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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 12 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. 14