Proceedings of 10th Conference on Artiﬁcial Life and Robotics (AROB05) (CD-ROM)

1

Rule ecology dynamics for studying dynamical andinteractional nature of social institutions
T. Hashimoto
hash@jaist.ac.jp

M. Nishibenishibe@hucc.hokudai.ac.jp

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

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

Abstract

even only one, from our daily life.” As we can easily
imagine, each institution has diﬀerent importance orinﬂuence for individuals’ behavior.
Aoki and Okuno-Fujiwara[2] discusses that diversity of institutions exists in diﬀerent social systems.
But, as quoted above, there are great diversity of institutions in a society. Aoki and Okuno-Fujiwara[2]point out the important feature of institutions, thatis an institutional complementarity. When an institution provides a reason to exist for another institution,these two institutions are refereed to as in the relationship of institutional complementarity. Since only
one of such institutions cannot change independently,
Aoki and Okuno-Fujiwara insist that they are stable.
However, social institutions often changes and many
institutions interact with each other. Accordingly, achange of an institution induces changes of other institutions. The chain of change may go on to all institutions.
This chain of change through interaction among
institutions is like an ecological system of biologicalspecies. In this paper, we conceptualize such dynamic
and interactional nature of institutions as an ecology of rules. Here we use a term ‘rule’ as a more
abstract version of institutions. We propose a newframework to formalize the ecology of rules and its
dynamics as an extension of evolutionary game theory. This framework is named “Rule Ecology Dynamics (RED)1 ”. This framework can integrate two gametheoretical treatment of institutions, one treats institutions as game rules, and the other as equilibria ofgames[3].

In this paper, we address a new concept to view asocial system as consisting of diversiﬁed institutions
interacting with each other and dynamically changing
through actions of individuals in a society. A mathematical framework for the concept is formalized. Theframework, called Rule Ecology Dynamics (RED), is
an extension of multi game dynamics, in which players
play plural games simultaneously, by introducing timedependent weights for the plural games and a “metarule”. The meta-rule is a rule to determine the changeof rules. We show simulation results of two kind ofmeta-rules , average payoﬀ type and inverse standard
deviation type. We discuss that this framework is a realization of rule dynamics and it has certain relevanceto describe real phenomena of institutional dynamics.
Keywords: Rule Ecology Dynamics, Formation andChange of Institutions, Replicator Equation, MultiGame Dynamics, Evolutionary Game Theory, RuleDynamics

1Introduction

Social institutions are rules for individuals’ behavior and cognition broadly accepted and used in asociety. In our society, there exist various institutions/rules ubiquitously. Seiyama[1] describes suchsituation as “We are living with institutions, suchas family, commuter passes, trains, universities, telephones, postal services, E-mails, meetings, lectures,laws, norms, conventions and so on. There is no one
that is not institutional. We engage in our daily life
postulating such institutions. They are omnipresent.
It is very diﬃcult to pick out non-institutional aspect,

1 The previous version of the framework was called “MetaEvolutionary Game Dynamics” since it is an extension of game
dynamics (replicator dynamics) introducing a meta-rule[3].

2where wg is the weight of the g-th game, satisfying∑Mgg
g=1 w = 1, λ is an evaluation of the g-th game,∑M
g g
and λ̄ =g=1 w λ is the weighted average of theevaluations of games, τ is a time coeﬃcient for thechanging velocity of the games relative to that of thestrategy populations, x = ({xi1 ···iM }) (ig = 1 · · · rg )is the strategy proﬁle that is a vector of population
share of each strategy, and ug (x) = ({ugi1 ···iM (x)})
is the combined payoﬀ or the payoﬀ proﬁle that is avector of the payoﬀ of each strategy.
The equation (5) is a meta-rule to give evaluation ofeach game. The variables of the evaluation function isthe strategy proﬁle x and the combined payoﬀ ug (x).
This setting makes an interaction loop among individual behavior, the rules and the meta-rule closed.
Namely, the change of rules depends on the consequences of the behavior under the rules.

Rule Ecology Dynamics

In our framework, rules and individual behaviors
are modeled by games and strategies of players in thegames, respectively. We start the formalization of theRED from the replicator equation that governs thedynamics of population of each strategy[4]:ẋi = xi (ui − ū)(i = 1 · · · N ) ,(1)where xi∑
is a population share of the i-th strategy andN
satisﬁes i=1 xi = 1, ui is a payoﬀ of the i-th strategy,∑Nū = i=1 xi ui is the average of the payoﬀs, and N is
the number of strategies.
To represent various compositional rules, multiple
games that all players play are brought in, as proposedin Multi Game Dynamics[5]:ẋi1 ···iM=xi1 ···iMM∑(ugi1 ···iM − ūg )

(2)

3

g=1

(g = 1 · · · M ) ,We show simulation results of the system for tow
settings of the meta-rule. The meta-rule should beconsidered appropriately for the objective system tobe modeled.
The ﬁrst example is the average type meta-rule∑λgA (x, u) =xi1 ···iM ugi1 ···iM .
(6)

where xi1 ···iM is a population share of a strategy (i1 , i2 , · · · , iM ), which means that a player
plays the strategy i1 at the game 1, the strategy i2 at the game 2 and so on, satisfying∑r1 ∑r2∑rM
gi1 =1i2 =1 · · ·iM =1 xi1 ···iM = 1, where r is thegnumber of options at the game g, ui1 ···iM is a payoﬀof the strategy (i1 , i2 , · · · , iM ) at the game g, ūg is theaverage payoﬀ at the game g, and M is the number ofgames.
We introduce a weight for each game to treat thedegree of inﬂuence or importance of each rule. The
weight of each game change with time through individual behavior. The change of the game weights isgoverned by a meta-rule. A meta-rule is a more basic and stabler rule than focal rules. It is for determine which rules are relatively desirable or relatively
strong in inﬂuence for individual behavior. Introducing a meta-rule is a representation of the hierarchicalstructure of interaction and stability of rules. Thus,
the meta-rule corresponds to the constitution or social values and norms, which are relatively not easy tochange. In our framework, it is modeled as an evaluation function of games. The RED is deﬁned by thefollowing three equations:ẋi1 ···iM= xi1 ···iMM∑wg (ugi1 ···iM − ūg ) ,i1 ,···,iMThis is a model of a market economics and each ruleis thought of as describing a market. We suppose in
the market economics that the more proﬁt a market
or a rule gives for individuals/organizations, the more
important the market/rule is, as in stock markets.
The simulation result is depicted in Fig. 12 . In
this case, a phenomenon like globalization is observed.
Namely, the system is monopolized by a strategy andthen games in which the monopoly strategy wins develop their weights.
The second example of the meta-rule is to describe
the egalitarianism that is the doctrine of the equalityof mankind and the desirability of political, economic
and social equality. This meta-rule is formalized as
the inverse of standard deviation of agents’ payoﬀs:−1 ∑
()
2λgIV (x, u) =xi1 ···iM ugi1 ···iM − ūg. (7)i1 ,···,iM(3)2 In the following simulations, we simplify the RED (Eq.(3)-(4)(5)(5)) as follows: The number of options of all games are thesame, r g = N (g = 1 ∼ M ) and each strategy is limited totake the same one at the all games and thus the strategy shareis indicated as xi1 ···iM ≡ xi .

g=1g

τ ẇ
λg

= w (λ − λ̄) ,= λg (x, ug ) ,gg

Simulation Result

2

1

1
0.9

0.8

0.8
0.7

0.6

0.6
0.5

0.4

0.4
0.3

0.2

0.2
0.1

0

0
050

100

150

200

250

300

350

400

450

500

0.35

050

100

150

200

250

300

350

400

050

100

150

200

250

300

350

400

1

0.3
0.8

0.25
0.6
0.2
0.4
0.15

0.2
0.1

0.05

0050

100

150

200

250

300

350

400

450500Figure 1: An example of dynamics of RED with theaverage type meta-rule (Eq.(6)). The y axes are thestrategy distribution for the top graph and the weightsof games for the bottom, respectively. The x axis istime steps. The system size is N = 6, M = 7.

Figure 2: An example of dynamics of RED with thethe inverse standard deviation type type meta-rule(Eq.(7)). The y axes are the strategy distribution for
the top graph and the weights of games for the bottom,respectively. The x axis is time steps. The system size
is N = 3, M = 10.

The dynamics of the strategy share and the games’
weights are shown in Fig. 2. This case gives revolutionary changes of predominant rules. Namely, no one
game and no one strategy can dominate the systemstably and temporarily prevalent games and strategies change continually. It is a rather paradoxical situation that egalitarianism induces destabilization of
established rules and revolutions.

4

ical phenomena are found in evolution, development,learning, adaptation and emergence. We call such dynamic phenomena rule dynamics.
The social rule is also an example having undecomposability between operators and operands. Socialstructures such as institutions and norms are formed,maintained and changed with time. The remarkabledynamic nature in such systems are self-modiﬁcation.
Individuals behave under some institutions and change
the institutions.
Some eﬀorts to study and to describe rule dynamics
have been launched [7, 8, 9, 10]. The RED proposed
here is also a framework to describe the rule dynamics,
especially observed in social systems. Actually, RED(Eqs.(3)-(5)) can be written in a matrix form asDiscussion – RED as Framework for
Rule Dynamics

When we try to understand some object, it is often
described by a set of states and a system of static functions. The functions are rules to govern the change ofstates. In other words, the rules are operators andthe states are operands. Describing with static functions implies that the decomposability between rules
and states is presupposed. The decomposability may
occasionally not be able to be presupposed. This undecomposability between rules and states, or between
operators and operands is one of the main featuresof complex systems[6]. In complex systems, rules areoften not static but dynamically changing. The typ-ẋ = (GT (t)x − x · GT (t)x) x ,(8)∑M
where GT = g=1 wg Ag is a total game that is theweighted sum of all games and Ag (g = 1, · · · , M ) is
the payoﬀ matrix of the g-th component game. It isclearly seen that RED is a replicator equation withtime dependent interaction matrix GT (t).
The RED can be resolved into a multi-population
replicator system[11] of the individual strategies and3

the rules:ẋi1 ···iMτ ẇgbe a tool of designing institutions from the viewpointof evolutionary economics.
= [(Aw)i1 ···iM − x · Aw] ẋi1 ···iM , (9)[]= λg − λ̄ wg(10)Acknowledgements
This research is partly supported by Canon Foundation Europe; and a Grant-in-Aid for Scientiﬁc Research (No.15700183) from the Ministry of Education,Culture, Sports, Science and Technology of Japan andby the Japan Society for the Promotion of Science.

where the element of a matrix A is (A)i1 ···iM g =ugi1 ···iM . This representation clearly show that RED
is interactions between the individual strategies andthe rules.

5Conclusion

References

We have proposed a new model for studying the dynamics of social institutions. The model called Rule
Ecology Dynamics (RED). The key concept of RED
is that rules in a society is diverge and ubiquitous, interact with each other and change through behavior
of individuals acting under the rules. The ecologicalinteraction of rules is described by a modiﬁcation of
the replicator equation. The notable point is to incorporate not only the interactions among individuals
but also those of rules. In RED, an interaction loop
between individual behavior and rules is realized byintroducing a meta-rule. Accordingly, RED is a mathematical model of the micro-macro loop proposed by
Shiozawa[12] to understand economic dynamics.
The formation and the change of social institutions
is a representative of rule dynamics phenomena, which
is a key feature of complex systems. We show that ourmodel is a framework to describe rule dynamics. We
also show that RED describes the interaction (game)between individual behavior and rules.
By setting two kind of meta-rules, the average type
and the inverse standard deviation type, we show thesimulation results that show actually the dynamicsof rules and support certain eﬀectiveness of RED to
study the dynamics of institutions.
We need to promote the relevance of the RED for
describing the real phenomena of rule dynamics. We
can suggest a concrete example of the rule dynamics tobe described by RED. It is changes of monetary credit
systems in Argentina where a local currency, GRT,
has been used. Since the crash of national currency,
Peso, caused by the governmental default in 2002, thecredit of the national and the local currencies, mechanisms to establish and support the currencies, relationship between them, users’ impression about them
have changed. All of these factors constitute institutions and interact with each other ecologically. Thisphenomena may be able to be modeled with RED.
Through such eﬀort, the framework of RED come to[1] K. Seiyama, Seidoron no kouzu (The plot of institutiontheory) (in Japanese), Soubunsha, 1995.
[2] M. Aoki, M. Okuno-Fujiwara (Eds.), Comparative Institutional Analysis: A New Approach to Economic
Systems (in Japanese), University of Tokyo Press,1996.
[3] T. Hashimoto, Y. Kumagai, “Meta-evolutionary gamedynamics for mathematical modelling of rules dynamics”, Advances in Artiﬁcial Life, W. Bnzhaf, et al.
(Eds.), Springer, pp. 107-117, 2003.
[4] P. Taylor, L. Jonker, “Evolutionary stable strategies
and game dynamics”, Mathematical Biosciences, 40,
145–156, 1978.
[5] K. Hashimoto, “Multi-game dynamics” (in Japanese),
Suri-Kagaku, 495, 75-83, 2004.
[6] T. Hashimoto, “Complex systems”, Knowledge Science(in Japanese), K. Sugiyama et al. (Eds.), Kinokuniya
Shoten, pp.126–131, 2002.
[7] N. Kataoka and K. Kaneko, “Functional dynamics I:articulation process”, Physica D, 138, 225-250, 2000.
[8] N. Kataoka and K. Kaneko, “Functional dynamics II:syntactic structure”, Physica D, 149, 174-196, 2001.
[9] G. Masumoto, T. Ikegami, “Lambda game system –an approach to a meta game–”, Advances in Artiﬁcial
Life, J. Kelemen, P. Sosik (Eds.), Springer, pp.695-699,
2001.
[10] J. Namikawa, T. Hashimoto, “Dynamics and computation in functional shifts”, Nonlineality, 17, 13171336, 2004.
[11] J. W. Weibull, Evolutionary Game Theory, MITPress, 1995.
[12] Y. Shiozawa, Fukuzatsusa no kiketsu (Consequencesof Complexity) (in Japanese), NTT Shuppan, 1997.

4