Optimal strategies and cost-benefit analysis of the n${\boldsymbol{n}}$-player weightlifting game

Abstract

The study of cooperation has been extensively studied in game theory. Especially, two-player two-strategy games have been categorized according to their equilibrium strategies and fully analysed. Recently, a grand unified game covering all types of two-player two-strategy games, i.e., the weightlifting game, was proposed. In the present study, we extend this two-player weightlifting game into an nn-player game. We investigate the conditions for pure strategy Nash equilibria and for Pareto optimal strategies, expressed in terms of the success probability and benefit-to-cost ratio of the weightlifting game. We also present a general characterization of nn-player games in terms of the proposed game. In terms of a concrete example, we present diagrams showing how the game category varies depending on the benefit-to-cost ratio. As a general rule, cooperation becomes difficult to achieve as group size increases because the success probability of weightlifting saturates towards unity. The present study provides insights into achieving behavioural cooperation in a large group by means of a cost–benefit analysis.

Full text

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Optimal strategies and cost‑benet
analysis of the
n
‑player
weightlifting game
Diane Carmeliza N. Cuaresma1,2*, Erika Chiba3, Jerrold M. Tubay2, Jomar F. Rabajante2,
Maica Krizna A. Gavina2, Jin Yoshimura1,4,5,6, Hiromu Ito4, Takuya Okabe7 & Satoru Morita1*
The study of cooperation has been extensively studied in game theory. Especially, two‑player two‑
strategy games have been categorized according to their equilibrium strategies and fully analysed.
Recently, a grand unied game covering all types of two‑player two‑strategy games, i.e., the
weightlifting game, was proposed. In the present study, we extend this two‑player weightlifting
game into an
n
‑player game. We investigate the conditions for pure strategy Nash equilibria and for
Pareto optimal strategies, expressed in terms of the success probability and benet‑to‑cost ratio
of the weightlifting game. We also present a general characterization of
n
‑player games in terms of
the proposed game. In terms of a concrete example, we present diagrams showing how the game
category varies depending on the benet‑to‑cost ratio. As a general rule, cooperation becomes
dicult to achieve as group size increases because the success probability of weightlifting saturates
towards unity. The present study provides insights into achieving behavioural cooperation in a large
group by means of a cost–benet analysis.
Competition and cooperation in human or animal society are prevalent1–5. e existence and evolution of coop-
eration have been an interest in various disciplines1,2,6–10. Studies in game theory aim to develop criteria for
selecting a strategy that maximizes gains and promotes cooperation11–17. Any situation can be considered a game
if agents maximize their own gains by anticipating the actions of their opponents18,19. A game requires only a
set of players, a set of strategies for each player, and corresponding pay-os for each strategy in response to the
strategies of other players. Rationality plays a strong role in determining what strategy a player should choose.
Rational players maximize their expected gains without caring about societal goals20–22. Under the assumption
of rationality, game theory nds an equilibrium of players’ strategies at the point where no player can gain from
changing his or her own strategy20. Game theory has received considerable attention from researchers as well as
decision makers seeking to solve problems of conict or cooperation18. Especially, the concept of the equilibrium
strategy has been applied in behavioural science and psychology2,8,23–25, computer science26,27, economics and
investments6,28–30, evolutionary biology3,4,10, and other elds.
Self-interest without regard to societal goals is best represented by the game known as the prisoner’s dilemma
(PD). In PD, two prisoners are to be convicted of a minor crime since prosecutors lack evidence to convict them
of a major crime. Separated and with no way to communicate, the prisoners are oered a reduced sentence if
they testify against each other. Rationality urges the two prisoners to betray one another, even though it is in
their best interest to remain silent2,31. e stag hunt (SH) game also presents a social dilemma. In the SH, two
hunters hunt for either a stag or a hare. ey depend on each other in terms of which animal to hunt since they
cannot kill the stag alone8. is results in two equilibria, one where both hunt a stag and another where both hunt
a hare, but the best outcome is the former8,32. In the hawk-dove game (HD), which is equivalent to the chicken
game (CH), the hawks are ready to ght for resources to drive the doves away, while the doves retreat whenever
the hawks are around. is game has two equilibria of (Dove, Hawk) and (Hawk, Dove), while the highest pay-
o is achieved for (Dove, Dove)2. Since cooperation and exploitation are prevalent in animals, game theory has
OPEN
1Graduate School of Science and Technology, Shizuoka University, Hamamatsu, Shizuoka 423-8561,
Japan. 2Institute of Mathematical Sciences and Physics, College of Arts and Sciences, University of the
Philippines Los Baños, 4031 Laguna, Philippines. 3Graduate School of Informatics, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464-8601, Japan. 4Department of International Health and Medical Anthropology, Institute
of Tropical Medicine, Nagasaki University, Nagasaki 852-8523, Japan. 5Department of Biological Sciences,
Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan. 6The University Museum, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan. 7Graduate School of Integrated Science and Technology, Shizuoka University,
Hamamatsu, Shizuoka 423-8561, Japan. *email: dncuaresma@up.edu.ph; morita.satoru@shizuoka.ac.jp
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been applied to study the evolution of animal behaviour, i.e., evolutionary game theory (EGT)2,3. In EGT, the
assumptions of the rationality of players and the equilibrium of strategies in classical game theory are replaced
with self-interest via Darwinian tness and evolutionarily stable strategies (ESSs), respectively3. Strategies in
EGT are behavioural phenotypes3,4.
e above three games and two trivial cases, C-dominant trivial (CT) and D-dominant trivial (DT), com-
prise the ve classes of two-by-two games (two players, two strategies: cooperation and defection), represented
by a
2×2
matrix (Table1)33. e pay-os in two-by-two games are represented by four quantities,
R,T,S
, and
P
. e reward
R
is received when the two players cooperate. e temptation
T
is experienced by a player who
then betrays the other player. e sucker
S
is the experience of the betrayed player. e punishment
P
is the
pay-o when both players betray each other. Depending on the values of
R,T,S
and
P
, two-by-two games are
categorized into the aforementioned ve types: prisoner’s dilemma (PD;
T≥R>P>S
), chicken game (CH;
T≥R≥S>P
), stag hunt game (SH;
R>T≥P>S
), D-dominant trivial (DT;
T≥P>R>S
) and C-dom-
inant trivial (CT;
R>T≥S>P
)7,33–35. DT and CT have equilibria of no dilemma, where all players defect or
cooperate, respectively7,34. Recently, Yamamoto etal.35 introduced the two-person weightliing game to unify
all the ve classes of dyadic games. In this game, each player either cooperates or defects in carrying a weight.
Studies on two-by-two games have contributed to understanding cooperation and dilemma in a social sys-
tem. However, many societal concerns require cooperation and decisions of not just two individuals5. Multiple-
player (or
n
-player) games have been studied extensively by researchers in various elds12,26,36–40, especially in
behavioural science41,42 and other application areas2,12,23,28,29,44,45. e most studied
n
-player cooperative game is
the public goods game (PGG)32,41, which is the
n
-player PD32,33. PGG models a society where members benet
equally from voluntary contributions (see refs.33 and43 for more discussion). Being an extension of PD, self-
interest causes individuals to make non-cooperative decisions. e
n
-player CH is typically used to model social
dilemmas caused by selsh individuals depleting a common resource33. Being equivalent to the
n
-player HD and
snowdri game, this game results in the coexistence of people who cooperate and people who free-ride on the
work of others (see refs.4,5). e
n
-player SH still gives equilibria where all hunters cooperate to take down a stag
or all defect to hunt hares instead (see ref.8). In these
n
-player games, it is generally expected that cooperation
will diminish as the group size increases owing to the rational behaviour of self-interested individuals4,20,38,41.
e two-person two-strategy weightliing game of Yamamoto etal.35 suggests a new way of investigating
n
-player games. In the present study, we extend this two-player game to an
n
-player game. Multiple-player games
have now become possible to study in a unied manner. We investigate the conditions for pure strategy equilibria
and optimal strategies, which we express with the success probability and the benet-to-cost ratio of this model.
Moreover, we provide the
n
-player extension of the classication conditions of two-by-two games according to
the equilibrium and optimal strategies. In the nal section, we discuss concrete examples of how the weightliing
game can explain behavioural cooperation in a large group.
Model and results
Preliminaries. To unify all the ve classes of two-by-two games, Yamamoto etal.35 introduced the weight-
liing game. In this game, each player either cooperates or defects in carrying a weight. Players who carry the
weight pay a cost,
c≥0
. e weight is successfully lied with probability
pi
, where
i=0,1,2
is the total number
of cooperators and
pi
increases with the number of cooperators
i
. If the cooperators succeed, both players receive
a benet
b>0
. However, in case of failure, both players gain nothing. e pay-o of the cooperators is
bpi−c
,
and the pay-o of the defectors is
bpi
(Table2). In terms of the parameters
p
1
=p
1
−p0
and
p
2
=p
2
−p1
,
which represents the increase in the probability of success due to an additional cooperator, the following ine-
qualities are obtained for the pay-os
R,T,S
, and
P
(Table1):
(i)
�p
1
>c/b
for
S>P
,
(ii)
�p
2
>c/b
for
R>T
, and
(iii)
�p
1
+�p
2
>c/b
for
R>P
.
Table 1. Pay-o table of the two-person two-strategy game.
Row\
Column
C
D
C
(R,R)
(S,T)
D
(T,S)
(P,P)
Table 2. Pay-o table of two-person weightliing game.
Row\
Column
C
D
C
R:bp2−c
S:bp1−c
D
T:bp1
P:bp0
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PD satises only (iii), CH satises (i) and (iii), SH satises (ii) and (iii), DT satises none of the three condi-
tions, and CT satises all three. In 2021, Chiba etal.1 studied the evolution of cooperation in society by incor-
porating environmental value in the weightliing game. ey found that the evolution of cooperation seems to
follow a DT to DT trajectory, which can explain the rise and fall of human societies.
The
n
‑player weightlifting game. In this study, we generalize the weightliing game to
n
-players. Sup-
pose
n
self-interested and rational individuals selected from a population of innite size. e
n
players are asked
to li a weight. Each individual (or player) can decide to either carry the weight (cooperate,
C
) or not carry/
pretend to carry the weight (defect,
D
). Players who decide to carry the weight can either succeed or fail. e
probability of successful weightliing is denoted by
pi
,
i=0, 1, ...,n
, where
i
indicates the number of coopera-
tors (henceforth,
i
always represents the number of cooperators). e probability of success increases with the
number of individuals cooperating, and it may remain less than unity even if all
n
individuals cooperate. Players
who decide to carry the weight pay a cost,
c≥0
, regardless of the outcome, while those who defect need not pay
anything. If the cooperators succeed, all
n
individuals receive a benet
b≥0
. ere is no penalty for failure. We
use the expected gains/losses of the players as the pay-o. If there are
i−1
cooperative players, then the pay-o
of
j
is
BC(i)=bpi−c
when
j
cooperates and
BD(i−1)=bpi−1
when
j
defects. e number of cooperators
diers by one, since in
BC(i)
, there is an additional cooperator, which is
j
him- or herself. To decide whether
to cooperate or defect, all players weigh their expected gain and rationally choose the option with the highest
expected gain. e graphical outline of this game is illustrated in Fig.1 (see also Supplementary Figure S1 for
the ow of the game). e pay-o table for a four-player game is shown as an example in Table3. Here, player
1
is the innermost row (strategies are listed in the second column of the table), player
2
is the innermost column
(strategies are listed in the second row of the table), and the succeeding players take the succeeding rows or
columns (we enter the rst player as a row player and the following player as a column player and continue in
this order). Each cell represents players’ pay-os, with the rst component being the pay-o for the rst player,
the second for the second player, and so on. For instance, consider the entry in the rst row and third column,
where players
1, 2
and
3
cooperate but player
4
defects. e pay-os of players
1
to
3
are
BC(3)
, while the pay-o
Figure1. A schematic diagram of the n-player weightliing game. In this game, players decide whether to
cooperate or defect in carrying the weight. Cooperators need to pay a cost. e weightliing can either succeed
or fail. In case of success, all players receive a benet. In case of failure, all players receive nothing. e player’s
pay-o depends on the benet, cost and probability of success. Each player decides whether to cooperate or
defect so as to maximize the expected gain.
Table 3. Pay-o table of four-player weightliing game.
Row\
Column
C
D
C
D
C
D
C
C
(BC(4),BC(4),BC(4),BC(4))
(BC(3),BD(3),BC(3),BC(3))
(BC(3),BC(3),BC(3),BD(3))
(BC(2),BD(2),BC(2),BD(2))
D
(BD(3),BC(3),BC(3),BC(3))
(BD(2),BD(2),BC(2),BC(2))
(BD(2),BC(2),BC(2),BD(2))
(BD(1),BD(1),BC(1),BD(1))
D
C
(BC(3),BC(3),BD(3),BC(3))
(BC(2),BD(2),BD(2),BC(2))
(BC(2),BC(2),BD(2),BD(2))
(BC(1),BD(1),BD(1),BD(1))
D
(BD(2),BC(2),BD(2),BC(2))
(BD(1),BD(1),BD(1),BC(1))
(BD(1),BC(1),BD(1),BD(1))
(BD(0),BD(0),BD(0),BD(0))
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of player
4
is
BD(3)
. In the above example, there are as many row players as column players because the number
of players is even. However, we can have one more player in the rows than in the columns if there is an odd
number of players.
Nash equilibrium and pareto optimal strategies. Here we present the Nash equilibrium and Pareto
optimal strategies of the
n
-player weightliing game in terms of the cost-to-benet ratio
c/b
and probability of
success
pi
. e Nash equilibrium consists of the best responses of each player. Players have no incentive to devi-
ate from this strategy prole since deviation will not increase an individual’s pay-o if the other players maintain
the same strategy. If
BC(i)≥BD(i−1)
, the best response of player
j
is to cooperate, but if
BC(i)≤BD(i−1)
,
the best response is to defect.
We have
pi=pi−pi−1≥0
for the increase in the probability of success because the probability
pi
increases
with the number of cooperators
i
. It is convenient to divide cases depending on whether
�pi>c/b
or
�pi<c/b
.
We obtain the following results (see Supplementary Text for the derivations):
Result 1 If
�p1≤c/b
, there is a Nash equilibrium at
(D,D,...,D)
. e Nash equilibrium at
(D,D,...,D)
is
unique if and only if
�pi<c/b
, for all
i=1, 2, ...,n
.
Result 2 If
�pn≥c/b
, there is a Nash equilibrium at
(C,C,...,C)
. e Nash equilibrium at
(C,C,...,C)
is
unique if and only if
�pi>c/b
, for all
i=1, 2, ...,n
.
Result 3 ere is a Nash equilibrium in the combination of strategies where
i−1
players choose
C
and the rest
of the players choose
D
if and only if
�pi<c/b<�pi−1
, for some
i=2, 3, ...,n
.
Result 1 shows that players have no incentive to cooperate when the cost relative to the benet is (very) high,
so much so that
�pi<c/b
, for all possible values of
i
. is case of all defection is a unique equilibrium, where no
player can improve the pay-o by cooperating. In contrast, Result 2 shows that all players cooperate when the cost
is suciently smaller than the benet. Results 1 and 2 indicate that cooperation is determined by the relationship
between the cost and the benet; raising the benet or lowering the cost can increase cooperation. ere may be
cases where full defection or cooperation is not a unique equilibrium (see cases 3 or 10, for example, in Table4).
e reason for this is covered by Result 3. is result shows the conditions for the existence of equilibria where
only some individuals cooperate, which we will refer to as anti-coordination equilibria. Result 3 also implies
the signicance of an individual in promoting cooperation. For instance, when
�p2<c/b<�p1
, we have an
equilibrium with a single cooperator. While there is a small chance of success, if an individual’s contribution to
the probability of success is substantial, cooperation will exist. ese three results cover all possible cases of pure
equilibrium. e equilibria at
(D,D,...,D)
and at
(C,C,...,C)
are covered by Results 1 and 2, respectively, and
the anti-coordination equilibria are covered by Result 3.
Result 4 e number of equilibria of an
n
-player weightliing game is at most
⌊n
2
⌋
i=0
C(n,2i
)
if
n
is even and
⌊n
2
⌋
i=0
C(n,2i)+
1
if
n
is odd, where
C(n,2i)
denotes the combination of
2i
out of
n
.
Result 4, on the other hand, gives the maximum number of equilibria in a weightliing game. To illustrate
this result, the equilibrium strategies (marked with X) of a four-player game are presented in Table4. Notably,
the one X in case 2 means not just one equilibrium but four equilibria:
(C,D,D,D)
,
(D,C,D,D),
(D,D,C,D)
and
(D,D,D,C)
. e same applies to the other cases (except 1 and 16). As shown in Table4, there can be at most three
types of equilibrium (case 11): all-
D
, anti-coordination, and all-
C
. ere is exactly one all-
D
and exactly one all-
C
strategy. However, there are
(2+2)!/(2!2!)=C(4, 2)
anti-coordination equilibria of two players cooperating and
two players defecting; thus, there are at most eight equilibria in a four-player game. is nding is in accordance
with Result 4:
2
i=0
C(4, 2i)=C(4, 0)+C(4, 2)+C(4, 4)=
8
.
Table 4. Equilibrium strategies of a four-player weightliing game.
Case
p4
p3
p2
p1
all-
D
1C,3D
2C,2D
3C,1D
all-
C
1 <
c/b
<
c/b
<
c/b
<
c/b
X
2 <
c/b
<
c/b
<
c/b
>
c/b
X
3 <
c/b
<
c/b
>
c/b
<
c/b
X X
4 <
c/b
<
c/b
>
c/b
>
c/b
X
5 <
c/b
>
c/b
<
c/b
<
c/b
X X
6 <
c/b
>
c/b
<
c/b
>
c/b
X X
7 <
c/b
>
c/b
>
c/b
<
c/b
X X
8 <
c/b
>
c/b
>
c/b
>
c/b
X
9 >
c/b
<
c/b
<
c/b
<
c/b
X X
10 >
c/b
<
c/b
<
c/b
>
c/b
X X
11 >
c/b
<
c/b
>
c/b
<
c/b
X X X
12 >
c/b
<
c/b
>
c/b
>
c/b
X X
13 >
c/b
>
c/b
<
c/b
<
c/b
X X
14 >
c/b
>
c/b
<
c/b
>
c/b
X X
15 >
c/b
>
c/b
>
c/b
<
c/b
X X
16 >
c/b
>
c/b
>
c/b
>
c/b
X
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In Pareto optimal strategies, players cannot increase their pay-os by changing their strategy without also
decreasing the other players’ pay-os. Owing to
pi≤pi+1
,
BD(i)≤BD(i+1)
and
BC(i)≤BC(i+1)
. us, if
a defector cooperates, the rest of the players will enjoy an increased pay-o. Moreover, some players will suer
from a decreased pay-o if cooperators decrease. In this case, we only have to check the condition that makes a
strategy prole Pareto-dominated, i.e., when defectors cooperate.
Result 5 Strategy
(C,C,...,C)
is Pareto optimal if and only if
n
j=
1�pj>c/
b
.
Result 6 e strategy prole with
i
defectors,
i=0, 1, ...,n−1
, is Pareto optimal if and only if
n
j
=
i
+1�pj<c/b
.
In
(C,C,...,C)
, the only way a player can deviate is to defect; thus, it is sucient to check the condition where
all-
D
Pareto-dominates all-
C
. However, in the following result, which covers the remaining strategies, all-
D
does
not Pareto-dominate these strategies since defectors are disadvantaged. Furthermore, we know that
n
j=
1p
j
saturates towards unity. us, intuitively, cooperation is Pareto optimal unless
c
is close to or greater than
b.
General properties of the
n
‑player games. While Yamamoto etal.35 considered only the conditions
that encourage cooperation, the violation of these conditions implicitly implies the satisfaction of the converse
conditions. us, PD, SH and DT satisfying
�p1<c/b
assures equilibrium at
(D,D)
. Moreover, PD and DT
satisfying
�p2<c/b
makes this equilibrium unique, according to Result 1. On the other hand, SH satisfying
�p2>c/b
leads to another equilibrium at
(C,C)
(Result 2). e anti-coordination equilibrium of CH is cov-
ered by the condition
�p2<c/b<�p1
of Result 3. In addition, the condition
�p1+�p2>c/b
(condition
iii), which PD, CH, SH and CT satisfy, indicates that all-
C
is more benecial than all-
D
. As in Result 5, the
counterpart of this condition for the
n
-player game is
n
j=
1�pj>c/
b
. Similarly, the inequality
n
j=
1�pj<c/
b
indicates that all-
D
is more benecial than all-
C
in Result 6 when
i=0
.
e ve classes of two-by-two games are characterized by their equilibria and optimal strategies. All these
games are unied under a single structure in the two-player weightliing game. As an extension of the
n
-player
game, the following correspondence occurs: With all-
C
being the optimal strategy, PD has a unique equilibrium
at all-
D
, CH has an anti-coordination equilibrium, SH has an equilibrium at both all-
C
and all-
D
, CT has a unique
equilibrium at all-
C
, and DT has a unique and optimal equilibrium at all-
D
. In the above results, we have shown
the existence and uniqueness of an equilibrium and the existence of optimal strategies. In summary, we present
the conditions and characterization of the
n
-player games in Table5.
Illustration. Let us consider a concrete example of liing a weight of
W=100
by four individuals (Figs.2
and 3). e weight that each individual cay carry is normally distributed with mean
µ
and standard deviation
σ
.
For
µ=10
and
σ=50
, we obtain
p1=0.013, p2=0.019, p3=0.026
and
p4=0.034
(Figs.2a1, 2a2).
e n-player CT obtains for
0<c/b<0.013
, SH for
0.013 <c/b<0.034
, PD for
0.034 <c/b<0.092
, and DT
for
0.092 <c/b<1
. In Figs.2a1 and 2a2, we show the parameter regions for Nash equilibria and Pareto optimal
strategies as hatched in the i-
c/b
plane, where i is the number of cooperators and
c/b
is the cost-to-benet ratio.
As
c/b
increases, the number
i
of cooperators drops from four to zero in Nash equilibria (Fig.2a1). In Pareto
optimal strategies, the number
i
decreases from four to zero, while the range of
c/b
for
i=0∼3
reaches the
right end point
c/b=1
(Fig.2a2). Note that the boundary values for the hatched bars are dierent for Nash
equilibria (Fig.2a1) and Pareto optimal strategies (Fig.2a2). Similarly, we obtain Figs.2b–e and 3a–e for
µ
from
20 to 100. e range for each game category varies depending on
µ
. As
µ
increases, SH ceases to exist (Fig.2e)
while the coexistence CH&PD begins to appear (Fig.2d) and disappear (Fig.3e). A pure CH appears aerwards
(Fig.3b).
Discussion
e present game is related to
n
-player Prisoner’s Dilemmas (
n
PDs), or Public Goods games (PGG)33,43. Consider
the public goods game played by
i
cooperators and
j=n−i
defectors. Each cooperator contributes
c
to the pub-
lic pool. Total contributions
ic
is equally distributed among all players aer multiplied by a factor
R
. us, each
player gains
icR/n
. Since this quantity is compared with
bpi
of the weightliing game, we see the correspondence
of
cR
to
b
and
i/n
to
pi
. Accordingly, the success probability
pi
of the weightliing game corresponds to the ratio
of cooperators among all players in the public goods game. Unlike this specic case, however, in general, and in
principle, the dependence of
pi
on the ratio
i/n
can be nonlinear. e eect of this nonlinearity is properly taken
into consideration in a general model of the weightliing game. For instance, as a second example, let us consider
n
-player stag hunt dilemmas (
n
SH). Pacheco etal.32 studied evolutionary dynamics in
n
SH, where it is assumed
that the “public goods” increases with the number of cooperators
i
inasmuch as
i
exceeds a certain threshold
value
M
while it is zero for
i<M
. is game is formally equivalent to replacing
R
of PGG with
Rθ(i−M)
, where
Table 5. Conditions for the
n
-player extension of two-by-two games.
Conditions for equilibrium strategy Conditions for optimal strategy
PD
�pi<c/b,∀i∈P
n
j=
1�pj>c/
b
Results 1 and 5
CH
�pi<c/b<�pi−1,∃i∈{2, 3, ...,n}
n
j=
1�pj>c/
b
Results 3 and 5
SH
�p1≤c/b
,
�pn≥c/b
n
j=
1�pj>c/
b
Results 1, 2 and 5
CT
�pi>c/b
,
∀i∈P
n
j=
1�pj>c/
b
Results 2 and 5
DT
�pi<c/b,∀i∈P
n
j=
1�pj<c/
b
Results 1 and 6
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Figure2. Equilibria and optimal strategies of the four-player weightliing game. Nash equilibria
(a1,b1,c1,d1,e1) and Pareto optimal strategies (a2,b2,c2,d2,e2). (a1,a2)
µ
=
10
. (b1,b2)
µ
=
20
. (c1,c2)
µ
=
30
.
(d1,d2)
µ
=
40
. (e1,e2)
µ
=
50
. e parameter regions for Nash equilibria and Pareto optimal strategies are
as hatched in the i-
c/b
plane, where i is the number of cooperators and
c/b
is the cost-to-benet ratio. We set
σ=50
in all cases. All players cooperate for a small value of
c/b
(CT), while they defect for a large value (DT).
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θ(x)
is the Heaviside step function satisfying
θ(x)=0
for
x<0
and
θ(x)=1
for
x≥0
. Consequently,
n
SH is
recovered by the weightliing game under the assumption
pi=iθ(i−M)/n
, i.e., we need at least
M
coopera-
tors for the weightliing to be successful (or to produce any benets). A third example is provided by a
n
-player
Figure3. Equilibria and optimal strategies of the four-player weightliing game. Nash equilibria
(a1,b1,c1,d1,e1) and Pareto optimal strategies (a2,b2,c2,d2,e2). (a1,a2)
µ
=
60
. (b1,b2)
µ
=
70
. (c1, c2)
µ
=
80
. (d1, d2)
µ
=
90
. (e1, e2)
µ
=
100
. See Fig.2 and the text for details.
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snowdri game46, while it is more sophisticated. Souza etal.46 studied the
n
-person snowdri game in which the
cost paid by a cooperator depends on the number of cooperators,
i
, i.e., it is given by
C/max(i,M)
, where
M
is
a minimum number of cooperators required for achieving a benet and
max(i,M)=i
if
i≥M
and
M
if
i<M
.
Each player receives a benet
Bθ(i−M)
, which is zero for
i<M
and
B
for
i≥M
. is game is obtained by
assuming
pi=θ(i−M)×max(i,M)/n=iθ(i−M)/n
as above, but with
b/c=nB/C
depending on
n
because
each cooperator’s cost decreases with the total number. is explicit
n
-dependence of the benet-to-cost ratio
may reinforce the impact of group size, as mentioned just below.
It has been generally acknowledged that cooperation becomes dicult to achieve as group size increases
4,20,37,41,46–48. e eects of group size may be discussed from a static perspective40. For instance, the size depend-
ence comes in through the decrease in the benet-to-cost ratio as the number of players increases. When the total
gain
W
does not increase in proportion to the number of players
n
, the benet of each player
b=W/n
decreases
as the number of players
n
increases. us, the inequality
�pi<c/b
should be met for all
i
eventually, because
the right-hand side increases in proportion to
n
. In other words, the larger the group, the less cooperative people
will be. However, when the total gain
W
increases in proportion to
n
, the right-hand side
c/b
stays constant even
if
n
increases. In this case, the impact of group size on cooperation can be positive (or, to be precise, the negative
eect of group size is mitigated when the benets reaped by one individual do not reduce the benets received
by another). In fact, the impact in the latter case (
W∝n
) has been studied as compared specically against the
former case (
W=
const.) (see, e.g., refs.49,50 ). Recently, the emergence of cooperation in a large group has been
extensively studied by means of dynamical models32,48,50. In this context, it should be remarked that the size eect
may also come about as a result of dynamical, stochastic processes of how the numbers of players with dierent
strategies vary, namely a genetic dri in evolutionary biology47. While assessing if the size eect due to genetic
dri is positive or negative requires further assumptions than necessary for the present ‘static’ results, we made
a calculation to nd that the size eect, as evaluated from Eq.(2.5) of Kurokawa and Ihara47, is negative (Sup-
plementary Text). us, we consider it an interesting future research direction to investigate population dynamics
of the present game, especially to make a more specic comparison with these prior studies.
Several studies8,32,46 discuss a minimum number
M
of players for cooperation, specically anti-coordination,
to exist. Our study can also supply this concept of threshold using the parameter
pi
. A good example is provided
by the concept of a ‘threshold’ in joining a strike, which is dened as the number of people in the strike for a given
employee to join the strike (see ref.51). is number (threshold) may be dierent for a dierent individual. In fact,
it is evaluated according to Result 3; an employee will join the strike under the condition
�pi<c/b<�pi−1
when
i−2
employees are in the strike. In the present model, the probability of success is used instead of the risk
preference to evaluate the threshold value. When each individual has his/her own success probabilities
pi
, t he
threshold can be dierent for each individual if the cost-to-benet ratio
c/b
is a xed constant. Specically, if
�p2<c/b<�p1
, a single employee (‘instigator’) will decide to start the strike, while it can be that the threshold
becomes so high that the condition
�pi<c/b<�pi−1
is not met for any
i
.
is ‘threshold’ behaviour is not unique to humans. Conradt and Roper52 studied social animals making
communal decisions. e animals decide how long they conduct a communal activity, which is benecial to
the group but takes time away from their own personal activities46,52. Conradt and Roper52 named this loss of
personal time the ‘synchronization cost’. ey noted that the animals that stop communal activity earlier should
have twice as much motivation as the others (‘double motivation’). If
i−1
animals pursue communal activities,
the animals to stop earlier (defect) are those that satisfy the condition
�pi<c/b<�pi−1
. In the present model,
the motivation for communal activity is modelled with the probability of success.
In the previous section (Section “Illustration”), we presented that the game follows CT-SH-PD-DT, CT-SH-
CH&PD–PD-DT, CT-CH&PD–PD-DT, CT-CH-CH&PD–PD-DT, and CT-CH-PD-DT (Figs.2 and 3). is
is consistent with the previous result1, while there are slight dierences in the order and where SH appears.
ese trajectories may be used to explain the dynamical process of joining a strike51. We may regard
µ
as the
employee’s rank in the company. e larger
µ
, the higher the rank. When the consequences for joining the
strike are minor, all employees are persuaded to join the strike (game type CT), regardless of the rank. As the
consequences become severe, they are inclined not to join the strike, especially for those with a low rank
µ
(SH).
If the consequences become more severe, those with a high rank
µ
are encouraged to leave the strike (PD). Not
surprisingly, no employees will join the strike when faced with much more severe penalties (DT). is is just an
example. Many other cases can be analysed from the perspective of the present study.
e weightliing game does not only pertain to the physical act of carrying a load, as we have seen in the
examples provided above. e probability of successful weightliing
pi
can also be interpreted in dierent ways.
For example, it can be interpreted as the probability of not depleting the public resource, the probability of driv-
ing the other species away, or the probability of taking down the stag. We can also regard
pi
as the probability of
i
individuals to reproduce and for its species to prevent extinction. Many other interpretations are possible. is
study can be extended in several ways: (1) by incorporating environmental eects, such as spatial and temporal
parameters; (2) by generalizing the cost
c
and benet
b
to depend on the players; (3) by considering the risk
aversion of the players; and (4) by considering pre-play communication. e evolution from one class of game
to another is also worth studying.
Data availability
is study is theoretical and does not use any data. In the illustration, all results can be computed directly from
the values and formulas presented in the text.
Received: 5 March 2022; Accepted: 27 April 2022
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References
1. Chiba, E. et al. Improving environment drives dynamical change in social game structure. R. Soc. Open Sci. 8, 201166 (2021).
2. Chen, W., Gracia-Lázaro, C., Li, Z. & Wang, L. Evolutionary dynamics of N-person hawk-dove games. Sci. Rep. 7, 4800 (2017).
3. Smith, J. Evolution and the theory of games (Cambridge University Press, 1982).
4. Nowak, M. Evolutionary dynamics: exploring the equations of life (Harvard University Press, Cambridge, MA, 2006).
5. Sui, X., Cong, R., Li, J. & Wang, L. Evolutionary dynamics of N-person snowdri game. Phys. Lett. A 379, 2922–2934. https:// doi.
org/ 10. 1016/j. physl eta. 2015. 08. 029 (2015).
6. Amaldoss, W., Meyer, R., Raju, J. & Rapoport, A. Collaborating to compete. Market. Sci. 19, 105–126. https:// doi. org/ 10. 1287/
mksc. 19.2. 105. 11804 (2000).
7. Ito, H. & Tanimoto, J. Scaling the phase-planes of social dilemma strengths shows game-class changes in the ve rules governing
the evolution of cooperation. R. Soc. Open Sci. 5, 181985. https:// doi. org/ 10. 1098/ rsos. 181085 (2018).
8. Luo, Q., Liu, L. & Chen, X. Evolutionary dynamics of cooperation in the N-person stag hunt game. Physica D 424, 132943. https://
doi. org/ 10. 1016/j. physd. 2021. 132943 (2021).
9. Tanimoto, J. & Sagara, H. Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-
player symmetric game. BioSystems 90, 105–114. https:// doi. org/ 10. 1016/j. biosy stems. 2006. 07. 005 (2007).
10. Wang, Z., Szolnoki, A. & Perc, M. Rewarding evolutionary tness with links between populations promotes cooperation. J. eoret.
Biol. 349, 50–56. https:// doi. org/ 10. 1016/j. jtbi. 2014. 01. 037 (2014).
11. Askari, G., Gordiji, M. & Park, C. e behavioral model and game theory. Palgrave Commun. 5, 57 (2019).
12. Parsons, S. & Wooldridge, M. Game theory and decision theory in multi-agent systems. Autonom. Agents Multi-agent Syst. 5,
243–254. https:// doi. org/ 10. 1023/A: 10155 75522 401 (2002).
13. Perc, M. Phase transitions in models of human cooperation. Phys. Lett. A 380, 2803–2808. https:// doi. org/ 10. 1016/j. physl eta. 2016.
06. 017 (2016).
14. Perc, M. & Wang, Z. Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game. PLoS ONE 5, e1117. https://
doi. org/ 10. 1371/ journ al. pone. 00151 17 (2010).
15. Rand, D. & Nowak, M. Human cooperation. Trends Cognit. Sci. 17, 413–425. https:// doi. org/ 10. 1016/j. tics. 2013. 06. 003 (2013).
16. Sigmund, K. e calculus of selshness (Princeton University Press, 2010).
17. von Neumann, J. & Morgenstern, O. e theory of games and economic behavior (Princeton University Press, 1944).
18. Poncela-Casasnovas, J. et al. Humans display a reduced set of consistent behavioral phenotypes in dyadic games. Sci. Adv. 2,
e1600451. https:// doi. org/ 10. 1126/ sciadv. 16004 51 (2016).
19. Ross, D. Game theory in e Stanford encyclopedia of philosophy (Winter 2019 ed) (ed. Zalta, E.N.); https:// plato. s tanf ord. edu/ archi
ves/ win20 19/ entri es/ game- theory/ (2019).
20. Fleishman, J. e eects of decision framing and others’ behavior on cooperation in a social dilemma. J. Conict Resolut. 32,
162–180 (1988).
21. Rabin, M. Incorporating fairness into game theory and economics. Am. Econ. Rev. 83, 1281–1302 (1993).
22. Skyrms, B. 1997. Game theory, rationality and evolution in Structures and norms in science. Synthese library (studies in epistemology,
logic, methodology, and philosophy of science) 260 (eds. Chiara, M., Doets, K., Mundici, D., &Van Benthem, J.) 73–74; https:// doi.
org/ 10. 1007/ 978- 94- 017- 0538-7_5 (Dordrecht: Springer, 1997).
23. Archetti, M. et al. Economic game theory for mutualism and cooperation. Ecol. Lett. 14, 1300–1312l. https:// doi. org/ 10. 1111/j.
1461- 0248. 2011. 01697.x (2011).
24. Camerer, C., Ho, T. & Chong, J. Behavioural game theory: thinking, learning and teaching in Advances in understanding strategic
behaviour (ed. Huck, S.) 120–180; https:// doi. org/ 10. 1057/ 97802 30523 371_8 (London: Palgrave Macmillan, 2004).
25. Costa-Gomez, M., Crawford, V. & Broseta, B. Cognition and behavior in normal-form games: an experimental study. Econometrica
69, 1193–1235. https:// doi. org/ 10. 1111/ 1468- 0262. 00239 (2003).
26. Abraham, I., Dolev, D., Gonen, R. & Halpern, J. Distributed computing meets game theory: robust mechanisms for rational secret
sharing and multiparty computation. Proceedings of the 25th ACM symposium on principles of distributed computing, 53–62; https://
doi. org/ 10. 1145/ 11463 81. 11463 93 (2006).
27. Durlauf, S. & Blume, L. Computer science and game theory in Game eory. e New Palgrave Economics Collection (eds. Durlauf,
S. & Blume, L) 48–65 (London: Palgrave Macmillan, 2010). https:// doi. org/ 10. 1057/ 97802 30280 847_5.
28. Allen, F. & Morris, S. Game theory models in nance in Game theory and business applications. International series in operations
research and management science, 35 (Chatterjee, K. & Samuelson, W.), pp. 17–48 (Boston, MA: Springer, 2002). https:// doi. org/
10. 1007/0- 306- 47568-5_2.
29. Arasteh, A. Combination of real options and game-theoretic approach in investment analysis. J. Ind. Eng. Int. 12, 361–375. https://
doi. org/ 10. 1007/ s40092- 016- 0144-z (2016).
30. De Bondt, W. & aler, R. Chapter13 Financial decision making in markets and rms: a behavioral perspective in Handbooks in
operations research and management science 9 (eds. Jarrow, R.A., Maksimovic, V. & Ziemba, W.T.) 385–410; https:// doi. org/ 10.
1016/ S0927- 0507(05) 80057-X (Elsevier,1995).
31. Kuhn, S. Prisoner’s Dilemma in e Standford encyclopedia of philosophy (Winter 2019 ed.) (ed. Zalta, E.N.); https:// plato. stanf ord.
edu/ archi ves/ win20 19/ entri es/ priso ner- dilem ma/ (2019).
32. Pacheco, J.M., Santos, F.C., Souza, M.O., & Skyrms, B. Evolutionary Dynamics of Collective Action in N-Person Stag Hunt Dilem-
mas. Proceedings: Biological Sciences 276(1655), 315–321; http:// www. jstor. org/ stable/ 30244 860 (2009).
33. Tanimoto, J. Fundamentals of evolutionary game theory and its applications (Springer, 2015).
34. Wang, Z., Kokubo, S., Jusup, M. & Tanimoto, J. Universal scaling for the dilemma strength in evolutionary games. Phys. Life Rev.
14, 1–30. https:// doi. org/ 10. 1016/j. plrev. 2015. 04. 033 (2015).
35. Yamamoto, T. et al. A single “weight-liing” game covers all kinds of games. R. Soc. Open Sci. 6, 1902. https:// doi. org/ 10. 1098/
rsos. 191602 (2019).
36. Aleta, A. & Moreno, Y. e dynamics of collective behavior in a crowd controlled game. EPJ Data Sci. 8, 22. https:// doi. org/ 10.
1140/ epjds/ s13688- 019- 0200-1 (2019).
37. Aumann, R. Acceptable points in general cooperative n-person games in Contribution to the theory of games IV (eds. Luce, R. &
Tucker, A.) 287–324 (Princeton University Press, 1959).
38. Bornstein, G. A classication of games by player type in New issues and paradigms in research on social dilemmas (eds. Biel, A.,
Eek, D., Garling, T. & Gustafsson, M.) 27–42; https:// doi. org/ 10. 1007/ 978-0- 387- 72596-3_3 (Boston, MA: Springer, 2008).
39. Kiss, H., Rosa-Garcia, A. & Zhukova, V. Conditional cooperation in group contests. PLoS ONE 15, e0244152. https:// doi. org/ 10.
1371/ journ al. pone. 02441 52 (2020).
40. Powers, S. T. & Lehmann, L. When is bigger better? e eects of group size on the evolution of helping behaviours. Biol. Rev.
92(2), 902–920. https:// doi. org/ 10. 1111/ brv. 12260 (2017).
41. Brewer, M. & Kramer, R. Choice behavior in social dilemmas: Eects of social identity, group size, and decision framing. J. Pers.
Soc. Psychol. 50(3), 543–549. https:// doi. org/ 10. 1037/ 0022- 3514. 50.3. 543 (1986).
42. Sabato, I., Rabinovich, Z., Obrztsova, S. & Rosenschein, J. Real candidacy games: a new model for strategic candidacy. Proceedings
of the 16th international conference on autonomous agents and multiagent systems, pp. 867–875. https:// doi. org/ 10. 5555/ 30911 25.
30912 48 (2017).
Content courtesy of Springer Nature, terms of use apply. Rights reserved

10
Vol:.(1234567890)
Scientic Reports | (2022) 12:8482 | https://doi.org/10.1038/s41598-022-12394-z
www.nature.com/scientificreports/
43. Wu, Z., Huang, H. & Liao, Q. e study on the role of dedicators on promoting cooperation in public goods game. PLoS ONE 16,
e0257475. https:// doi. org/ 10. 1371/ journ al. pone. 02574 75 (2021).
44. Chu, F. & Halpern, J. A decision-theoretic approach to reliable message delivery. Distrib. Comput. 14, 1–16. https:// doi. org/ 10.
1007/ PL000 08922 (2001).
45. Zhou, W., Koptyug, N., Ye, S., Jia, Y. & Lu, X. An extended n-player network game and simulation of four investment strategies on
a complex innovation network. PLoS ONE 1, e0145407. https:// doi. org/ 10. 1371/ journ al. pone. 01454 07 (2016).
46. Souza, M. O., Pacheco, J. M. & Santos, F. C. Evolution of cooperation under N-person snowdri games. J. eor. Biol. 260(4),
581–588. https:// doi. org/ 10. 1016/j. jtbi. 2009. 07. 010 (2009).
47. Kurokawa, S. & Ihara, Y. Emergence of cooperation in public goods games. Proc. Biol. Sci. 276(1660), 1379–1384. https:// doi. org/
10. 1098/ rspb. 2008. 1546 (2009).
48. Joshi, N. V. Evolution of cooperation by reciprocation within structured demes. J. Genet. 66, 69–84. https:// doi. org/ 10. 1007/ BF029
34456 (1987).
49. Taylor, M. e possibility of cooperation (Cambridge University Press, 1987).
50. Boyd, R. & Richerson, P. J. e evolution of reciprocity in sizable groups. J. eor. Biol. 132(3), 337–356. https:// doi. org/ 10. 1016/
S0022- 5193(88) 80219-4 (1988).
51. Granovetter, M. reshold models of collective behavior. Am. J. Sociol. 83, 1420–1443 (1978).
52. Conradt, L. & Roper, T. Group decision-making in animals. Nature 421, 155–158. https:// doi. org/ 10. 1038/ natur e01294 (2003).
Acknowledgements
is work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (grants no.
17J06741, 17H04731, 19KK0262 and 21H01575 to H.I.; grants no. 18K03453 and 21K03387 to S.M.; grants no.
15H04420 and 26257405 to J.Y.; grant no. 21K12047 to T.O.).
Author contributions
D.C.N.C., E.C., J.Y., S.M. and T.O. conceived the study and developed the original model. D.C.N.C., J.F.R., J.M.T.,
M.K.A.G., H.I., S.M. and T.O. analysed and nalized the model. D.C.N.C., J.Y., S.M. and T.O. wrote the dra
manuscript. All authors revised and nalized the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 022- 12394-z.
Correspondence and requests for materials should be addressed to D.C.N.C.orS.M.
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