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Reward and punishment in climate change dilemmas

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Mitigating climate change effects involves strategic decisions by individuals that may choose to limit their emissions at a cost. Everyone shares the ensuing benefits and thereby individuals can free ride on the effort of others, which may lead to the tragedy of the commons. For this reason, climate action can be conveniently formulated in terms of Public Goods Dilemmas often assuming that a minimum collective effort is required to ensure any benefit, and that decision-making may be contingent on the risk associated with future losses. Here we investigate the impact of reward and punishment in this type of collective endeavors — coined as collective-risk dilemmas — by means of a dynamic, evolutionary approach. We show that rewards (positive incentives) are essential to initiate cooperation, mostly when the perception of risk is low. On the other hand, we find that sanctions (negative incentives) are instrumental to maintain cooperation. Altogether, our results are gratifying, given the a-priori limitations of effectively implementing sanctions in international agreements. Finally, we show that whenever collective action is most challenging to succeed, the best results are obtained when both rewards and sanctions are synergistically combined into a single policy.
Gradient of selection (top panels, A and B) and stationary distribution (bottom panels, C and D) for the different values of per-capita budget δ indicated, using either pure-Reward (w = 1, left panels) or purePunishment (w = 0, right panels). The black curve is equal on the left and right panels, since in this case δ = 0. As δ increases, the behaviour under Reward and Punishment is qualitatively similar, by displacing the (unstable) coordination equilibrium towards lower values of k/Z, while displacing the (stable) coexistence equilibrium towards higher values of k/Z. This happens, however, only for low values of δ. Indeed, by further increasing δ one observes very different behaviours under Reward and Punishment: Whereas under Punishment the equilibria are further moved apart (in accord with what happened for low δ) under Reward the coordination equilibrium disappears, and the overall dynamics becomes characterized by a single coexistence equilibrium which consistently shifts towards higher values of k/Z with increasing δ. This difference in behaviour, in turn, has a dramatic impact in the overall prevalence of configurations achieved by the population dynamics, as shown by the stationary distributions: On panel C (pure-Reward) the population spends most of the time on intermediate states of cooperation. On panel D (pure-Punishment) the population spends most of the time on both extremes (high and low cooperation) but especially on low cooperation states. Other parameters: Z = 50, N = 10, M = 5, c = 0.1, B = 1, r = 0.5, a = b = 1, β = 5 and µ = 0.01 (see Methods for details).
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Reward and punishment in climate
change dilemmas
António R. Góis1,2,3, Fernando P. Santos4,1,2, Jorge M. Pacheco5,6,2 & Francisco C. Santos
1,2,7*
Mitigating climate change eects involves strategic decisions by individuals that may choose to limit
their emissions at a cost. Everyone shares the ensuing benets and thereby individuals can free ride
on the eort of others, which may lead to the tragedy of the commons. For this reason, climate action
can be conveniently formulated in terms of Public Goods Dilemmas often assuming that a minimum
collective eort is required to ensure any benet, and that decision-making may be contingent on the
risk associated with future losses. Here we investigate the impact of reward and punishment in this type
of collective endeavors — coined as collective-risk dilemmas — by means of a dynamic, evolutionary
approach. We show that rewards (positive incentives) are essential to initiate cooperation, mostly
when the perception of risk is low. On the other hand, we nd that sanctions (negative incentives)
are instrumental to maintain cooperation. Altogether, our results are gratifying, given the a-priori
limitations of eectively implementing sanctions in international agreements. Finally, we show that
whenever collective action is most challenging to succeed, the best results are obtained when both
rewards and sanctions are synergistically combined into a single policy.
Climate change stands as one of our biggest challenges in what concerns the emergence and sustainability of
cooperation1,2. Indeed, world citizens build up high expectations every time a new International Environmental
Summit is settled, unfortunately with few resulting solutions implemented so far. is calls for the development of
more eective incentives, agreements and binding mechanisms. e problem can be conveniently framed resort-
ing to the mathematics of game theory, being a paradigmatic example of a Public Goods Game3: at stake there is
a global good from which every single individual can prot, irrespectively of contributing to maintain it. Parties
may free ride on the eorts of others, avoiding any eort themselves, while driving the population into the trag-
edy of the commons4. Moreover, since here cooperation aims at averting collective losses, this type of dilemmas
is oen referred as public bad games, in which achieving collective goals oen depends on reaching a threshold
number of cooperative group members58.
One of the multiple obstacles attributed to such agreements is misperceiving the actual risk of future losses,
which signicantly aects the ensuing dynamics of cooperation5,9. Another problem relates to both the incapac-
ity to sanction those who do not contribute to the welfare of the planet, and/or to reward those who subscribe
to green policies10. Previous cooperation studies show that reward (positive incentives), punishment (nega-
tive incentives) and the combination of both1123 have a dierent impact depending on the dilemma in place.
Assessing the impact of reward and punishment (isolated or combined) in the context of N-person threshold
games — and in the particular case of climate change dilemmas — remains, however, an open problem.
Here we study, theoretically, the role of both institutional reward and punishment in the context of climate
change agreements. Previous works consider the public good as a linear function of the number of contribu-
tors12,17,21,22 and conclude that punishment is more eective than reward (for an optimal combination of pun-
ishment and reward see ref.12). We depart from this linear regime by modeling the returns on the public good
as a threshold problem, combined with an uncertain outcome, represented by a risk of failure. As a result – and
as detailed below – the dynamical portrait of our model reveals new internal equilibria9, allowing to identify the
dynamics of coordination and coexistence typifying collective action problems. As discussed below, the reward
and punishment mechanisms will impact, in a non-trivial way, those equilibria.
1INESC-ID and Instituto Superior Técnico, Universidade de Lisboa, IST-Taguspark, 2744-016, Porto, Salvo, Portugal.
2ATP-group, P-2744-016, Porto, Salvo, Portugal. 3Unbabel, R. Visc. de Santarém 67B, 1000-286, Lisboa, Portugal.
4Department of Ecology and Evolutionary Biology, Princeton University, Princeton, USA. 5Centro de Biologia
Molecular e Ambiental, Universidade do Minho, 4710 - 057, Braga, Portugal. 6Departamento de Matemática e
Aplicações, Universidade do Minho, 4710 - 057, Braga, Portugal. 7Machine Learning Group, Université Libre de
Bruxelles, Boulevard du Triomphe CP212, 1050, Bruxelles, Belgium. *email: franciscocsantos@tecnico.ulisboa.pt
OPEN
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We consider a population of size Z, where each individual can be either a Cooperator (C) or a Defector (D),
when participating in a N-player Collective-Risk dilemma (CRD)5,9,10,2430. In this game, each participant starts
with an initial endowment B (viewed as the asset value at stake) that may be used to contribute to the mitigation
of the eects of climate change. A cooperator incurs a cost corresponding to a fraction c of her initial endowment
B, in order to help prevent a collective failure. On the other hand, a defector refuses to have any cost, hoping to
free ride on the contributions of others. We require a minimum number of 0 < M N cooperators in a group of
size N before collective action is realized; if a group of size
N
does not contain at least M Cs, all members lose their
remaining endowments with a probability r, where r (0 r 1) stands as the risk of collective failure. Otherwise,
everyone will keep whatever she has. is CRD formulation has been shown to capture some of the key features
discovered in recent experiments5,24,3133, while highlighting the importance of risk. In addition, it allows one to
test model parameters in a systematic way that is not possible in human experiments. Moreover, the adoption of
non-linear returns mimics situations common to many human and non-human endeavors6,3441, where a mini-
mum joint eort is required to achieve a collective goal. us, the applicability of this framework extends well
beyond environmental governance, given the ubiquity of such type of social dilemmas in nature and societies.
Following Chen et al.12, we include both reward and punishment mechanisms in this model. A xed group
budget Nδ (where δ 0 stands for a per-capita incentive) is assumed to be available, of which a fraction w is
applied to a reward policy and the remaining 1-w to a punishment policy. We assume the eective impact of both
policies to be equivalent, meaning that each unit spent will directly increase/decrease the payo of a cooperator/
defector by the same amount. For details on policies with dierent eciencies, see Methods.
Instead of considering a collection of rational agents engaging in one-shot Public Goods Games32,42, here we
adopt an evolutionary description of the behavioral dynamics9, in which individuals tend to copy those appearing
to be more successful. Success (or tness) of individuals is here associated with their average payo. All individu-
als are equally likely to interact with each other, causing all cooperators and defectors to be equivalent, on average,
and only distinguishable by the strategy they adopt. erefore, and considering that only two strategies are avail-
able, the number of cooperators is sucient to describe any conguration of the population. e number of indi-
viduals adopting a given strategy (either C or D) evolves in time according to a stochastic birth–death process43,44,
which describes the time evolution of the social learning dynamics (with exploration): At each time-step each
individual (X, with tness fX) is given the opportunity to change strategy; with probability μ, X randomly explores
the strategy space45 (a process similar to mutations in a biological context that precludes the existence of absorb-
ing states). With probability (1-μ), X may adopt the strategy of a randomly selected individual (Y, with tness
fY), with a probability that increases with the tness dierence (fY–fX)44. is renders the stationary distribution
(see Methods) an extremely useful tool to rank the most visited states given the ensuing evolutionary dynamics
of the population. Indeed, the stationary distribution provides the prevalence of each of the populations possible
conguration, in terms of the number of Cs (k) and Ds (Z-k). Combined with the probability of success charac-
terizing each conguration, the stationary distribution can be used to compute the overall success probability of
a given population – the average group achievement, ηG. is value represents the average fraction of groups that
will overcome the CRD, successfully preserving the public good.
Results
In Fig.1 we compare the average group achievement ηG (as a function of risk) in four scenarios: (i) a reference
scenario without any policy (i.e., no reward or punishment, in black); and three scenarios where a budget is
applied to (ii) rewards, (iii) punishment and (iv) a combination of rewards and sanctions (see below). Our results
are shown for the two most paradigmatic regimes: low (Fig.1A) and high (Fig.1B) coordination requirements.
Naturally ηG improves whenever a policy is applied. Less obvious is the dierence between the various policies.
Applying only rewards (blue curves in Fig.1) is more eective than only punishment (red curve) for low values of
risk. e opposite happens when risk is high. On scenarios with a low relative threshold (Fig.1A), rewards play
the key role, with sanctions only marginally outperforming them for very high values of risk. For high coordina-
tion thresholds (Fig.1B) reward and punishment portray comparable eciency in the promotion of cooperation,
with pure-Punishment (w = 0) performing slightly better than pure-Reward (w = 1).
Justifying these dierences is dicult from the analysis of ηG alone. To better understand the behavior dynam-
ics under Reward and Punishment, we show in Fig.2 the gradients of selection (top panels) and stationary distri-
butions (lower panels) for each case and dierent budget values. Each gradient of selection represents, for each
discrete state k/Z (i.e., fraction of Cs), the dierence
=−
+−
GkTkTk() () ()
among the probability to increase
(T+(k)) and decrease (T(k)) the number of cooperators (see Methods) by one. Whenever G(k) > 0 the fraction
of Cs is likely to increase; whenever G(k) < 0 the opposite is expected to happen. e stationary distributions
show how likely it is to nd the population in each (discrete) conguration of our system. e panels on the
le-hand side show the results obtained for the CRD under pure-Reward; on the right-hand side, we show the
results obtained for pure-Punishment.
Naturally, both mechanisms are inoperative whenever the per-capita incentives are inexistent (δ = 0), creat-
ing a natural reference scenario in which to study the impact of Reward and Punishment on the CRD. In this
case, above a certain value of risk (r), decision-making is characterized by two internal equilibria (i.e., adjacent
nite population states with opposite gradient sign, representing the analogue of xed points in a dynamical
system characterizing evolution in innite populations). Above a certain fraction of cooperators the population
overcomes the coordination barrier and naturally self-organizes towards a stable co-existence of cooperators
and defectors. Otherwise, the population is condemned to evolve towards a monomorphic population of defec-
tors, leading to the tragedy of the commons9. As the budget for incentives increases, using either Reward or
Punishment leads to very dierent outcomes, as depicted in Fig.2.
Contrary to the case of linear Public Goods Games12, in the CRD coordination and co-existence dynam-
ics already exist in the absence of any reward/punishment incentive. Reward is particularly effective when
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cooperation is low (small k/Z), showing a signicant impact on the location of the nite population analogue of
an unstable xed point. Indeed, increasing δ lowers the minimum number of cooperators required to reach the
cooperative basin of attraction (as well as increasing the prevalence of cooperators in co-existence point on the
right), which ultimately disappears for high δ (Fig.2A). is means that a smaller coordination eort is required
before the population dynamics start to naturally favor the increase of cooperators. Once this initial barrier is
surpassed, the population will naturally tend towards an equilibrium state, which does not improve appreciably
under Reward. e opposite happens under Punishment. e location of the coordination point is little aected,
yet once this barrier is overcome, the population will evolve towards a more favorable equilibrium (Fig.2B). us,
while Reward seems to be particularly eective to bootstrap cooperation towards a more cooperative basin of
attraction, Punishment seems eective in sustaining high levels of cooperation.
As a consequence, the most frequently observed congurations are very dierent when using each of the
policies. As shown by the stationary distributions (Fig.2C,D), under Reward the population visits more oen
states with intermediate values of cooperation (i.e., where Cs and Ds co-exist). Intuitively, this happens because
the coordination eort is eased by the rewards, causing the population to eectively overcome it and reach the
coexistence point (the equilibrium state with an intermediate amount of cooperators) thus spending most of the
time near it. On the other hand, Punishment will not ease the coordination eort, and thus the population will
spend most of the time in states of low cooperation, failing to overcome this barrier. Notwithstanding, once sur-
passed, the population will stabilize on higher states of cooperation. is is especially evident for high budgets,
as shown with δ = 0.02 (blue line). Moreover, since Nδ corresponds to a xed total amount which is distributed
by the existing cooperators/defectors, this causes the per-cooperator/defector budget to vary depending on the
number of existing cooperators/defectors (i.e., each of the j cooperators receives wδN/j and each defector loses
(1 w)δN/(N j)). In other words, positive (negative) incentives become very protable (or severe) if defection
(cooperation) prevails within a group. In particular, whenever the budget is signicant (see, e.g., δ = 0.02 in Fig.2)
the punishment becomes so high when there are few defectors within a group, that a new equilibrium emerges
close to full cooperation.
e results in Fig.2 show that Reward can be instrumental in fostering pro-social behavior, while Punishment
can be used for its maintenance. is suggests that, to combine both policies synergistically, pure-Reward (w = 1)
should be applied at rst, when there are few cooperators (low k/Z); above a certain critical point (k/Z = s) one
should switch to pure-Punishment (w = 0). In the Methods section, we demonstrate that, similar to linear Public
Goods Games12, in CRDs this is indeed the policy which minimizes the advantage of the defector, even if we con-
sider the alternative possibility of applying both policies simultaneously. In Methods, we also compute a general
expression for the optimal switching point s*, that is, the value of k above which Punishment should be applied
instead ofReward to maximize cooperation and group achievement. By using such policy — that we denote by s*
— we obtain the best results shown with an orange line in Fig.1. We propose, however, to explore what happens
in the context of a CRD when s* is not used. How much cooperation is lost when we deviate from s* to either of
the pure policies, or to a policy which uses a switching point dierent from the optimal one?
Figure 1. Average group achievement ηG as a function of risk. Le: Group relative threshold M/N = 3/10. Right:
Group relative threshold M/N = 7/10. In both panels, the black line corresponds to a reference scenario where
no policy is applied. e red line shows ηG in the case where all available budget is applied to pure-Punishment
(w = 0), whereas the blue line shows results for pure-Reward (w = 1). Pure-Reward is most eective at low risk
values, while pure-Punishment is marginally the most eective policy at high risk. ese features are more
pronounced for low relative thresholds (le panel), and only at high thresholds does pure-Punishment lead
to a sizeable improvement with respect to pure-Reward. Finally, the orange line shows the results using the
combination of Reward and Punishment, leading (naturally) to the best results. In this case, we adopt pure-
Reward (w = 1) when there are few cooperators and, above a certain critical point k/Z = s = 0.5, we switch to
pure-Punishment (w = 0). As detailed in the main text (see Fig.3 and Methods), s = 0.5 provides the optimal
switching point s* for cooperation to thrive. Other parameters: Population size Z = 50, group size N = 10, cost
of cooperation c = 0.1, initial endowment B = 1, budget δ = 0.025, reward eciency a = 1, punishment eciency
b = 1, intensity of selection β = 5, mutation rate µ = 0.01.
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Figure3 illustrates how the choice of the switching point s impacts the overall cooperation, as evaluated by
ηG, for dierent values of risk. For a switching point of s = k/Z = 1.0 (0.0) a static policy of always pure-Reward
(pure-Punishment) is used. is can be seen on the far right (le) of Fig.3. Figure3 suggests that, for low thresh-
olds, an optimal policy switching (which, for the parameters shown, occurs for s = 50%, see Methods) is only mar-
ginally better than a policy solely based on rewards (s = 1). Figure3 also allows for a comparison of what happens
when the switching point occurs too late (excessive rewards) or too early (excessive sanctions) in a low-threshold
scenario. A late switch is signicantly less harmful than an early one. In other words, our results suggest that
when the population conguration cannot be precisely observed, it is preferable to keep rewarding for longer.
is said, whenever the perception of risk is high (an unlikely situation these days) an early switch is slightly less
harmful than a late one. In the most dicult scenarios, where stringent coordination requirements (large M) are
combined with a low perception of risk (low r), the adoption of a combined policy becomes necessary (see right
panel of Fig.1).
Discussion
One might expect the impact of Reward and Punishment to lead to symmetric outcomes – Punishment would
be eective for high-cooperation the same way that Reward is eective for low-cooperation. In low-cooperation
scenarios (under low risk, threshold or budget) Reward alone plays the most important role. However, in the
opposite scenario, Punishment alone does not have the same impact. Either a favourable scenario occurs, where
any policy yields a satisfying result, or Punishment cannot improve outcomes on its own. In the latter case, the
synergy between both policies becomes essential to achieve cooperation. Such optimal policy involves a combi-
nation of the single policies, Reward and Punishment, which is dynamic, in the sense that the combination does
not remain the same for all congurations of the population. It corresponds to employing pure Reward at rst,
when cooperation is low, switching subsequently to Punishment whenever a pre-determined level of cooperation
is reached.
Figure 2. Gradient of selection (top panels, A and B) and stationary distribution (bottom panels, C and D)
for the dierent values of per-capita budget δ indicated, using either pure-Reward (w = 1, le panels) or pure-
Punishment (w = 0, right panels). e black curve is equal on the le and right panels, since in this case δ = 0.
As δ increases, the behaviour under Reward and Punishment is qualitatively similar, by displacing the (unstable)
coordination equilibrium towards lower values of k/Z, while displacing the (stable) coexistence equilibrium
towards higher values of k/Z. is happens, however, only for low values of δ. Indeed, by further increasing
δ one observes very dierent behaviours under Reward and Punishment: Whereas under Punishment the
equilibria are further moved apart (in accord with what happened for low δ) under Reward the coordination
equilibrium disappears, and the overall dynamics becomes characterized by a single coexistence equilibrium
which consistently shis towards higher values of k/Z with increasing δ. is dierence in behaviour, in turn,
has a dramatic impact in the overall prevalence of congurations achieved by the population dynamics, as
shown by the stationary distributions: On panel C (pure-Reward) the population spends most of the time on
intermediate states of cooperation. On panel D (pure-Punishment) the population spends most of the time on
both extremes (high and low cooperation) but especially on low cooperation states. Other parameters: Z = 50,
N = 10, M = 5, c = 0.1, B = 1, r = 0.5, a = b = 1, β = 5 and µ = 0.01 (see Methods for details).
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e optimal procedure, however, is unlikely to be realistic in the context of Climate Change agreements.
Indeed, and unlike other Public Goods Dilemmas, where Reward and Punishment constitute the main policies
available for Institutions to foster cooperative collective action, in International Agreements it is widely recog-
nized that Punishment is very dicult to implement2,42. is has been, in fact, one of the main criticisms put
forward in connection with Global Agreements on Climate Mitigation: ey suer from the lack ofsanctioning
mechanisms as it is practically impossible to enforce any type of sanctioning at a Global level. In this sense, the
results obtained here by means of our dynamical, evolutionary approach, are gratifying, given these a-priori limi-
tations of sanctioning in CRDs. Not only do we show that Reward is essential to foster cooperation, mostly when
both the perception of risk is low and the overall number of engaged parties is small (low k/Z), but also we show
that Punishment mostly acts to sustain cooperation, aer it has been installed. Given that low-risk scenarios are
more common and harmful to cooperation than high-risk ones, our results in connection with rewards provide
a viable way to explore in the quest for establishing Global cooperative collective action. Reward policies may
also be very relevant in scenarios where Climate Agreements are coupled with other International agreements
from which parties are not interested to deviate from2,42. Finally, the fact that rewards ease coordination towards
cooperative states suggests that positive incentives should also be used within intervention mechanisms aiming at
fostering pro-sociality in articial systems and hybrid populations comprising humans and machines4649.
e model used takes for granted the existence of an institution with a budget available to implement either
Reward or Punishment. New behaviours may emerge once individuals are called to decide whether or not to
contribute to such an institution, allowing for a scenario where this institution fails to exist10,28,50,51. At present,
and under the Paris agreement, we are witnessing the potential birth of an informal funding institution, whose
goal is to nance developing countries to help them increase their mitigation capacity. Clearly, this is just an
example pointing out to the fact that the prevalence of local and global institutional incentives may depend and
may be inuenced by the distribution of wealth available among parties, in the same way that it inuences the
actual contributions to the public good10,29,33. Finally, several other eects may further inuence and/or aect
the present results. Among others, if intermediate tasks are considered33, or if individuals have the opportunity
to pledge their contribution before their actual action7,40,52, it is likely that pro-social behavior may be enhanced.
Work along these lines is in progress.
Methods
Public goods and collective risks. Let us consider a population with Z individuals, where each individual
can be a cooperator (C) or a defector (D). For each round of this game, a group of N players is sampled from the
original nite population of size Z, which corresponds to a process of sampling without replacement. e proba-
bility of a group comprising any possible combination of Cs and Ds is given by the hypergeometric distribution.
In the context of a given group, a strategy is associated with a payo value corresponding to an individuals earn-
ings in that round, which depend on the action of the rest of group. Fitness is the expected payo of an individual
in a population, before knowing to which group he was assigned. is way, for a population with k out of Z Cs and
each group containing j out of N Cs, the tness of a D and a C can be written as:
Figure 3. Average group achievement ηG as a function of the location of the switching point s. e switching
point s corresponds to the conguration (fraction of Cs in the population, k/Z) above which w suddenly
switches from pure-Reward (w = 1) to pure-Punishment (w = 0). Assuming both policies are equally ecient,
the optimal switching point occurs at 50% of cooperators (k/Z = 0.5). e far-le values of s correspond
to a static policy of always pure-Punishment – the switch from pure-Reward to pure-Punishment occurs
immediately at 0% of cooperators. On the far-right (switching point = 100%) a pure-Reward policy is depicted.
We can also see what happens when the switch occurs too late or too early, for dierent values of risk. For low
values of risk, it is signicantly less harmful to have a late switch from Reward to Punishment than an early one,
meaning that when the population conguration cannot be precisely observed, it is preferable to keep rewarding
for longer. See Methods for the calculation of the optimal switching point (s*) that maximizes cooperation
tness relative to defection – and consequent group achievement. Other parameters: Z = 50, B = 1, µ = 0.01,
β = 5, N = 10, M = 3, c = 0.1, and δ = 0.025.
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=
−−
−−
Π
=
()
fZ
Nk
jZk
Nj j
1
11
1()
(1)
Dj
N
1
0
1
D
=
−−
Π+
=
()
fZ
NkjZk
Nj j
1
111(1)
(2)
Cj
N
1
0
1
C
where
Πj()
C
and
Πj()
D
stand for the payo or a C and a D in a single round, in a group with N players and j Cs. To
dene the payo functions, let
θx()
be a Heaviside step-function distribution, where θ(x) = 0 if x < 0 and θ(x) = 1
if x 0. Each player can contribute with a fraction c of her endowment B (with 0 c 1), and in case a group
contains less than M cooperators (0 < M N) there is a risk r of failure (0 r 1), in which case no player obtains
her remaining endowment. e payo of a defector (
Πj()
D
) and the payo of a cooperator (
Πj()
C
), before incor-
porating any policy, can be written as9:
θθΠ= −+−−jBjM rjM() {( )(1)[1 ()]} (3)
D
Π=Π−jjcB() () (4)
CD
Reward and punishment. To include a Reward or a Punishment policy, let us follow ref.12 and consider
a group budget Nδ which can be used to implement any type of policy. e fraction of Nδ applied to Reward
is represented by the weight w, with 0 w 1. Parameters a and b correspond to the eciency of Reward and
Punishment (for all Figures above it was assumed that a = b = 1).
δ
Π=Π−
jj
bwN
Nj
() ()
(1 )
(5)
D
PD
δ
Π=Π+jj
awN
j
() ()
(6)
C
RC
Naturally, these new payo functions can be included into the previous tness functions (
replaces
ΠD
and
replaces
ΠC
), letting tness values account for the dierent policies.
Evolutionary dynamics in nite populations. e tness functions written above allow us to setup the
(discrete time) evolutionary dynamics. Indeed, the congurations of the entire population may be used to dene
a Markov Chain, where each state is characterized by number of cooperators9,44. To decide in which direction
the system will evolve, at each step a player i and a neighbour j of her are drawn at random from the population.
Player i decides whether to imitate her neighbour j with a probability depending on the dierence between their
tness43,44. is way, a system with k cooperators may stay in the same state, switch to k 1 or to k + 1. e prob-
ability of player i imitating player j can be given by the Fermi function:
≡+
β−−
()
pk e() [1 ]
(7)
ji ff
,1
ji
where β is the intensity of selection. Using this probability distribution, we can fully characterize this Markov
process. Let k be the total number of cooperators in the population and Z the total size of the population.
+
T k()
and
T k()
are the probabilities to increase and decrease k by one, respectively44:
=
+β±−
Tk
k
Z
Zk
Z
e() [1 ]
(8)
fk fk[()()
1
CD
e most likely direction can be computed using the dierence
≡−
+−
GkTkTk() () ()
. A mutation rate can be
introduced by using transition probabilities
μμ
=− +
μ
++
T
kTk() (1 )()
Zk
Z
and
μμ
=− +
μ
−−
TkT
k() (1 )()
k
Z
.
In all cases we used amutation rate μ = 0.01, this way avoiding the population to xate in a monomorphic congu-
ration. In this context, the stationary distribution becomes a very useful tool to analyse the overall population
dynamics, providing the probability =
p
P
()
kk
Z
for each of the Z + 1 states of this Markov Chain to be occupied53,54.
For each given population state k, the hypergeometric distribution can be used to compute the average fraction of
groups that obtain success aG(k). Using the stationary distribution and the average group success, the average
group achievement (ηG) can then be computed, providing the overall probability of achieving success:
η=
=pa k()
Gk
ZkG
0
.
Combined policies. By allowing the weight w to depend on the frequency of cooperators, we can derive
the optimal switching point s* between positive and negative incentives by minimizing the defector’s advantage
(fD fC). is is done similarly to ref.12, but using nite populations and therefore a hypergeometric distribution
(see Eqs (1), (2), (5), and (6)), to account for sampling without replacement. From Eqs (1) and (2), we get
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δ
δ
=
−−
−−
Π−
=
−−
−−
Π+−+ +
=
=
()
()
f
k
j
Zk
Nj
Z
N
jbwN
Nj
f
kjZk
Nj
Z
N
jc
awN
j
1
1
1
1
() (1 )
11( 1)
1
1
1
(( 1) 1)
Dj
N
D
Cj
N
C
0
1
0
1
from which we aim at nding the value of w (with respect to k) that minimizes F = fD fC. Since
Πj()
D
,
Π+j(1)
C
and c do not depend on w, these quantities do not aect the choice of the optimal w, leaving us with the problem
of minimizing the following expression:
∑∑
δδ=−
−−
−−
−−
+
=
=
() ()
FN
k
j
Zk
Nj
Z
N
bw
Nj N
k
j
Zk
Nj
Z
N
aw
j
1
1
1
1
(1 )
1
1
1
1
1
j
N
j
N
0
1
0
1
Since
=
−−
−−
=
−−
−− −−
k
j
k
j
Zk
Nj
Zk
Nj
1
and
1
11
,
k
kj
Zk
Nj
Zk
(1)
δ
δ
δ
′=
−−
++
−−
−− ++
=−
−−
+−−
−− ++
−−
−−
−− ++
=
=
=
()
()
()
FN
kj
Zk
Nj
Z
N
aw
j
bw
Nj
k
kj
ZkNj
Zk
N
kjZk
Nj
Z
N
wa
j
b
Nj
k
kj
ZkNj
Zk
N
kjZk
Nj
Z
N
b
Nj
k
kj
ZkNj
Zk
11
1
11
(1 )1
11
1
11
1
11
1
1
1
j
N
j
N
j
N
0
1
0
1
0
1
e second summation does not depend on w; thus the optimal policy is given by the minimization of:
δ″=
−−
+−−
−− ++
=
()
FN
kjZk
Nj
Z
N
wa
j
b
Nj
k
kj
ZkNj
Zk
11
1
11
1
j
N
0
1
Since N and δ are always positive, the whole expression can be divided by Nδ without changing the optimiza-
tion problem. Moreover, by multiplying the expression by (1), it can nally be shown that minimizing fD fC is
equivalent to maximizing the following expression:
Figure 4. Optimal switching point s* as a function of the ratio a/b, for dierent values of N (see Methods).
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−−
+−−
−− ++
=
()
w
kj
Zk
Nj
Z
N
a
j
b
Nj
k
kj
ZkNj
Zk
11
1
11
1
j
N
0
1
where j represents the number of Cs in a group of size N, sampled without replacement from a population of size
Z containing k Cs. Now, let us consider that the optimal switching point s* depends on k. Since this sum decreases
as k increases, containing only one root, the solution to this optimization problem corresponds to having w set to
1 (pure Reward) for positive values of the sum, suddenly switching to w = 0 (pure Punishment) once the sum
becomes negative. e optimal switching point s* depends on the ratio
a
b
, group size N and population size Z. e
eect of population size (Z) and group size (N) on s* is limited, while the impact of the eciency of reward (a)
and punishment (b) is illustrated in Fig.4. For
=1
a
b
the switching point is s* = 0.5 (see Fig.4). Interestingly, we
note that, also in the CRD, s* is not impacted by the group success threshold (M) or the risk associated with los-
ing the retained endowment when collective success is not attained (r). is is the case as we assume that the
decision to punish or reward is independent on M or r. Notwithstanding, the model that we present can, in the
future, be tuned to test more sophisticated incentive tools, such as rewarding or punishing depending on (i) how
far group contributions remained from (or surpassed) the minima to achieve group success or (ii) how so/strict
is the dilemma at stake, given the likelihood of losing everything when collective success is not accomplished.
Received: 20 May 2019; Accepted: 15 September 2019;
Published: xx xx xxxx
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Acknowledgements
This research was supported by Fundação para a Ciência e Tecnologia (FCT) through grants PTDC/EEI-
SII/5081/2014 and PTDC/MAT/STA/3358/2014 and by multiannual funding of INESC-ID and CBMA (under
the projects UID/CEC/50021/2019 and UID/BIA/04050/2013). F.P.S. acknowledges support from the James S.
McDonnell Foundation 21st Century Science Initiative in Understanding Dynamic and Multi-scale Systems -
Postdoctoral Fellowship Award.All authors declare no competing nancial or non-nancial interests in relation
to the work described.
Author contributions
A.R.G., F.P.S, J.M.P. and F.C.S. designed and implemented the research; A.R.G., F.P.S, J.M.P. and F.C.S prepared
all the Figures; A.R.G., F.P.S, J.M.P. and F.C.S. wrote the manuscript; A.R.G., F.P.S, J.M.P. and F.C.S reviewed the
manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to F.C.S.
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... We now analyse the alignment between promoting cooperation and social welfare in minimal models of institutional reward and punishment (Duong and Han, 2021, Góis et al., 2019, Sasaki et al., 2015. Namely, we consider a well-mixed population consisting of individuals playing the one-shot PD game who can adopt either C or D in the interactions. ...
... shows that, as expected(Duong and Han, 2021, Góis et al., 2019, Han, 2022b, both types of incentives lead to the same level of cooperation assuming the are equally effective (i.e. a = b) and costly. However, institutional reward leads to positive social welfare (red areas in panels a and b) for a much wider range of incentive impact and cost. ...
... There is an optimal intermediate δ for highest social welfare. This is a notable observation as previous models of institutional incentives(Góis et al., 2019, Sasaki et al., 2012, 2015 do not and are unable to provide insights on how strong punishment is strong enough, focusing on promoting high levels of cooperation. In fact, previous works only consider that strong punishment is needed to ensure cooperation. ...
Preprint
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Understanding the emergence of prosocial behaviours among self-interested individuals is an important problem in many scientific disciplines. Various mechanisms have been proposed to explain the evolution of such behaviours, primarily seeking the conditions under which a given mechanism can induce highest levels of cooperation. As these mechanisms usually involve costs that alter individual payoffs, it is however possible that aiming for highest levels of cooperation might be detrimental for social welfare -- the later broadly defined as the total population payoff, taking into account all costs involved for inducing increased prosocial behaviours. Herein, by comparatively analysing the social welfare and cooperation levels obtained from stochastic evolutionary models of two well-established mechanisms of prosocial behaviour, namely, peer and institutional incentives, we demonstrate exactly that. We show that the objectives of maximising cooperation levels and the objectives of maximising social welfare are often misaligned. We argue for the need of adopting social welfare as the main optimisation objective when designing and implementing evolutionary mechanisms for social and collective goods.
... Некоторые исследователи не смогли выявить значимую связь между склонностью к риску и готовностью к сотрудничеству (de Heus, Hoogervorst, van Dijk, 2010). Однако ряд экспериментальных и теоретических исследований показал, что выраженная готовность к риску стимулирует индивидов к сотрудничеству, тем самым способствуя достижению коллективных целей (Milinski, Sommerfeld, Krambeck, Reed, Marotzke, 2008;Liu, Chen, 2018;Góis, Santos, Pacheco, Santos, 2019;Sun, Liu, Chen, Szolnoki, Vasconcelos, 2021). A. Glöckner и B.E. Hilbig (2012) обнаружили, что влияние склонности к риску при выборе сотрудничества зависит также от окружающей среды: неприятие риска увеличивает частоту выбора сотрудничества только в среде, которая благоприятствует таким решениям. ...
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An important role in decision-making is taking a risk, which can direct a person towards one of a number of alternatives. Research involving the study of risk-taking in the context of social dilemmas considers cooperation or non-cooperation as possible decision-making options. A classic example of this choice is the decision-making process in the context of the "Prisoner's Dilemma". The choice between cooperating or not cooperating is characterized by risk, since such decisions have uncertain consequences. Despite the widespread popularity of the "Prisoner's Dilemma," the influence of the factor of collective interaction on an individual's behavior in this context has not been sufficiently studied. The aim of this research is to investigate attitudes towards risk taking in solving social dilemmas, taking into account the process of social interaction. The research methods used were the "Iterated prisoner's dilemma", including a stage of socialization through the formation of small social groups, and economic games to identify emotional and cognitive indicators of attitudes of risks in conditions of uncertainty. The analyzed sample consisted of 124 individuals aged 18 to 36 years (M = 20.39, SD = 2.80), of whom, 100 were female (80.65%), and 24 were male (19.35%). The results of the study did not demonstrate a connection between risk-taking and decisionmaking in solving social dilemmas in collective action. However, there was a slight increase in risk tolerance in individuals who chose defection in most rounds prior to socialization. This indicates the need for anticipating the future decisions of others in order to maximize rewards. An additional analysis was also conducted on a female sample, which confirmed the existence of gender-specificity in solving a social dilemma under conditions of collective interaction.
... Некоторые исследователи не смогли выявить значимую связь между склонностью к риску и готовностью к сотрудничеству (de Heus, Hoogervorst, van Dijk, 2010). Однако ряд экспериментальных и теоретических исследований показал, что выраженная готовность к риску стимулирует индивидов к сотрудничеству, тем самым способствуя достижению коллективных целей (Milinski, Sommerfeld, Krambeck, Reed, Marotzke, 2008;Liu, Chen, 2018;Góis, Santos, Pacheco, Santos, 2019;Sun, Liu, Chen, Szolnoki, Vasconcelos, 2021). A. Glöckner и B.E. Hilbig (2012) обнаружили, что влияние склонности к риску при выборе сотрудничества зависит также от окружающей среды: неприятие риска увеличивает частоту выбора сотрудничества только в среде, которая благоприятствует таким решениям. ...
Article
An important role in decision-making is taking a risk, which can direct a person towards one of a number of alternatives. Research involving the study of risk-taking in the context of social dilemmas considers cooperation or non-cooperation as possible decision-making options. A classic example of this choice is the decision-making process in the context of the "Prisoner's Dilemma". The choice between cooperating or not cooperating is characterized by risk, since such decisions have uncertain consequences. Despite the widespread popularity of the "Prisoner's Dilemma," the influence of the factor of collective interaction on an individual's behavior in this context has not been sufficiently studied. The aim of this research is to investigate attitudes towards risk taking in solving social dilemmas, taking into account the process of social interaction. The research methods used were the "Iterated prisoner's dilemma", including a stage of socialization through the formation of small social groups, and economic games to identify emotional and cognitive indicators of attitudes of risks in conditions of uncertainty. The analyzed sample consisted of 124 individuals aged 18 to 36 years (M = 20.39, SD = 2.80), of whom, 100 were female (80.65%), and 24 were male (19.35%). The results of the study did not demonstrate a connection between risk-taking and decisionmaking in solving social dilemmas in collective action. However, there was a slight increase in risk tolerance in individuals who chose defection in most rounds prior to socialization. This indicates the need for anticipating the future decisions of others in order to maximize rewards. An additional analysis was also conducted on a female sample, which confirmed the existence of gender-specificity in solving a social dilemma under conditions of collective interaction.
... For example, commitments can be implemented by repeated play [51,52], the costs of one's reputation [53][54][55], a contract or a physical constraint. Another method is to 'change the game', which can be achieved by the creation of institutions [56][57][58][59][60][61]. These institutions are effective in facilitating collective cooperation, if they promote cooperative understanding, communication or commitments [62,63]. ...
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As artificial intelligence (AI) systems are increasingly embedded in our lives, their presence leads to interactions that shape our behaviour, decision-making and social interactions. Existing theoretical research on the emergence and stability of cooperation, particularly in the context of social dilemmas, has primarily focused on human-to-human interactions, overlooking the unique dynamics triggered by the presence of AI. Resorting to methods from evolutionary game theory, we study how different forms of AI can influence cooperation in a population of human-like agents playing the one-shot Prisoner's dilemma game. We found that Samaritan AI agents who help everyone unconditionally, including defectors, can promote higher levels of cooperation in humans than Discriminatory AI that only helps those considered worthy/cooperative, especially in slow-moving societies where change based on payoff difference is moderate (small intensities of selection). Only in fast-moving societies (high intensities of selection), Discriminatory AIs promote higher levels of cooperation than Samaritan AIs. Furthermore, when it is possible to identify whether a co-player is a human or an AI, we found that cooperation is enhanced when human-like agents disregard AI performance. Our findings provide novel insights into the design and implementation of context-dependent AI systems for addressing social dilemmas.
... To choose π i , we focus on a game coined Collective Risk Dilemma (CRD; Milinski et al., 2008;Santos & Pacheco, 2011), suitable to study mechanism design (Góis et al., 2019). Each round requires a critical mass of cooperators to achieve success and prevent collective losses. ...
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Predictions often influence the reality which they aim to predict, an effect known as performativity. Existing work focuses on accuracy maximization under this effect, but model deployment may have important unintended impacts, especially in multiagent scenarios. In this work, we investigate performative prediction in a concrete game-theoretic setting where social welfare is an alternative objective to accuracy maximization. We explore a collective risk dilemma scenario where maximising accuracy can negatively impact social welfare, when predicting collective behaviours. By assuming knowledge of a Bayesian agent behavior model, we then show how to achieve better trade-offs and use them for mechanism design.
... The current work focuses on institutional incentives (Sasaki et al. 2012;Sigmund et al. 2010;Wang et al. 2019;Duong and Han 2021;Cimpeanu et al. 2021;Sun et al. 2021;Van Lange et al. 2014;Gürerk et al. 2006;Góis et al. 2019;Sun et al. 2021;Liu and Chen 2022;Flores and Han 2024;Wang et al. 2024;Hua and Liu 2024), which are a plan of action involving the use of reward (i.e., increasing the payoff of cooperators), punishment (i.e., decreasing the payoff of defectors), or a combination of the two, by an external decision-maker. More precisely, we study how the aforementioned institutional incentives can be used in a cost-efficient way for maximising the levels of cooperative behaviour in a population of self-regarding individuals. ...
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In this paper, we study the problem of cost optimisation of individual-based institutional incentives (reward, punishment, and hybrid) for guaranteeing a certain minimal level of cooperative behaviour in a well-mixed, finite population. In this scheme, the individuals in the population interact via cooperation dilemmas (Donation Game or Public Goods Game) in which institutional reward is carried out only if cooperation is not abundant enough (i.e., the number of cooperators is below a threshold $$1\le t\le N-1$$ 1 ≤ t ≤ N - 1 , where N is the population size); and similarly, institutional punishment is carried out only when defection is too abundant. We study analytically the cases $$t=1$$ t = 1 for the reward incentive under the small mutation limit assumption and two different initial states, showing that the cost function is always non-decreasing. We derive the neutral drift and strong selection limits when the intensity of selection tends to zero and infinity, respectively. We numerically investigate the problem for other values of t and for population dynamics with arbitrary mutation rates.
... As is the case with such experiments in general, our experiment differs from the real climate change situation in a number of ways, such as 1) everyone in the experiment knows exactly how much contribution is needed to avoid disaster and when it will 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 occur (but see Raihani and Aitken (2011); Dannenberg et al (2015)), 2) the game is between only a few players (but see Milinski et al (2016)), 3) costs and payoffs are immediate (but see Jacquet et al (2013)), 4) there is no way to punish or regulate noncooperative behaviour (but see Fehr and Gächter (2019); Góis et al (2019)), 5) players start at the same monetary level and have equal impact on reaching the target (but see Tavoni et al (2011);Burton-Chellew et al (2013); Kroll (2017, 2021)), 6) players participate in the game anonymously, thus their actions don't affect their reputation (but see Rockenbach and Milinski (2006); Milinski et al (2006)), 7) there is no way of making negotiations and commitments and setting intermediate targets (but see Milinski et al (2011); Barrett and Dannenberg (2012); Milinski et al (2016)). However, the effects of the differences listed here have already been investigated in previous experimental works, as indicated in the introduction and here with the cited references. ...
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Using different variants of the classic climate game, we investigate the role of competition and the source of endowment (windfall vs. earned). Participants completed a detailed personality test (including climate attitudes and economic preferences) before the experiment and were asked about their strategies after-wards. We find that competition did not significantly affect whether groups 1 reached the target, even though the probability of achieving the common goal was lower in the presence of competition. Participants cooperated more when they had to earn the endowment. Based on the pre-experiment questionnaire, participants who viewed their personal actions as more important and effective in combating climate change were more likely to cooperate in the climate game, while the rest of the measured personality items did not exhibit a consistent pattern. Analysis of the post-experiment survey indicates that those who aimed to maximise earnings contributed less to the common pool. In contrast, those who believed the goal was achievable and aimed to achieve it contributed more to the common pool throughout the game.
... De Nadai et al. [67] investigated whether intellect places constraints on the digital world in contrast to those that we are aware exist in the real world, such as preserving consistency over the number of acquaintances and favorite locations over time. This is undoubtedly the case when examining the results of a fascinating image that connects human behavior in the physical and digital spheres and establishes a connection between the study of human mechanics, computer science, and computational social science. ...
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Networks form an integral part of our daily lives, comprising structures known as nodes connected via ties. These networks exhibit a complex interplay of random and structured features. We encounter networks regularly in our daily activities, whether it is logging into websites, commuting by train, going to work, or attending school. Networks are prevalent in various academic disciplines and practical applications, spanning biology, physics, engineering, social science, and numerous other fields. The early 21st century has witnessed a rapid expansion of techniques facilitating human interaction, significantly impacting behavior and social bonding in both urban and rural settings. The availability and accessibility of extensive population datasets have spurred ongoing research into understanding and shaping human inclinations and sociological phenomena in unconventional ways. This review delves into the nuanced aspects of social physics, exploring the rationale behind choosing this research topic, various conceptual frameworks, categorizations, and distinctions between machine learning and deep learning. The article briefly outlines the advantages of leveraging big data in social physics, highlighting its transformative potential. Lastly, the authors provide insights into several applications of social physics in subsequent sections, offering a glimpse into its real-world implications.
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Risks, such as climate change, disease outbreak, geopolitical tension, may exacerbate food insecurity by negatively impacting crop yield. Additional agricultural nitrogen input may partly offset yield losses, with a corresponding increase in nitrogen pollution. The problems of food insecurity and nitrogen pollution are urgent and global but have not been addressed in an integrated fashion. Current efforts to combat food insecurity occur primarily through the United Nations’ World Food Program at the international level, and, at the local community level, through food banks. The international program to monitor and reduce global nitrogen pollution is in its early stage. Food provision and nitrogen pollution reduction from agriculture presents a dual challenge that requires an integrated solution. Here, we propose a cooperative food bank, where membership is a matter of choice and is not coerced. Membership requires participants to reduce nitrogen pollution in agriculture but creates a risk-buffering system, providing food compensation when participants are affected by risk factors. We delineate the structure of the cooperative food bank, its operation, from the short-term mobilization of resources to long-term capacity building. Lastly, we assess the feasibility of its implementation and highlight the potential major roadblocks to its implementation within the current socio-political context. The cooperative food bank showcases a novel solution that simultaneously tackles food insecurity and nitrogen pollution via governance. We hope this proposal will stimulate a research agenda and policy discussions focused on integrated approaches to effective governance regimes for linked socio-environmental problems.
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We study the evolution of cooperation in the spatial public goods game in the presence of third-party rewarding and punishment. The third party executes public intervention, punishing groups where cooperation is weak and rewarding groups where cooperation is strong. We consider four different scenarios to determine what works best for cooperation, in particular, neither rewarding nor punishment, only rewarding, only punishment or both rewarding and punishment. We observe strong synergistic effects when rewarding and punishment are simultaneously applied, which are absent if neither of the two incentives or just each individual incentive is applied by the third party. We find that public cooperation can be sustained at comparatively low third-party costs under adverse conditions, which is impossible if just positive or negative incentives are applied. We also examine the impact of defection tolerance and application frequency, showing that the higher the tolerance and the frequency of rewarding and punishment, the more cooperation thrives. Phase diagrams and characteristic spatial distributions of strategies are presented to corroborate these results, which will hopefully prove useful for more efficient public policies in support of cooperation in social dilemmas.
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The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.
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Machines powered by artificial intelligence increasingly mediate our social, cultural, economic and political interactions. Understanding the behaviour of artificial intelligence systems is essential to our ability to control their actions, reap their benefits and minimize their harms. Here we argue that this necessitates a broad scientific research agenda to study machine behaviour that incorporates and expands upon the discipline of computer science and includes insights from across the sciences. We first outline a set of questions that are fundamental to this emerging field and then explore the technical, legal and institutional constraints on the study of machine behaviour. Understanding the behaviour of the machines powered by artificial intelligence that increasingly mediate our social, cultural, economic and political interactions is essential to our ability to control the actions of these intelligent machines, reap their benefits and minimize their harms.
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This paper envisions a future where autonomous agents are used to foster and support pro-social behavior in a hybrid society of humans and machines. Pro-social behavior occurs when people and agents perform costly actions that benefit others. Acts such as helping others voluntarily, donating to charity, providing informations or sharing resources, are all forms of pro-social behavior. We discuss two questions that challenge a purely utilitarian view of human decision making and contextualize its role in hybrid societies: i) What are the conditions and mechanisms that lead societies of agents and humans to be more pro-social? ii) How can we engineer autonomous entities (agents and robots) that lead to more altruistic and cooperative behaviors in a hybrid society? We propose using social simulations, game theory, population dynamics, and studies with people in virtual or real environments (with robots) where both agents and humans interact. This research will constitute the basis for establishing the foundations for the new field of Pro-social Computing, aiming at understanding, predicting and promoting pro-sociality among humans, through artificial agents and multiagent systems.
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Fairness plays a fundamental role in decision-making, which is evidenced by the high incidence of human behaviors that result in egalitarian outcomes. This is often shown in the context of dyadic interactions, resorting to the Ultimatum Game. The peculiarities of group interactions – and the corresponding effect in eliciting fair actions – remain, however, astray. Focusing on groups suggests several questions related with the effect of group size, group decision rules and the interrelation of human and agents’ behaviors in hybrid groups. To address these topics, here we test a Multiplayer version of the Ultimatum Game (MUG): proposals are made to groups of Responders that, collectively, accept or reject them. Firstly, we run an online experiment to evaluate how humans react to different group decision rules. We observe that people become increasingly fair if groups adopt stricter decision rules, i.e., if more individuals are required to accept a proposal for it to be accepted by the group. Secondly, we propose a new analytical model to shed light on how such behaviors may have evolved. Thirdly, we adapt our model to include agents with fixed behaviors. We show that including hardcoded Pro-social agents favors the evolutionary stability of fair states, even for soft group decision rules. This suggests that judiciously introducing agents with particular behaviors in a population may leverage long-term social benefits.
Book
Building on the author's more than 35 years of teaching experience, Modeling and Analysis of Stochastic Systems, Third Edition, covers the most important classes of stochastic processes used in the modeling of diverse systems. For each class of stochastic process, the text includes its definition, characterization, applications, transient and limiting behavior, first passage times, and cost/reward models. The third edition has been updated with several new applications, including the Google search algorithm in discrete time Markov chains, several examples from health care and finance in continuous time Markov chains, and square root staffing rule in Queuing models. More than 50 new exercises have been added to enhance its use as a course text or for self-study. The sequence of chapters and exercises has been maintained between editions, to enable those now teaching from the second edition to use the third edition. Rather than offer special tricks that work in specific problems, this book provides thorough coverage of general tools that enable the solution and analysis of stochastic models. After mastering the material in the text, readers will be well-equipped to build and analyze useful stochastic models for real-life situations.