Discussion Paper

No. 2008-12

The Evolution of Fairness under an Assortative Matching Rulein the Ultimatum Gameby
Yasuhiro Shirata

December, 2008

The Evolution of Fairness under an Assortative
Matching Rule in the Ultimatum Game ∗Yasuhiro Shirata †December, 2008

Abstract
This paper studies how a matching rule aﬀects the evolution of fairness inan ultimatum mini game. Gale et al. [1995] show that only selfish behaviour
survives in the deterministic replicator dynamics under the random matchingrule. In contrast, this paper shows that, under an assortative matching rule,
the fair behaviour may survive at an asymptotically stable state.

1

Introduction

Why would people behave in a fair manner, sacrificing their own monetary payoﬀs. In the ultimatum game, selfish individuals propose to exploit almost the total
surplus, and accept these unfair oﬀers as responders. Contrary to this standard
game theoretical prediction, many experimental data show that people tend to divide the total surplus equally (e.g. Güth et al. [1982] and Binmore et al. [2002]).
This paper studies this paradox in the framework of the evolutionary game theory,
focusing on a matching rule.
Gale et al. [1995] consider the ultimatum mini game given in Figure 1. In thegame, agent 1 proposes either a high oﬀer (H) or a low oﬀer (L). If she adopts
strategy H, it is assumed that agent 2 (responder) always accepts it. If she adopts
strategy L, the responder decides to accept (Y) or reject (N) it. Let x1 and x2 be
proportions of individuals adopting actions L and Y, respectively, in the populationof proposers and responders, respectively.
∗

I am very grateful to Professor Akira Okada for his guidance and encouragement. I also thankparticipants in Hitotsubashi Game Theory Workshop 2007 for helpful comments. This work is
supported by Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows.
†
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan.
E-mail: ed081004@g.hit-u.ac.jp

1

1b

HH
H
HL

HH 2

Hr
r
Y @ N
2,2
@
@r
r
3, 1

0, 0

Figure 1: The ultimatum mini game.
The standard replicator dynamics is described asx˙1 =g1 (x) = x1 ( fL − ϕ1 )x˙2 =g2 (x) = x2 ( fY − ϕ2 ),where fk is the fitness of individuals adopting action k (k = L, Y), and ϕi is the average fitness of population i = 1, 2. In the ultimatum mini game given in Figure 1,
it holds thatg1 (x) = x1 (1 − x1 )(3x2 − 2)g2 (x) = x2 (1 − x2 )x1 .
Gale et al. [1995] show that it is a unique asymptotically stable state of the deterministic replicator dynamics that all proposers will make the selfish oﬀer (L)and all responders will accept it (Y). That is, only the unfair behaviour survives in
the replicator dynamics under the random matching rule. Thus, the paradox stillholds.
To represent an evolutionary drift, Gale et al. [1995] and Binmore and Samuelson [1999] introduce the following perturbed selection dynamics,ẋ = g(x) + h(x).
If a drift function h is strictly decreasing in a diﬀerence between the largest andthe smallest expected payoﬀs, there exists an asymptotically stable state which isan imperfect Nash equilibrium leading to the fair allocation. The asymptoticallystable state, however, critically depends on the form of h (Binmore and Samuelson[1999]). For example, if h is not sensitive to payoﬀs, then only the subgame
perfect equilibrium is asymptotically stable.
Unlike these previous approaches, this paper studies how a matching rule affects the evolution of fairness. Especially, we consider an assortative matchingrule which is introduced by Becker [1973, 1974]. The assortative matching rule2

is a matching rule under which similar types of individuals are paired more often
than under the random matching rule.
Under the assortative matching rule, an interaction rate between individuals
depends on their own actions in contrast to the random matching rule. This property leads to the replicator dynamics with non-linear fitness functions (Taylor and
Nowak [2006]). In symmetric 2 × 2 strategic form games, Taylor and Nowak[2006] introduce a generalized matching rule with non-uniform interaction rates.
They show that the non-uniform interaction rates generate interior equilibria even
if one strategy dominates another. Bergstrom [2003] introduces another kind of
the assortative matching rule in the prisoners’ dilemma game. Taylor and Nowak[2006] and Bergstrom [2003] show that cooperation survives under the assortative
matching rule in the prisoners’ dilemma game.
The main result of this paper is that, under an assortative matching rule, there
exist asymptotically stable states at which the fair allocation may prevail. Especially, if the matching rule is completely assortative, there exist only two asymptotically stable states, the fully fair equilibrium and selfish equilibrium. The results provide an evolutionary support for the fair allocation which has been observed by many experiments in the ultimatum game.
The paper is organized as follows. Section 2 defines an assortative matchingrule and the selection dynamics of the ultimatum mini game. Section 3 presents
the main results. Section 4 gives an example of an assortative matching rule.
Section 5 discusses the results.

2

The Model

There are two populations, proposers (population 1) and responders (population
2). The sizes of the two populations are equal. In each period, an agent in one
population matches with an agent in the other population according to a predetermined matching rule. Each pair of agents plays the ultimatum mini game given in
Figure 1.
We call a proposer selfish if she adopts strategy L, and fair if she adopts H.
Similarly, we call a responder selfish if he adopts strategy Y, and fair if he adopts
N. Let x1 denote the proportion of selfish proposers in population 1, and x2 denotethe proportion of selfish responders in population 2, respectively. A state of thesystem is represented by a pair x = (x1 , x2 ). For a technical reason, we assume thatthe range of xi is [ϵ, 1 − ϵ] for all i = 1, 2 where ϵ is a suﬃciently small positive
number 1 .
If this assumption does not hold, the matching rule (pi , qi )i=1,2 are not Lipschitz continuous(see Definition 1). Section 3 and 4 discuss the set of asymptotically stable states as ϵ goes to zero.
1

3

p1

L

Y
1-q 1

1-p 1
H
q1

N

Figure 2: The matching probability in population 1: p1 = Pr(L meets Y), q1 =Pr(H meets N).
Now, we define a matching rule as a pair (pi (x), qi (x))i=1,2 of two functions on[ϵ, 1−ϵ]×[ϵ, 1−ϵ], where pi (x) is the probability that a selfish agent in population
i meets a selfish agent in population j and qi (x) is the probability that a fair agent
in population i meets a fair agent in population j at state x (Figure 2).
Definition 1. (pi (x), p j (x))i=1,2 is a matching rule if for all i = 1, 2, pi (x) and qi (x)are Lipschitz continuous on [ϵ, 1 − ϵ] × [ϵ, 1 − ϵ] 2 , and satisfies
lim pi (x) = 0,

x j →0lim qi (x) = 0,

x j →1

lim pi (x) = 1,

x j →1

lim qi (x) = 1,

x j →0lim pi (x) = x j ,xi →1

lim qi (x) = 1 − x j ,xi →0

and for all x ∈ [ϵ, 1 − ϵ] × [ϵ, 1 − ϵ],xi pi = x j p j ,(1 − xi )qi = (1 − x j )q j ,(1 − pi )xi = (1 − q j )(1 − x j ).

(1)
(2)
(3)

Equations (1), (2), and (3) are parity equations which imply that probability
functions pi and qi are consistent as a matching rule. All players can be paired
as long as these equations are satisfied. Note that if one of p1 , p2 , q1 , or q2 isdetermined, then the other valuables is automatically determined by using these
conditions. The random matching rule assumes that pi (x) = qi (x) = x j for all iand j (i , j).
Under an assortative matching rule, similar types of agents are matched more
often than under the random matching rule.
The function p is Lipschitz continuous if for any x, y ∈ [ϵ, 1 − ϵ] × [ϵ, 1 − ϵ], there exists a
constant k such that | p(y) − p(x) |< k | y − x |.
2

4

Definition 2. A matching rule (pi , qi )i=1,2 is assortative, if for all i = 1, 2, pi andqi satisfy the following conditions:(i) pi (x) is monotonically non-increasing in xi and monotonically non-decreasing
in x j .
qi (x) is monotonically non-increasing in x j and monotonically non-decreasingin xi .
(ii) pi (x) > x j and qi (x) > 1 − x j .
(iii) lim xi →0 pi (x) > x j and lim xi →1 qi (x) > 1 − x j .
An assortative matching rule has two characteristics. First, by (i) and (ii), theprobability pi has a strictly higher value than the one under the random matchingrule and it increases as the frequency of their same type opponents increases.
Second, the increase of the selfish type frequency xi causes the decrease of theprobability pi . Let AM be the set of assortative matching rules.
Next, we define the replicator dynamics, where the state space is [ϵ, 1 − ϵ] ×[ϵ, 1 − ϵ].
Definition 3. The replicator dynamics with a matching rule (pi , qi )i=1,2 on [ϵ, 1 −ϵ] × [ϵ, 1 − ϵ] is defined byx˙1 = (x1 − ϵ)( fL − ϕ1 )x˙2 = (x2 − ϵ)( fY − ϕ2 ),

(4)
(5)

where ϕi is the average payoﬀ of population i such that
ϕ1 = (x1 + ϵ) fL + (1 − x1 − ϵ) fH
ϕ2 = (x2 + ϵ) fY + (1 − x2 − ϵ) fN ,and f j ( j = L, H, Y, N) is the expected payoﬀ for strategy j; fL = 3p1 (x), fH = 2,
fY = p2 (x) + 2(1 − p2 (x)), and fN = 2q2 (x).
Thus, the system is described asx˙1 = g1 (x) = (x1 − ϵ)(1 − ϵ − x1 )ϕ̄1 (x)x˙2 = g2 (x) = (x2 − ϵ)(1 − ϵ − x2 )ϕ̄2 (x),whereϕ̄1 (x) = 3p1 (x) − 2
ϕ̄2 (x) = 2 − p2 (x) − 2q2 (x).
5

(6)
(7)

Although this dynamics is not the standard replicator dynamics, it has the same
properties, regularity and monotonicity (Binmore and Samuelson [1999]). In thisselection dynamics, the growth rate is continuous on state space [ϵ, 1−ϵ]×[ϵ, 1−ϵ](regularity) and that a growth rate of a relatively low-payoﬀ action is smaller thanthat of a relatively high-payoﬀ action (monotonicity).
Finally, we define some standard concepts of dynamic stability (e.g. VegaRedondo [2003]).
Definition 4. (1) A state x = (x1 , x2 ) is a rest point of (4) and (5) if and only if
x˙1 = 0 and x˙2 = 0.
(2) A state x∗ is an asymptotically stable point of (4) and (5) if and only if thefollowing two conditions hold:(i) (Liapnov stability) Given any neighborhood U1 of x∗ , there exists someneighborhood U2 of x∗ , such that for any path x = x(t), x(0) ∈ U2 impliesx(t) ∈ U1 for all t > 0.
(ii) There exists some neighborhood V of x∗ such that for any path x = x(t),x(0) ∈ V implies limt→∞ x(t) = x∗ .

3

Results

Proposition 1. Let R(a) be the set of all rest points of system (6) and (7) under∪
an assortative matching rule a = (pi , qi )i=1,2 . As ϵ goes to 0, R = a∈AM R(a)converges to{(1, 0), (1, 1), (x1′ , x2′ )} ∪ {(0, c)|c ∈ [0, 1]},
where (x1′ , x2′ ) satisfies ϕ̄1 (x1′ , x2′ ) = 0 and ϕ̄2 (x1′ , x2′ ) = 0.
Proof. Obviously, (ϵ, ϵ), (ϵ, 1 − ϵ), (1 − ϵ, ϵ), (1 − ϵ, 1 − ϵ) are rest points of (6) and(7). Next, we show that there exists no a ∈ AM under which there exists some x1
satisfying ϕ¯1 (x1 , 1 − ϵ) = 0. Suppose that ϕ̄1 (x1 , 1 − ϵ) = 0. Then, p1 (x1 , 1 − ϵ) = 32 .
2
x1 by parity equations (1) and (3). However, thisThis implies p2 (x1 , 1 − ϵ) = 3(1−ϵ)
contradicts condition (ii) in Definition 2 (p2 > x1 ) when ϵ is suﬃciently small.
We can construct an assortative matching rule under which there exist x1′ , x2′ ,x1′′ , x2′′ , and x2′′′ satisfying ϕ̄1 (x1′ , x2′ ) = 0, ϕ̄2 (x1′ , x2′ ) = 0, ϕ̄1 (x1′′ , ϵ) = 0, ϕ̄2 (ϵ, x2′′ ) =0, and ϕ̄2 (1 − ϵ, x2′′′ ) = 0. Define an assortative matching rule a′ = (pi , qi )i=1,2 as
p1 (x1 , x2 ) = (1 − α)x2 + α(min[6x2, 1]),x1(8)

where α ∈ (0, 1]. Under this a′ , when α = 1,3 (x1′ , x2′ ), (x1′′ , ϵ), (1 − ϵ, x2′′′ ), (x1′ , x2′ )exist, and when α = 1/2, (ϵ, x2′′ ) exists.
By (6) and (7), there exist no other rest points under any assortative matchingrule. Thus, for suﬃciently small ϵ,
R = {(ϵ, ϵ), (ϵ, 1 − ϵ), (1 − ϵ, ϵ), (1 − ϵ, 1 − ϵ)}∪{(x1′ , x2′ ), (x1′′ , ϵ), (ϵ, x2′′ ), (1 − ϵ, x2′′′ )}.
Since (1), ϕ̄1 (x1′′ , ϵ) = 0, and p2 ≤ 1, we obtain
x1′′ =3ϵp2 (x1′′ , ϵ) ≤ ϵ.
′′p1 (x1 , ϵ)2Then, x1′′ converges to 0 as ϵ goes to 0. Since ϕ̄2 (1 − ϵ, x2′′′ ) = 0 and p2 , q1 ≤ 1, we
obtain2 = p2 (1 − ϵ, x2′′′ ) + 2q2 (1 − ϵ, x2′′′ )= p2 (1 − ϵ, x2′′′ ) + 2q1 (1 − ϵ, x2′′′ )
≤ 1+2

ϵ
.
1 − x2′′′ϵ1 − x2′′′

Then, x2′′′ ≥ 1−2ϵ. Hence, x2′′′ converges to 1 as ϵ goes to 0. Since limϵ→0 p2 (ϵ, x2′′ ) =0 and limϵ→0 q2 (ϵ, x2′′ ) = 1, any x2′′ ∈ [ϵ, 1 − ϵ] satisfies limϵ→0 ϕ̄2 (ϵ, x2′′ ) = 0.
Thus, as ϵ goes to 0, R converges to{(1, 0), (1, 1), (x1′ , x2′ )} ∪ {(0, x2 ) | x2 ∈ [0, 1]}.

Since any x with x1 = 0 is in R, the equal allocation is supported as a rest
point of the replicator dynamics (6)-(7). The following proposition shows that theequal allocation is asymptotically stable.
Proposition 2. Let A(a) be the set of asymptotically stable points under an assor∪
tative matching rule a = (pi , qi )i=1,2 . As ϵ goes to 0, A = a∈AM A(a) converges
to{(1, 1)} ∪ {(0, x2 )|x2 ∈ [0, 1/2)}.
Proof. We first prove the two claims.
3This rule is equal to a rule in section 4.

7

Claim 1. There exists no assortative matching rule under which either (ϵ, 1 − ϵ)or (1 − ϵ, ϵ) is asymptotically stable.
We will show only x = (ϵ, 1 − ϵ) < A. The Jacobian matrix of g at (ϵ, 1 − ϵ) is∂g(ϵ, 1 − ϵ)∂x (1 − 2ϵ)(3p1 (ϵ, 1 − ϵ) − 2)

0
= 

0
(2ϵ − 1)(2 − p2 (ϵ, 1 − ϵ) − 2q2 (ϵ, 1 − ϵ))(1 − 2ϵ)(3p1 (ϵ, 1 − ϵ) − 2)
0
 .
= 
p
(ϵ,
1
−
ϵ))
0
(2ϵ − 1)(2 − 2−ϵ1−ϵ 1
It is well-known that a rest point of the system is asymptotically stable if andonly if the real parts of both eigenvalues of the Jacobian are negative (e.g. Arnold[2006]). Thus, (ϵ, 1−ϵ) is asymptotically stable if and only if 2(1−ϵ)< p1 (ϵ, 1−ϵ) <2−ϵ2. Since p1 (ϵ, 1 − ϵ) > 1 − ϵ by (ii) in Definition 2, p1 (ϵ, 1 − ϵ) < 23 is impossible
3
for suﬃciently small ϵ. We can show that (1 − ϵ, ϵ) is not asymptotically stable by
the same procedure.
Claim 2. If there exists x′ = (x1′ , x2′ ) which satisfies ϕ̄1 (x′ ) = 0 and ϕ̄2 (x′ ) = 0,then x′ is a saddle point under any assortative matching rule.
2x′

x′

1
Since ϕ̄1 (x′ ) = 0, p1 (x′ ) = 32 . Then, p2 (x′ ) = 3x1′ and q2 (x′ ) = 1 − 3(1−x′ ) by the22(1)-(3). Thus, we obtain x2′ = 21 by ϕ̄2 (x′ ) = 0. The Jacobian matrix of g at (x1′ , x2′ )
is


′
′
(x1′ (1 − x1′ ) − ϵ(1 − ϵ))( ∂ϕ̄1 (x ) ) (x1′ − ϵ)(1 − ϵ − x1′ )( ∂ϕ̄1 (x ) ) ∂g ′ ′∂x1∂x2
(x , x ) =  ′′ ′∂x 1 2(x2 − ϵ)(1 − ϵ − x2′ )( ∂ϕ̄∂x2 (x1 ) ) (x2′ (1 − x2′ ) − ϵ(1 − ϵ))( ∂ϕ̄∂x2 (x2 ) )(x1′ − ϵ)(1 − ϵ − x1′ )(3 ∂p1 (x′ )) (x1′ − ϵ)(1 − ϵ − x1′ )(3 ∂p1 (x′ ))
∂x1
∂x2
 .
= 
∂p1 ′
1
1
′
2 ′ 16( 21 − ϵ)2 (−6x1′ ∂p
(x
))
(
−
ϵ)
(x
(
−
6
(x
)))
1 3
∂x1
2
∂x2
In this matrix, one eigenvalue is negative and the other is positive. Therefore,(x1′ , x2′ ) is a saddle point.
By claims 1 and 2, it suﬃces us to construct assortative matching rules underwhich x = (x1′′ , ϵ), (ϵ, x2′′ ), (1 − ϵ, x2′′′ ), (ϵ, ϵ), and (1 − ϵ, 1 − ϵ) are asymptoticallystable for suﬃciently small ϵ. First, we consider x = (ϵ, ϵ). The Jacobian matrixof g at (ϵ, ϵ) is(1 − 2ϵ)(3p1 (ϵ, ϵ) − 2)

∂g
0

(ϵ, ϵ) = ∂x0(1 − 2ϵ)(2 − p2 (ϵ, ϵ) − 2q2 (ϵ, ϵ))(1 − 2ϵ)(3p1 (ϵ, ϵ) − 2)

0

= 
1+ϵ 
2ϵ
− p1 (ϵ, ϵ) 1−ϵ)0(1 − 2ϵ)( 1−ϵ82ϵby (1)-(3). If an assortative matching rule satisfies 1+ϵ< p1 (ϵ, ϵ) < 32 , then botheigenvalues are negative. If we consider the rule (8) with α = 1/10, these conditions are satisfied. Hence, (ϵ, ϵ) is asymptotically stable if these conditions are satisfied. Assortative matching rules, under which other points, (x1′′ , ϵ), (1 − ϵ, 1 − ϵ),(1 − ϵ, x2′′′ ), and (ϵ, x2′′ ) are asymptotically stable for suﬃciently small ϵ, are calculated by the same procedure. Therefore, we obtain

A = {(ϵ, ϵ), (1 − ϵ, 1 − ϵ), (x1′′ , ϵ), (1 − ϵ, x2′′′ ), (ϵ, x2′′ )},where x1′′ , x2′′ , and x2′′′ satisfy ϕ̄1 (x1′′ , ϵ) = 0, ϕ̄2 (ϵ, x2′′ ) = 0, and ϕ̄2 (1 − ϵ, x2′′′ ) = 0.
For each x ∈ A, the conditions that x is asymptotically stable for suﬃciently
small ϵ are given by the followings:(i) (ϵ, ϵ) is asymptotically stable if an assortative matching rule satisfies
2ϵ< p1 (ϵ, ϵ) < 2/3.
1+ϵ(ii) (1 − ϵ, 1 − ϵ) is asymptotically stable if an assortative matching rule satisfies
ϵ2< p1 (1 − ϵ, 1 − ϵ) < 1 −.
32−ϵ(iii) (x1′′ , ϵ) is asymptotically stable if an assortative matching rule satisfies∂p1 ′′(x , ϵ) < 0
∂x1 1
2ϵ
2
< p1 (x1′′ , ϵ) = .
1+ϵ3(iv) (1 − ϵ, x2′′′ ) is asymptotically stable if an assortative matching rule satisfiesx′′′2< p1 (1 − ϵ, x2′′′ ) = 2 2 ′′′3
1 + x2
1 − x2′′′
∂p1
′′′
)
>
2
(1 − ϵ, x2.
∂x2
1 + x2′′′(v) (ϵ, x2′′ ) is asymptotically stable if an assortative matching rule satisfies
x2′′
2
′′
> p1 (ϵ, x2 ) = 2
3
1 + x2′′
1 − x2′′∂p1(ϵ, x2′′ ) > 2.
∂x2
1 + x2′′9

By Proposition 1, limϵ→0 (x1′′ , ϵ) = (0, 0) and limϵ→0 (1 − ϵ, x2′′′ ) = (1, 1). By
parity equations (1)-(3), ϕ̄2 (x) = 0 impliesx2.
p1 (x1 , x2 ) = 2
1 + x2Since p1 (ϵ, x2′′ ) < 23 , x2′′ < 1/2. Then, limϵ→0 (ϵ, x2 ) ∈ {(0, c)|c ∈ [0, 1/2)}. Thus, as
ϵ goes to 0, A converges to {(1, 1)} ∪ {(0, x2 )|x2 ∈ [0, 1/2)}.
Proposition 2 shows that imperfect Nash equilibria in (0, c), c ∈ [0, 1/2), areasymptotically stable under some assortative matching rules. We call each (0, c)with c ∈ (0, 1/2) a partially fair equilibrium where all proposers are fair but there
exist some selfish responders, (0, 0) the fully fair equilibrium where all agentschoose fair actions, and (1, 1) the selfish equilibrium where all agents choose selfish actions.
Proposition 3. The fully fair equilibrium and the selfish equilibrium are asymptot1
ically stable under any assortative matching rule satisfying ∂p(x′′ , ϵ) < 0 where∂x1 1
ϕ̄2 (ϵ, x2′′ ) = 0.
2ϵProof. When p1 (ϵ, ϵ) < 2/3, (ϵ, ϵ) is asymptotically stable since 1+ϵ< ϵ <∂p1 ′′′′p1 (ϵ, ϵ). When p1 (ϵ, ϵ) ≥ 2/3, by ∂x1 (x1 , ϵ) < 0, (x1 , ϵ) is asymptotically stable.
Thus, the fully fair equilibrium (0, 0) is asymptotically stable under any assorta1
tive matching rule satisfying ∂p(x′′ , ϵ) < 0, while a partially fair equilibrium is∂x1 1
not always asymptotically stable4 .
ϵWhen 1 − 2−ϵ> p1 (1 − ϵ, 1 − ϵ), (1 − ϵ, 1 − ϵ) is asymptotically stable sinceϵp1 (1 − ϵ, 1 − ϵ) > 1 − ϵ > 2/3. When 1 − 2−ϵ≤ p1 (1 − ϵ, 1 − ϵ), since q2 (x) is
non-decreasing in x2 ,1 − p1 (1 − ϵ, x2′′′ )
1
∂p1
=
,
(1 − ϵ, x2′′′ ) ≥′′′∂x2
1 − x2
1 + x2′′′where ϕ̄2 (1 − ϵ, x2′′′ ) = 0. This implies, by x2′′′ ≥ 1 − 2ϵ,1 − x2′′′∂p1(1 − ϵ, x2′′′ ) > 2.
∂x2
1 + x2′′′
Then, (1 − ϵ, x2′′′ ) is asymptotically stable. Thus, the selfish equilibrium (1, 1) is
asymptotically stable under any assortative matching rule.
Intuitively, fair responders are easy to encounter fair proposers than selfish
responders by assortativity. Hence, if proposers are almost fair (x1 ≈ 0), thenstrategy N generates higher average payoﬀ than strategy Y. If proposers are almost
selfish (x1 ≈ 1), in contrast, an average utility of strategy N is smaller than an
average payoﬀ of strategy Y. Therefore, the fully fair equilibrium and the selfish
equilibrium coexist.
4

A counter-example is given in section 4.

10

4

An Example

In this section, we give an example of a completely assortative matching rule
under which only the fully fair equilibrium and the selfish equilibrium are asymptotically stable.
Definition 5. A matching rule (pi , qi )i=1,2 is called completely assortative if thematching probability in population 1 is defined as{ x2
if x1 > x2
p 1 = x1
1 otherwise{ 1−x2
if x1 ≤ x2
q1 = 1−x1
1otherwise,
and the matching probability in population 2 is defined as{ x1
if x1 ≤ x2
p 2 = x2
1 otherwise{ 1−x1
if x1 > x2
q2 = 1−x2
1otherwise.
Figure 3 shows the probability with which selfish proposers may meet selfish
responders under this assortative matching rule and under the random matchingrule. As can be seen in Figure 3, fair proposers meet more likely fair responders
than selfish proposers. The completely assortative matching rule maximizes anumber of pairs which consist of a fair proposer and a fair responder.
Under the completely assortative matching rule, the selection dynamics is described as follows:case 1: x1 > x2
x2
− 2)
x1
1 − x1x˙2 = g2 (x1 , x2 ) = (x2 − ϵ)(1 − ϵ − x2 )(1 − 2).
1 − x2x˙1 = g1 (x1 , x2 ) = (x1 − ϵ)(1 − ϵ − x1 )(3

(9)
(10)

case 2: x1 ≤ x2
x˙1 = g1 (x1 , x2 ) = (x1 − ϵ)(1 − ϵ − x1 )(3 − 2)x1x1x˙2 = g2 (x1 , x2 ) = (x2 − ϵ)(1 − ϵ − x2 )( + 2 − 2 − 2).
x2
x2
11

(11)
(12)

p1
1

p1

0.8

0.8

assortative matching
0.6

0.4

0.6

0.4

random matching

0.2

O

assortative matchingrandom matching

0.2

0.2

0.4

0.6

0.8

1 x1

O

0.2

0.4

0.6

0.8

1

x2

Figure 3: The probability p1 = Pr(L meets Y). The graph in left side is p1 (x1 , 0.5).
The graph in right side is p1 (0.5, x2 ).

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

Figure 4: Phase diagram under the completely assortative matching rule.

12

This selection dynamics is a nonlinear dynamic system. Figure 4 shows the phase
diagram of the system (9)-(12).
Proposition 4. Let Ā be the set of asymptotically stable points of the system (9)(12). Ā = {( 32 ϵ, ϵ), (1 − ϵ, 1 − 2ϵ)}.
Proof. It is straightforward to see that the system (9)-(12) has the following set ofrest points,
R̄ = {(ϵ, ϵ), (ϵ, 1 − ϵ), (1 − ϵ, ϵ), (1 − ϵ, 1 − ϵ)}3∪{( ϵ, ϵ), (1 − ϵ, 1 − 2ϵ)(3/4, 1/2)}.
2
To prove the proposition, we examine the eigenvalues of the Jacobian matrix.
In the case of x = (1 − ϵ, ϵ), the Jacobian matrix at (1 − ϵ, ϵ) is( 3ϵ − 2)(2ϵ − 1)∂g0
1−ϵ(1 − ϵ, ϵ) = 
 .
2ϵ
∂x
0
(1 − 1−ϵ )(1 − 2ϵ)Therefore, (1 − ϵ, ϵ) is not asymptotically stable for suﬃciently small ϵ. Similarly,
we can show that x = (ϵ, 1 − ϵ) is not asymptotically stable, either.
In the case of x = (1 − ϵ, 1 − ϵ), it is not suﬃcient to consider the system in
only one case, since two case of the system is surely included the neighborhoodof (1 − ϵ, 1 − ϵ). The both Jacobian at (1 − ϵ, 1 − ϵ) in cases 1 and 2 are2ϵ − 1

∂g
0
 .
(1 − ϵ, 1 − ϵ) = ∂x0
1 − 2ϵTherefore, (1 − ϵ, 1 − ϵ) is a saddle point for suﬃciently small ϵ. We can show thatx = (3/4, 1/2), (ϵ, ϵ) is not asymptotically stable but a saddle point in the samemanner.
Finally, we examine the case of x = ( 32 ϵ, ϵ). The Jacobian at ( 32 ϵ, ϵ) is− 2 + 5 ϵ 1 − 5 ϵ 
∂g 3
2 .
( ϵ, ϵ) =  3 3
−1+2ϵ ∂x 201−ϵThus, ( 32 ϵ, ϵ) is asymptotically stable for suﬃciently small ϵ. We can show that
(1 − ϵ, 1 − 2ϵ) is asymptotically stable in the same manner.
Proposition 4 shows that only the selfish equilibrium and the fully fair equilibrium survive in the replicator dynamics with the completely assortative matching
rule.
13

5

Discussion

In this paper, we have studied the role of matching rules in the replicator dynamics
in ultimatum mini game. If encounters are random, then Gale et al. [1995] showsthat the subgame perfect equilibrium is the only one asymptotically stable point.
However, many experimental observations do not support this result.
There are several possible explanations for fair actions. One explanation isinequity aversion. Subjects’ preferences depend not only on their own monetary
payoﬀs but on fairness or equity (Bolton and Ockenfels [2000], Fehr and Schmidt[1999]).
Another explanation is the reputation eﬀect. In repeated situation, people
worry about bad reputations and would act fairly (Nowak and Sigmund [1998],
Nowak et al. [2000], Ohtsuki and Iwasa [2004]). If responders accept any unfair
oﬀer, this may become known and the next proposer will make unfair oﬀers. To
act fairly improves their long-term payoﬀs even if their short-term payoﬀs decrease.
Here we have considered the evolution of fair actions. An assortative matchingrule (Becker [1973, 1974], Shimer and Smith [2000], Atakan [2006]) can be regarded as an alternative explanation of fair actions. The assortative matching rule
make replicator dynamics to be a nonlinear system, and thus expands the set ofstable points. For the ultimatum mini game, there exist some assortative matchingrule supporting imperfect Nash equilibria. The average payoﬀ of fair actions may
become higher than selfish actions depending on the mass of fair agents. Therefore, the fair actions may survive. These assortative matching rules support fairactions as asymptotically stable states without fair preference or reputation.
Our study has some limitations. First, the selfish equilibrium is also asymptotically stable. The dynamic path depends on an initial state and an assortative
matching. Second, it is not known that the same result holds for other types of dynamics. The ultimatum “mini” game is surely restricted. The analysis of generalgames in left for future works.

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