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OPEN
Evolution of fairness
in the divide‑a‑lottery game
Jeong‑Yoo Kim 1* & Kyu‑Min Lee 2*
In this paper, we show that fairness can evolve in the divide‑a‑lottery game which is more general than
the divide‑a‑dollar game by using an indirect evolutionary approach. In the divide‑a‑lottery game, the
size of a pie is uncertain. Two players sequentially bid for a share and they get their bid if the allocation
based on the bids turns out to be feasible and otherwise neither gets anything. In this game, rational
players over‑compete for a higher share, resulting in a high probability of failure in agreement,
whereas fair players who dislike the disparity between shares lower their bids thereby reducing the
failure probability and thus increasing the expected payoff. As a result, fairness strictly dominates
rationality. This is the mechanism through which fairness evolves. However, this result is not robust
against even a slight uncertainty about the opponent’s type. Surprisingly, we show a contrasted
simulation result that only rational players who are strictly dominated by fair players survive
evolutionarily for most of the parameter values if players have even a slight chance of not knowing the
opponent’s type. Our simulation results in a local interaction model in which players only know the
type of closer neighbors capture both insights and demonstrate that moderate proportions of both
types coexist evolutionarily over time, and that the population average fitness of this polymorphic
population is higher than monomorphic population consisting only of fair types or rational types.
Although most economists and game theorists assume that material self-interest is the sole motivation of people,
there is overwhelming counter-evidence gathered by psychologists and experimental economists. This evidence
indicates that a substantial percentage of human beings are strongly motivated by other-regarding preferences
including fairness, altruism etc. (For recent theoretical developments in evolution of prosocial cooperative
behavior in various situations, e.g., in heterogeneous network structures, directional networks, and multilayer
interactions, see McAvoy et al.1, Su et al.2,3). Considering that the selfish behavior, by definition, maximizes the
individual’s utility or fitness and thus only homo economicus appears to be able to survive in the long run, it
is rather puzzling that fair behavior survives in the long run in an evolutionary environment. The evidence of
fairness is, however, well documented. For example, in the ultimatum game, a robust result across hundreds of
experiments is that the vast majority of the offers are between 40 and 50 percent of the available surplus (see, for
example, Güth et al.4, Camerer and Thaler5, Roth6, Camerer7).
In this paper, we show that fairness can evolve in the divide-a-lottery game which is more general than the
divide-a-dollar game by using an indirect evolutionary approach. (In the indirect evolutionary approach, which
was developed by Güth and Yaari8 and Güth9, preferences are treated as endogenous in an evolutionary process,
while actions are still determined by Nash equilibrium). A divide-a-dollar game, which is also known as a Nash
demand game10 is one of the most widely used bargaining games, describing a procedure of how to split a dollar.
Unlike the ultimatum game that is sequential in the sense that one player proposes a share and then the other
player decides whether to accept or reject it, a divide-a-dollar game is simultaneous. The game goes as follows.
Player 1 and player 2 simultaneously bid the amount of x and y respectively to divide a dollar. If the bids turn out
to be a feasible division in the sense that x + y ≤ 1, each of them gets the share of their own bid, but otherwise
neither gets anything. In this game, both players have a chance to bid, unlike the ultimatum game.
In real situations, bargaining is usually proceeded sequentially and both players have a chance to bid. Therefore, we consider a combination of the two bargaining games in which the two players offer bids sequentially.
Unlike the ultimatum game, however, the value of the pie is uncertain. We call this a divide-a-lottery game. So,
in a divide-a-lottery game, players bargain for a lottery by bidding sequentially. (Wang et al.11 also introduce the
randomness associated with the size of pies into the model, but they consider a variant of an ultimatum game,
not a variant of a divide-a-dollar game; hence, no sequential bidding in their model).
1
Department of Economics, Kyung Hee University, 1 Hoegi-dong, Dongdaemun-gu, Seoul 137-701, Republic
of Korea. 2College of Business, Korea Advanced Institute of Science and Technology, Seoul 02455, Republic of
Korea. *email: jyookim@khu.ac.kr; kyuminlee@kaist.ac.kr
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If two rational players bid sequentially, there is a first mover advantage. So, the first bidder bids 1 and the sec2
ond bidder bids 41 when the value of the lottery is uniformly distributed on [0, 1]. It results in a high probability
of disagreement (i.e., infeasible allocation) due to severe bidding competition. However, a fair player who feels
disutility from disparate bargaining shares makes the other player reduce the bid, increasing the probability of
agreement. The upshot is that fairness has the role of lowering the bid thereby increasing the expected payoff. As
a result, fairness strictly dominates rationality. This is the mechanism through which fairness evolves. However,
this result is not robust against even a slight uncertainty about the opponent’s type. Surprisingly, we show a
contrasted simulation result that only rational players who are strictly dominated by fair players survive evolutionarily for most of the parameter values if players have even a slight chance of not knowing the opponent’s type.
Also, through simulations, we show that moderate proportions of both types coexist evolutionarily over time,
and that the population average fitness of this polymorphic population is higher than monomorphic population
consisting only of fair types or rational types.
Many authors have demonstrated that fairness evolves in the ultimatum game, not in the divide-a-dollar game
nor in the divide-a-lottery game. Nowak et al.12 highlighted the role of reputation in the evolution of fairness.
If players interact repeatedly, accepting low offers as rational players do can induce the next proposer to make
a low offer, so a fair strategy offering and demanding a high share can fare better than a rational strategy. Rand
et al.13 introduced the possibility of making mistakes to explain the evolution of fairness. Ichinose and Sayama14
considered a game which they call not quite ultimatum game in a spatial interaction. Bethwaite and Tompkinson15
considered players who are concerned about equity of the allocation similar to our model but did not investigate
the evolutionary process of fairness.
Our result that fairness can survive evolutionarily in a more general bargaining situation than in the ultimatum game is a novel finding that does not rely on modeling artifacts in the sense that it is not due to repeated
interaction (reputation effect) nor spatial structure of interaction.
Methods
Model.
We consider a population consisting of a continuum of players of finite measure. Players are classified
into two types: rational players (R) and fair players (F). At any time, they are pairwise matched and play a dividea-lottery game. The value of a lottery, denoted by v, is uncertain. We assume that it is uniformly distributed on
[0, 1]. The divide-a-lottery game goes as follows. First, one player (player 1) of the matched pair bids x and then
the other player (player 2) bids y. After that, the value of v is realized. If it turns out that x + y ≤ v , they get x
and y respectively and if x + y > v , neither gets anything. We assume that each player of the matched pair can
be either the first player or the second player with equal probability. Also, we assume symmetric role assignment
to make our analysis isolated from role assignment. For (reputation-based) role assignment in the dictator game,
see Yang et al.16 and Li et al.17.
We assume that when a pair is matched, the preference types of the players are known to each other. Since
there are two possible types of players, it implies that four pairing combinations are feasible in a stage game.
Let the material payoff of player i be πi (x, y) for i = 1, 2. We assume that each player’s material payoff is
defined as
π1 (x, y) = x(1 − x − y),
(1)
π2 (x, y) = y(1 − x − y).
(2)
This is the expected value of player i’s share, since 1 − x − y = P(x + y ≤ v), which is the probability of agreement, i.e., the probability that the allocation by bids is feasible. Although the payoff functions in the divide-alottery game are similar to those in the duopoly game, we do not believe that our results will be straightforwardly
applied to behavior of firms. In fact, to the best of our knowledge, there is no empirical finding that most firms
behave fairly in the duopoly game without maximizing their profits. For endogenous sequencing in the duopoly
game, see Dowrick18, Boyer and Moreaux19, and Hamilton and Slutsky20.
When players choose their bids in a stage game, they maximize their subjective utility, not the material payoff.
Let UiF represent the subjective utility of fair player i. It is defined by
UiF (x, y) = πi (x, y) − αd(x, y),
(3)
where α ≥ 0 is a parameter to represent how much this individual cares about fairness and d(x, y) is a difference
between the shares of the two players. (Some authors assume that the disutility is asymmetric, i.e., disutilities
when x > y and x < y are different. However, this preference is not really fair). As α → ∞, the player is fairer.
(Conceptually, it is possible that α = ∞, but we will restrict our attention to α ∈ [0, 1] to avoid the case that a
fair player’s payoff is −∞.) If α → 0, the player is almost rational. Throughout the article, we assume a simple
functional form of d(x, y) = (x − y)2.
Let UiR be the subjective utility of a rational player. We assume that the subjective utility of a rational player
is the same as his material payoff:
UiR (x, y) = πi (x, y).
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Results
In this section, we analyze the two cases, the case in which each player is informed of the type of the opponent
he is facing against (complete information case) and the other case in which each player is not informed of the
opponent’s type (incomplete information case).
Complete information about the opponent’s type. We consider two symmetric matching cases,
rational player vs. rational player and fair player vs. fair player, and one asymmetric matching case, rational
player vs. fair player.
Rational player vs. rational player. Since a stage game is sequential, we use backward induction to obtain the
subgame perfect equilibrium. If rational player 1 plays against rational player 2, given the bid of player 1, x,
player 2 seeks to maximize his material payoff by choosing
y R (x) = arg max π2 (x, y) = y(1 − x − y),
y
(5)
so we obtain player 2’s best response as a function of x:
y R (x) =
1−x
.
2
(6)
Taking account of this response, player 1 chooses x to maximize
1−x
.
π1 (x, y R (x)) = x(1 − x − y R (x)) = x 1 − x −
2
(7)
Therefore, equilibrium bids are x ∗ = 12 and y ∗ = y R (x ∗ ) = 41.
Let π RR be the material expected payoff of a rational player playing against another rational player. The
1
material payoffs of rational player 1 and player 2 are π1 = 18 and π2 = 16
respectively. This shows the first mover
advantage. The first mover can choose a higher bid than the second mover who is passive. By increasing his bid
before player 2, he can enjoy a strategic advantage. Since a player can be player 1 or player 2 with equal probability,
1
3
= 32
their expected value of the material payoff is π RR = 12 π1RR + π2RR = 12 18 + 16
.
Fair player vs. fair player. If a fair player plays against another fair player, player 2 chooses his bid to maximize
his subjective utility, taking his opponent’s bid x as given:
U2FF (x, y) = y(1 − x − y) − α(x − y)2 .
(8)
From the first-order condition for maximizing (8), we obtain fair player 2’s best response function as
y F (x) =
1 + (2α − 1)x
.
2(1 + α)
(9)
Taking this response of player 2 into account, player 1 chooses x to maximize
U1FF (x, y F (x)) = x(1 − x − y F (x)) − α(x − y F (x))2
1 + (2α − 1)x 2
1 + (2α − 1)x
−α x−
.
=x 1−x−
2(1 + α)
2(1 + α)
Therefore, we obtain equilibrium bids as x ∗ (α) =
∂x ∗ (α)
∂α
(10)
4α 2 +14α+1
2α 2 +6α+1
and y ∗ (α) = y F (x ∗ (α)) = 2(8α
2 +19α+2) where
8α 2 +19α+2
9
19
= 58 > 29 . As α gets larger, i.e., players get fairer, player 1
9
< 0, x ∗ (0) = 12 and x ∗ (1) = 29
< 31 and y ∗ (1)
bids less to reduce the first mover advantage, and if α = 1, the first mover advantage disappears completely,
because x ∗ (1) < y ∗ (1). (It is easy to check that y ∗ (α) is not monotonic with respect to α).
Let π FF be their expected value of the material payoff. Then, we can compute π FF as
2
2 +12α+1)
FF
.
π = 12 (π1FF + π2FF ) = (8α +26α+3)(8α
8(8α 2 +19α+2)2
Rational player vs. fair player. If player 1 is rational and player 2 is fair, we know from (9) that y F (x) =
Calculating this response of player 2, player 1 chooses x to maximize
1 + (2α − 1)x
RF
F
F
.
U (x, y (x)) = x(1 − x − y (x)) = x 1 − x −
2(1 + α)
1+(2α−1)x
2(1+α) .
(11)
2
2α+1
4α +8α+1
Therefore, we obtain x ∗ (α) = 2(4α+1)
and y ∗ (α) = y F (x ∗ (α)) = 4(α+1)(4α+1)
. Note that limα→0 x ∗ (α) = 12 ,
3
3
limα→0 y ∗ (α) = 41 , limα→1 x ∗ (α) = 10
, and limα→1 y ∗ (α) = 13
40 > 10 . (It is interesting to note that
1
1
1
∗
∗
x ( 2 ) = y ( 2 ) = 3 , which is identical to the equilibrium outcome of the simultaneous divide-a-lottery game).
(2α+1)2
Again, the first mover advantage disappears in the case of fair player 2. Also, we obtain π1RF = 8(4α+1)(α+1)
and
π2RF =
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1+3α
2+9α and
(3α+1)(6α+1)
(6α+1)2
FR
and π2 = 4(9α+2)
2.
2(9α+2)2
On the other hand, if player 1 is fair and player 2 is rational, equilibrium bids are x ∗ (α) =
1+6α
y ∗ (α) = 2(2+9α)
, and accordingly the equilibrium material payoffs are π1FR =
Let πR and πF be the equilibrium expected material payoffs of a rational player and a fair player respectively
when a rational player and a fair player are matched. Then, they can be computed as
πR =
1 RF
π + π2FR ,
2 1
(12)
πF =
1 FR
π + π2RF .
2 1
(13)
Table 1 shows the computation results of material payoffs.
If the fitness of a player is determined by his material payoff, we can see from Table 1 that fairness is the
dominant strategy for any α ∈ [0, 1]. For the numerical proof, Fig. 1A shows that πF > πRR and πFF > πR for
any α ∈ (0, 1]. So, if strategies evolve in proportion to the material payoffs, only fairness can survive evolutionarily in the long run. Here, as a dynamic solution concept, we are using a long-run asymptotic (local) attractor
that can be roughly defined by the population distribution to which an initial distribution converges over time
whenever it starts from the neighborhood.
Proposition 1 Only fair players can survive evolutionarily if a randomly matched pair in a population plays
a divide-a-lottery game and the players know each other’s type.
The analytic proof is omitted, because it is well known that a strict Nash equilibrium is an evolutionarily
stable strategy (ESS) by Maynard Smith and Price21 which is a long-run asymptotic attractor. In this game, (F, F)
is a strict Nash equilibrium.
At this moment, it is worthwhile to compare this game with the prisoners’ dilemma (PD) game. In a PD game,
cooperation (C) is strictly dominated by defection (D), but (C, C) yields higher fitness than (D, D). In other
words, (C, C) is the collectively rational outcome (socially efficient outcome), whereas (D, D) is the individually
rational outcome (privately optimal outcome). The discrepancy is where social dilemma comes from. In our
divide-a-lottery game, fairness strictly dominates rational behavior, but unlike the PD game, the individually
rational outcome (F, F) yields higher fitness than (R, R). This is the main difference from the PD game. Also, it
is interesting to note that the collectively rational outcome in this game is (F, R) and (R, F), not (F, F), for most
parameter values except for very small values of α ∈ (0, 0.04) (Fig. 1B). This implies that for most values of α,
a polymorphic population is socially better than a monomorphic population consisting only of fair players in
terms of the population average fitness.
Since the complete information assumption that drives this result is too strong to properly capture the real
world phenomenon, we relax the assumption and consider the incomplete information case in the next section.
Incomplete information about the opponent’s type. In this section, we assume that players cannot
tell the type of the opponent but only know the proportion of each type. Let p be the proportion of fair players
in the population. A rational player 1 chooses x R to maximize
E(π1 ) = (1 − p)x(1 − x − y R (x)) + px(1 − x − y F (x))
1 + (2α − 1)x
1−x
= (1 − p)x 1 − x −
.
+ px 1 − x −
2
2(1 + α)
(14)
The first order condition leads to
A
B
SD\RII
πF
πRR
πFF
πR
πF πR
πFF
α
πF πR
πFF
α
α
Figure 1. Values of material payoffs for complete information cases. For the comparison of material payoffs, we
observe the values of each material payoff (A), and (πF + πR)-vs. 2 ∗ πFF (B, inset), according to the parameter
α.
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xR =
αp + α + 1
.
6αp + 2α + 2
A fair player chooses x F to maximize
2
E(π1 ) = (1 − p) x(1 − x − y R (x)) − α x − y R (x)
+ p x(1 − x − y F (x)) − α(x − y F (x))2
1−x 2
1−x
− α(x −
)
= (1 − p) x 1 − x −
2
2
1 + (2α − 1)x
1 + (2α − 1)x 2
+p x 1−x−
−α x−
.
2(1 + α)
2(1 + α)
(15)
(16)
Thus, we obtain
3α 3 (1 − p) + α 2 (7 − 5p) + α(p + 5) + 1
.
9α 3 (1 − p) + 4α 2 (5 − 3p) + α(6p + 13) + 2
xF =
(17)
Substituting (15) and (17) into (6) and (9), we get
5αp + α + 1
,
4(3αp + α + 1)
(18)
6α 3 (1 − p) + α 2 (13 − 7p) + α(8 + 5p) + 1
,
2[9α 3 (1 − p) + 4α 2 (5 − 3p) + α(6p + 13) + 2]
(19)
2α 2 p + 2α 2 + 5αp + 3α + 1
,
4(α + 1)(3αp + α + 1)
(20)
6α 4 (1 − p) + 4α 3 (5 − 4p) + α 2 (23 − 5p) + 5α(p + 2) + 1
.
2(α + 1)[9α 3 (1 − p) + 4α 2 (5 − 3p) + α(6p + 13) + 2]
(21)
y R (x R ) =
y R (x F ) =
y F (x R ) =
y F (x F ) =
Let πiIR and πiIF be the equilibrium material payoffs of a rational (or fair respectively) player i where i = 1, 2 in the
case of incomplete information. Then, we can compute the expected value of the material payoff of each type as
π IR =
1 IR
(π + π2IR ),
2 1
(22)
π IF =
1 IF
(π + π2IF ),
2 1
(23)
where
π1IR = (1 − p)x R (1 − x R − y R (x R )) + px R (1 − x R − y F (x R )),
(24)
π2IR = (1 − p)y R (x R )(1 − x R − y R (x R )) + py R (x F )(1 − x F − y R (x F )),
(25)
π1IF = (1 − p)x F (1 − x F − y R (x F )) + px F (1 − x F − y F (x F )),
(26)
π2IF = (1 − p)y F (x R )(1 − x R − y F (x R )) + py F (x F )(1 − x F − y F (x F )).
(27)
Let us consider the following replicator dynamics
pt+1 = pt
π IF
,
π¯I
(28)
where π¯I = (1 − pt )π IR + pt π IF . Then, we can find the limiting distribution of R and F. Figure 2A shows our
simulation results that only the monomorphic population consisting only of rational players emerge as a result
of evolution for most parameter values (black region), while a polymorphic population can emerge for high
values of α (degree of fairness) and p0 (initial proportion of fairness) (yellow region). Figure 2B shows the average of material payoffs in the limiting states for various combinations of (p0 , α). It implies that the polymorphic
population consisting of mixture of rational players and fair players is better than the monomorphic population
consisting solely of rational players in terms of the population average fitness. This implies that the population
distribution is very unlikely to converge to the monomorphic population distribution that yields the highest
population average fitness when players interact with each other globally with equal probabilities.
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A
SILQDO
B
πILQDO
0.6
S
S
pfinal
0.4
0.2
0
α
α
Figure 2. Numerical observation of final fraction of fair players ( pfinal ) and the average fitness of the population
(π̄final ) from the replicator dynamics of incomplete case. (A) For each pair of parameter values p0 and α, the
final fractions of fair players in equilibrium are presented as color scale. The black region means only rational
players can survive in final state while both types of players can coexist in the yellow region. (B) For each pair
of parameter values p0 and α, the average fitness of the population (π̄final ) in equlibrium are presented as color
scale, too. The yellow region denotes a higher average fitness compared to the brown region. It indicates that
evolution does not favor the population distribution with the highest average fitness, when players interact
globally.
R
F
R
3 3
32 , 32
πR , π F
F
πF , π R
π FF , π FF
612α 4 +924α 3 +441α 2 +86α+6
,π
16(α+1)(4α+1)(9α+2)2
F
(8α 2 +26α+3)(8α 2 +12α+1)
8(8α 2 +19α+2)2
Table 1. Payoff matrix (material payoffs). πR =
1224α 5 +3492α 4 +3106α 3 +1169α 2 +196α+12
, π FF
32(α+1)2 (4α+1)(9α+2)2
=
=
This result is quite puzzling. In this game, fairness strictly dominates rationality in the case of complete information, as shown in Tab.1. It means that it is better for a player to play fairly, regardless of the opponent’s type.
This seems to imply that a player does not need to know the opponent’s type, because he will get a better payoff
when he plays fairly than when he plays rationally. Then, how can rational players still survive evolutionarily if
players cannot be sure of the opponent’s type? Specifically, when p0 ≈ 0, how can it be possible that π IR > π IF in
the case of incomplete information, although p0 ≈ 0 means that the population consists only of rational players
3
so Table 1 seems to suggest that only fair players can survive because πF > 32
? The answer for this puzzle can
1
FR
RF
be found from the difference between πF = 2 (π1 + π2 ) given in (13) and π IF = 12 (π1IF + π2IF ) given in (23)
when p0 ≈ 0 . Note that π2RF = y F (x R )(1 − x R − y F (x R )) and π2IF ≈ y F (x R )(1 − x R − y F (x R )) given in (27)
when p0 ≈ 0. Although they look the same, the values of x R in the two formulas are different, depending on what
the opponent is. In the former (in the complete information case), it is computed from the assumption that the
second mover is fair. (We used the notation x ∗ (α) instead of x R in the analysis of complete information case to
distinguish them). In the latter (in the incomplete information case), however, the rational player chooses x R ,
expecting that the second mover is highly likely to be rational when p0 ≈ 0. (In other words, the true opponent
type and the expected opponent type can be different in the case of incomplete information, whereas it is not
possible in the case of complete information). So, x R in this case is larger and thus it is more likely to be rejected.
Hence, π2RF is lower, so is the fitness of a fair player in the case of incomplete information.
The overall intuition for the case of incomplete information goes as follows. If players have complete information about the opponent’s type, a rational player (player 1) bids very high to the rational opponent, so rational
player 2 is severely exploited, while he bids lower to the fair opponent, because he knows that the fair opponent
(fair player 2) will bid so high that his high bid would be very likely to lead to a failure in bargaining. However, if
players have incomplete information about the opponent’s type, a rational player who is unsure of the opponent’s
type must bid lower than when he faces a rational opponent, and so a rational player 2 is not very much exploited.
This is one of the main reasons why rational players fare better under incomplete information. Similarly, a fair
player who is unsure of the opponent’s type can bid higher if there is a high probability that the opponent is a
rational type. So, incomplete information can have the role of making a rational player play more like a fair player
and making a fair player play more like a rational player. (If a player is the second mover, his decision does not
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depend on the opponent’s type. So, the incomplete information of the opponent’s type does not affect his choice
under complete information).
Under complete information, a rational player earns a low material payoff because he is significantly exploited
when he is the second mover. On the other hand, under incomplete information, he is not so much exploited
because the opponent is not sure whether he is rational or not. However, if the proportion of fair players is very
high, it becomes an almost complete information game, and the advantage of fairness in the case of complete
information is almost balanced with the advantage of rationality in the case of incomplete information, so both
of the two types can survive and evolve over time.
Local interaction on a network. In this section, we consider a simple network structure on which players
interact locally to play a divide-a-lottery game. For recent studies on local interaction in other situations such as
the prisoners’ dilemma game with social diversity or the snowdrift game, see Perc and Szolnoki22 and Hauert and
Doebeli23. Initially, there are n(= 100) players on a circle. The type of each player (R or F) is assigned randomly
according to the pre-assigned ratio of fair players ( p0). Here, we introduce two parameters, interaction radius
(rinter ) and information radius (rinfor ). Each player interact only with neighbors within the given rinter . As we
investigated the cases of complete and incomplete information about the opponent’s type in previous subsections, players know the type of her neighbor within the given rinfor and does not know outside of the length rinfor .
Then, players observe their own fitness and their neighbors’ within the interaction radius after they play the
divide-a-lottery game with the neighbors. Finally, each player decides to change her type by imitating the type of
her neighbors when the average fitness of her neighbors of different types is greater than the fitness of her own
type. The simulation continues until the dynamics becomes stable. We observe the fraction of fair players at the
final step of simulation ( pfinal ) with varying two parameters of initial fraction of fair players ( p0) and fairness
careness α as the average of 103 times of ensembles (see Fig. 3).
For given rinter = 10 and rinfor = 1, we can find that there are mainly two phases of final states: (1) At higher
α and higher p0, only fair players survive (white region) and (2) rational players are predominant with higher α
and lower p0 values. This means that the emergence of fairness can be determined by the given condition of α
and p0. Note that if α is too high, it may not be good for a fair player because his bid becomes lower (Fig. 3A).
For different combinations of rinter and rinfor , the results are qualitatively similar (see Fig. S1). At lower values
of α, both types of players can coexist, as depicted in yellow color in Fig. 3A. This is also confirmed in Fig. 3B
which illustrates how some initial distribution reaches a stationary spatial distribution over time by simulations.
Note that in the case of some nodes, they oscillate unstably at first, and then become stable and maintain their
types over time. Also, Fig. 3C shows that a polymorphic population consisting of both of rational players and
fair players is better than the monomorphic population consisting only of fair players in terms of population
average fitness. This is mainly because πF + πR > 2πFF for most parameter values of α. This figure implies that
the evolution process in a local interaction favors the ultimate population distribution that yields high population average fitness.
Before closing this section, we highlight the intuition for why fair players can survive evolutionarily. If two
rational players play the game, there is a first mover advantage. Player 1 preempts an advantageous position by
making a high bid which makes player 2 makes a low bid. However, if player 2 is fair, player 1 cannot bid high
because he knows that the fair opponent will not reduce his bid very much due to his concerns for fairness. This
makes player 1 reduce his bid. So, as α is larger, i.e., player 2 is fairer, player 1 reduces his bid more so that player
2 increases his bid, and thus player 1’s payoff gets smaller and player 2’s payoff gets larger, until they bid the same
A
SILQDO
πILQDO
C
B
0
50
S
100
simulation
time step
S
150
200
R
F
R
F
R F
R
α
α
Figure 3. Results of local interaction of a network. (A) Simulation result of final fraction of fair players
( pfinal ) according to the parameter α and p0 for the given interaction radius rinter = 10 and information radius
rinfor = 1. The white region indicates that only fair players can survive while both types of players can coexist
in the yellow and orange region. (B) For the given α = 0.1 and p0 = 0.1, we illustrate how individual players
change their types according to simulation steps. The blue (red) circle denotes fair (rational) players respectively.
(C) Simulation result of average final fitness of the population (π̄final ) according to the parameter α and p0 for
the given interaction radius rinter = 10 and information radius rinfor = 1. The yellow region indicates that the
population average fitness is very high compared to the other regions shown in red and black.
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Figure 4. Comparison of bids and payoffs with rational opponent and fair opponent. Black lines denoted by
x R (y) and y R (x) are reaction curves when both players are rational, and the red line denoted by y F (x) is the
reaction curve when the player is fair. Blue curves are the two players’ indifference curves that yield same utility
to each player. The cap-shaped curve is player 1’s indifference curve. His utility increases as it moves downward
towards x-axis, while player 2’s utility increases as it moves inward towards y-axis. If the second player is
rational, his reaction curve is y R (x), so player 1 chooses the point that gives the maximum utility on y R (x). It is
the tangent point of y R (x) and the indifference curve, (1/2, 1/4) if α = 0. If the second player is fair, the tangent
point of y F (x) and the indifference curve is (3/10, 13/40) if α = 1.
Figure 5. Equilibrium bids when both players are fair If player 1 is fair, his indifference curve has the zero slope
on x F (y), not on x R (y), because x F (y) is the optimal point for him given y. This figure shows the equilibrium
bids (9/29, 19/58), which is the tangent point of player 1’s indifference curve and player 2’s reaction curve y F (x),
when both are fair, if α = 1.
when α = 12 . If α exceeds 12 , player 2 begins to reduce his bid, although he still bids more than player 1. So, player
1’s payoff begins increasing as α gets larger. Figure 4 shows this intuition.
If both players are fair, the situation is similar. In this case, the best response curves are both upward sloping.
Since player 1 takes the best response function of player 2 as given, he will choose the optimal point along the
best response curve of player 2 which is lower than x = 13 , the intersection of the two best response curves. Since
the best response curve of player 1 is upward sloping, low x means low y. Since both x and y are reduced, both
fair players get higher payoffs than rational players. It is illustrated in Fig. 5.
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Discussion
In this paper, we demonstrated that fair players can survive evolutionarily in a divide-a-lottery game. Moreover,
we showed that rational players can also survive in the environment in which the bargaining players do not
know each other’s type until they play the bargaining game with the opponent, depending on the initial population distribution. Considering the reality that players often compete with their local neighbor whose type is
not known (until they interact) and for the pie the value of which is uncertain, we believe that this result gives
a sensible prediction in the real world.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Received: 29 December 2022; Accepted: 25 April 2023
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Acknowledgements
This work was supported by the Ministry of Education of the Republic of Korea and the National Research
Foundation of Korea (NRF-2022S1A5A2A0304932311).
Author contributions
J.Y.K. conceptualization, methodology, formal analysis, writing-original draft, writing-review & editing, supervision. K.M.L. methodology, numerical simulation, formal analysis, writing-original draft, writing-review &
editing, visualization, project administration.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary Information The online version contains supplementary material available at https://doi.org/
10.1038/s41598-023-34131-w.
Correspondence and requests for materials should be addressed to J.-Y.K. or K.-M.L.
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