Evolut Inst Econ Rev (2017) 14:1–27
DOI 10.1007/s40844-017-0071-8
ARTICLE

Theoretical model of institutional ecosystems
and its economic implications
Takashi Hashimoto1•

Makoto Nishibe2,3

Published online: 3 May 2017
Ó Japan Association for Evolutionary Economics 2017

Abstract Previous game-theoretic studies of institutions have viewed institutionalchanges as either exogenous changes in game form or changes in the game equilibrium
through exogenous shocks. Both views of institutions are static and cannot express
endogenous changes in institutions. The latter approach states that multiple institutional
systems can be kept stable through institutional complementarity and that the changes ininstitutional systems only arise from exogenous shocks that are sufficiently large tooverturn such complementarity. However, they cannot account for the aspects of competition and co-existence where multiple institutions change their relative frequency
through endogenous changes. In this article, we model the ecological systems of institutions, as an extensive synthesis of replicator dynamics and evolutionary games, to describe
institutional systems that evolve phylogenetically associated with changes in population
structure or a pool of rules as replicators, which corresponds to a gene pool. A mathematical model of rule ecosystem dynamics describes rule dynamics wherein multiple ruleschange their relative weights through evaluations by individuals. In this model, the concept ‘‘a meta-rule = an individual value consciousness’’ is introduced for the ruleThis article is the revised English version of the Japanese original version (Hashimoto and Nishibe2012).
& Takashi Hashimoto
hash@jaist.ac.jp
Makoto Nishibenishibe@econ.hokudai.ac.jp
1

School of Knowledge Science, Japan Advanced Institute of Science and Technology (JAIST),
1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan

2

Graduate School of Economics and Business Administration, Hokkaido University, Kita 9 Nishi
7, Kita-ku, Sapporo 060-0809, Japan

3

Present Address: School of Economics, Senshu University, 2-1-1 Higashimita,
Tama-ku, Kawasaki, Kanagawa 214-8580, Japan

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evaluation. Depending on the setting of the meta-rule, the dynamics of the game rules andindividual strategic rules change. We can thus comprehend the endogenous formation,
alteration, and extinction (i.e., the evolution of institutions) through the interactions among
the game rules as well as those between the game rules and strategic rules. Many otherstudies focus on how rational individuals select strategies to maximize their payoffs
without considering the bounds of rationality. Even when considering these, individual
cognitive frameworks and values are typically given. By contrast, this study assumes that
individuals have internal rules that express cognitive frameworks and values as meta-rules
and analyzes the dynamic interactions between institutions, as social external rules at themeso level, and strategic rules and value consciousness, as individual internal rules at the
micro level. In our model, institutional changes do not arise as game equilibria (i.e.,
players’ selection of strategies in a game), but rather as the rise and fall of game forms, asvarious rules, in multi-games based on a meta-rule. This view is based on an evolutionary
approach where socio-economic evolution is considered to be a selection of rules andinstitutions rather than that of individuals or their strategies. We discuss the implications of
the model of institutional ecosystems on the description of the socio-economy and its
evolutionist institutional design.
Keywords Replicator as rule  Institutional variety and endogenous change Institutional ecosystem  Rule ecosystem dynamics  Meta-rules  Evolutionist
institutional design  Micro–meso–macro loops
JEL Classification B25  B52  C73  E14  E42

1 Introduction
Institutions, as ways of thought and behavior common in society (Veblen 1899),
consciously or unconsciously regulate our thought and behavior in some way and enable
our social lives. Most of these institutions are not given to us a priori, but they are rather
spontaneously formed from our behavior. Institutional changes refer to the ruledynamics (Hashimoto and Nishibe 2005; Hashimoto 2006) that arise from theinteractions between institutions and individual behavior regulated by these institutions.
In regard to institutions, there are two viewpoints from a game-theoretic
perspective: ‘‘institutions as game forms’’ (North 1990; Hurwicz 1996) and‘‘institutions as equilibria of strategies’’ (Lewis 1969; Pagano 1992; Pagano andRowthorn 1994; Young 1998; Aoki 2001).
On the one hand, the former states that institutions are the rules of interactions. Here, aninstitution is described by a game form that consists of a set of players, options, andconsequence functions (payoff matrices); individual behaviors are then described as
options in the game. In this case, as long as the game form is not changed externally, there
will be no change in the institution. On the other hand, the latter regards an institution as the
Nash equilibria of a game and individual behaviors as the moves chosen on the basis of thestrategies in the game. In addition, an established institution is characterized by the
combinations of strategies at the equilibria. To make this static framework dynamic
through the introduction of an evolutionary game, we can treat the establishment of123

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3

institutions as a starting point for strategy combinations out of an equilibrium state,moving toward those in an equilibrium state. However, once an institution has been
established, it is in an equilibrium; therefore, as long as there are no exogenous shocks todisplace the state from the equilibrium, the institution will not change.
Accordingly, from both perspectives, an established institution shows no endogenouschanges and it will remain static as long as no exogenous disruptions exist. In this paper,
we integrate these two game-theoretic viewpoints of institutions and introduce amathematical model that can explicitly deal with rule dynamics. Institutions are formed,
maintained, altered, and eliminated through the meso-level interactions amongindividual cognition, thought, and behavior (micro) and social consequences (macro),and are formed by alternative or complementary relationships in those processes.
Institutions are of two types: ‘‘outer institutions’’, as explicitly enforced social
rules such as laws, regulations and game forms, and ‘‘inner institutions’’, asspontaneously formed rules such as values, norms, ethics, collective consciousness,and standard tactics. We suppose that both types of institutions are composed of‘‘if–then’’ rules (replicators) shared by many individuals (interactors) located at the
micro level. It is then ontologically seen that such rules shared within a group orsociety exist at the meso level (i.e., above individuals at the micro level and below
social consequences at the macro level). Accordingly, both outer and innerinstitutions could be called ‘‘external rules’’ because they act on individuals fromthe outside. On the contrary, individual-specific if–then rules (replicators) are called‘‘internal rules’’ since they determine the cognition, thoughts, and behaviors ofindividuals from the inside. Those rules and institutions are arranged as follows:
Meso level: external rules = institutions (outer institutions and inner institutions).
Micro level: internal rules.
Each individual (interactor) exhibits routinized cognition, thoughts, and behaviors based on replicators as both external rules and internal rules (Hayek 1967;
Nishibe 2005, 2006, 2010a).1 Accordingly, these types of individuals are notrational entities that pursue optimality. While there are limits on their ability to
1

The definition and distinction of external/internal rules and outer/inner institutions by treating meso levelinstitutions as game rules (outer institutions) and meta-rules as game rule evaluations (inner institutions) are
different from previous ones (Nishibe 2006). While the game rules (outer institutions) set the range of behavior
for each individual ‘‘from the outside’’, or in a top-down manner, meta-rules (inner institutions) set the nature of
the total game ‘‘from the inside’’ by using the relative frequency of each game rule, or in a bottom-up manner.
While the ‘‘external/internal’’ in external/internal rules represents the static ‘‘boundaries of a set’’ or ‘‘areas’’,‘‘outer/inner’’ implies a dynamic ‘‘causal direction’’ or ‘‘determined relationship direction.’’ For this reason, we
use the terms inner/outer institution. Note that outer and inner institutions are not distinguished by the explicit or
implicit sharing of rules among individuals. If that were the case, then it would be more appropriate to use theterms ‘‘explicit/implicit institution’’ or ‘‘formal/informal institution.’’ In that case, the classifications would
become closer to legal and regulatory/ethical and customary. Then, ‘‘implicit’’ or ‘‘informal’’ institutions would
merely be unexpressed conventions or codes that arise within the domain of freedom set by ‘‘explicit’’ or‘‘formal’’ institutions, and eventually fill the space left by ‘‘explicit’’ or ‘‘formal’’ institutions. The problem is thatif ‘‘implicit’’ or ‘‘informal’’ institutions are only implicit parts of the game rules, they cannot be meta-rules forvalue consciousness on the fairness and appropriateness of the game rules at the meta level; thus, it would be
impossible to show mutual determination between the outer and inner institutions found in this paper. Here,meta-rules that regulate the game rules are given for a provisional treatment. However, this does not mean thatthey exist transcendentally from the outset. They can change dynamically and endogenously. A meta-rule is

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reason, calculate, and act, they are realistic in how they adapt to actual changes. An
institutional ecosystem is a type of system that continuously maintains a variety of
institutions within the process of multiple institutions co-existing and becoming
extinct in an unending micro–meso–macro loop (Nishibe 2005, 2006, 2010a, 2012;
Hashimoto et al. 2010; Kobayashi et al. 2013). We aim to construct a mathematicalmodel for the institutional ecosystem, named rule ecosystem dynamics (RED),
which describes the dynamics of this type of system (Hashimoto and Nishibe 2005).
The model outlined in this paper is, mathematically, an extension of the replicator
dynamics commonly used in evolutionary game theory. First, we treat external
institutions as the interaction rules (game rules hereafter) and represent the game form
(payoff matrix). Replicators (internal rules), which take the form of individual behavior,
are regarded as the game strategies and called ‘‘strategy rules’’. Moreover, to show theresults of the interactions among multiple outer institutions, we introduce multiple
games along with their respective weights. From now on, weights are used to express thesignificance of the game rules. Each player plays all games and earns a payoff multipliedby the weight of each game. The strategy rule distribution varies with time depending on
the payoff. Along with the strategy rule distribution, the game rule weights also varywith time, with the variations depending on the payoff and strategy rule distribution.
In this study, we introduce ‘‘meta-rules’’ anew to determine how the game rule weights
change. These meta-rules are used to evaluate the game rules and are thought to be at amore basic level than the game rules when viewing changes. Such meta-rules, in general,are not necessarily expressed explicitly, but rather should be considered to be existing
implicitly as a collective representation of the game rule evaluations made by gameparticipants. In other words, these meta-rules are external rules as inner institutions. Notethat this type of evaluation does not evaluate the personal payoff or utility gained as aresult of the game but rather the game rules themselves. This evaluation thus differs fromso-called consequentialism. The meta-rules discussed in this paper are not a type of
aggregate rules set to obtain a social welfare function composed of each member’s utilityor the social preferences from each member’s preferences to evaluate society’s overallsatisfaction from a utilitarian standpoint.2 In addition, these meta-rules are exogenously
given; we consider them to be neither (game) rules nor the consequences of rules.
Footnote 1 continued
synthesized as value consciousness from individuals’ evaluation in their minds. During the process, it isthen transferred from the internal to the external.
2

Games that explicitly introduce bidirectional causality between the game rules and meta-rules are
generally understood to be ‘‘triangular game processes’’ (Nishibe 2006). Meta-rules herein are of two
types: ‘‘definition and modification meta-rules’’, which predefine the rules and modify them on the basisof ex post facto evaluations, and ‘‘standard meta-rules’’, which evaluate the rules on the basis of the game
results. In RED introduced herein, meta-rules are first understood to be ‘‘standard meta-rules.’’ This is thegame evaluation function kg ¼ kg ðx; ug Þ given in Eq. (21). Each game rule is evaluated from high to low
on the basis of this function. In general, game rule g is modified after having been evaluated low by thesestandard meta-rules (or perhaps by violating a meta-rule standard). For example, if a certain law or rule isdetermined to be unconstitutional, it will likely be altered. However, in RED, game rule g itself is notaltered or edited after receiving a low evaluation, but rather its weight, wg, will decrease. In this manner,‘‘definition and alteration meta-rules’’ include situations where only the weight, wg, of game rule g ismodified; game rule g itself is not modified. Within triangular game processes, it is possible to include
meta-meta-rules, even higher-level processes that can alter meta-rules. As outlined in this paper, within
RED, meta-rules are assumed to be exogenous and do not include the processes wherein they are revised.

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An example of a meta-rule is profit orientation in the market economy. If we saythat each market is evaluated by the sum of all the market transactions within the
broad principles of the market economy, then the markets that generate a greater
profit to participants overall are evaluated higher and their weights increase,
whereas the ones that do not will be evaluated lower and their weights decrease.
Because of this, transactions in weaker markets diminish until these markets haveno choice but to disappear eventually. Alternatively, norms and value consciousness, such as those related to ethical foundations in utilitarianism or egalitarianism,
express meta-rules. In the former, the rules closest to producing the greatest
happiness for the greatest numbers are evaluated higher, while in the latter rules that
produce the fewest disparities among people are evaluated higher.
In RED as introduced herein, meta-rules do not change and are set exogenously.
Accordingly, one should be empirically led toward reasonable meta-rules because we
can expect it to be difficult or even impossible to discuss a priori what type of metarule is normatively the most desirable [as in Arrow’s (1951) impossibility theorem].
In addition, meta-rules are collective representations of the norms and values formed
by game participants according to the game rules and hence cannot be easily madeexplicit. In other words, they must be discovered by empirical research.
While each game rule (or game form) does not change, the overall rules ofsociety, as a set of games, are modified through changes in the weights of the game
rules. There are many levels of game rule plasticity (or ease of change), so we must
assume that the game rules have a hierarchy. For example, let us consider thesystem of law. We have, in a constitutional state such as Japan, constitutional law,criminal law, civil law, regulations, and many other types of rules. Within these areeasily changeable types such as regulations or civil law as well as difficult-tochange types such as criminal law. We also have constitutional law, whichconstrains both criminal and civil law; this is harder to change, and it sets forth theprinciple to change the lower-level rules. Within the hierarchy of plasticity, metarules are seen as more universal and less plastic compared with the rules on which
we are focusing. If we are interested in changes at the civil law level, we treat theconstitutional law level and its unchanging nature as given.
Both the strategy rule distribution and the game rule weight distribution change
in this framework. These distributions may be in equilibria or show metastability
with certain periods of steady states. The case of metastability is thought of as astate with a provisional institution.
In this paper, we first derive RED from replicator dynamics and then perform asimple simulation analysis. Next, we consider examples of the institutional
ecosystem and meta-rules and discuss future directions and economic implications.

2 Introduction to RED
RED is defined as an extension of replicator dynamics. In this section, we explain
replicator dynamics as the foundation, multi-game dynamics, which extend
interactions into multiple games, and finally RED, wherein we introduce thedynamics of the games themselves.

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2.1 Replicator dynamics
People have certain cognition, thought, and behavioral patterns in everyday life.
These patterns are the ‘‘internal rules’’ that peoples’ cognition, thoughts, andbehaviors follow within customs, habits, inclinations, and values. They are different
from ‘‘external rules’’, which constrain peoples’ range of cognition, thoughts, andbehaviors externally as societal laws and regulations (outer institutions) or values
and norms (inner institutions). These patterns are typically expressed as sets of if–then rules. They are formed through education and traditions at home, at school, andin the community; they are then spread by imitation and studied via media or directinteraction. Such patterns of cognition, thoughts, and behaviors are propagated andspread in this manner to families, communities, schools, corporations, markets,states, and other groups of people. We consider these ‘‘patterns’’ to be replicators(Nishibe 2006, 2010b).
In general, the propagation dynamics of replicators can be described by replicator
dynamics. The following differential equation describes the changes in thefrequency of N-type replicators (as a ratio of population) in interactions as a game:
x_i ¼ xi ðui  uÞ;ði ¼ 1  NÞ;

ð1Þ

where the variable xi is the i-th replicator’s frequency and this satisfies the following
equation:NX

xi ¼ 1;

ð2Þ

i¼1

the variable ui is the payoff gained by the i-th replicator,
ui ¼

N
X

Eði; jÞxj ;

ð3Þ

j¼1

and u is the average payoff written as
u ¼

N
X
i¼1

x i ui ¼

N X
NXEði; jÞxi xj ;

ð4Þi¼1 j¼1

by supposing3 Eði; jÞ as the payoff resulting from the interactions between replicators i and j.
Here, we make the following simplification. First, each individual interacts withthe others in a certain game under the condition that the outcome of the individual’s
behavior depends on others’ behavior. In general, individuals interact with theothers within various games, but here we use only one game. Second, eachindividual has only one internal rule for cognition, thought, and behavior within thegame. Accordingly, the individual’s characteristic is expressed through an internal
3Eði; jÞ need not be given prior to interactions; it only needs to be the sum of the interaction results.
Replicators are behavioral patterns and not thoughts to be used to select behaviors on the basis of the precalculation of the game results.

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rule as a replicator. This individual imitates a replicator used by another wellbehaved individual (i.e., a replicator highly gained in the game). Thus, while eachindividual is characterized by one replicator, one replicator can be used by many
individuals. In this manner, the relative frequencies of successful replicators
increase. Within replicator dynamics, the rate of imitation is proportional to thedegree of success. More concretely, replicators with a higher-than-average payoff
increase in proportion to the difference from the average, while those with a lowerthan-average payoff decrease.4
We set no particular specifications for Eði; jÞ, the function that determines thepayoff (i.e., the outcome function of the game). We do, however, assume that two
individuals gain some payoff as an outcome of behaving and interacting according
to their own replicators.
2.2 Multi-game dynamics
An institutional ecosystem conceptualizes social reality not as independent
institutions but as the co-existence of and interactions among multiple institutions.
In other words, in an institutional ecosystem, individuals simultaneously sharegame-theoretic interactions with other individuals in multiple games rather thanplaying only one game. Multi-game dynamics is thus an extension of replicator
dynamics for describing an individual as playing multiple games simultaneously.
To make our explanation easy to understand, we first use the example of twogames being played simultaneously and then expand that example into a generalone. By representing the payoff matrices of two games by A and B, where thenumber of possible moves (matrix rank) can be different, and considering a strategy
rule as a set of replicators (a combination of the two replicators used in each game),
we rewrite the strategy rule as follows:
Strategy rule ði; jÞ: replicator i is used in game rule A and replicator j in game
rule B.5
We suppose that xij is the frequency, which is the population
Pshare relative to thewhole, of an individual with strategy rule ði; jÞ, wherei;j xij ¼ 1, and that
4

In general, explanations of replicator dynamics in the evolutionary game theory literature call these
replicators ‘‘strategies’’ . Strategies are easily confused with moves in games, so some care must be taken.
A strategy is a method for selecting a move. In individuals, replicators correspond to strategies andbehaviors to moves. Different strategies (replicators) may make the same move (behave in the same way).
The term ‘‘strategy rules’’ used in this paper refers to this type of strategy. For example, the tit-for-tat
strategy in the iterated prisoner dilemma game is an if–then rule that states ‘‘if the other player cooperated
in the last game, then I will cooperate with him/her this time; otherwise, if the other player betrayed me inthe last game, then I will betray him/her this time’’. This is a replicator of an individual. The specificmoves are ‘‘cooperate’’ and ‘‘betray’’. The All-C strategy can be written as ‘‘if *, then cooperate’’, where‘‘*’’ is a wild card representing any behavior. In other words, no matter what the past history between
another player and an individual, this strategy is to always cooperate with the others. These two strategies
execute the same ‘‘move’’, cooperate, when the other player has cooperated in the last game.

5

Note that this is different from a strategy profile, which is a set of strategies selected by players for one
game rule and represents a state of a society. Strategy rule ði; jÞ herein expresses one’s way of deciding
behavior, namely the internal rule, for two game rules.

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E½ði; jÞ; ðk; lÞ is the payoff gained by the individual with strategy rule ði; jÞ in acompetition between strategy rules ði; jÞ and ðk; lÞ. Thus,
E½ði; jÞ; ðk; lÞ ¼ Aik þ Bjl ;ð5Þ

where Aik represents the element of row i and column k of matrix A. We assume thatthe two games are independent and that the payoff from each game independently
contributes to the total payoff of each player.
By using these expressions, we can now write the frequency changes (replicator
dynamics) of an individual with strategy rule ði; jÞ as
8
9
<X
=
XX
x_ij ¼ xij
ð6ÞðAik þ Bjl Þxkl 
xi0 j0 ðAi0 k þ Bj0 l Þxkl :
: k;l
;
0 0
k;l
i ;j

This is organized as follows:
80
1 0
19
< X
=
X
XX
XX
x_ij ¼ xij @Aik xkl 
xi0 j0 Ai0 k xkl A þ @Bjl xkl 
xi0 j0 Bj0 l xkl A
: k;l
;
0 0
0 0
k;l
k;l
k;l
i ;j

i ;j

2
X
¼ xij
ðugij  ug Þ;
g¼1

ð7Þ
where ugij is the payoff that strategy rule ði; jÞ gains in game rule g, namely
u1ij ¼

X

Aik xkl ;u2ij ¼

k;lX

Bjl xkl ;and ug is the average payoff in game rule g,XXXX
xi0 j0 Ai0 k xkl ; u2 ¼
xi0 j0 Bj0 l xkl :u1 ¼
i0 ;j0ð8Þk;li0 ;j0

k;l

ð9Þ

k;l

As multi-game dynamics with two games can be simply written as in Eq. (7), we
can obtain the generalized form of multi-game dynamics by playing M games asfollows:x_i1 iM ¼ xi1 iM

M
X

ðugi1 iM  ug Þ;

ð10Þ

g¼1

where a strategy rule ði1 ; i2 ;    ; iM Þ is the internal rule or a set of replicators,‘‘playing game rule 1 with replicator i1 ;    ; game rule g with replicator ig ;    ;game rule M with replicator iM ’’, and its frequency is xi1 iM . This also satisfies thefollowing equation:

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9

r1 X
r2
X



i1 ¼1 i2 ¼1

rM
X

xi1 iM ¼ 1;

ð11Þ

iM ¼1

where r g is the number of options within game rule g, that is, the number of
replicators and the rank of payoff matrix Ag . The variable ugi1 iM is the payoff of thestrategy rules ði1 ; i2 ; . . .; iM Þ in game rule g:
X g
ugi1 iM ¼
Aig kg xk1 kM ;ð12Þ
k1 kMg

and the variable u is the average payoff in game rule g:
X X
xj1 jM Agjg kg xk1 kM :
ug ¼

ð13Þ

j1 jM k1 kM

2.3 RED
Researchers have analyzed institutions by using the multi-game framework that
illustrates individual social behavior within multiple institutions (Gagen 2000, 2003;
Sallach et al. 2010). How such a multi-game framework changes the equilibrium
structures and dynamics of elemental games has also been analyzed (Hashimoto
2009; Hashimoto and Aihara 2009). Within the multi-game dynamics introduced inthis paper, individuals are expressed as the strategy rules (internal rules), that is, setsof replicators, playing multiple games. From the standpoint of the game rules (gameforms) as institutions, this framework illustrates the replicator dynamics in multiple
institutions where the populations of the strategy rules (replicators) that have higherthan-average payoffs increase their relative frequencies, while those that do notdecline. However, the conception so far, namely, multi-game dynamics, still
considers institutions to be static.
As stated above, institutions are not static entities within the institutional
ecosystem. At the meso level, they intermediate between individual cognition,
thought, and behavior at the micro level and societal consequences at the macrolevel, interacting with these two levels. Through these interactions, institutions
emerge, alter, and diminish; further, an institutional ecosystem with substitutive andcomplementary relationships among institutions changes over time. This is the
dynamics of the micro–meso–macro-loop.
Let us now extend the multi-game dynamics to express these dynamics. We
consider that each game rule in the multiple games has its weight that changes overtime with the time development of the strategy rule distribution. By introducing
weight wg of game rule g, we rewrite Eq. (10) as
x_i1 iM ¼ xi1 iM

M
X



wg ugi1 iM  ug :

ð14Þ

g¼1

This represents the dynamics of the strategy rule distribution.

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The weights of the game rules also have dynamics, which are assumed to dependon how individuals evaluate each game rule (outer institutions of external rules)
including the payoff structure (or payoff matrix). Game rules with higher-thanaverage evaluations increase their weights, while those with lower-than-average
evaluations decrease their weights. In other words, the weights of the game ruleschange just as in replicator dynamics according to the evaluations of the game rules.
The dynamics can be written as
sw_ g ¼ wg ðkg  kÞ;

ð15Þ

g

where k is the evaluation of game rule g, and
M
X

k ¼

wg kg ;

ð16Þg¼1

is the weighted average of the evaluations. The parameter s, a time constant,determines the rate of change of the game rules (i.e., the ratio of the changing speedof all institutions to that of individual strategies). The weight of game rule g satisfies
the following equation:
MX

wg ¼ 1:ð17Þ

g¼1

Now, the problem is how to evaluate the game rules. We believe that they can beevaluated by using the meta-rules discussed above. A meta-rule is the collective
representation of individuals’ value consciousness to evaluate the game rules and is,
in RED, given as an evaluation function for the game rules. The evaluations of gameform (external rule) arise as a result of the cognition, thought, and behavior
according to the particular strategy rules (internal rules or replicators) within games.
Accordingly, the evaluation of the game rules must be a function of the strategy ruleprofile and the payoff profile. In other words, they take the following form:
kg ¼ kg ðx; ug Þ;

ð18Þ

where x ¼ ðx1;1;...;1 ; x1;1;...;2 ; . . .; xr1 ;r2 ;...;rM Þ is a strategy rule profile (a vector of the
population
ratio
of
each
strategy)andug ðxÞ ¼ ðug1;1;...;1ggðxÞ; u1;1;...;2 ðxÞ; . . .; ur1 ;r2 ;...;rM ðxÞÞ is a payoff profile (a payoff vector for each strategy rule) within game g.
In summary, RED is defined by the following three equations:
x_i1 iM ¼ xi1 iM

M
X

wg ðugi1 iM  ug Þ;

ð19Þ

g¼1


sw_ g ¼ wg ðkg  kÞ;

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kg ¼ kg ðx; ug Þ:ð21Þ

Equations (19)–(21) represent the changes in individual replicators (strategy rule
distribution), the dynamics of the weights of the game rules, and the evaluation of
the games, respectively. Table 1 presents the characteristics of this framework, incomparison with those of replicator dynamics and multi-game dynamics, as shownin Table 1.
In addition to replicators (or behavioral patterns), as with replicator dynamics,
that express the internal rules of individuals, or the strategy rules used byindividuals, we consider, in RED, the game rules (external rules) as a kind of
replicator in the multi-game framework and introduce their selection dynamics. The
selection dynamics of the two types of replicators, micro-level strategy rules(internal rules) and meso-level game rules (an external rule as an outer institution),can be seen in Eqs. (19) and (20), respectively. Each type exists simultaneously at
different levels. Individuals are seen as mere vehicles (interactors) for these twocategories of replicators. The meta-rule in Eq. (21) determines the rise and fall of
the game rule replicators. Although we say that individuals are vehicles for these
two types of replicators, they are collectively engaged in forming the gameevaluation functions in Eq. (21) through value consciousness.

3 Examples of RED
We now discuss two examples, kept as simple as possible, of the evaluation
functions of the game rules and meta-rules and show their dynamics by using
simulations in order to demonstrate how they actually act.
3.1 Examples of meta-rules
One example is the average-payoff-type meta-rule defined by
Table 1 Comparison of the three frameworks
Replicator
dynamics

Multi-game dynamics

Rule ecosystem dynamics

Individual
characterization

Replicators(strategies)

Replicator groups(strategy rules)

Replicator groups (strategy
rules)

Games played

One game

Multiple games

Multiple games

Game rules

No weights

No weights

Weighted

Payoffs

No weights

No weights

Weighted

Game evaluations

None

None

According to meta-rules

Meta-rules

None

None

Distribution of strategy rules
and scores

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kgA ðx; uÞ ¼

X

xi1 iM ugi1 iM :ð22Þ

i1 iM

The higher the average payoff gained from the game rules, the higher those rules
are evaluated. Note that kgA ðx; uÞ is not the total of each individual’s payoff or eachstrategy rule’s payoff, but rather the total payoff of the strategy rules for each game
rule. For example, among diversified exchanges, which have various regulations andprovide various financial products, we can assume a meta-rule in financial markets
that regards exchanges giving a large profit on average highly. This is similar to thecase that the total of each individual’s utility functions represents social welfare
under the supposition of cardinal utility.
Another example is the inverse-variance-type meta-rule defined as follows:o1
XnkgIV ðx; uÞ ¼
xi1 iM ðugi1 iM  ug Þ2:
ð23Þ
i1 iM

This is an egalitarian meta-rule, namely a game rule with a larger variance inindividuals’ payoffs that is lower evaluated. This corresponds, for example, to asocial democratic meta-rule in which it is thought that a government should correct
economic disparity by income redistribution.
3.2 Simplifications and simulation settings
The two types of replicators, strategy rules and game rules, make RED a highdimensional dynamical system. To show the basic behavior of RED by using thetwo meta-rules introduced above, we simplify the system by posing several
constraints. First, we set the number of options of all the game rules to be the same,
expressed herein as N:
r g ¼ N;ðg ¼ 1  MÞ:

ð24ÞThis means that the types of thoughts and behaviors that can be selected are thesame in number in a situation of interactions (i.e., in all the elemental game rules
being played simultaneously). We also limit the strategy rules to select an identicalstrategy number (strategies with the same row and column) for all the game rules:
xi1 iM ¼ xi :ð25ÞIndividuals that select strategy rule 1 for game rule 1 will choose strategy rule 1even for game rule 2, and so on. Because the numbering of the strategy rules isarbitrary and the game structure does not change under the simultaneous
permutation of rows and columns in the matrix, the generality of strategy rulenumbers does not decrease.6 There is, however, a constraint that individuals whochoose the same strategy rule for a game rule must choose the same strategy rule for6

When an individual choose strategy rule 1 for game rule 1 and strategy rule 2 for game rule 2, replacingrow 1 with 2 and column 1 with 2 in the game rule 2 matrix makes the individual select strategy rule 1 forgame rule 2.

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the other game rules. In other words, individuals who choose ‘‘strategy rule 1 forgame rule 1 and strategy rule 1 for game rule 2’’ cannot co-exist with those whochoose ‘‘strategy rule 1 for game rule 1 and strategy rule 2 for game rule 2’’. This
constraint means that all the strategy rules (internal rules) of an individual should beconsistent with each other and with his/her customs, habits, routines, norms, andvalues. For example, although a corporation may adjust its strategy rules dependingon each market, its core strategy rule remains the same; one’s personality is not sofree to behave differently depending on the differing game rules, but its basic
personality is nevertheless the same. Mathematically, this limits the possible
diversity of replicators to a subspace, but may be useful as the first steps toward
understanding the nature of RED.
3.3 Simulation results
We determine the number of game rules M and strategy rules N for each game. In
the following simulations, the elemental game rules are randomly generated withthe average payoff 0 and variance 1. The time constant s = 1000. The initial sharesare the same for all the strategy rules.
With these settings, we first show a typical example of the average-payoff-typemeta-rule (Eq. (22)). Figure 1 shows a typical example of the changes over time forthe population shares of the strategy rules (a) and weight shares of the game rules(b) for the average-payoff-type meta-rule. In this case, one particular strategy rule
rapidly dominates. Following the domination, a game rule in which the dominant
strategy rule earns the highest payoff grows the most to obtain the highest weightshare. This phenomenon is a kind of globalization.
When a particular strategy rule becomes successful to gain population share, thegame rules in which that strategy rule gains more become highly evaluated andregarded as being relatively important. Then, the frequency of the successful
strategy rule that succeeds in the game rules increases again. They form a positive
feedback loop, and finally the strategy rule and game rule, respectively, monopolize.
The rate of following the strategy rule monopolization by the game rulemonopolization depends on the parameter s to decide the time constant. The
phenomenon of monopolization by both a strategy rule and a game rule, however,always occurs with the average-payoff-type meta-rule.
Figure 2 shows an example of the inverse-variance-type meta-rule, with N ¼ 3
and M ¼ 10.7 While there are periods in which a strategy rule dominates, those
periods do not last long, and the dominant strategy rules are continually replaced instark contrast to the case of the average-payoff-type meta-rule. Likewise, there are
periods in which a game rule dominates, but the dominant game rules arecontinually changing place as well. This phenomenon of continual replacement iseverlasting; while the period of replacement becomes long, as a saddle network, thecharacteristic of replicator dynamics is formed. The sudden changing of thedominant game rules, such as in revolutionary changes in social structure, is afeature of this meta-rule.
7

The other settings for the simulation are the same as in Fig. 1.

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Population share

(a)

1
0.8
0.6
0.4
0.2
0
0

200

400

600

800

1000

800

1000

Generation(b)

1

Weight share

0.8
0.6
0.4
0.2
0
0

200

400

600

Generation
Fig. 1 Typical example of the dynamics of the strategy rules (a) and game rules (b) within the averagepayoff-type meta-rule (six strategy rules, seven game rules, s = 1000)

In the case of inverse-variance-type meta-rules, when a particular game rule anda particular strategy rule have relatively high scores, the variance of the score in thatgame rule will increase and the evaluation of the game rule will decrease.
Accordingly, the game rules in which many strategy rules have similar scores will
have their weight share increase. While one might think this type of ‘‘egalitarian’’
game becomes dominant and an egalitarian society will emerge, any game rule has adifference in scores from the strategy rules unless all the strategy rules areequivalent. Because of the characteristics of replicator dynamics (i.e., strategy ruleswith higher-than-average scores increase and those with lower-than-average scores
decrease their population share), the disparities in the score distribution are

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(a)

15

1

Population share

0.8

0.6

0.4

0.2

0
0

50

100

150

200

250

300

350

250

300

350

Generation(b)

1

Weight share

0.8

0.6

0.4

0.2

0
0

50

100

150

200

Generation
Fig. 2 Typical example of the dynamics of the strategy rules (a) and game rules (b) within the inversevariance-type meta-rule (three strategy rules, 10 game rules, s = 1000)

magnified. When the disparity expands, the variance in that game rule increases andits evaluation decreases. The weight of the game rules with small variances for the
strategy rule distribution at that point will increase. This phenomenon will thenrepeat. In this manner, both types of replicators, game rules (external rules) and

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strategy rules (internal rules), form cycles under the inverse-variance-type metarule. The cycle of the strategy rules precedes that of the game rules. While periodsof both cycles are more or less the same, the game rules change more dramatically.

4 Implications
4.1 Rule ecosystem as interactions between the population of strategy rule
replicators (internal rules) and that of game rule replicators (external
rules as outer institutions)
RED can be written in the form of a multi-population replicator system. A replicatorsystem with two replicator populations, x and y, is generally written as the following
system of equations:x_i ¼ ½uðxi ; yÞ  uðxi ; yÞxi ;ð26Þ

y_i ¼ ½uðyi ; xÞ  uðyi ; xÞyi ;ð27Þ

where uðxi ; yÞ is the payoff gained by the i-th individual in population x in competition with population y, and uðxi ; yÞ is the average payoff in population x.
Likewise, uðyi ; xÞ is the payoff gained by the i-th individual in population y incompetition with population x, and uðyi ; xÞ is the average payoff in population y. In
other words, competition as a game occurs between these two populations, and theresults of this competition are compared with those of other individuals in each
population. Individuals with a higher-than-average payoff increase their share,
whereas those with a lower-than-average payoff decrease their share.
Now, let us consider the case of RED. First, we define matrix A whoseði1    iM ; gÞ element is ðAg xÞi1 iM , which represents payoff ugi1 iM gained by strategy
rule xi1 iM in elemental game rule g, where matrix Ag expresses elemental game ruleg. By introducing the vector expression of the weights (the weight profile)w ¼ ðw1 ;    ; wg ;    ; wM Þ, the payoff of strategy rule xi1 ...iM taking account ofweight is ðAwÞi1 ...iM and the average within population x is x  Aw. As a result,
Eqs. (19) and (21) can be rewritten as follows8:
x_i1 iM ¼ ½ðAwÞi1 iM  x  Awxi1 iM ;ð28Þ

 ug Þwg :sw_ g ¼ ½kg ðx; ug Þ  kðx;ð29ÞThese equations indicate that RED comprises multi-population replicator
dynamics with the interactions between a population of the strategy rules (internal
rules) and a population of the game rules (external rules as outer institutions). In
other words, each strategy rule interacts with the population of the game rules andcompetes within the population of the strategy rules, and each game rule interacts
8Equation (29) is not represented in matrix form as Eq. (28) because the meta-rule is generally notlimited to a linear function.

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with the population of the strategy rules and competes within the population of thegame rules. This interaction causes the frequency and weight of each to change. In
this manner, RED is a model that describes co-evolutionary processes as theinteractions between the replicators of the game rules (external rules as outerinstitutions) at the meso level and the replicators of the strategy rules (internal rules)at the micro level via the vehicles of replicators (i.e., individual cognition, thought,
and behavior). In these co-evolutionary processes, both types of replicators not only
rise and fall by altering their relative frequencies but also form mutuallycomplementary or alternative relationships.
4.2 Institutional ecosystem from the RED viewpoint
How are institutions understood from the viewpoint of RED? We define institutions
as a group of replicators (internal or external rules) stably shared among a relatively
large number of individuals (interactors) including organizations (Nishibe
2006, 2010a).
The phrase ‘‘stably shared among a relatively large number of individuals(interactors)’’ within RED means that the relative frequencies of the strategy rules
and relative weights of the game rules exceed a certain value for a certain period.
The value of the relative frequencies (weights) can be 0.5, 0.1, or 0.01, depending
on interest. The issue of how reasonable these values are is reduced to a matter ofdegree, becoming the point at which we can call these replicators ‘‘institutions’’ on
the basis of their dissemination. For example, if we say there is only one institution,
the value will be greater than 0.5. If we are interested in understanding theconditions in which many institutions co-exist, the value will be smaller, perhaps
0.1. Furthermore, when we are observing the formation of a new institution, thevalue may be set to 0.01. We wish to understand the evolution of economic society
as an institutional ecosystem wherein institutions are formed, persisted, altered, andextinguished and where multiple institutions co-exist. In this case, we must consider
a broad range of relative frequencies for replicators (rules). The period they share isalso reduced to a matter of degree and needs to be set appropriately depending on
the problem under consideration.
RED can be regarded as a system of the interactions between two types ofreplicators, strategy rules (internal rules) and game rules (external rules as outerinstitutions), as described in the previous subsection. Within institutional ecosystem
dynamics, game rules such as laws, rules, and regulations become ‘‘outerinstitutions’’ when they are shared among a relatively large number of individuals.
Further, strategy rules such as customs, practices, and values become ‘‘innerinstitutions’’ when they are shared among a relatively large number of individuals.
Just as there are two types of replicators, there are also two kinds of institutions,which interact with each other within the institutional ecosystem, mutually
supporting and inducing changes in one another. Traditional views of institutions
have focused exclusively on outer institutions and ignored the existence and roles of
inner institutions in an attempt to understand institutions. However, within theinstitutional ecosystem, understanding the interaction dynamics between outer andinner institutions is critical.

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What is an institutional ecosystem, where multiple game rules (outer institutions)exhibit complementarity and substitutability relationships and change as theyinteract along with individual strategies? One example can be found in theArgentine monetary system in the first decade of this century. The Argentine
government at the time had defaulted on its foreign debt; the peso, the national
currency, subsequently devalued. Multiple currencies were being used in parallel:
the key currency (the US dollar), the national currency (peso), bond currencies(patacón and LECOP), and community currencies (such as the RGT) (Gómez 2009).
These various currencies were used in different domains, differing in the class they
were used, what they could purchase, and in which markets they were used.
By viewing multiple currencies as trying to expand their domain of use, we
consider this to be a monetary ecosystem with two parties interacting with eachother: currencies as outer institutions (game rules) and the portfolio mixes of thesecurrencies as inner institutions (strategy rules). In the monetary ecosystem, users
were vehicles both for currencies as replicators representing the outer institutions
and for replicators representing the inner institutions (internal rules) such as thevalue consciousness toward currency. Both aimed to get as many people using thatcurrency as possible. Users viewed currencies as resources and wished to get as
many different types of currencies as possible. At the same time, users contributed
to the formation of meta-rules by evaluating the institutions of currencies. This
forms the interaction of the replicator dynamics at the two levels in the institutional
ecosystem, namely game rules and strategy rules, as shown in Eqs. (28) and (29).
4.3 Meta-rules in the monetary institutional ecosystem
In a monetary institutional ecosystem as seen in the Argentina example, the region
and amount of currency distribution change along with users’ strategy rules (internal
rules); further, the monetary institutional ecosystem itself, with its multiple coexisting currencies, changes, too. The internal rules corresponding herein to thestrategy rules include the habits, idiosyncrasies, routines, norms, and values that
determine an individual’s cognition, thought, and behavior. We believe thatpeople’s value consciousness is at the root of the evolutionary processes that changeboth outer institutions as the game rules (e.g., currency) and inner institutions (e.g.,currency portfolios). Value consciousness plays the role of evaluation rules formonetary institutions (i.e., meta-rules in RED). Each individual evaluates themonetary institutions (game rules) by deciding the weight of each currency to be
used. Participation in a game is decided by referring to one’s internal rules (strategy
rules) and payoff, including not only the currency income but also the social andcultural values that one might gain owing to these internal rules. Meta-rules are thebasis for determining those weights, but what are they really?
To help answer this question, a money consciousness survey was conducted(Kobayashi et al. 2010, 2013). This survey included 164 subjects consisting of theoperators, issuers, and participants of community currencies as well as those who
were or had been employed in financial organizations and others (e.g., working
people in other sectors, undergraduate and graduate students). They were asked torespond to 27 randomly ordered question items on money consciousness by123

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selecting the most suitable option from five options (strongly agree, weakly agree,
neutral, weakly disagree, strongly disagree). A factor analysis of the results
suggested three factors: ‘‘currency diversity’’, ‘‘fairness’’, and ‘‘profit orientation’’(Kobayashi et al. 2010). Such question items as ‘‘Do you think it is good that wehave different moneys from national currencies to live with?’’ (Q. 22), ‘‘Do you
think it is good that money can be created or issued freely by people?’’ (Q. 4), and‘‘Do you think it is good that money can be issued or created not only by the central
bank or commercial banks but also by people or communities?’’ (Q. 25) loaded
highly onto ‘‘currency diversity’’. Such question items as ‘‘Do you think thegovernment should provide every adult beyond a certain age with basic income for
their minimum standard of living?’’ (Q. 10), ‘‘Do you think it is good that we
accommodate each other with money?’’ (Q. 19), and ‘‘Do you think moneylenders
should not be in such financial institutions as commercial banks, but the
government?’’ (Q. 13) with respect to the fair distribution of money loaded highly
onto ‘‘fairness.’’ Such question items as ‘‘Do you think it is good for money to beable to buy anything you want?’’ (Q. 7), ‘‘Do you think it is better to earn moremoney?’’ (Q. 15), and ‘‘Do you think it is good that money can be created or issued
for the purpose of profit?’’ (Q. 12) loaded highly onto ‘‘profit orientation.’’Considering the relationship with the meta-rules exemplified in Sect. 3, we thinkthat ‘‘fairness’’, which highly evaluates a fair distribution, corresponds to ‘‘theinverse-variance-type meta-rule’’, which aims toward the equality of results. This
disposition is in contrast to ‘‘profit orientation’’, which favors having more moneyfor one’s convenience, which corresponds to ‘‘the average-payoff-type meta-rule.’’This intends to increase average income, an idea similar to a doctrine of economic
growth.
Returning to the results of the simulation analysis above, if everyone had the‘‘profit orientation’’ meta-rule, then monetary institutions would converge on asingle currency that could be used to earn money and spent to buy anything. Thisprediction implies that currencies, as a platform institution, would be locked in one
currency as a de facto standard, which is similar to that forecasted in the argument
for standardization competition based on increasing returns and network externalities (Arthur 1994). Even if a monetary institution that aimed for ‘‘fairness’’ (i.e.,equality of results) was established, economic activities using that currency would
always result in income disparities. Our simulation results thus indicate that themonetary institution would continue to be revised in an effort to achieve true
fairness.
How would ‘‘currency diversity’’, the first factor in the money consciousness
survey (Kobayashi et al. 2010), be formulated as a meta-rule (game evaluation
function) within the RED framework? If we think of ‘‘diversity’’ as the range ofscore distribution, then we can take a function of variance (a payoff-variance-typemeta-rule) thus:X
kgV ðx; uÞ ¼
xi1 iM ðugi1 iM  ug Þ2 :ð30Þ
i1 iM

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This meta-rule evaluates the variety in the payoff profile of each strategy rule as aresult of interactions. However, question items in the money consciousness survey
dealing with ‘‘currency diversity’’ included ‘‘Do you think it is good that we havedifferent moneys from national currencies to live with?’’ (Q. 22), ‘‘Do you think it isgood that money can be created or issued freely by people?’’ (Q. 4), and ‘‘Do you
think it is good that money can be issued or created not only by the central bank orcommercial banks but also by people or communities?’’ (Q. 25). This fact indicatesthat the co-existence of multiple monetary institutions is considered to besignificant. In other words, the diversity of opportunity, not of interaction results,
is evaluated. Accordingly, the payoff-variance-type meta-rule that evaluates thediversity of payoff profiles can only be used as an indirect result (i.e., the 0th orderapproximation) of the diversity of the game rules (outer institutions).
Unfortunately, the diversity of the game rules cannot be directly evaluated byusing the RED formulated thus far. Therefore, we now try to extend the RED
framework. To do so, Eq. (21) needs to be redefined to include a weight profile, w,
into the meta-rule evaluation function:kg ¼ kg ðx; ug ; wÞ:ð31Þ

Then, we introduce a function to determine the evaluation of the game rules (i.e.,a meta-rule) by considering their diversity with the weight variance or distance of
weights from the average. We cannot use the weight variance solely to evaluate thediversity of the game rules because, as seen in the following equation:
kgWV ðx; u; wÞ ¼

MXg Þ 2 ;ðwg  w

ð32Þ

g¼1

which is determined by the weight profile of all the game rules rather than used toevaluate each game rule. This meta-rule is not an evaluation of each monetary
institution, but rather a basis for assessing the state of the overall monetary
ecosystem (or the diversity of the institutions in an ecosystem). Therefore, we
consider the evaluation to be the distance from the largest weight, denoted as wMAX :kgWD ðx; u; wÞ ¼ wMAX  wg :ð33ÞThis meta-rule highly evaluates the game rules with small weights. In otherwords, it works to evaluate the game rules that have not been evaluated.
Accordingly, we expect the weights to become homogenized over time.
Alternatively, we can also suppose a meta-rule that evaluates the closeness of theweight of each game rule to the average weight, instead of the overall variance of
the weight profile:g Þ2 :kgWD ðx; u; wÞ ¼ ðwg  w

ð34Þ

In addition, the fact that three meta-rules were found through the money
consciousness survey suggests that all three co-existed simultaneously. The
allocations of those meta-rules likely varied by person. Indeed, Kobayashi et al.

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21(2010) showed the different emphases of people related to community currencies
and people working in the financial sector. Therefore, we must pursue our analysis
by combining the multiple meta-rules in RED as below:
kg ðx; uÞ ¼ akgA þ bkgIV þ ckgWD

ða þ b þ c ¼ 1Þ:

ð35Þ

These meta-rules are a combination of functions with different behavior such as
the variance- and the inverse-variance-types, and their dynamics are likely to becomplex.
4.4 RED as micro–meso–macro loops
A dynamic society can be viewed as a micro–meso–macro loop: institutions mediate
between individuals’ cognition, thought, and behavior (micro) and societal
consequences (macro) at the meso level, and they are formed, maintained, altered,and diminished through the interactions among individuals. Substitutive andcomplementary relationships among institutions are formed in that process. The
micro, with the meso as an intermediary, determines the macro and, at the sametime, the macro, with the meso as an intermediary, determines the micro (Nishibe2006). Let us reconsider RED from this perspective.
First, we schematize the replicator system from this perspective (see Fig. 3). The
strategy rules (internal rules) of individuals at the micro level interact through thegame rules (external rules as outer institutions). A payoff distribution is then formed
through these interactions as a societal consequence at the macro level. Replicator
dynamics determine the change in the distribution of the strategy rules in responseto the payoff distribution, and a regulative relationship from macro to micro isformed. The game rules are thought to be at a dimension that mediates between the
micro and macro (i.e., the meso level). If the game rules (game forms) are viewed asexternal rules, we can make the following contrasts: micro: meso: macro = strategy
rules (internal rules): game rules (external rules as outer institutions): societal
consequences. In usual replicator dynamics, however, there are no regulatory
relationships from macro to meso; thus, there are no changes in the game. This
makes it impossible to describe and analyze the dynamics of institutions, namely theformation and alterations through micro–meso–macro loops.
Within arguments that consider the equilibria of strategic forecasting as
institutions (Lewis 1969; Pagano 1992; Pagano and Rowthorn 1994; Young 1998;
Aoki 2001), a strategy distribution at an equilibrium becomes an institution when
positioned at the meso level as a shared belief (Fig. 4). From this standpoint, micro,
meso, and macro are all maintained statically, and regulatory relationships frommacro to micro work to stabilize and self-maintain the institution.
Next, we consider the micro–meso–macro loops within a rule ecosystem (Fig. 5).
Because we have introduced meta-rules herein, the states of the micro and the macro
influence the game rules via the meta-rules. Thus, two regulatory relationships areadded: one is a relationship from the macro to meta-rules at the meso, while theother is that from the micro to the meta-rules. Both these relationships influence the

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Macro
PayoffDistribution

Meso
Game Rule

Micro
Strategy Rules
Strategy Rules
Strategy Rules

Fig. 3 Micro–meso–macro loop in a replicator system. The dotted line represents inputs, while the blacklines show effects. Gray areas are dynamic, while the white area is static

evaluation of the game rules through the meta-rules. As a result, loops among
micro–meso–macro levels are formed, and the game rules change dynamically.
That the framework of the rule ecosystem is a replicator system with timevarying games is more easily understood by the following formulation. We
characterize the state of a society consisting of individuals playing multiple gameswith the weighted average of elemental games. We introduce GT to represent this,called the total game:
GT ¼

M
X

wg Ag ;

ð36Þ

g¼1

where Ag ðg ¼ 1; . . .; MÞ is the payoff matrix of elemental game g. As clearly shownby wg ¼ wg ðtÞ, which changes over time, GT is a function that depends on time. The
dynamics of the strategy rules distribution in RED (19) can be shown in matrix form
by using the total game:

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Macro

Equilibrium
Distribution of
payoffs

Meso
Game Rules
Equilibrium
Distribution ofStrategy Rules

Micro
Strategy Rules
Strategy Rules
Strategy Rules
Fig. 4 Micro–meso–macro loop of a replicator system at an equilibrium state. The dotted line represents
inputs, while the black lines show effects. The double line means equivalent. The white area is static

x_ ¼ ðGT ðtÞ  x  GT ðtÞÞx:ð37Þ

In other words, this system is an extended replicator equation that has two layersof sets of replicators, a set of the strategy rules and that of the game rules, whichinteract and develop over time through an interaction matrix GT ðtÞ that also changes
over time.
As with the equilibrium distribution of the strategy rules, we can also consider an
equilibrium of the game rules. Under strategy rule profile x and meta-rule kg , whengame rule g (or a total game) satisfies
00

kg ðx; ug Þ [ kg ðx; ug Þð38Þfor any game rule g0 , game rule g is stable in response to invasion by any game rule.
Further, if strategy rule profile x is an evolutionary stable strategy rule (ESSR)

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Macro
PayoffDistribution

Meso
Game Rules
Game
Rules
Meso
Game Rules

Meta-rules

Micro
Strategy Rules
Strategy Rules
Strategy Rules
Fig. 5 Micro–meso–macro loop of a rule ecosystem. The dotted lines represent inputs, while the blacklines show effects. Gray areas are dynamic. The white area is static

profile, then there is no incentive for either the game rules or the strategy rules tochange. Thus, that system will be in a state of equilibrium for both types of rules.
We call game rule g under ESSR profile x evolutionary stable game rules (ESGRs).
When both ESGRs and ESSRs exist, the overall rule ecosystem is in an
equilibrium and the micro, meso, and macro will not change; Fig. 6 presents the
micro–meso–macro loop. As with the average-payoff-type meta-rule, simple metarules also manifest in this type of equilibrium state. Further, there are meta-rules
such as the inverse-variance-type meta-rule in which the dominant game rules areconstantly invaded by the other game rules. As noted previously, in the real world, it
is typical for these types of meta-rules to be combined and seldom for ESGRs such00

as kg ðx; ug Þ [ kg ðx; ug Þ for all games g0 to exist.

5 Conclusion
In this paper, we introduced RED as an integration and extension of evolutionary
game theory and replicator dynamics and observed the realization of dynamicchanges in both the strategy rules and the game rules. By treating both strategies andgames in a unified manner as rules, two game-theoretic viewpoints of institutions

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Macro

25

Equilibrium
Distributionof Payoff

Meso
Evolutionary
StableGame Rules

Metarules

Evolutionary Stable
Strategy Rules

Micro
Strategy Rules
Strategy Rules
Strategy Rules
Fig. 6 Micro–meso–macro loop of a rule ecosystem at an equilibrium state with ESSRs and ESGRs. The
dotted lines indicate inputs, while the black lines show effects. The double line means equivalent. The
white areas are static

can be successfully integrated. We also observed intermittent changes among thequasi-stable states of the strategy rules and game rules, as in the case of the inversevariance-type meta-rule.
Because we can view the rule ecosystem as a multi-population replicator system,
the game rules may also be considered to be replicators, or external rules, that are
replicated through interactions with individuals’ strategy rules. To deal with theduality of replicators (i.e., game rules and shared strategy rules), or internal rules,
the concept of the micro–meso–macro loop is necessary. Here, we introduce themeso level, which connects the micro and macro levels.
By discussing the monetary institutional ecosystem and results from a money
consciousness survey, we formulate a meta-rule that expresses ‘‘diversity’’ as theprimary consciousness for money. To directly evaluate the diversity of institutions,
we examine an extension of the RED framework, finding that an analysis of systems
that integrate three meta-rules, namely diversity, fairness, and profit orientation, is

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necessary. Analyzing the dynamics of such a system is, therefore, a topic for future
research.
Meta-rules are expressed collectively as individuals’ weighted game evaluations
and are difficult to manipulate. If we do not assume a priori rationality according toneoclassical economics, we should consider micro-level replicators, or strategy
rules, as internal rules that are determined by individual cognitive frameworks,
practices, routines, and values. These become cultures, traditions, norms, andsocietal consciousness (money consciousness), or inner institutions, when they areshared. Working on these internal rules and inner institutions consciously to change
is not necessarily impossible but it becomes a pressing policy matter. For example,
Japan implemented a policy of promoting energy conservation by appealing topeople’s sense of danger in response to a nuclear accident. However, it is, of course,
difficult to directly manipulate and design people’s sense for this purpose.
Evolutionist institutional design has been proposed to indirectly influence these
replicators through the institutional design of games, or external rules, as outerinstitutions, notably through the design of platform institutions such as currencies
and accounting systems (Nishibe 2006, 2010b). Further developing and substantiating the RED framework would thus enable us to make simulation models that
could be used in evolutionist institutional design.
Acknowledgements This research was partly supported by JSPS Kakenhi Nos. 17651088 and 21330063.

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