Role of commitment and emotions for long-term relations

Role of commitments and
emotions for long-term relations
a modelling perspective
Tom Lenaerts
MLG AI lab

© Tom Lenaerts, 2018 FutureICT2.0 Tallinn
coordination
cooperation
Human society

Simple interactions
Complex patterns

Global
local

Catastrophes
Errors
Attacks
Mistakes

Why don’t things fail more often?

Mechanisms that induce
Trust
Translation to ICT solutions
Inherent to every human
being

Good Agreements Make Good Friends
The Anh Han1,2
, Luı´s Moniz Pereira3
, Francisco C. Santos4,5
& Tom Lenaerts1,2
1
AI lab, Computer Science Department, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium, 2
MLG, De´partement
d’Informatique, Universite´ Libre de Bruxelles, Boulevard du Triomphe CP212, 1050 Brussels, Belgium, 3
Centro de Inteligeˆncia
Artificial (CENTRIA), Departamento de Informa´tica, Faculdade de Cieˆncias e Tecnologia, Universidade Nova de Lisboa, 2829-516
Caparica, Portugal, 4
INESC-ID and Instituto Superior Teńico, Universidade de Lisboa, IST-Taguspark, 2744-016 Porto Salvo,
Portugal, 5
ATP-group, CMAF, Instituto para a Investigaçaõ Interdisciplinar, P-1649-003 Lisboa Codex, Portugal.
When starting a new collaborative endeavor, it pays to establish upfront how strongly your partner commits
to the common goal and what compensation can be expected in case the collaboration is violated. Diverse
examples in biological and social contexts have demonstrated the pervasiveness of making prior agreements
on posterior compensations, suggesting that this behavior could have been shaped by natural selection.
Here, we analyze the evolutionary relevance of such a commitment strategy and relate it to the costly
punishment strategy, where no prior agreements are made. We show that when the cost of arranging a
commitment deal lies within certain limits, substantial levels of cooperation can be achieved. Moreover,
these levels are higher than that achieved by simple costly punishment, especially when one insists on
sharing the arrangement cost. Not only do we show that good agreements make good friends, agreements
based on shared costs result in even better outcomes.
C
onventional wisdom suggests that cooperative interactions have a bigger chance of surviving when all
participants are aware of the expectations and the possible consequences of their actions. All parties then
clearly know to what they commit and can refuse such a commitment whenever the offer is made. A
classical example of such an agreement is marriage1,2
. In that case mutual commitment ensures some stability in
the relationship, reducing the fear of exploitation and providing security against potential cataclysms. Clearly
such agreements can be beneficial in many situations, which are not limited to the type of formal and explicit
contracts as is the case for marriage. Commitments may even be arranged in a much more implicit manner as is
the case for members of the same religion3,4
, or by some elaborate signaling mechanism as is the case in primates’
use of signaling to synchronize expectations and the consequences of defaulting on commitment in their different
ventures5
.
Here we investigate analytically and numerically whether costly commitment strategies, in which players
propose, initiate and honor a deal, are viable strategies for the evolution of cooperative behavior, using the
symmetric, pairwise, and non-repeated Prisoner’s Dilemma (PD) game to model a social dilemma. Next to
the traditional cooperate (C) and defect (D) options, a player can propose its co-player to commit to cooperation
before playing the PD game, willing to pay a personal cost ð Þ to make it credible. If the co-player accepts the
arrangement and also plays C, they both receive their rewards for mutual cooperation. Yet if the co-player plays D,
then he or she will have to provide the proposer with a compensation at a personal cost (d). Finally, when the co-
player does not accept the deal, the game is not played and hence both obtain no payoff.
Although there is a kind of punishment associated with the agreement, the notion of compensating a partner
when not honoring a negotiated deal is not entirely equivalent to the general notion of punishment as has been
studied in Evolutionary Game Theory6–13
so far. In the current work, both parties are aware of the stakes before
they start the interaction: the person who accepts to commit knows upfront what to expect from the person that
proposes the commitment and what will happen if he or she does not act appropriately. Even more, the co-player
has the possibility not to accept such an agreement and continue interacting without any prior commitment with
the other players, and with no posterior repercussions from commitment proposers. In the current literature,
punishment (with or without cost) is imposed as a result of ‘‘bad’’ behavior, which can only be escaped by not
participating in the game at all9,12,14
. As such, the present work differs from peer and pool punishment9,12
in that
the latter imposes the commitment to the other players, i.e. defectors will always be punished even when they did
not want to play with punishers. In addition, there is no notion of compensation incorporated in the model that
remunerates the proposer when her accepted deal is violated, in contradistinction to our own model. Moreover,
because the creation of the agreement occurs explicitly in the current work, players can behave conditionally
(even without considering previous interactions, whether direct or indirect), and that is not considered in the
OPEN
SUBJECT AREAS:
BIOLOGICAL PHYSICS
BEHAVIOURAL METHODS
EVOLUTIONARY THEORY
SOCIAL EVOLUTION
Received
2 July 2013
Accepted
30 August 2013
Published
18 September 2013
Correspondence and
requests for materials
should be addressed to
T.A.H. (h.anh@ai.vub.
ac.be) or T.L. (Tom.
Lenaerts@ulb.ac.be)
SCIENTIFIC REPORTS | 3 : 2695 | DOI: 10.1038/srep02695 1
Our (cognitive) capacity for
commitment may have evolved
by natural selection
Emotions manage mutually
beneficial relationships

Commitment ?
“A commitment is an act or signal that gives up options in
order to inﬂuence someone’s behaviour by changing
incentives and expectations”
“They can be enforced by external incentives,
but also by some combination of reputation
and emotion”
“Commitments can be promises to help, or
threats to harm”
“Our (cognitive) capacity for commitment
may have evolved by natural selection”

prisoners’ dilemma
R = reward
S = suckers payoff
T = temptation to defect
P = punishment
fear = P >S
greed = T > R
R
D
C
C
D
S
PT
T > R > P > S
C.H. Coombs (1973) A reparameterization of the prisoner’s dilemma
game. Behavioral Science 18:424-428

prisoners’ dilemma
R
D
C
C
D
S
PT
T > R > P > S
C.H. Coombs (1973) A reparameterization of the prisoner’s dilemma
game. Behavioral Science 18:424-428
7
3
3 7
0
1
1
0
C
D
C D
7
1
1
7
Nash equilibrium

Q1
Q2
Propose
(at a cost ε) NOT propose
accept
NOT
accept
IN OUT
OUT
Adding commitment
Propose = promise to play C or
threat to not play at all if not playing C
Accept = agree to also play C
(following promise or threat)
COMMIT
C
D
C D
FREE
FREE

IN OUT
propose accept
NOT propose
Q3
C D C D
Q3 COMMIT
NOT
play
propose NOT accept
COMMIT
propose/
accept
C
C
accept
D
FREE
FREE
FAKE
pay compensation
(δ)
accept
FAKE

R-ε/2 R-ε 0 S+δ-ε R-ε
R R S S S
0 T P P P
T-δ T P P P
R T P P P
COMMIT
FAKE
FREE
COMMIT FAKE FREE
C
D
C D
T=b;
R=b-c;
P=0;
S=-c
Translation to Donation
game
b=beneﬁt of cooperation
c=cost of cooperation

b-c
-ε/2
b-c-ε 0 -c+δ-ε b-c-ε
b-c b-c -c -c -c
0 b 0 0 0
b-δ b 0 0 0
b-c b 0 0 0
COMMIT
FAKE
FREE
COMMIT FAKE FREE
C
D
C D
δ > c+3ε/4
ε < 2(b-c)/3
Cooperation will evolve
when …
T=b;
R=b-c;
P=0;
S=-c
Translation to Donation
game
b=beneﬁt of cooperation
c=cost of cooperation

Frequencyofstrategy
ε
δ=4
COMMIT
FREE
D
δ=4
ε=0.25
FAKE
Same conclusions hold for 2 or more players
β=0.1
F
P
F
P
A
DC
P
D
F
P

Commitments are often
long-term…
Iterated prisoners’ dilemma

With a probability ω the interaction continues
time
C C C C
C C C
P1
P2 C C C C
C C C …
…

How to deal with mistakes ? (intentional or not)
Should we collect the compensation or continue
the agreement?
Should one take revenge or apologise and
forgive?
With a probability ω the interaction continues
time
C D C C
C C C
P1
P2 D D C C
C C D …
…

Individuals prefer to
take revenge
time
C C C D
C C D
P1
P2 C C C C
D D D …
…
0.4
−0.2
0
0.2
0.4 a
t
)−Fr(NC,−,AllD)
NN
AllD
PN
AllD
TFT
abundancerelative
tothedefectors
0.0
0.2
0.4
-0.2
log(noise)
-3 -2 -1
D
P-C-D
P-C-TFT
pay compen-
sation δ
When apologising is
not possible

time
C C C C
C C D
P1
P2 C C C C
C C C …
…
Oeps!
sorry
really sorry
no
problem
apology
cost (γ)
0 2 4 6 8
−0.1
0
0.1
0.2
0.3
0.4
γ
Fr(S)−Fr(NC,−,AllD,−)
α=0.01
0 2 4 6 8
−0.1
0
0.1
0.2
0.3
0.4
γ
Fr(S)−Fr(NC,−,AllD,−)
α=0.1
(P,C,AllD,q=1) (P,C,AllD,q=0) (P,D,AllD,q=1) (A,D,AllD,q=1)
abundancerelative
tothedefectors
0.0
0.1
0.3
0.2
0.4
-0.1
0 2 4 6 8
D
P-C-D+A
P-C-D + NA
apology cost
P-D-D+A
A-D-D+A
These are like
the fake players
When apologising is
possible
Apology needs to
be sincere

To err is humane, to forgive …
Alexander Pope, An essay on Criticism (1711)
evolutionary viable when
the apology is sincere

Science=collaboration
The Anh Han (UK) Luis Moniz Pereira (P) Luis Martinez (I)

Produces equivalent results as in the mathematical
model
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
−3b/c=1.5, =0.01
0 2 4 6 8
0
2
4
6
8
b/c=2, =0.01
0 2 4 6 8
0
2
4
6
8
b/c=4, =0.01
0 2 4 6 8
0
2
4
6
8
b/c=1.5, =0.1
0 2 4 6 8
0
2
4
6
8
b/c=2, =0.1
0 2 4 6 8
0
2
4
6
8
b/c=4, =0.1
0 2 4 6 8
0
2
4
6
8
Agent-based modelling
An agent’s
genotype is
deﬁned by:
apology cost (γ)
forgiveness
threshold (τγ)
© Tom Lenaerts, 2017 AISB in Bath

S
T
R+1RR-1
P-1
P+1
P
7
3
3 7
0
1
1
0
C
D
C D T > R > P > S
7
3
3 7
1
0
0
1
C
D
C D T > R > S > P
3
7
7 3
0
1
1
0
C
D
C D
R > T > P > S
3
7
7 3
1
0
0
1
C
D
C D
R > T > S > P
Space of social dilemmas
© Tom Lenaerts, 2017 AISB in Bath

S
T
R+1RR-1
P-1
P+1
P
C D
C
D
S-P
R-S-T+P
C
D
An evolutionary game theory
perspective ….
REPLICATOR DYNAMICS
C D

Finite population dynamics
D
D
C
D
CD
DC
DD
t µ
CD
O
1-µ
C
C
p=[1+eβ(f(D)-f(C))]-1
D
C
D
CD
DC
DD
t+1
D
A moran process (birth-death process)
CC
selection strength
© Tom Lenaerts, 2016

Assuming that mutations are rare, we have either …
n(C)=Z
C
C
C
C
C
CC
CC
C
C
C
n(C)=0
D
D
D
D
DD
DD
D
D
D
D
n(C)=1
C
D
D
D
D
DD
DD
D
D
D
...
ﬁxation probability
ρ of C in D
population
n(C)=Z-1
C
C
C
C
C
CD
CC
C
C
C
ﬁxation probability
ρ of D in C
population
Finite population dynamics

Which produces a reduced Markov chain
For which the stationary distributions can
be calculated = how likely it is to end up in either
monomorphic state
DC
Fudenberg, D., & Imhof, L.A. (2006). Imitation processes with small mutations. J. Econ.Theo, 131,251–262.
Imhof, L.A., Fudenberg, D., & Nowak, M.A. (2005). Evolutionary cycles of cooperation and defection. Proc
Nat Acad Sci USA, 102(31), 10797–10800.
ρCD
ρDC
Dynamics in ﬁnite populations
x% y%=1-x

Dynamics in ﬁnite populations
4
3
3 4
0
1
1
0
C
DC
D
DC
1.6ρN
0.6ρN
26.5%
73.5%
with β=0.01
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Selection strength (β)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stationarydistribution
C
D
for varying β
ρN=drift=1/N
How can we
achieve
cooperation?

Rules for the evolution of
cooperation
Nowak, M.A. Five Rules for the Evolution of Cooperation. Science 314, 1560–1563 (2006).
provided that
b/c > 1 + (n/m) (5)
Evolutionary Success
Before proceeding to a comparative analysis of
the five mechanisms, let me introduce some
The entries denote the payoff for the row
player. Without any mechanism for the evolution
of cooperation, defectors dominate cooperators,
which means a < g and b < d. A mechanism for
the evolution of cooperation can change these
inequalities.
1) If a > g, then cooperation is an evo-
lutionarily stable strategy (ESS). An infinitely
large population of cooperators cannot be in-
vaded by defectors under deterministic selec-
tion dynamics (32).
2) If a + b > g + d, then cooperators are
risk-dominant (RD). If both strategies are
ESS, then the risk-dominant strategy has the
bigger basin of attraction.
3) If a + 2b > g + 2d, then cooperators are
advantageous (AD). This concept is important
for stochastic game dynamics in finite pop-
ulations. Here, the crucial quantity is the fix-
ation probability of a strategy, defined as the
probability that the lineage arising from a
single mutant of that strategy will take over
the entire population consisting of the other
strategy. An AD strategy has a fixation proba-
bility greater than the inverse of the popu-
lation size, 1/N. The condition can also be
expressed as a 1/3 rule: If the fitness of the in-
vading strategy at a frequency of 1/3 is greater
than the fitness of the resident, then the fix-
ation probability of the invader is greater than
1/N. This condition holds in the limit of weak
selection (52).
We have encountered
evolution of cooperati
mathematical formali
mechanisms are very
each theory is a simp
coherent mathematica
the derivation of all fi
is that each mechanis
game between two st
payoff matrix (Table
can derive the relevan
of cooperation.
For kin selection
inclusive fitness prop
(31). The relatedness
Therefore, your payof
to mine. A second me
to a different matrix
direct reciprocity, the
while the defectors
expected number of r
tit-for-tat players coop
tat versus always-defe
first move and then
iprocity, the probabili
reputation is given b
unless the reputation
dicates a defector. A
network reciprocity, i
expected frequency o
by a standard replicat
formed payoff matrix
the payoff matrices o
Kin selection
Network reciprocity
Direct reciprocity
Indirect reciprocity
1 r
the five mechanisms, let me introduce some inequalities.
1) If a > g, then coope
lutionarily stable strategy (E
large population of cooperat
vaded by defectors under de
tion dynamics (32).
2) If a + b > g + d, the
risk-dominant (RD). If bo
ESS, then the risk-dominant
3) If a + 2b > g + 2d, the
advantageous (AD). This con
for stochastic game dynami
ulations. Here, the crucial q
ation probability of a strateg
probability that the lineage
single mutant of that strateg
the entire population consis
strategy. An AD strategy has
bility greater than the inve
lation size, 1/N. The condi
expressed as a 1/3 rule: If the
vading strategy at a frequenc
than the fitness of the resid
ation probability of the invad
1/N. This condition holds in
selection (52).
Kin selection
Network reciprocity
Direct reciprocity
1 r
Table 1. Each mechanism ca
interaction between cooperato
essary conditions for evolution
Before proceeding to a comparative analysis of
the five mechanisms, let me introduce some
the evolution of cooperation ca
inequalities.
1) If a > g, then cooperat
lutionarily stable strategy (ESS
large population of cooperators
vaded by defectors under deter
tion dynamics (32).
2) If a + b > g + d, then c
risk-dominant (RD). If both
ESS, then the risk-dominant s
3) If a + 2b > g + 2d, then
advantageous (AD). This conce
for stochastic game dynamics
ulations. Here, the crucial quan
ation probability of a strategy,
probability that the lineage a
single mutant of that strategy
the entire population consistin
strategy. An AD strategy has a
bility greater than the inverse
lation size, 1/N. The conditio
expressed as a 1/3 rule: If the fi
vading strategy at a frequency o
than the fitness of the residen
ation probability of the invader
1/N. This condition holds in th
selection (52).
Kin selection
Network reciprocity
Direct reciprocity
1 r
Table 1. Each mechanism can
interaction between cooperators
Network reciprocity
Group selection
Cooperators Defectors
Fig. 3. Five mechanisms for the evolution of
cooperation. Kin selection operates when the
probability that the lineage arising from a
single mutant of that strategy will take over
the entire population consisting of the other
strategy. An AD strategy has a fixation proba-
bility greater than the inverse of the popu-
lation size, 1/N. The condition can also be
expressed as a 1/3 rule: If the fitness of the in-
vading strategy at a frequency of 1/3 is greater
than the fitness of the resident, then the fix-
ation probability of the invader is greater than
1/N. This condition holds in the limit of weak
selection (52).
tit-for-tat players cooperate
tat versus always-defect co
first move and then defec
iprocity, the probability of
reputation is given by q.
unless the reputation of
dicates a defector. A defe
network reciprocity, it can
expected frequency of coo
by a standard replicator e
formed payoff matrix (54)
the payoff matrices of the
Network reciprocity
Group selection
Cooperators Defectors
Fig. 3. Five mechanisms for the evolution of
cooperation. Kin selection operates when the
donor and the recipient of an altruistic act are
genetic relatives. Direct reciprocity requires re-
peated encounters between the same two individ-
uals. Indirect reciprocity is based on reputation; a
helpful individual is more likely to receive help.
Network reciprocity means that clusters of coop-
erators outcompete defectors. Group selection is
Table 1. Each mechanism can be described by a simple 2 × 2 payoff matri
interaction between cooperators and defectors. From these matrices we can di
essary conditions for evolution of cooperation. The parameters c and b denote,
for the donor and the benefit for the recipient. For network reciprocity, we us
[(b − c)k − 2c]/[(k + 1)(k − 2)]. All conditions can be expressed as the
exceeding a critical value. See (53) for further explanations of the underlyin
Cooperation is…
0
)1)((
rcbD
cbrrcbC
0
)1/()(
bD
cwcbC
0)1(
)1(
qbD
qccbC
0HbD
cHcbC
)())(( cnmcbnmcbC
DC
Kin
selection
Direct
reciprocity
Indirect
reciprocity
Network
reciprocity
Group
rc
b 1
rc
b 1
rc
b 1
r…
w…
q…
k…
n…
wc
b 1
w
w
c
b 2
w
w
c
b 23
qc
b 1
q
q
c
b 2
q
q
c
b 23
k
c
b
k
c
b
k
c
b
nb
1
nb
1
ESS RD AD
nb
1
Payoff matrix
Assortment

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