_B_ Micro-Micro Dynamics _MMD_ by malj

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									Complexity of Collective Decision



       UQR2202 Simplicity
         Semester 1 2003




        Sylvia Chua Siew Tien
          Hui Wen Wei Joy
       Karunananethy Kalaivani




                                    1
                                           Introduction


In their paper, The Complexity of Collective Decision, Saori Iwanaga and Akria Namatame from
the Department of Computer Science at the National Defense Academy at Yokosuka, sought to
model the dynamics of collective decision by relating the micro-variables that influence the
individual agent’s behaviour to the final collective decision of the organisation of such agents that
emerges. Their model departs from previous such studies in that they do not assume that the
individual agents have the same payoff matrices with the same utilities. Instead, their model
utilises heterogeneous agents who have their own idiosyncratic payoffs. These agents are
assumed to be rational and pursue their individual self-interest only. Each heterogeneous agent
has to make a decision between two choices- either to vote for or to vote against. Furthermore,
each agent may or may not be influenced by the decisions of other agents depending on the type
of agent one is: conformist or non-conformist. Altogether, we then have four different types of
agents: agents termed hard cores who have a strong preference towards their own choices, either
of voting for or against regardless of the other agents’ decisions and then we have agents who are
strongly influenced by others and are either conformist in following the majority decision or
nonconformist in voting contrary to the majority. The classification of each type agent would
depend on their idiosyncratic pay off structure.


By defining a certain critical value termed the threshold value θ, the authors were able to simplify
and categorise the numerous resulting payoff matrices and thereby, relate the individual agent’s
preferences to the resulting group decision, gaining an understanding of the dynamics of the
organisation’s decision making. In our report, we present and critically review the methods that
were used and the conclusions derived from the results. The Micro-Marco dynamics thus
obtained that presents the relationship between the collective decision and the underlying
individual decisions are called emergent properties of the system. Emergent properties are
fascinating as it is difficult for anyone to anticipate the full consequences of even simple forms of
interaction. In this case, the authors were able to show that the individual agent’s rational decision
combined with others’ decisions produces stable orders and sometimes complex cyclic patterns.




                                                                                                    2
                              Organisation with Heterogeneous Agents


An organisation of agents is termed G and contains an N number of agents. Each agent has two
choices, either to vote for (S1) or vote against (S2). The binary decision faced by an agent has
externalities. Here, an externality occurs when external influences, for example the decisions
other’s reach, have an impact on an agent’s decision. Therefore, an agent’s decision depends on
two factors: their own individual payoffs as well as others’ decisions. The following table shows
the typical payoff matrix of each agent Ai played against every other agent. The terms Ui1, Ui2,
Ui3 and Ui4 are that agent’s utility values


                                                                                              Table 1.


The first assumption that is made is that at any given moment, each agent has the opportunity to
observe the exact distribution of the number of agents who choose each strategy at time t and he
can make his own decision at the time period t+1. This is a valid assumption since the elections
are held again and again. When people make a decision, they usually consult previous decisions
and they also consider the outcome of those decisions.


The proportion of agents in an organisation G that have chosen S1 at time t is given by p(t).
Therefore, the expected utility of an agent choosing S1 or S2 is denoted by U i and is given as

follows:

                                    Ui (S1 )  p(t )Ui1  (1  p(t ))Ui 2

                                     Ui (S2 )  p(t )Ui 3  (1  p(t ))Ui 4             (1)

An agent will choose S1 if

                                              Ui (S1 )  Ui (S2 )

                         p(t )Ui1  (1  p(t ))Ui 2  p(t )Ui 3  (1  p(t ))Ui 4
                                                                                                   (2)
Simplifying the above expression and bringing the term p(t) to the right hand side, we have
therefore:

                             p(t )  (Ui 4  Ui 2 ) /((Ui1  Ui 2 )  (Ui 3  Ui 4 ))

                          Let   (Ui 4  Ui 2 ) /((Ui1  Ui 2 )  (Ui 3  Ui 4 ))

                                                                                                   (3)


                                                                                                    3
We can then define this expression to be θ and it is the critical value that determines if an agent
will choose S1 or not. When an individual’s threshold is reached, he is activated and acts in a way
that will maximize his utility under the existing conditions. In this way, there is no need to deal
with the numerous resulting payoff matrices. The threshold value θ allows us to simplify the
micro-variables and relate them to the macro behaviour of the organisation. However, we do lose
the knowledge of whether the final collective decision is Pareto optimal, that is, whether it
maximises the utility of the whole organisation. We only know that each agent maximises his
own utility.


Purposive and Contigent Behaviours
An agent exhibits purposive behaviour when he acts in a way that maximises his interests. An
agent exhibits contingent behaviour when his decision is influenced by others or constrained by
other agents’ decisions.


Considering our specific example where each agent faces two choices, either S1 to vote for or S2
to vote against. An agent’s decision to vote for a particular alternative may depend heavily on
how many others decide to do so likewise. This could be partly due to social influences or it
could be due to the agent trying to avoid the wastage of his single vote by voting for the side most
likely to succeed.


The values of the utility in the pay off matrix in Table 1 are defined as follow:

                              U i1   i  i ,U i 2   i ,U i 3  0,U i 4  i
                              (1   i  1, 0  i  1)
                                                                                                  (4)
αi represents the agent’s personal preference. If αi < 0, the agent has a personal preference to vote
for S1. If αi >0, the agent has a personal preference to vote against. βi represents the consistency
level of an agent’s decision with others. If βi <0, the agent does not prefer the choice of the
majority and chooses the opposite. We call this agent non-cooperative or perverse. If βi > 0, the
agent prefers the choice of the majority. We call this agent cooperative.


Although the paper models the outcome of the decisions by the individual agents, it cannot be
used to measure the utility one derives from the decision as it assumes that one automatically has
negative utility if one prefers S1 or is a non- cooperative agent. It is not a valid penalty.




                                                                                                   4
      1.         The Rational Decision Rule of a Cooperative Agent


      A cooperative agent is one with i > 0 and has a threshold value given as follows:
      i  (i   i ) /( i  i   i  i )  (1   i / i ) / 2 1                                                   (5)

      Therefore, the rational decision of a cooperative agent will be:
                  i.      To vote for S1 when p(t) ≥ θi
                 ii.      To vote for S2 when p(t) < θi
      Therefore, it can be seen that a cooperative agent’s decision is significantly influenced by
      aggregate information of p(t). The following diagram allows us to easily visualise the nature of a
      cooperative agent and how he votes according to what the majority decision is. Let us consider
      two agents, agent A and agent B with threshold values of θA and θB respectively. The Y axis
      represents p(t) and the solid line represents the threshold as the function  i /  i . Since αAis

      negative, agent A desires to vote against. In contrast, agent B has a positive α B which represents
      his desire to vote for. If p1 is the proportion of agent who voted for at time t, agent A will vote
      against and agent B will vote for at the next time period time t+1. However, if the proportion of
      agents who voted for at time t increased to p2, both agent A and B will vote for. Similarly, at p 3,
      both agents will vote against.




                                                                                                                Diagram 1


      2.         The optimal Decision Rule of a Perverse Agent


      The perverse agent has a payoff matrix and threshold as follows:

      U i1  1 ,U 2i   i  i ,U 3i  i ,U i 4  0
      (1   i  1, 0  i  1)

      i  (i  i ) /(i  i  i  i )  (1  i / i ) / 2                                                        (6)

      Therefore, the rational decision of a perverse agent is:
 i.          To vote for S1 when p(t) ≤ θi
ii.          To vote for S2 when p(t) > θi


      1
          Actually in the paper the expression is given as   i  (i   i ) /( i  i   i  i )  (1   i / i ) / 2 .
      We presume this could only be a typo.


                                                                                                                            5
Similarly, to explain this diagrammatically we shall consider two agents, agent A and agent B
with their corresponding threshold values. At a level p(t) equal to p 1, agent A will desire to vote
against and agent B will vote for S1. When p(t) increases to p2, both agent A and B will vote for.
At p3, agent A will vote for and agent B will vote against.




Diagram 2


3.      The four types of Agents


The author further subdivides these two categories we have already outlined to yield four types of
agents. Firstly, we have the Hardcore of S1 or S2, agents who vote for their own preferences of
either S1 or S2 and are little influenced by the majority’s decisions. This occurs when the absolute
value of α is bigger than the absolute value of β. However, a hardcore of S1 will have a positive α
value and a hardcore of S2 will have a negative α value.


Then we have the Conformist and Non-Conformist who, on the contrary are strongly influenced
by the majority decision but in two different ways. The conformist or coordinate agent goes along
with the majority whereas the non-conformist or perverse agent goes contrary. This occurs when
the absolute value of α is smaller than the absolute value of β. However, a conformist will have a
positive β value and a non-conformist will have a negative β value. The next diagram summaries
these categorisations.


                                                                                         Diagram 3




                                                                                                  6
                                Micro-Macro Dynamics (MMD)


MMD describes the dynamics relating the collective decision of a given population of N agents
and the agents’ individual decisions. Growth starts when a set of unstructured decisions (rational
decisions of individual agents) are made or adjusted under the influence of other agents. In the
long run, stable patterns emerge and there is a dynamic order. This emergence (macro perspective)
is unanticipated and uniquely different from the micro perspective, which is represented by the
variety of decisions individual local agents can make.

MMD of Coordinate Agents


Coordinate agents are those who prefer what the majority does. The proportion of coordinate
agents with threshold values which are equal to or less than  is given by


                                       F1 ( )   n1 (i ) / N1
                                                i 
                                                                                                  (7)

Where
   n1(i) is the number of coordinate agents (that is I  0) with the same threshold i in the
      organisation G.
   N1 is the total number of coordinate agents in G

The authors assume two properties on the part of the agents. The first is inertia, that is that the
decision at next time step is strongly influenced by the decision at the previous time step, in other
words a reluctance to change the decision reached previously. The other assumption is myopia
which means that the agents only think ahead to the next time step and do not attempt any long
term or global strategy. The proportion of agents p(t+1) who will vote for at the next time period,
t+1 is then related to the proportion at time t, p(t) (Hansarnyi & Selten 1988). Therefore we have,
                                          p(t+1) = F1(p(t))                                 (8)

Hardcore agents of S1 will always go for a particular decision and hardcore for S2 agents are
always against it while the conformists’ decisions will depend on N1.


1. Hardcore of S1 (i  0) -- extremely low threshold
Even when there is no one for a particular decision, these agents will continue to vote for since
their threshold i is very low. For example, Jack, a little school boy has to decide whether to play
soccer or not. If his threshold i (representing the number of friends he has) is equal to zero and


                                                                                                   7
he is a hardcore of S1 (for playing games rather than anything else) then even if there are no
friends around him playing he will still decide to play soccer alone.


2. Hardcore of S2 (i  1) -- extremely high threshold
Even if everyone is for it, these agents are likely to be against since their threshold, i is very
high. Using the example of Jack again, if he has i equal to 1, then even if all his friends are
playing football he is unlikely to join them because his threshold is at the maximum number.


3. Conformist (i  0.5) --- mid range
The conformist’s final decision depends on the majority of the population’s decision. Considering
the example of Jack who is a conformist, when he has (2X+1) friends of whom X is for soccer, X
against playing soccer and one friend still undecided, the resultant decision Jack makes will
depend on that one friend. If that friend (represented by the white circle in Fig 1a) chooses for
(pink circle) then X will choose “pink” X as well. The converse is true when he chooses against
(blue)



                             1                                   1



                     X               X                   X                X


                                                                                    Fig 1a




In a given scenario where N1 is sufficiently large, the conformist’s decision based on the
threshold difference of “for” and “against” agents is marginal. Therefore we can assume that the
threshold for conformist will lie approximately at the half way mark.


4. Non Conformist (i  0.5) --- mid range.
The difference is illustrated by Fig 1b below




                                                                                                 8
                            1                                       1



                                                                X                   X
               X                X



                                                                                          Fig 1b
Jack will choose against when his friend chooses for if he is a perverse agent.
The converse is also true.

Organisation With Hardcores
Hardcores are agents who care about what they actually want to do personally and they do not
care so much about what the others decide.



                                                                  For a coordinate agent population N1,
                                                                  consisting of hardcores and no
In Fig 2a the population is normalised. Hence, a symetrical looking function results. This is
                                                                  conformists,
assuming that the threshold population is sufficiently large. Given that this is not always the case,
                                                                  Coordinates are as follows
                                                                       Hardcore given point as
an assymetrical Fig 2a will result in a change in the rate of convergence from a of S1 (0,max)
                                                                       Hardcore of S2 (0,max)
compared to a symetrical figure.Correspondingly to a smaller n(i)/N, arrows as shown in Fig 2b
                                                                       Conformists (0,0)
will be shorter and lesser. Hence, convergence occurs more quickly. This implies that when thereas in Fig
                                                                  Hence, the shape of the function
           proportion of agents, a consensus (equilibrium point) 2a. be reached more quickly. In
is a lowerFig 2a: The distribution function of threshold          can
spite of this the equilibrium position will remain at i = 0.5 regardless of the of rate convergence.
                                                                          Fig 2b is derived from adding the area
                                                                          under the graph of fig 2a.

                                                               Since the population consists of
Organisation With Conformists                                  coordinate agents, at
                                                               p(t) = 0, p(t+1) = 0.
Conformists are agents who are more influenced by how other agents behave rather than by what
                                                               Similarly, at p(t) = 1, p(t+1) = 1
they actually prefer to do.
                                                                          Therefore, Fig 1b is upwards sloping.

                                                                      (0,0) and E2 are unstable fixed points
                                                                      because it converges to Ei, a stable fixed
                                                                      point, as shown by the arrows when
                                                                    For a coordinate agent population N1,
           the analysis for the organisation with and the
Similar toFig 2b: Accumulated function of threshold Hardcores, the rate of divergence depends onno
                                                                    consisting of applied. and the
                                                                      iteration is conformist
          convergence of the collective decision of hardcores
                                                                hardcores,
shape of Fig 3a. An unsymmetrical shape results in an unequal rate of divergence between p(t) (0,
          and conformists.
0.5) and (0.5, 1). Eventually, the divergence results in the same equilibrium points at E1 & E3.
                                                                   Coordinates are as follows
                                                                            Hardcore of S1 (0,0)
                                                                            Hardcore of S2 (0,0)
                                                                            Conformists (0,max)
                                                                                                        9
                                                                        Hence, the shape of function as in Fig
                                                                        3a
MMD of Perverse Agents
Perverse agents are those who prefer to be inconsistent with the decisions of other agents’
decisions. The proportion of coordinate agents with threshold values less than  is given by

                                        F2 ( )   n2 (i ) / N 2
                                                 i 
                                                                                                  (9)
            Fig 3a: The distribution function of threshold
Where
   n1(i) is the number of coordinate agents (I  0) with the same threshold i in the
      organisation G.
   N2 is the total number of perverse agents in G

Assuming the properties of inertia and myopia again on the part of each agent, the proportion of
perverse agents who will vote for at the next time period, t+1, will be as given
                                           p(t+1) = 1 – F2(p(t))
                                                                                                 (10)

Similar to the MMD for Coordinate agents, hardcores of S1 will always go for a particular
decision while hardcore for S2 agents are always against it and the non-conformists’ decision will
depend on N2.

Organisation With Hardcores                                          Since population consists of perverse
                                                                     agents then at
                                                                     p(t)=0, p (t+1)=1.
                                                                     Similarly, at p(t)= 1, p(t+1)=0
Organisation With Non-Conformists
                                                                     Therefore, Fig 4a is downwards sloping.
Organisation With Non-Conformists                                      Since the population consists of
                                                                     Since4a, within the range (0, 0.5) perverse
                                                                        fig the
                                                                     Inagents atpopulation consists of
                                                                     perversedecreases
                                                                     gradient 0,agents atdecreasingly because
                                                                       p(t) = p(t+1) = 1.
                                                                     with = 0, p(t+1) = 1. 2a,p(t+1) = 0.
                                                                     p(t) reference to Fig
                                                                       Similarly, at p(t) = 1, gradient
                                                                     Similarly, at p(t) = 1,
                                                                     increases decreasingly. p(t+1) = 0.
                                                                       Therefore, Fig 4b is downwards sloping.
                                                                     Therefore, range is downwards
                                                                     Similary forFig 4b(0.5, 1), gradient
                                                                     follows thatwithin theand hence0.5) the
                                                                     sloping. in Fig 2a range (0,
                                                                       In fig 4b,
                                                                     decreases increasingly
                                                                        ient decreases increasingly because with
          Fig 4a: The accumulated function of threshold and the          fig 4b, within 2a, range (0, 0.5)
                                                                     Inreference to Fig the the gradient the
          convergence of the collective decision of hardcores and    (0,1) and (1,0) are unstable fixed points
                                                                        increases increasingly.       because
                                                                     ient decreases increasinglystable fixed
                                                                     because it converges to E1, a
          non conformists.
                                                                     point, as shown torange (0.5, 1), the
                                                                                               2a, the gradient
                                                                     with reference by Fig arrows when
                                                                        Similary for the the
                                                                     increases increasingly.
                                                                        gradient applied.
                                                                     iteration is follows that in Fig 2a and hence
                                                                       decreases decreasingly.
 Fig 4b: The accumulated function of threshold and the               Similary for the range (0.5, 1), the
 convergence of the collective decision of non conformists.            E1 are follows that points because
                                                                     gradient unstable fixedin Fig 2a and it
                                                                     hence decreases decreasingly. stable
                                                                       diverges to (0,1) and (1,0), the
                                                                       fixed points, as shown by the arrows
                                                                        when the iteration is points
                                                                     E1 are unstable fixedapplied.because
                                                                                                  10
                                                                     it diverges to (0,1) and (1,0), the
                                                                     stable fixed points, as shown by the
                                                                     arrows when the iteration is applied.
                                 Dynamics of Collective Decision



Having considered the individual behaviour of both coordinate and perverse agents, we now can
examine the collective behaviour of an organization composed of varying proportions of each
type of agents. Iwanaga and Namatame wished to determine whether as a result of the
interdependence between each agent’s decision and the whole organization’s there arises any
emergent macro behaviour. They assert that given certain conditions, they were able to obtain
stable orders and even cyclic behaviour. They called this phenomenon Micro-Macro Dynamics
(MMD).


There are four types of agents: hardcores who choose S1, hardcores who choose S2, conformists
and noncomformists. The proportion of the coordinate agents, that is whose i > 0, is equal to k
where k = N1/N and the proportion of perverse agents, whose i < 0, is equal to (k – 1) = N2/N
where N = N1 + N2 is the number of agents in the organization G. The dynamics of the system is
as modeled by the equations given on page 148 where p(t) represents the proportion of coordinate
agents who vote for S1 at time t. As a result of the inertia of the agents and their myopia, each
agent assumes that the value of p(t) at time (t+1) is related to the value of p(t) at time t. Therefore,
the proportion of agents voting S1 at (t+1), F1(p(t)), is as given by the following relation:
                                          p(t+1) = F1(p(t))
                                                               n1 i 
where                                      F1 ( )  
                                                       i      N1
This gives a form of iteration. Now in order to determine the MMD, we need to predict the final
proportion of agents who make each decision and this is equivalent to determining the
equilibrium of this system. The collective decision at the equilibrium is described by the fixed
point
                                              p* = F1(p*)


Similarly for the MMD of perverse agents, p(t+1) is given by the following equation:
                                        p(t+1) = 1 – F2(p(t))
                                                    n2 i 
where                              F2 ( )  
                                            i      N2
and with the fixed points give by: p* = 1 – F2(p*). The resulting dynamics for each type of agent
has already been described earlier in the paper. Now, we shall consider various organisations


                                                                                                     11
which contain the different types of agents in different proportions and try to determine if any
correlation between the collective decision and the underlying agent’s motivations exists.


Accordingly therefore, the authors claim that the following relation allows us to describe the
MMD of the organization:
                         p(t+1) = [k  F1(p(t))] + [(1 − k)  (1− F2(p(t)))]
                                                                                                  (11)
The term k is the portion of coordinate agents in the organization. By varying this term, we will
be able to study how this factor influences the collective behaviour of the organisation given the
various proportions of the types of agents already described. The authors then looked at three
cases: firstly, the organisation composed primarily of hardcore agents who vote without regarding
the decisions of others; secondly, an organisation made up of conformists and non-conformists;
and lastly, one consisting of hardcore and non-conformists. The results they obtained are as
follows.


Case 1. Majority of hardcore of both S1 and S2 in the organisation
For three different values of k, the collective decision always converged to p(t) = 0.5. The authors
state that for any given value of k and any initial value of p(t), the final value always become 0.5.
This is shown in the following diagrams. However, the authors failed to mention why in each
case with different values of k (0.3, 0..5 and 0.8), the convergence should proceed in an
individualistic manner as can be seen. In Diagram 4(a), there are two oscillations before the
convergence is reached while in the third one, there are two rates of convergence. However, the
final result is still 0.5 meaning that there is an equal number that prefers S 1 to S2 with no majority
preference dominating.




Diagram 4


Case 2. Organisation consisting of conformists and non-conformists
In this case, three distinct situations emerges at different values of k. In the first one, there is an
oscillation between 0.4 and 0.6 for k = 0.3. When this value is increased to k = 0.5 Diagram 5(b)
however, the oscillations disappears and the collective decision converges to 0.5 for any initial
value of p(t). Upon a further increase in the k value to 0.8, there results two convergent values at
0.2 and 0.8 depending on the given initial values as shown in the diagram. Therefore, in this


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particular example there is a situation of sensitivity to initial conditions. If in the beginning, even
the slightest majority preferred S1, then p(t) quickly converges to a value of 0.8 ensuring that S1 is
selected as the collective decision. Whereas, if p(t) was initially even slightly less than 0.5, then
the final collective decision will be S2.


Case 3. Consisting of hardcore and nonconformists
Here we have a totally different scenario. Most cooperative agents are hardcore of either S 1 or S2
and most perverse agents are non-conformists as shown earlier. When the ratio of cooperate
agents is small, for example k = 0.3, there is again an oscillation between 0.2 and 0.8. The degree
of oscillation decreases as k is increased until finally, at k = 0.8, there is convergence to p(t) = 0.5.
These results are shown in Diagram 6.


The results described are not usually expected from what we know of the underlying micro-
structure of the organisations. In most cases, there is convergence to a value of p(t) that is equal to
0.5, that is, there is no clear dominant strategy and a stalemate results. It is not obvious why this
should be so. Perhaps the categorisation of agents into four types resulted in an oversimplification.
All the results shown were for three distinct values of k, 0.3, 0.5 and 0.8 and no explanation was
provided why these particular values were chosen though it does show the situation when
coordinate agents are either dominating or are present in equal numbers or are in the minority.


                                                                                     Diagrams 5 and 6




                                                                                                     13
                                           Conclusion


Such simple but organised global behaviour, oscillations and convergence, arising from the
individual micro-motives of the agents with no coordination among them is a clear example of
emergent behaviour. The authors have shown through this simulation that knowing the
preferences and motives of agents gives us only a necessary but not a sufficient condition
explaining the various outcomes, which are not easily adduced from knowledge of the micro-
structure of the system alone. The results as shown are surprising and not obvious.


Previous work in this respect though also focusing on the relation between the micro-and macro-
behaviour assumed that the payoffs for each individual agent were homogenous. However, in this
work, the authors allowed for heterogeneous payoffs which greatly complicate the system.
Nevertheless, they were able to show that even with the infinite variations afforded by relaxing
this assumption, predictable self-organised behaviour can emerge. They were able to classify
three types of behaviours resulting from three different types of organisations. The resulting
dynamics then depend on such macro-variables as the distribution pattern of the threshold and the
proportions of hardcore, conformists and non-conformists.




Reference


Iwanaga, Saori and Namatame, Akira. The Complexity of Collective Decision. Nonlinear
Dynamics, Psychology and Life Sciences, Vol. 6, No. 2, April 2002.




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