# Evolution of commitment in the spatial Public Goods Game
through institutional incentives

Lucas S. Flores  Instituto de Física, Universidade Federal do Rio Grande do
Sul, CP 15051, CEP 91501-970 Porto Alegre - RS, Brazil  The Anh Han  School of
Computing, Engineering and Digital Technologies, United Kingdom

###### Abstract

Studying social dilemmas prompts the question of how cooperation can emerge in
situations where individuals are expected to act selfishly. Here, in the
framework of the one-shot Public Goods Game (PGG), we introduce the concept
that individuals can potentially adjust their behaviour based on the
cooperative commitments made by other players in the group prior to the actual
PGG interaction. To this end, we establish a commitment threshold that group
members must meet for a commitment to be formed. We explore the effects of
punishing commitment non-compliant players (those who commit and defect if the
commitment is formed) and rewarding commitment-compliant players (those who
commit and cooperate if the commitment is formed). In the presence of
commitment and absence of an incentive mechanism, we observe that conditional
behaviour based on commitment alone can enhance cooperation, especially when
considering a specific commitment threshold value. In the presence of
punishment, our findings suggest that the survival of cooperation most likely
happen at intermediate commitment thresholds. Notably, cooperation is
maximised at high thresholds, when punishment occurs more frequently. We also
see that when cooperation rarely survives, a cyclic behaviour emerges,
facilitating the persistence of cooperation. For the reward case, we found
that cooperation is highly frequent regardless of the commitment threshold
adopted.

##  I Introduction

Prior to embarking on a collective project, individuals involved may solicit
commitment from group members and estimate how interested they are in
contributing to the group’s efforts. This assessment helps them determine
whether it is worthwhile to initiate the endeavour and/or if it would be
beneficial to join. Commitment mechanisms for enhancing cooperation are
widespread in nature, which exist in various forms and contexts, including
legal contracts and pledges [32], marriage [47], deposit-refund schemes [10],
emotion-based [19] or reputation-based commitment [18, 30]. Both empirical and
theoretical studies demonstrated that high levels of cooperation can be
achieved through reliable commitments [34, 12, 52, 23, 9]. They enable
individuals to reach mutual cooperation even when there is little knowledge
about others’ past behaviours [22, 24, 42, 33], as it requires them to reveal
their preferences or intentions [25, 50, 44].

However, prior models of commitment have mainly focused on well-mixed
population settings, potentially overlooking the significant influence of
commitment dynamics within a population’s actual network structure. This
structure dictates who may form commitments with whom [48, 4], ultimately
determining the global worthiness of arranging conditional commitments.
Network reciprocity plays a crucial role in shaping human social interactions,
fostering the creation of cooperative clusters and thereby influencing
cooperation dynamics [48, 36, 49, 4]. Here, our study reveals that this
critical aspect also significantly influences how commitments contribute to
the emergence of cooperation.

Moreover, the initiation of many collective projects hinges on the majority of
participants making a commitment to contribute towards a common good. For
instance, for a cooperative hunting endeavor to take place, it typically
requires a sufficient number of participants ready and willing to participate
[1, 45]. While some international agreements require ratification by all
parties before entering into force, most (especially global treaties) require
a minimum of less than the total number of negotiating countries [5, 10]. In
general, it appears that the necessary level of commitment is contingent on
the specific nature of the problem at hand. However, this issue has been
under-explored in theoretical modeling, especially in the context of spatial
group-interaction settings.

As motivated, herein we investigate the potential of a conditionally applied
commitment strategy, contingent on the required commitment level from the
group, to promote the evolution of cooperation in a structured population. Our
analysis is carried out in the context of the one-shot Public Goods Game (PGG)
[29, 26, 35, 43, 14]. In this game, a group of $G$ players have the choice to
invest (cooperate, paying a cost $c$) or not (defect, paying nothing) in a
common pool of the group. All their contributions are then multiplied by a
factor $r\ (1<r<G$) and in the end the result is divided equally among all
players, regardless of their initial choice. Before engaging in a PGG game,
players can choose whether or not to join a commitment and cooperate in the
game. The commitment is formed if a threshold $\tau$ ($0\leq\tau\leq G$)
regarding the number of committed players is met. Subsequently, players make
decisions in the game based on whether the commitment is formed or not.

In addition, for understanding optimal incentive mechanisms that enable
commitment compliance [22], we assess the comparative effectiveness of
institutional reward and punishment in promoting cooperation, given a
commitment formation threshold. Our findings indicate that both punishment and
reward mechanisms can positively impact cooperation. Furthermore, even in the
absence of these mechanisms, the potential for a cooperator to switch to
defection can still contribute to cooperation, especially with a specific
commitment threshold. Notably, in the presence of punishment, we observe that
high punishment and high commitment thresholds are most effective. However,
intermediate thresholds can sustain cooperation for difficult PGGs (i.e. those
with small values of $r$). In the case of reward, high levels of cooperation
can be achieved regardless of the commitment threshold.

##  II Model

Players interact following the one-shot PGG of size $G$. First, each player
decides to commit or not to cooperate. If the total number of committed
players exceeds a given threshold, denoted by $\tau$, the commitment is
formed. In addition, players can choose to cooperate or not if the commitment
is formed, contributing to the common pool of the group. We establish the
possibility that players who committed but then defect may have to pay a fine
and those who committed and cooperated may receive a reward. If the commitment
is not formed (when the commitment threshold is not reached), players can
decide to change their choice in the PGG (compared to what they would have
played if the commitment was formed).

We denote the steps above as follows: $i$ the decision to accept ($i=A$) or
not ($i=N$) to commit; $j$ the decision to cooperate ($j=C$) or not ($j=D$) if
the commitment is formed; and $k$ the decision to cooperate ($k=C$) or not
($k=D$) if the commitment is not formed. For example, the $NCD$ strategy does
not commit ($i=N$), cooperates if the commitment is formed ($j=C$), and
defects if it is not formed ($k=D$). In total, there are eight possible
strategies, see Table 1.

Strategies |  Accept commitment? |  Cooperate in presence of commitment? |  Cooperate in absence of commitment?
---|---|---|---
ACC |  Yes |  Yes |  Yes
ACD |  Yes |  Yes |  No
ADC |  Yes |  No |  Yes
ADD |  Yes |  No |  No
NCC |  No |  Yes |  Yes
NCD |  No |  Yes |  No
NDC |  No |  No |  Yes
NDD |  No |  No |  No
Table 1: The eight strategies with commitment formation.

For one group $X$, an individual with strategy $ijk$ has the following payoff

$\Pi_{ijk}=\frac{r}{G}\,\sum_{x\in X}c_{x}-c_{ijk}+incentive,$ (1)

where $c_{x}$ is the contribution from group member $x$ (including the focal
player $ijk$), $c_{ijk}$ is the contribution from the focal player, and the
incentive is a reward gained or a punishment incurred by the focal player
(explained below, see also Table 2). A player contributes to the common pool,
i.e. $c_{ijk}=c$, if they cooperate when the commitment is formed ($j=C$) and
the commitment was actually formed, or if they cooperate when the commitment
is not formed ($k=C$) and the commitment was actually not formed. Otherwise,
the player does not contribute. In that case, $c_{ijk}=0$. Without loss of
generality, we set $c=1$ in our analysis.

The total budget for providing incentives, i.e., punishment against commitment
non-compliant players (i.e., $ADC$ and $ADD$) or reward for commitment-
compliant players (i.e., $ACC$ and $ACD$) is given by $G\delta$ per group,
where $\delta$ is the per capita incentive. This budget $G\delta$ is then
divided into two parts based on a relative weight $\omega$ with
$0\leq\omega\leq 1$. The reward part $\omega G\delta$ is equally shared among
$n_{com}$ committed compliant players, who received a reward $\omega
G\delta/n_{com}$. The punishment part $(1-\omega)G\delta$ is equally impacted
by $n_{non-com}$ committed non-compliant players, who incur a punishment
$(1-\omega)G\delta/n_{non-com}$. We summarise all incentives in Table 2

Strategies |  incentives
---|---
ACC |  $\omega G\delta/n_{com}$
ACD |  $\omega G\delta/n_{com}$
ADC |  $-(1-\omega)G\delta/n_{non-com}$
ADD |  $-(1-\omega)G\delta/n_{non-com}$
NCC |  0
NCD |  0
NDC |  0
NDD |  0
Table 2: The eight strategies with commitment formation and their incentives
if the commitment is formed. If the commitment is not formed, no incentives is
provided to any strategy.

A player’s accumulated payoff is the sum of payoffs it obtained from the
interactions in all groups it participates in (in case of a square lattice, it
receives payoffs from five groups). Note that for $\tau=\delta=0$ we have the
classical PGG.

In an evolutionary step, first a random player ($ijk$) is selected from the
population. Its payoff is calculated according to Equation 1. Then, a random
neighbor ($i^{\prime}j^{\prime}k^{\prime}$) of ($ijk$) is selected, and we
repeat the same calculation for its payoff. Player ($ijk$) will adopt
($i^{\prime}j^{\prime}k^{\prime}$) strategy according to a probability given
by the Fermi update rule,

$W_{ijk\rightarrow
i^{\prime}j^{\prime}k^{\prime}}=\frac{1}{1+e^{-(\Pi_{i^{\prime}j^{\prime}k^{\prime}}-\Pi_{ijk})/K}}\,\,,$
(2)

where $K$ is a noise related to irrationality. It is important to notice that
commitment coevolves with cooperation and defection.

This evolutionary step is repeated $N$ times where $N$ is the population size,
characterising one Monte Carlo step (MCS). We set the total simulation time as
$t=10^{5}$ MCS which is enough for the equilibrium to be reached. In line with
previous works [48, 39, 36, 37], we perform our simulations on a square
lattice with von Neumann neighborhood, with size $N=100^{2}$ and $K=0.1$. We
initialise the population with an equal number of each strategy. Since each
player can cooperate or not according to their neighborhood, we define the
cooperation frequency as the number of times that an interacting player
cooperates per group.

##  III Results

As the baseline, we begin by exploring the effect of commitment on the
evolution of cooperation in the spatial PGG in the absence of any incentive
mechanism (Section A). We then study the impact of punishment of commitment
non-compliant behaviour (Section B) and reward of commitment-compliant
behaviour (Section C).

###  III.1 Commitment without incentives ($\delta=0$)

We show in Fig. 1 the cooperative density for varying $r$ for all threshold
values. For $\tau=0$, we reproduce the the classical PGG (i.e., without
commitment), where every player has a fixed choice in the PGG since the
commitment is always formed. For all other thresholds, except $\tau=3$, we
observe a similar cooperation outcome to the classical case. An important
remark is that spatial reciprocity plays a crucial role in games in structured
populations. In such games, cooperators can form clusters to survive, by
avoiding defecting neighbours. We notice that only some commitment-based
strategies (Table 1) have spatial reciprocity, meaning that when they cluster
together they will cooperate with each other. They are $NDC$, $NCC$, $ACC$,
and $ACD$. Nevertheless, if the commitment threshold is low and thus easily
formed, $NDC$ players would have an advantage over the committing strategies
above, since they can cluster and cooperate with each other while at the same
time defecting against them. Moreover, they are not exploitable by defectors
that commit, because they would defect in such players’ presence. Note that,
unlike $NDC$, $NCC$ possesses neither of those two advantages. Thus, defectors
that do not commit have a benefit and cooperators that commit or have
unconditional strategies are exploitable. On the other hand, if the commitment
threshold is high, it would be harder to be formed, and therefore $ACD$ would
have spatial reciprocity at the same time exploiting the other strategies that
also have it. They are not exploited by defectors who do not commit since they
would defect in their presence. Thus, defectors that commit have a benefit and
cooperators that do not commit or have unconditional strategies are
exploitable.

In summary, low thresholds select for the non-committing strategies ($i=N$),
while high thresholds for the committing ones ($i=A$). Thus, strategies will
always cooperate or defect unconditionally, resulting in the classical PGG.

Interestingly, for $\tau=3$, we have a different scenario from the classical
game, even without punishment or reward. We observe that non-committing
strategies ($NDD$ and $NDC$) coexist with committing ones ($ADD$ and $ACD$).
The intermediate threshold $\tau=3$ is high enough for $ACD$ to exploit $NDC$
(capable of invading a cluster without reaching the commitment thresholds) and
low enough for the $NDC$ to exploit the $ACD$ (able to invade a cluster and
still meet the commitment). In the end, we see that their densities always
converge to the same value when interacting alone, independently of $r$.
Therefore, we end up with committing and non-committing strategies being able
to coexist. If we set only two strategies at a time in the population, we
observe that $ACD$ performs better against $NDD$ than $NDC$. The same is true
for $ADD$, where $NDC$ performs better than $ACD$. This can be understood by
the fact that, when we have an interaction between $NDD$ and $NDC$, we have
the classical scenario, where they never change strategy. When $NDD$ interacts
with $ACD$, a cooperator can sometimes defect and avoid exploitation.
Therefore, acting conditionally to avoid exploitation is better than
unconditionally cooperating, for low $r$ values. For high $r$ values,
unconditional cooperation becomes viable and therefore can outperform
defective strategies.

We illustrate the observations above with snapshots of the population when the
equilibrium is reached, see Fig. 2. We observe that for low threshold values
($\tau<3$), only non-committers survive by clustering ($NDC$ and $NCC$) in a
sea of $NDD$ and $NCD$ players, whereas for high threshold values ($\tau>3$)
only committers survive by clustering ($ACD$ and $ACC$) in a sea of $ADC$ and
$ADD$ players. For $\tau=3$ there are no longer clearly formed clusters as in
the previous cases. This is because $ACD$ and $NDC$ can invade one another.

Figure 1: Frequency of cooperation as a function of the public goods
multiplication factor, $r$. For all $\tau\neq 3$, a similar outcome is
observed to that of the classical PGG game ($\tau=0$). For $\tau<3$, only non-
committers survive whereas for $\tau>3$, only committers do. For $\tau=3$, a
coexistence of committers and non committers, allowing conditional strategies
($ACD$ and $NDC$) to gain an advantage over unconditional defectors ($NDD$ and
$ADD$).

Figure 2:  Snapshots of the population at equilibrium for $r=4$ and different
threshold values: $\tau=2$ (left), $3$ (middle) and $4$ (right). For $\tau=2$,
only non-committers are present, where $NDC$ players (blue) cluster around
$NDD$ (red). For $\tau=4$ we see that only committers survive, where $ACD$
players (yellow) cluster in a sea of $ADD$ (black). For $\tau=3$ there are not
anymore clear clusters and we see that $ACD$ and $NDC$ keep invading one
another in the presence of defectors ($NDD$ and $ADD)$.

###  III.2 Punishment ($\omega=0$, for varying $\delta$)

Here we explore the effect of punishment, where players who committed to
cooperate yet defect after the formation of a commitment are punished and thus
have a decrease in their payoff. Indeed, Fig. 3 shows the frequency of
cooperation for different thresholds, in the presence of punishment. We
observe that, for a sufficiently high threshold, namely, $\tau\geq 2$,
punishment leads to improved cooperation compared to the classical scenario
(note that for $\tau=0$ the model is equivalent to the classical PGG). For
$\tau=1$, there is no improvement compared to the classical scenario. This is
because in this case a commitment is never formed, therefore no punishment is
applied. This results in the same classical scenario since cooperators and
defectors who do not commit are possible strategies. We observe significant
improvements only for high thresholds ($\tau=4$ and $5$), commitments being
formed and therefore the punishment being applied.

We also observe that for all threshold values eventually increasing punishment
strength stops to affect cooperation. The lower the threshold, the sooner this
behaviour occurs, resulting in cooperation enhancement in high thresholds and
low $r$ values for stronger punishments. Therefore, high threshold values are
most conducive to the emergence and dominance of cooperation. Interestingly,
cooperation is not dominant in the population for $\tau=3$ in our parameters
range. This is due to the coexistence between $ACD$ and $NDC$ players. For the
lowest $r$ values where cooperation survives, we observe that only $ACD$
players coexist with $NDD$ ones. Increasing $r$ benefits cooperative
strategies and therefore, increases cooperative density. But at the same time,
the increase of $r$ allows $NDC$ players to survive in the population. Their
presence is detrimental for the overall cooperative density since they are
less effective against non-committing defectors, as discussed in the previous
section. Despite that, intermediate thresholds can sustain cooperation for
smaller $r$ values if the punishment is low enough. We illustrate this in Fig.
4, where we plot the critical values of $r$ for the survival (for low
punishment) of cooperation, as a function of the threshold $\tau$. In general,
cooperation is enhanced in the sense of survival (minimal $r$ that cooperation
survives) for intermediate thresholds. Moreover, cooperation is dominant
(minimal $r$ that cooperation dominates) when the threshold is high and
punishment is strong.

Now, we examine which strategies contribute to the presence of cooperation in
Fig. 3. Fig. 5 shows the frequency of each behaviour as a function of the
threshold for $r=3.6$ and $\delta=0.19$. The same trend is observed from the
one-shot PGG when changing thresholds. For low thresholds, only non-committers
survive, while for high thresholds, only committers do. But now, this is a
more continuous transition since we observe the coexistence of committers and
non-committers for more thresholds, such as 2, 3, and 4 (depending on the $r$
value). It can also be seen that for all thresholds, the cooperative density
matches the density of those strategies that would change their choice in the
PGG if the commitment is not formed. The case $\tau=0$ is an exception because
the commitment is always formed.

Recalling that for the case of commitment without incentives (Section A), we
observed that high thresholds selected for defectors that commit and low
thresholds for defectors that do not commit. As such, a commitment is likely
being formed for high thresholds, thus, defectors are always be punished. This
is why high punishments under high thresholds are highly effective (for
promoting cooperation). For lower thresholds, a commitment is less likely to
be formed, and thus defectors are punished less often. Therefore even high
punishments are not highly effective. Another interesting observation is that
for $\tau=5$, cooperation can even survive for very low values of $r$
(corresponding to PGG games where it is very hard for cooperation to survive,
e.g. $r=1.1$), due to a cyclic behaviour that will be explored in the next
section.

Figure 3:  Phase diagram $r\times\delta$ for the density of cooperative
behaviour for all threshold values, in the presence of punishment. We observe
that for $\tau\geq 2$, cooperation benefits from the presence of punishment,
in the sense of a reduced critical $r$ for cooperation to prevail. The effect
of stronger punishments stagnate slower for high thresholds, allowing
cooperation to survive for lower $r$ values. Despite that, intermediate
thresholds (i.e. $\tau=3,\ 4$) can sustain cooperation for the same $r$ value
with a smaller punishment value. Another interesting observation is that for
$\tau=5$ cooperation can even survive for very hard PGG (i.e. small $r$,
$1<r\leq 1.1$), due to a cyclic behaviour. Figure 4: We show the critical
value of $r$ that sustains cooperation, as a function of the threshold $\tau$,
for different costs of punishment, $\delta$. We observe that weaker
punishments are more effective for intermediate thresholds to sustain
cooperation.  Figure 5:  Density of strategies for varying the threshold, for
$r=3.6$ and $\delta=0.19$. Increasing threshold selects for $ACD$ strategy. We
observe that for low thresholds cooperation evolves due to the conditional
non-committers, while for high thresholds, it evolves due to the conditional
committers. Note that the density of cooperation is slightly smaller than the
density of committers because if committers and non-committers coexist the
former can sometimes defect.

#### Cyclic behaviour

For very low values of $r$ and $\delta$, we observe that a cyclic behaviour is
possible, where $ADD>ACD>NDD>ADD$ (here $X>Y$ or $Y<X$ means X invades Y). One
interesting observation is that all strategies involved defect if the
commitment is not formed. This reinforces the idea that cooperating when a
commitment is not formed is disadvantageous since it indicates the players’
intention to cooperate is unlikely. Each step of the cycle $ADD>ACD>NDD>ADD$
can be explained as follows. For low enough $r$, $ADD$ invades $ACD$ while for
high values of $r$, the opposite occurs. It is because the commitment is
always formed and thus the former defects in the interactions. Next, $ACD$
players invade $NDD$ ones. Both defect if the commitment is not being formed.
Despite that, $ACD$ has spatial reciprocity while $NDD$ doesn’t. Thus, even if
they defect when interacting with each other the $ACD$ players have higher
payoffs as they contribute in the groups where there are only $ACD$ players.
Now, $NDD$ players invade $ADD$ ones, due to the fact that the latter are
always being punished when a commitment is formed, while the former are not.

The cycle also occurs when replacing $NDD$ with $NCD$, where the same
explanation applies. Since the cycle has a $ADD$ strategy, for a strong enough
punishment, they become extinct, and only $ACD$ survive, breaking the cycle.
However, if the punishment is sufficiently low, the invasion $NDD>ADD$ is
slow, which is detrimental for $ACD$ players’ success and results in the
dominance of defection. The cycles can happen for a variety of $r$ values but
with a negligible frequency of cooperation. The most relevant regions are for
extremely low $r$ values if $\tau=5$ and for small punishments for $\tau=3$
and $4$ (below the plateau).

###  III.3  Reward ($\omega=1$, for varying $\delta$)

We now consider the situation where commitment-compliant players, i.e. those
who commit to cooperate and actually cooperate when the commitment is formed,
are rewarded. The rewarded players gain an increase in their payoff according
to Table 2. In Fig. 6, we show the phase diagrams $r\times\delta$ of
cooperative intensity for varying the threshold $\tau$. We observe that the
effect of reward is almost invariant among the threshold values. Recall that
in the absence of incentives (Section A), low thresholds selected for non-
committers due to $NDC$ clustering and exploiting the other clustering
strategies (namely, $NCC$, $ACC$, and $ACD$). Now, in the presence of reward,
$ACD$ and $ACC$ are rewarded for committing and cooperating, thus even when
being exploited by $NDC$ strategy, they can prevail.

We show, in Fig. 7 (a), the density of cooperative committing ($ACC+ACD$) and
non-committing ($NCC+NDC$) strategies for varying $r$ (fixing $\tau=2$ and
$\delta=0.1$). An interesting dynamics takes place now, where for $r<4$ only
committers survive, including defectors; while for $r>4$, the population
transitions into the dominance of non-committing strategies only. This happens
due to the fact that for low thresholds, $NDC$ can exploit and invade $ACD$.
But if $r$ is low enough, non-committing cooperators cannot survive in the sea
of defectors, since we would recover the classic PGG. Therefore, for low
incentives there occurs a similar phase diagram to the punishment case for
$\tau=1$ and $2$, while increasing the reward results in a new region of
survival of cooperation due to committers. This transition between committing
and not committing ends up harming cooperation momentarily since cooperators
were being rewarded but it ended for high $r$.

Now, in Fig. 7 (b), we show the cooperative densities as function of $\delta$
(fixing $r=4.1$ and $\tau=2$). We observe that for low rewards we cannot have
coexistence between committers and non-committers, as shown in (a). But for
sufficiently high reward values the committing cooperators are selected for
and outperform the non-committing ones while they coexist. A sufficiently high
reward benefits committers and therefore cooperation for low $\tau$. For
higher thresholds, committers are selected for even without a reward, as
previously explained.

Figure 6: Phase diagram $r\times\delta$ for the density of cooperative
behaviour for all thresholds, in the presence of reward. We observe that for
all thresholds, the critical values of $r$ for the survival of cooperation are
similar. Despite that, some thresholds differ for high $r$ values.

Figure 7:  Frequency of cooperation, committers ($ACC+ACD$) and non-committers
($NCC+NDC$) for $\tau=2$, as functions of: $r$ (Left panel, a), for
$\delta=0.1$, and $\delta$ (Right panel, b), for $r=4.1$. In the former,
similar to the behaviour discussed in the punishment case for $\tau=3$, we
observe that for a high enough $r$ value non-committers can survive, thereby
jeopardizing cooperation. For the latter (right panel), we observe that for a
sufficiently high reward value, committers can be favored where non-committers
would have an advantage.

Overall, given the analyses above, we arrive, first of all, at the conclusion
that reward is preferred to punishment for promoting cooperation when the
commitment threshold is low. When it is high, punishment and reward have a
similar effect on the cooperation outcomes. Nevertheless, there exists clear
difference between punishment and reward, which is most evident for $\tau=4$,
where punishment ensures a higher advantage to cooperation. Another notable
finding from our analyses is that conditional strategies were essential for
the maintenance of cooperation under punishment. This is due to the fact that
punishment does not necessarily mean a benefit to cooperators since non-
committing defectors could still exist. For the reward case, unconditional
cooperators become more viable if the reward is high enough. Finally, another
noticeable finding is that $ADC$ and $NCD$ strategies are the least favorable
in general. They can be deemed irrational or contradictory since there is no
clear benefit to commit and only cooperate if the commitment is not met.
Despite that, $NCD$ could still survive in the punishment case due to possible
patterns of cyclic dominance.

##  IV Discussion

In this paper, we have explored evolutionary dynamics in the spatial PGG with
the introduction of commitment. In this setting, behavioural strategies can be
conditional, choosing cooperation or defection based on the level of
commitment from the group members. Interestingly, with only this assumption,
we found an increase in cooperation for a specific commitment threshold. This
happens because cooperators that can change to defection in some situations
outperform unconditional cooperators In this case, we also found that having
low commitment thresholds selected for only non-committing strategies, while
having high commitment thresholds selected for only committing ones.

We have also studied the effect of punishing commitment non-compliant players,
showing that intermediate commitment thresholds required the least severe
punishment to sustain cooperative behaviour. Nevertheless, to achieve highest
levels of cooperation, it is necessary to strictly impose high commitment
thresholds and strong punishment.

When considering the effect of rewarding commitment-compliant players, we
found that all commitment thresholds led to similar cooperation outcomes. This
is because, for any threshold, a high enough reward could sustain cooperators
that committed. This was not the case with punishment since defectors could
still avoid punishment by not committing. The insight gleaned from this
finding provides valuable implications for designing institutional mechanisms
that foster pro-social behavior, particularly in situations where pre-
communication is permitted to establish mutual cooperative agreements [9, 10,
32, 33, 23].

It is noteworthy that institutional incentives have been studied as an
important pathway for promoting the emergence of cooperation in social dilemma
situations, both in well-mixed and spatial settings [41, 43, 8, 15, 11, 20,
46, 6, 28, 16]. However, they have not explored the repercussions of
introducing a commitment before interactions in networks, a common occurrence
in real-world personal and business settings [5, 32, 21, 18, 10, 3]. Moreover,
an issue with pro-social incentives aimed at promoting cooperation is the
potential for antisocial reward and punishment dynamics. In this scenario,
defectors may choose to punish cooperators or reward fellow defectors, thereby
impeding the evolutionary progress of cooperation [27, 39, 13]. This concern
diminishes when a prior commitment is established, as it clarifies the
expected behaviour of all parties involved in the interaction. As such, only
those who commit to cooperation can face repercussions for defection or
receive rewards for cooperation [22].

Furthermore, many previous evolutionary game models have shown that different
heterogeneous aspects, including but not limited to network structures, update
mechanisms and incentives, play an important role for the emergence of
cooperation [38, 51, 31, 2, 7, 40, 17, 37]. As a future work, it would be
interesting to incorporate heterogeneity into our model. This would
specifically involve examining the influence of individualised commitment
thresholds, for each player, rather than treating them as global parameters;
as well as that of different heterogeneous population structures.

## CRediT authorship contribution statement

L. S. Flores contributed with Software and all authors contributed equally in
Conceptualization, Formal analysis and Writing.

## Acknowledgments

L.S.Flores thanks the Brazilian funding agency CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior) for the Ph.D. scholarship. The
simulations were performed on the IF-UFRGS computing cluster infrastructure.

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