quan i ica ion o he p ocesses in he
con ex o he pa adigm o in e es . Mo eo e , MPT models allow o a s aigh o wa d
es o di e ences in pa ame e alues ac oss expe imen al condi ions (Klaue e al.,2011;
Rie e & Ba chelde ,1988). C ucially, MPT models equi e he esea che o o mula e
p ecise hypo heses, as he o malized model canno be de eloped i he numbe , na u e,
and composi ion o he p ocesses a e no clea ly speci ied. Addi ionally, he mos impo an
p ocesses playing a ole in he pa adigm o in e es mus be included o he model o wo k
p ope ly. The e o e, model speci ica ion mus be d i en by ca e ul heo e ical analysis.
Once he model has been speci ied, i can be i ed using he maximum likelihood me hod,
de e mining he pa ame e alues ha make he esponse da a mos likely.
The pa ame e s can be in e p e ed as he p obabili y o each p ocess o d i e esponses.
Mo e speci ically, hey a e he condi ional p obabili ies depending on he p e ious p ocess
(Hü e & Klaue ,2016). Each b anch in he ee ep esen s he ope a ion o a single p ocess
o a succession o p ocesses leading o an obse able esponse. High pa ame e alues
deno e ha he p ocess has a s ong in luence on esponse p oduc ion. C ucially, he
es ima ion o he pa ame e s is dependen on he o de o he p ocesses. Such dependency
has wo main consequences. Fi s , pa ame e compa ison ac oss models mus be ca ied ou
on models en o cing he same pa ame e o de . Second, pa ame e s mus be in e p e ed as
p obabili ies condi ioned on p e ious p ocesses. Once he pa ame e s ha e been es ima ed,
he model i can be e alua ed using a chi-squa e es (Hu & Ba chelde ,1994) and mo e
ine-g ained me ics such as Cohen’s w (Cohen,1988). Rega ding he chi-squa e es , a
canonical h eshold o signi icance es ing can be assumed,
α
= 0.05, bu addi ional ca e
is wa an ed, as he chi-squa e es is known o be o e sensi i e o la ge sample sizes
(Foldnes & Henning Olsson,2015;Powell & William,2001). Rega ding Cohen’s w, we
assume
w = 0.10
as a easonable h eshold o a small magni ude. Any de ia ion om
he ideal i la ge han ha will be conside ed oo much, leading o he conclusion ha
sa is ac o y i has no been eached.
Se e al esea ch p og ams, such as he in es iga ion o mo al dilemmas (Conway
& Gaw onski,2013;Hennig & Hü e ,2020) and he au oma ici y o a i ude acquisi ion
(Hü e & Sweldens,2018;Hü e e al.,2012), ha e al eady bene i ed om he applica-
ion o hese models, and ou goal is o apply such a amewo k in he con ex o he
Ul ima um Game.
Games 2025,16, 2 5 o 21
1.3. MPT Model o he Ul ima um Game
To he bes o ou knowledge, modeling he Ul ima um Game’s esponde s’ decisions
using MPT models has ne e been a emp ed. The exis ing li e a u e al eady in es iga es
he d i e s o p oposals gene a ion and has highligh ed he cen al ole o ai ness (Fo sy he
e al.,1994;Ha ison & McCabe,1996;Ho man e al.,1994). Al hough success ul, hese p io
a emp s ook he p opose ’s pe spec i e o elied on expe imen al designs o in es iga e
he decision’s d i e s in isola ion. We hink ha he li e a u e on he Ul ima um Game
can g ea ly bene i om he applica ion o MPT models, as p ope quan i ica ion o he
p ocesses d i ing esponses in his pa adigm will shed new ligh on he deba e a ound he
p ocesses hemsel es, and on he ole o ai ness in pa icula . Mo eo e , many economic
games, such as he Ul ima um Game, exhibi simila ea u es in e ms o s uc u e, esponse
ype, and d i ing p ocesses ha closely esemble o he asks (i.e., mo al dilemmas) ha
ha e al eady bene i ed om he use o MPT models (Conway & Gaw onski,2013;Hennig
& Hü e ,2020).
1.3.1. Unde lying P ocesses
To make ou app oach ui ul, howe e , ou o mal model mus be guided by ca e ul
heo e ical analysis, s a ing om he iden i ica ion o he d i ing p ocesses. Based on
he li e a u e abo e, we conside he no m o ai ness, maximiza ion o ela i e gain,
and maximiza ion o absolu e gain as he mos impo an p ocesses playing a ole in he
Ul ima um Game. The no m o ai ness is de ined as he implici and socially es ablished
p inciple ha unbalanced esou ces spli s should no be p oposed nei he accep ed. Rela i e
gain is de ined as he a io be ween he esou ces des ined o he ecei e o e he esou ces
des ined o he p opose . I an o e alloca es mo e esou ces o he ecei e han o he
p opose , i can be conside ed an un ai o e , wi h ela i e gain a o ing he ecei e . Finally,
absolu e gain is de ined as he absolu e payo ea ned by he ecei e i he o e is accep ed.
Ou heo e ical analysis does no di ide p ocesses in o cogni i e (Messick & Schell,1992;
Rabin,1993;Bol on & Ocken els,2000;Feh & Schmid ,1999) e sus emo i e (Pillu la
& Mu nighan,1996;San ey e al.,2003) bu a he me ges hese wo sides o he same
coin. Fo example, nega i e emo ions a ising om ecei ing an un ai , disad an ageous
o e a e equally embedded in he ai ness-based and ela i e-gain-based p ocess. Indeed,
nega i e emo ions migh a ise om bo h pe cei ing he un ai , disad an ageous o e as
o ensi e, due o he iola ion o he no m o ai ness, and as an a emp o educe he
ecei e ’s ela i e gain. On he o he hand, an un ai , ad an ageous o e migh igge
ewe nega i e emo ions, as only he la e , he a emp o educe he ecei e ’s ela i e
gain, is missing, while he o me , he iola ion o he no m o ai ness, is s ill a e si e and
po en ially igge ing. Simila ly, expec a ions can o e lay wi h such p ocesses, as long as
hey p o ide p edic ions ha a e cohe en wi h he p ocess i sel . The exis ing li e a u e,
o example (Ha ison & McCabe,1996;Suleiman,1996), pos ula es ha decision-make s
and ecei e s migh expec p opose s o ad ance ai o e s. In. I is no a heo e ical
p oblem i emo ions, expec a ions, o any o he phenomena on a di e en le el o analysis
a e con la ed wi h he p ocess o in e es , as long as hese p ocesses lead o clea , non-
con adic o y heo e ical p edic ions and show no logical inconsis encies, which is he case,
as de ailed in he upcoming pa ag aph.
To es ima e he model, he o de o he p ocesses mus be speci ied. Howe e , he o de
o he model does no ha e any heo e ical implica ions, as he p ocesses a e assumed o ac
in pa allel (Hü e & Klaue ,2016). Analy ically speaking, he o de speci ied imposes only
one cons ain , namely, ha he same p ocesses o de mus be speci ied o compa e wo
models. Simila ly, he pa ame e s should no be in e p e ed in isola ion bu in he la ge
con ex o he whole model. The e o e, we o de ed he p ocesses based on he heo e ical
Games 2025,16, 2 6 o 21
le el in which hey eside. We s a ed wi h ai ness, which is he mos ela ed o social
no ms. This no m exis s in he p esence o an agen , a leas one in e ac ion pa ne , and a
la ge social g oup en o cing he no m. Second, we in oduced ela i e gain maximiza ion,
which implies some le el o in e ac ion wi h a social o he . This p ocess esides a he
in e pe sonal le el, in be ween he decision-make s and he playe s o he game. Finally,
we added absolu e gain maximiza ion, which can be placed a he indi idual le el. This las
p ocess equi es only he decision-make and an indi idual p e e ence o each possible
ou come. This o de ing, om he mos social o he mos indi idual, makes sense om
he heo e ical poin o iew and, as long as model compa ison happens be ween models
en o cing he same o de ing, i is no analy ically p oblema ic (Hü e & Klaue ,2016).
To summa ize, we de eloped a model in which he esponse in he Ul ima um Game
is d i en by h ee p ocesses. These p ocesses a e embedded in an MPT model wi h he
ollowing pa ame e s:
o ai ness,
1,
2,
3, and
4 o he maximiza ion o ela i e gain,
and a o he maximiza ion o absolu e gain. Rela i e gain maximiza ion equi es ou
pa ame e s o encode he di e en in ensi ies o ela i e gain maximiza ion a a ying o e
le els. Theo e ically,
pa ame e s ep esen he same p ocess, bu he a ying in ensi y
mus be ep esen ed by sepa a e pa ame e s.
1.3.2. The Model and I s Theo e ical P edic ions
A e he iden i ica ion o he p ocesses o in e es , clea and es able heo y-d i en
p edic ions mus be speci ied o o malize he model. The expec ed decision ou come
o each o e le el mus be speci ied unde he assump ion ha each p ocess guides
he beha io (Hü e & Klaue ,2016). Mo eo e , such p edic ions mus a oid logical
inconsis encies and mus make he model iden i iable (Singmann & Kellen,2013).
The ai ness-based p ocess has ai ly s aigh o wa d p edic ions. E e y ime an o e
spli s esou ces une enly, his p ocess leads o ejec ion, while accep ance is expec ed i he
o e dis ibu es esou ces e enly. Howe e , his o maliza ion assumes ha he bounda ies
o ai ness a e clea -cu . Especially in epea ed-game e sions o he Ul ima um Game
( e sions in which he p opose and he ecei e go h ough se e al ba gaining ounds);
howe e , he bounda ies o ai ness migh be mo e lenien (Fo sy he e al.,1994;Ha ison
& McCabe,1996). A ecei e migh conside a sligh ly une en disad an ageous o e as ai
wi h he expec a ion ha he p opose will conside , and p o ide, he ad an ageous e sion
o he same sligh ly une en o e . C ucially, unde bo h o maliza ions he p edic ions a e
he same.
Rega ding he ela i e-gain-based p ocess, p edic ions can be easily de i ed along
he whole con inuum o possible o e s excep o a pe ec ly e en o e . Speci ically, i
he amoun des ined o he ecei e is lowe han he amoun des ined o he p opose ,
his p ocess imposes ejec ion, while accep ance is expec ed i he ela i e p opo ion o
esou ces a o s he ecei e . In he singula case o a pe ec ly e en o e , ela i e gain does
no a o any o he wo playe s. He e, his p ocess is silen , as i is logically impossible o
use ela i e gain o make a decision in he case o an o e ha does no show any ela i e
gain ad an age o ei he side. To e lec his, ou model does no ha e a ela i e-gain-based
p edic ion o pe ec ly e en o e s. Such an ins ance o a p ocess lacking p edic ions is
no an issue, as long as he combina ion o he emaining p ocesses and hei p edic ions
make he model iden i iable. Addi ionally, some e idence sugges s ha he mo e une en
he o e , he s onge he eac ion (San ey e al.,2003;Messick & Tho nga e,1967;Van’
Wou e al.,2006). To p ope ly cap u e his inc eased in ensi y, ela i e gain maximiza ion
will be modeled as se e al p ocesses wi h iden ical p edic ions (i.e., one pa ame e o each
le el o in ensi y).
Games 2025,16, 2 7 o 21
Finally, he absolu e gain p ocess is he one wi h he mos s aigh o wa d p edic ions.
In line wi h he classical economics accoun , his p ocess p edic s ha each o e ha
alloca es a leas some esou ces o he ecei e will be accep ed. Any ejec ions sugges
ha his p ocess has been o e u ned.
Collec ing all he p edic ions o each p ocess unde each o e le el, we can de i e
he o mal MPT model and i s pa ame e s. Speci ically, we ha e an
pa ame e modeling
he ai ness-based p ocess, mul iple
pa ame e s modeling he di e en s ages o he
ela i e-gain-based p ocess, and an
a
pa ame e modeling he absolu e-gain-based p ocess
(Figu e 1A).
Games 2025, 16, x FOR PEER REVIEW 7 o 21
Finally, he absolu e gain p ocess is he one wi h he mos s aigh o wa d p edic-
ions. In line wi h he classical economics accoun , his p ocess p edic s ha each offe ha
alloca es a leas some esou ces o he ecei e will be accep ed. Any ejec ions sugges
ha his p ocess has been o e u ned.
Collec ing all he p edic ions o each p ocess unde each offe le el, we can de i e
he o mal MPT model and i s pa ame e s. Speci ically, we ha e an 𝑓 pa ame e model-
ing he ai ness-based p ocess, mul iple 𝑟 pa ame e s modeling he diffe en s ages o he
ela i e-gain-based p ocess, and an 𝑎 pa ame e modeling he absolu e-gain-based p o-
cess (Figu e 1A).
Figu e 1. Mul inomial p ocessing ee models embedding (A) “S ic ” and (B) “Lenien ” ai ness
concep ualiza ion and hei espec i e p edic ions o all offe le els. Ve ical lines ep esen ai -
ness bounda ies. Shaded ex ep esen s offe accep ance, while non-shaded ex ep esen s offe
ejec ion.
The only heo e ical unknown is a wha offe le el ai ness bounda ies can be placed.
To allow o a mo e lenien o mula ion o ai ness (Fo sy he e al., 1994; Ha ison &
McCabe, 1996), which allows o less clea -cu bounda ies, a second model can be de i ed.
In such a model, he ai ness-based p ocess p edic s accep ance in a la ge ange o offe
le els cen e ed on he pe ec ly e en one. He e, ai ness and absolu e gain maximiza ion
con e ge wi h ela i e gain maximiza ion only on he igh -hand side, whe e offe s a e
ad an ageous o he ecei e , while i con adic s ela i e gain maximiza ion bu s ill
con e ges wi h absolu e gain maximiza ion (Figu e 1B).
I is wo h no ing ha , in addi ion o ha ing a ela i ely high numbe o pa ame e s,
bo h models yield heo e ically meaning ul and consis en p edic ions. Mo eo e , bo h
models ake all h ee p ocesses in o accoun while equi ing he lowes numbe o pa am-
e e s. Indeed, al e na i e models ha conside all h ee p ocesses a e possible a he cos
o an inc eased numbe o pa ame e s. The e o e, he p esen models a e he ones ha
capi alize he leas on chance due o he model complexi y in e ms o he numbe o pa-
ame e s.
Addi ionally, he model lexibili y in e ms o ai ness bounda ies and mul iple 𝑟
pa ame e s is heo e ically jus i ied and ne e yields logical inconsis encies. Less lexible
models (i.e., a model wi h a single pa ame e ) a e possible a he cos o iola ing heo-
e ically meaning ul p edic ions (i.e., assuming ha ela i e gain maximiza ion is he
same o sligh ly and hea ily une en offe s).
Figu e 1. Mul inomial p ocessing ee models embedding (A) “S ic ” and (B) “Lenien ” ai ness
concep ualiza ion and hei espec i e p edic ions o all o e le els. Ve ical lines ep esen ai ness
bounda ies. Shaded ex ep esen s o e accep ance, while non-shaded ex ep esen s o e ejec ion.
The only heo e ical unknown is a wha o e le el ai ness bounda ies can be placed.
To allow o a mo e lenien o mula ion o ai ness (Fo sy he e al.,1994;Ha ison &
McCabe,1996), which allows o less clea -cu bounda ies, a second model can be de i ed.
In such a model, he ai ness-based p ocess p edic s accep ance in a la ge ange o o e
le els cen e ed on he pe ec ly e en one. He e, ai ness and absolu e gain maximiza ion
con e ge wi h ela i e gain maximiza ion only on he igh -hand side, whe e o e s a e
ad an ageous o he ecei e , while i con adic s ela i e gain maximiza ion bu s ill
con e ges wi h absolu e gain maximiza ion (Figu e 1B).
I is wo h no ing ha , in addi ion o ha ing a ela i ely high numbe o pa ame-
e s, bo h models yield heo e ically meaning ul and consis en p edic ions. Mo eo e ,
bo h models ake all h ee p ocesses in o accoun while equi ing he lowes numbe o
pa ame e s. Indeed, al e na i e models ha conside all h ee p ocesses a e possible a
he cos o an inc eased numbe o pa ame e s. The e o e, he p esen models a e he ones
ha capi alize he leas on chance due o he model complexi y in e ms o he numbe
o pa ame e s.
Addi ionally, he model lexibili y in e ms o ai ness bounda ies and mul iple
pa ame e s is heo e ically jus i ied and ne e yields logical inconsis encies. Less lexible
models (i.e., a model wi h a single pa ame e ) a e possible a he cos o iola ing heo e i-
cally meaning ul p edic ions (i.e., assuming ha ela i e gain maximiza ion is he same o
sligh ly and hea ily une en o e s).
1.4. The P esen Resea ch
In he p esen esea ch, we aim o (a) use he MPT amewo k o p edic Ul ima um
Game esponses, ob aining sa is ac o y model i ; (b) alida e he model by in es iga ing
Games 2025,16, 2 8 o 21
how i pe o mances unde di e en heo e ically meaning ul condi ions (i.e., he s anda d
and Thi d-Pa y Ul ima um Game); and (c) loca e he bounda ies o pe cei ed ai ness by
es ing p edic ions o pa ame e s implemen ing s ic and lenien concep ualiza ions o
ai ness. We base ou in es iga ion on h ee s udies.
The i s goal (a) o he p esen esea ch is achie ed mainly by ou i s expe imen ,
and ma ginally by he emaining wo. In Expe imen 1, we aim a es ing he p oo o
concep o ou model. We simply un a s anda d Ul ima um Game o es i he MPT
amewo k can model he da a p ope ly. Speci ically, we es i he models (wi h “s ic ”
and “lenien ” ai ness bounda ies) i he obse ed da a.
The second goal (b) is achie ed by he compa ison o da a om he i s expe imen
wi h he da a om he Thi d-Pa y Ul ima um Game o Expe imen 2. He e, ou expec a ion
is ha he model i s p ope ly on bo h expe imen s sepa a ely, and ha he compa isons
o he models’ pa ame e s i ou heo e ical p edic ions. Rega ding model i , we expec
ha he model wi h he “lenien ” ai ness concep ualiza ion will i he da a be e han
he one wi h he “s ic ” concep ualiza ion. Rega ding model pa ame e s, we expec he
pa ame e o be lowe in he s anda d e sion o he pa adigm (Expe imen 1) han in he
hi d-pa y e sion (Expe imen 2). Simila ly, we expec he a pa ame e o be lowe in he
hi d-pa y e sion in compa ison o he s anda d e sion.
The hi d goal (c) is achie ed by e alua ing how di e en models (wi h “s ic ” o
“lenien ” ai ness concep ualiza ion) i obse ed da a ac oss all h ee expe imen s and, in
pa icula , in Expe imen 3. In Expe imen 3, pa icipan s ecei e he explici ins uc ion o
“no aking any side” be o e a Thi d-Pa y Ul ima um Game. The e o e, i is possible ha
bo h ou models show unsa is ac o y i , as an indi e en decision-make canno maximize
ela i e gain. I his is he case, Expe imen 3 may p o ide insigh s on he bounda y
condi ions unde which ou model does no apply.
2. Expe imen 1
In Expe imen 1, we p o ide a p oo o concep o ou models. Speci ically, we es
ou hypo hesis ha Ul ima um Game esponses can be modeled elying on h ee p ocesses,
namely ai ness, ela i e gain maximiza ion, and absolu e gain maximiza ion. Mo eo e ,
we aim a p obing po en ial ai ness bounda ies by es ing he “s ic ” and “lenien ” ai ness
concep ualiza ions embedded in he wo models p esen ed abo e. Da a and ma e ials
o Expe imen 1 a e a ailable a he OSF eposi o y (h ps://os .io/ zbpd/, accessed on
26 Decembe 2024).
2.1. P ocedu e
In Expe imen 1, pa icipan s we e in ol ed in a s anda d Ul ima um Game. The
expe imen was conduc ed online, and p io o he beginning o he p ocedu e, pa icipan s
p o ided hei in o med consen o ake pa in he expe imen . The expe imen began wi h
he collec ion o socio-demog aphic in o ma ion abou he pa icipan , hen ins uc ions
we e p esen ed. Pa icipan s we e in o med ha hey we e abou o ake pa in an Ul ima-
um Game. Speci ically, pa icipan s we e in o med ha some o hem would be assigned
o he ole o he p opose and ha hei ask was o come up wi h spli p oposals o a o al
o 10 Eu o. The emaining pa icipan would be assigned o he ole o he ecei e , and
hei ask was o decide whe he o accep o ejec each o e . Mo eo e , pa icipan s we e
in o med ha he p opose / ecei e pai s would be cons an h oughou he expe imen ,
meaning ha hey would always in e ac wi h he same pa ne . All pa icipan s we e as-
signed o he ecei e ole. A ansla ion o he inal ins uc ions ecei ed by he pa icipan
is a ailable in he online eposi o y (h ps://os .io/3x 4y, accessed on 26 Decembe 2024)
and Appendix A.
Games 2025,16, 2 15 o 21
hi d-pa y condi ion did no ecei e any mone a y u ili y and migh only ma ginally ake
he side o he ecei e (Ci ai e al.,2013).
4. Expe imen 3
In Expe imen 3, we aim o alida e he model e en u he . We implemen ed a
manipula ion o he Thi d-Pa y Ul ima um Game ha ins uc s pa icipan s o decide
based only on hei sense o ai ness. Indeed, we expec he
pa ame e o be highe han
he same pa ame e in bo h p e ious expe imen s. As be o e, we will p oceed one s ep a a
ime. Fi s , he global i o he model will be assessed. Second, i he global i is adequa e,
we will cons ain e e y pa ame e o be ze o, assessing he need o he pa ame e o be
in he model. Thi d, we will un an in eg a i e analysis, allowing us o compa e join
models including da a om Expe imen 3 and he da a om he p e ious expe imen s in
o de o compa e he
pa ame e ac oss expe imen s. I he la e analysis is inconclusi e,
we will conclude ha he
pa ame e is una ec ed by he manipula ion in Expe imen
3. I any o he model’s pa ame e s p o e o no be necessa y, o i he global i o he
model is no sa is ac o y, we will conclude ha he model we de eloped canno desc ibe
esponses p oduced in a si ua ion in which pa icipan s a e asked o base hei esponses
on one p ocess only ( ai ness in his case) bu ha e no s ake in he game. I all analyses
a e success ul and p edic ions a e me , his hi d expe imen will alida e ou model e en
u he ; howe e , i poo model i is ob ained, he hi d expe imen will be conside ed a
es o bounda y condi ions ou side o which ou model canno be applied.
I is wo h no ing ha ou ins uc ions o pa icipan s can be seen as con o e sial.
I a pa icipan is ins uc ed o beha e in a ce ain way, he esul ing beha io canno
be gene alized ou side o he labo a o y se ing, whe e he ins uc ions a e no imposed.
Howe e , gene alizing pa icipan s’ beha io o o he si ua ions is no he goal o his
expe imen . The a he a i icial si ua ion we c ea ed in he lab can p o ide insigh s in o
whe he pa icipan s can egula e hei beha io (i.e., base hei esponses on ai ness alone)
i hey a e ins uc ed o do so. Such an a i icial si ua ion would be unaccep able i he goal
o he expe imen was, o example, o es pa icipan s’ inna e disposi ion o comply wi h
ai ness. Howe e , as he expe imen is mean o s ess- es he model, he a i iciali y o
he expe imen al si ua ion is less p oblema ic.
4.1. P ocedu e
The p ocedu e o Expe imen 3 was iden ical o Expe imen 2, wi h one mino mod-
i ica ion. In Expe imen 3, pa icipan s we e asked o decide as a “ ai judge”. Tha is,
pa icipan s we e ins uc ed o a oid “ aking sides” and base hei judgmen only on hei
sense o wha is ai . The se ies o expe imen al ials, comp ehension checks, and inqui y
as o whe he he pa icipan s ecalled aking pa in a simila expe imen we e iden ical o
Expe imen s 2 and 1.
4.2. Pa icipan s
The equi ed sample size was de e mined simila ly o Expe imen 2. Again, da a
collec ion would ha e been in e up ed i i exceeded 27 days, o i pa icipan ec ui men
ell below 10 pa icipan s pe day.
A e i e days, he minimal h eshold o ec ui men equency was eached; howe e ,
as a ha ime he o al sample was composed o only N = 197, wi hou excluding hose
who ook pa in he p e ious expe imen s, we con inued da a collec ion while e aining
om analyzing he un eliable sample. Rec ui men equency emained low, p obably due
o he da a collec ion occu ing ou side he egula semes e . The e o e, on day 13, we sen
a eminde o he same mailing lis . The eminde wo ked, as on he day on which i was
Games 2025,16, 2 16 o 21
sen , we ec ui ed N = 142 new pa icipan s. Th ee days a e he eminde , ec ui men
equency d opped below he h eshold again, bu his ime, he o al sample size (wi hou
exclusion) was su icien ly la ge, N = 415.
A o al o N = 19 pa icipan s decla ed ha hey ook pa in a simila expe imen
and we e excluded. A o al o N = 353 pa icipan s comple ed he comp ehension checks
immedia ely, N = 36 a e one ins ance o eedback, and N = 7 a e addi ional eedback. No
pa icipan ook mo e han six a emp o comple e he comp ehension checks. The inal
sample was N = 396 pa icipan s (131 males, 256 emales, 8 di e se, and 1 undisclosed),
aged be ween 18 and 74 (M = 25.68, SD = 8.39).
4.3. Resul s
As in ou p e ious s udies, he analy ical s a egy s a ed wi h assessing he global i
o models encoding bo h he “s ic ” and “lenien ” ai ness concep ualiza ions. Howe e ,
in his case, bo h models equi ed one addi ional assump ion. Indeed, as he pa icipan is
ins uc ed o a oid “ aking sides”, p edic ions based on he ela i e gain maximiza ion a e
somewha misleading, as he p ocess subsumes asymme ical p e e ences. The e o e, o
i ing pu poses only, we assumed ha ela i e gain maximiza ion p esc ibes accep ance o
o e s a o ing he ecei e and ejec ion o hose a o ing he p opose . Wi h his in mind,
model i can be in es iga ed. Rega ding he i s model encoding he “s ic ” ai ness
concep ualiza ion, bo h i measu es poin ed owa d poo i , G
2
(3) = 418.89, p< 0.001,
w = 0.34. Simila ly, he model encoding he “lenien ” ai ness concep ualiza ion did no
i he da a adequa ely, G
2
(3) = 158.51, p< 0.001, w = 0.21. Gi en he lack o i o bo h
models, p oceeding o he es ima ion and in e p e a ion o indi idual pa ame e s as well
as he in eg a i e analysis compa ing Expe imen 3 wi h he p e ious wo is unwa an ed.
The in e es ed eade who wishes o explo e such analyses is di ec ed owa d he online
eposi o y con aining he aw da a and equi ed sc ip s.
4.4. Discussion
The esul s o Expe imen 3, al hough nega i e, a e s ill in o ma i e. Indeed, he
lack o model i can guide us o conclude ha , in he con ex o Thi d-Pa y Ul ima um
Games, he MPT model p oposed canno explain da a in which he decision-make is
o ally indi e en be ween a o ing he p opose and he ecei e . Mo e specula i ely, he
models’ mis i can be impu ed o he heo e ical sho coming o ela i e gain maximiza ion
p edic ions. He e, i is heo e ically impossible o decide a p io i which side o he o e ’s
con inuum is p e e ed; he e o e, he model’s p edic ed esponses canno be in o med by
ela i e gain maximiza ion.
In sum, he esul s o Expe imen 3 sugges ha ou MPT model, and i s unde lying
p ocesses, do no apply o si ua ions in which he decision-make has no s ake in he
game. The e o e, hese esul s can in o m us o he scope o ou model and i s bounda y
condi ions. Limi ing he scope o ou model based on empi ical e idence and heo e ical
cons ain s, howe e , is desi able as i makes ou heo y mo e p ecise.
5. Gene al Discussion
Since i s i s de elopmen , he Ul ima um Game un eiled he disc epancy be ween
he no ma i e beha io p esc ibed by s anda d economic heo y and ac ual beha io
documen ed by empi ical obse a ion. Many accoun s ha e been p oposed o explain such
a disc epancy, om he “wounded p ide/spi e model” (Pillu la & Mu nighan,1996) o
nega i e ecip oci y (Rabin,1993) and he no m o ai ness (Messick & Schell,1992;Messick,
1995). The deba e on which accoun is mos sui ed o such an explana o y pu pose is s ill
ongoing, and i gene ally di ides beha io d i e s in o wo ca ego ies, namely cogni i e
Games 2025,16, 2 17 o 21
heu is ics and emo ional eac ions (Ci ai,2013). In he p esen pape , we o e come such a
dicho omy, add essing he issue by elying on se e al p ocesses ha pa ially belong o
bo h ca ego ies. C ucially, he p esen in es iga ion is capable o quan i ying each p ocess’s
con ibu ion o he inal esponse p oduced. The mul inomial p ocessing ee models
in es iga ed combine he oles o ai ness, ela i e gain maximiza ion, and absolu e gain
maximiza ion and assign o each p ocess pa o he esponsibili y o he inal decision.
Based on h ee s udies, we ini ially es ed he models’ abili y o accoun o he
p ocesses a play, demons a ing ha a heo e ically meaning ul manipula ion (Thi d-Pa y
Ul ima um Game) leads o p edic able changes in he ela i e impo ance o each p ocess,
and p o ides bounda y condi ions delimi ing he scope o he models. Mo eo e , mo ing
mone a y u ili y away om he decision-make (i.e., Expe imen 2) success ully shi ed i s
conce ns owa d p ese ing a ai s a e o a ai s. Such a success ul manipula ion u he
alida es ou model.
5.1. Con ibu ions o he Bunda y Condi ions o he Exis ing Li e a u e
Se e al heo e ical accoun s p o ide eliable explana ions o he empi ical obse a-
ions on coope a i e and compe i i e games. Fo example, Feh and Schmid ’s wo k (Feh
& Schmid ,1999) p o ided a b oad pe spec i e on he de e minan s o decision-making.
They show ha coope a ion is main ained e en hough compe i i e beha io migh lead
o g ea e mone a y gain. Simila ly, ai ness is always pa o he equa ion. O he wo ks,
ocusing on he ole o ecip oci y, demons a e ha his is no a ma ginal cons uc (Bol on,
1991;Rabin,1993;Thale ,1988). Ou wo k ex ensi ely builds upon his li e a u e and
does no aim o c i icize hese heo ies. On he con a y, we embedded hese heo e ical
accoun s in o ou model. The key con ibu ions om ou wo k a emp o consolida e ou
suppo o he e iewed heo e ical accoun s and o p obe how a hey can go o explain
decision-making in he Ul ima um Game. Fo example, ou in es iga ion p o ides insigh
in o he heo ies’ bounda y condi ions. Fo example, he decision-make s’ indi e ence
causes heo e ical sho coming highligh ed by he impossibili y o he mode o each a
sa is ac o y i . I seems ha he scope o exis ing heo ies does no ex end o si ua ions in
which he decision-make is guided by ai ness alone. Howe e , his conclusion is based on
a nega i e inding and cau ion is wa an ed. I is ad isable ha u he esea ch explo es
o he po en ial bounda y condi ions highligh ing he scope o exis ing heo ies.
5.2. Fai ness (and O he P ocesses) in Compe i i e Games
Ano he con ibu ion o he p esen esea ch ela es o ai ness in he con ex o
compe i i e games. Speci ically, ou in es iga ion suppo s he no ion ha ai ness is no as
clea -cu as classical economic accoun s sugges . Indeed, ai ness bounda ies a e wide han
expec ed, as he model wi h he “lenien ” ai ness concep ualiza ion always ou pe o med
he one embedding a mo e “s ic ” concep o ai ness. The e o e, ou esul s a e in line
wi h a concep ualiza ion o ai ness ha is mo e liquid han wha pe ec ly dis ibu i e
jus ice would p esc ibe (Ha din,1968;Sch oede e al.,2003). Speci ically, we specula e ha
decision-make s a e willing o ole a e mild de iance om a pe ec ly e en o e (i.e., 5:5),
as he bounda ies o wha is “ ai ” include une en spli s. Mo eo e , ole a ion o such mild
de iance has also been documen ed in one-sho games (Suleiman,1996), sugges ing ha
such a ole ance is p esen e en wi hou he expec a ion o be compensa ed in he u u e.
Howe e , building expec a ions ha span ac oss se e al ( u u e) in e ac ions could be an
addi ional p ocess in epea ed games. As se e al Ul ima um Game ounds a e played,
a easonable decision-make migh accep an o e de ia ing sligh ly om an e en spli ,
expec ing a compensa o y o e in la e ounds. This explana ion, which is ad hoc o
epea ed games, allows o cons uing ai ness in a mo e in e ac i e way ha is sp ead o e
Games 2025,16, 2 18 o 21
ime (i.e., o e mul iple ounds). Such a heo iza ion esona es mo e wi h social no ms and
an implici o m o p ocedu al/ es o a i e jus ice (Sch oede e al.,2003). Howe e , u he
esea ch is equi ed o explici ly in es iga e his accoun .
5.3. Limi a ions and Fu u e Di ec ions
As wi h any esea ch, he p esen in es iga ion is no ee om limi a ions. Fo
example, one o such limi a ions conce ns he na u e o he ewa ds used. E en hough he
numbe o esou ces accumula ed by pa icipan s was indeed con e ed in o eal chances o
winning he a le, allowing pa icipan s o “compe e” o eal money migh ha e igge ed
al e na i e mo i a ions. Mo eo e , he o al amoun o money a s ake in ou expe imen is
qui e small. I emains unclea how he p ocesses we in es iga e migh un old when mo e
esou ces a e a s ake. Fu he esea ch migh in ol e g ea e amoun s o money di ec ly
p o ided o pa icipan s o es i ou esul s hold unde such condi ions. Addi ionally,
ou expe imen al si ua ion placed pa icipan s, bo h ecei e s and decision-make s, a a
“sa e” dis ance om he p opose s. Indeed, embodying p opose s wi h eal pa icipan s,
physically p esen in he same oom as es subjec s, migh ha e unexpec ed consequences.
Being in he p esence o a social o he ac i ely iola ing he no m o ai ness migh igge
s onge eac ance and consequen ly inc eased ejec ion. Simila ly, he same si ua ion migh
lead o inc eased accep ance due o pa icipan s’ in imida ion caused by bold p opose s ha
do no hesi a e o iola e ai ness i i se es hei own pu pose. Which o he wo opposing
eac ions is mo e likely o happen is an empi ical ques ion ha canno be add essed by
an online in e ac ion, which lacks he simul aneous p esence o bo h he p opose and he
ecei e /decision-make in he same physical space.
These limi a ions, howe e , a e qui e common in he e iewed li e a u e and all
ou side o he scope o he p esen esea ch. Add essing hese limi a ions is undoub edly a
iable a enue o ex end ou indings, which, howe e , al eady con ibu e o he exis ing
li e a u e on economic decision-making.
Au ho Con ibu ions: Concep ualiza ion, M.B., M.H. and L.O.; me hodology, M.B. and M.H.,
o mal analysis, M.B.; esou ces, M.B.; da a cu a ion, M.B.; w i ing—o iginal d a p epa a ion,
M.B.; w i ing— e iew and edi ing, M.H. and L.O., isualiza ion, M.B.; supe ision, M.H.; p ojec
adminis a ion, M.B. and L.O.; unding acquisi ion, M.B. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding: This esea ch was unded by he Eu opean Associa ion o Social Psychology ha awa ded a
Seedco n G an (2020) o Ma co Biella. The APC was unded by he Publica ion Fund o he Uni e si y
o Basel o Open Access.
Da a A ailabili y S a emen : Da a and ma e ials (R sc ip s o ep oduce he analysis) a e a ailable
on he OSF eposi o y (h ps://os .io/ zbpd/).
Acknowledgmen s: The au ho s hank Nihels Kukken o he inpu on he analysis.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Appendix A. T ansla ion o Pa icipan s’ Ins uc ions
Expe imen 1
You ask is o decide on he P opose ’s o e . I you accep he o e , you will bo h
be c edi ed wi h he amoun o money speci ied in he p oposal. I you ejec he o e ,
nei he o you will be c edi ed wi h any money in his ound. The P opose will only
ecei e eedback abou he money hey ecei ed a he end o he en i e game and no a e
each ound.
Games 2025,16, 2 19 o 21
You chances o winning he a le a he end o he s udy will be calcula ed based
on he amoun o money accumula ed. Fo e e y eu o you ecei e in he s udy, you will
ecei e one icke o he a le.
Expe imen 2
You ask is o decide on he P opose ’s o e on behal o he Recei e . I you accep
he o e , he P opose and he Recei e will bo h be c edi ed wi h he amoun o money
speci ied in he p oposal. I you ejec he o e , nei he he P opose no he Recei e will
be c edi ed wi h any money in his ound. The P opose will only ecei e eedback abou
he money hey ecei ed a he end o he en i e game and no a e each ound.
The Recei e ’s and he P opose ’s chances o winning he a le a he end o he
s udy will be calcula ed based on he amoun o money accumula ed. Fo e e y eu o he
Recei e s and he P opose ob ain in he s udy, hey will ecei e one icke o he a le.
Expe imen 3
You ask is o decide as a “ ai judge”. I you accep he o e , he P opose and he
Recei e will bo h be c edi ed wi h he amoun o money speci ied in he p oposal. I you
ejec he o e , nei he he P opose no he Recei e will be c edi ed wi h any money in
his ound. The P opose will only ecei e eedback abou he money hey ecei ed a he
end o he en i e game and no a e each ound.
The Recei e ’s and he P opose ’s chances o winning he a le a he end o he
s udy will be calcula ed based on he amoun o money accumula ed. Fo e e y eu o he
Recei e s and he P opose ob ain in he s udy, hey will ecei e one icke o he a le.
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