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Random Selection of Legislators for a More Efficient Parliament

A. Pluchino, M. Caserta, A. Rapisarda, C. Garofalo, S. Spagano,

University of Catania

This work moves in the context of Sociophysics, a recent interdisciplinary field of research that joins together contributions from physics, mathematics, social and political sciences, economics, engineering and others, aiming to propose a new approach finalized to decode the hidden complex dynamics of social and economical systems (Galam 2008, Bankes et al. 2002). The central point of this approach is that, when human beings are forced to live, move and act within the (physical or cultural) boundaries of a social system, subjected to its socio-economic rules, their degrees of freedom (i.e. their possible behaviors) are extremely reduced. Then, as it happens in physics, the complexity of the system is not due to the complexity of its elements, but to the complexity of the dynamical patterns emerging from their interaction. For this reason, Sociophysics makes an extensive use of mathematical models, based on very simplified hypothesis or assumptions, and of new powerful tools particularly suitable for dealing with complex systems, like network analysis or agent-based simulations (Epstein 2007). Following this line of research, in this paper we show that it is possible to reproduce, with simple mathematical models, the complex dynamics of the legislative activity of a Parliament, and also to explore, analytically or with the help of numerical simulations, the beneficial impact of randomly selected legislators on the Parliament efficiency. The paper is divided in three parts. In the first part we present a two dimensional mathematical model of a Parliament, where legislators are represented as points in a two dimensional plane, and we review some recent results (Pluchino et al. 2011) which demonstrate how the introduction of randomly selected legislators, considered independent from traditional Parties, can positively affect the efficiency of the Parliament in presence of only two Parties or Coalitions. In this case we also find an analytical “efficiency golden rule” which allows to fix the optimal number of such independent legislators. In the second part we extend this model by introducing several Parties in the Parliament and we analyze, as case study, the political scenario of the Italian Senate after the last national elections on February 24-25, 2013, also exploring the role of 5 Star Movement which could present some elements of similarity with independent legislators introduced in the first part. Finally, in the third part, we study the efficiency of a Parliament with a one dimensional version of the model presented in the first part, easier to handle analytically, where the role of the statistical distribution of both the independent legislators and the legislative proposals emerge as crucial. 1. A 2D model of a Parliament A Parliament can be considered a mirror of the population represented by its members. In 1976 the Italian economist Carlo M. Cipolla suggested a very simple diagram (that we call ”Cipolla diagram”) to describe how a given population could be characterized focusing on only two features of human behavior, i.e. benefits and losses that an individual causes to him or herself (personal gain, positive or negative) and benefits and losses that an individual causes to others (social gain, positive or negative). Representing the population on a 2D graph with the first feature on the x-axis and the second on the y-axis (see Fig.1), we obtain four groups of people:

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-In Quadrant 1 (top right) we find Intelligent people, i.e. individuals whose actions produce (on average) a gain for themselves and also for other people; -In Quadrant 2 (top left) we find Helpless / Naive people, i.e. individuals whose actions produce (on average) a loss for themselves but a gain for other people; -In Quadrant 4 (bottom right) we find Bandits, i.e. individuals whose actions produce (on average) a gain for themselves but a loss for other people; -In Quadrant 3 (bottom left) we find those who Cipolla calls Stupid people, i.e. individuals whose actions produce (on average) a loss for themselves and for other people; Of course, under certain circumstances a given person acts intelligently and under different circumstances the same person will act helplessly, and so on, therefore one has to calculate for each person his average position in the graph. For example, a helpless person may occasionally behave intelligently and on occasion he may also perform a bandit’s action. But since the person in question is fundamentally helpless most of his action will have the characteristics of helplessness. Thus the overall weighted average position of all the actions of such a person will place him in quadrant 2 of the Cipolla diagram. The basic idea of this study is to use the Cipolla classification in order to elaborate a prototypical model of Parliament with only one Chamber, consisting of N members and K Parties, and to evaluate its efficiency. The economic literature contains different assumptions about the behaviour of members of a Parliament after the elections. On one hand, they are assumed to be motivated by purely personal interests, which may coincide with the desire to be re-elected or with the temptation of deriving incomes and benefits from a position of power. On the other hand, one may safely assume that they are motivated by the aim of achieving specific public policies that maximize the collective interest, i.e. the social Welfare. Taking both motivations into account, it is possible to represent individual legislators as points in a Cipolla diagram, where we fix arbitrarily the range of both axes in the interval [1,-1], with the personal gain/interest on the x-axis and social gain/interest (understood as the final outcome of trading relations made possible and allowed by law) on the y-axis. In particular, all the points representing members of a Party Pi will lie inside a circle with a given radius ri and with a centre Pi(x, y) falling in one of the four quadrants. Of course we are not interested in classifying legislators or Parties as intelligent, bandits, helpless or stupid, but only in characterizing them according to their attitude to promote personal or social interest. In Fig.2, a possible realization of such a virtual Parliament has been represented, with N legislators divided in K=2 Parties P1 and P2. The centre of each Party indicates the average behaviour (in terms of Cipolla’s

Fig.1

Fig.2

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classification) of all its members, while the radius size (in the figure we have r=0.4 for both the Parties) tells us how strong the tolerance of that Party (considered as a whole) is with respect to the individual behaviour of its members: the smaller the radius, the smaller the degree of tolerance, i.e. the lower the independence of individual members, and vice-versa. Therefore we call the circle associated to each Party ”circle of tolerance”, whose radius will indicate the degree of legislative freedom inside the Party. During a Legislature L, each member (legislator) lik(x,y) of a given Party k (where i = 1,...,Nk, being Nk the number of members of Party k) proposes one or more acts of Parliament (an, with n = 1,...,Na, being Na the total number of acts proposed during the Legislature) with a given personal and social advantage, depending on his/her position on the diagram. Therefore, the size of the circles of tolerance indicates the extent to which the proposed act an (coinciding with the member who proposes it) may depart from the centre of the Party, while the position of the centre of the Party will characterize the average legislative behaviour of all its members, in terms of obtaining personal benefits or achieving social welfare through their acts of Parliaments. It is clear that, in real Parliaments, the fact of belonging to a Party increases, for a legislator, the likelihood that his/her proposals get approved. But it is also likely that the social advantage provided by a set of passed laws will be on average reduced if all the legislators lie within the circle of tolerance of some Party (more or less authoritarian), since their votes will be forced by the common position represented by the centres of their Parties on the Cipolla diagram (Party discipline). Therefore the main goal of this paper is to explore how the global efficiency of a Parliament may be affected by the introduction of a given number Nind of independent members, i.e. randomly selected legislators free from the influence of any Party. These legislators will be indicated with labels li(x,y) (with only one index i = 1,...,Nind), and will be represented as free points on the Cipolla graph. In Fig.3 we plot an example of Parliament with N=500 members, 250 of which are independent (represented by free black points) and the other 250 are distributed into the two Parties (whose circles of tolerance are explicitly drawn). Notice that some free point could apparently fall within the circle of tolerance of some Party, but of course the correspondent legislator will remain independent. Suppose now to have a legislator belonging to a Party, or an independent one, who proposes a given act an. How does every member of the Parliament decide whether to vote in favour or against it? We will show that each legislator, including the proponent, has several possible behaviours depending on his/her membership. In order to explain these behaviours we need to introduce a couple of new concepts, the ”acceptance window” and the ”voting point”. The acceptance window is a rectangular window of the Cipolla diagram into which a given proposed act has to fall in order to be accepted by any free legislator li(x,y) o by the members of a Party Pk (which vote as if all of them would be placed at the centre of the Party). Such a window, indicated with wj, derives from the fact that, in our model, each free legislator perceives from any act of Parliament a personal and a social advantage, therefore it can be likely assumed that he, or she, will fix a lower threshold, both for social and for personal advantage, below which no act of Parliament is accepted. This follows from the assumption that the agents only accept proposals better or equal to their own ones. Similarly, each Party will fix similar thresholds for its personal and social advantages,

Fig.3

Fig.4

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which will be shared by all its members. Of course each Party or each member of Parliament has a different acceptance window, depending on his or her position in the Cipolla diagram (quite clearly, all such windows intersect, having in common at least the point (1,1)), and all the acts of Parliament which fall into a given window are approved by the correspondent legislator or Party (see Fig.4). But the task of establishing if an act of Parliament an, proposed by a legislator, would meet the requirements expressed by the window’s definition associated to another legislator, or to a Party, is not as easy as it may seem. Actually, while the perception y(an) of the social advantage of a given proposal can be likely considered the same for each legislator or Party

and therefore can be assumed as unambiguously determined, the perception of the personal advantage x(an) cannot. Indeed, the fact that an would be beneficial for the proponent legislator, clearly does not imply that it should be beneficial also for any other voting legislator or for a given Party. Rather, the proponent will assign a personal advantage to an which will be in general not correlated with that one of any other. Therefore, the coordinate x(an) of any given proposal should be randomly re-extracted over the straight line y=y(an), parallel to the x-axis before checking if the proposal falls in the acceptance windows of the other legislators or Parties. For example, in Fig.4, for the act of Parliament an(x,y) proposed by a legislator lik(x,y) belonging to Party k, a new voting point an(x*, y) has been randomly extracted over this line (in grey) and compared with the acceptance window of legislator li(x,y): since the voting point falls within the window wi, li(x,y) should accept the act. If the same voting point an(x*, y) had been extracted for the Party Pk′, again the act would be accepted by all the members of the Party since it falls within the window wk′. From this point of view, since for a given proposal the x-coordinate of the voting point changes randomly for each legislator or Party, it is clear that the larger a given acceptance window is, the higher will be the probability that, for that legislator or that Party, the voting point will fall within the window, i.e. the higher will be the probability that the correspondent act would be accepted. More in general, with several Parties (K ≥ 2) and a certain number Nind of independent legislators, for a given act of Parliament an we have the following voting rules: 1) if the proponent is an independent legislator lj(x,y):

- another independent legislator li(x,y) (with i≠j) votes in favour of the act only if the correspondent voting point an(x*,y) (randomly extracted from a uniform probability distribution and different for each legislator) falls within his/her acceptance window wi (of course lj(x,y) will vote his own proposal in any case);

- a legislator lik(x,y), member of Party k, votes in favour of the act only if the correspondent voting point an(x*,y) (randomly extracted from a uniform probability distribution only one time for all the members of the Party) lies in the acceptance window of his/her Party wk;

2) if the proponent is a legislator lik(x,y), member of Party k: - an independent legislator li(x,y) votes in favour of the act only if the correspondent

voting point an(x*,y) (randomly extracted from a uniform probability distribution and different for each legislator) falls within his/her acceptance window wi;

- a legislator lik’(x,y) belonging to the same Party with respect to lik(x,y), i.e. with k′ = k, votes the act in any case (Party discipline);

Fig.4

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- a legislator lik’(x,y) not belonging to the same Party with respect to lik(x,y), i.e. with k′≠k, votes in favor of the act only if the correspondent voting point an(x*,y) (randomly extracted from a uniform probability distribution only one time for all the members of Party k′) lies in the acceptance window of his/her Party wk′ (Party discipline).

Once this rule has been repeated over all the members of the Parliament, one can verify if the majority of them (N/2 + 1, at least) has voted in favour of the act: only in such a case that act will be considered accepted by the Parliament and it will be possible to pass to the next one. At this point we need some global quantity which in some way can express the efficiency of the Parliament during a Legislature L, being the latter defined as a sequence (array) of Na acts of Parliament. An immediate measure of the Parliament activity is of course the number of accepted proposals (laws), Nacc(L), expressed as a percentage of the total number of proposed acts. But another important quantity is surely the average social welfare Y(L) ensured by all the accepted acts of Parliament. Therefore it is straightforward to take the product of these two quantities in order to obtain the efficiency of the Legislature: Eff(L)=Nacc(L)×Y(L). We can now describe the algorithm for simulating an entire Legislature L of our prototypical Parliament, composed by N members or legislators, Nind of which are independent and the remaining (N - Nind) are distributed over K ≥ 2 Parties (with a given radius, equal for all the Parties) in a given different percentage: 1 - The first step is to randomly distribute the independent legislators and the centres of the K Parties over the Cipolla diagram. The members of each Party, in the established percentage, are then randomly placed inside the respective circle of tolerance (as in Fig.3, for K = 2); 2 - The second step is to submit the array of Na acts of Parliament {an | n = 1, ..., Na} to the vote of all the legislators, following the rules previously explained and storing the number of accepted acts and their value of y(an); 3 - The last step is to calculate the final percentage of accepted proposal, Nacc(L), their average social welfare Y(L) and, finally, the efficiency Eff(L) of the legislature. Of course these final results strictly depend on the distribution of the K Parties and the independent legislators over the Cipolla diagram, distribution which plays the role of initial condition for a given legislature L. Therefore, in order to make our results independent of the particular realization of the initial conditions, we have to average them over a set of different Legislatures {Lh | h = 1,...,NL}, thus obtaining the average values AV(Nacc), AV(Y) and the global efficiency AV(Eff). In our previous paper (Pluchino et al. 2011) we focused on a Parliament with N = 500 members distributed over two Parties P1 and P2, with different sizes N1 and N2 (N1+N2=N), a configuration which is simple and interesting at the same time, being typical of the Anglo-Saxon world but also of other Countries with a bipolar political system. In the following we summarize the main results obtained in that paper. Let us consider, first, the two limiting cases with Nind = 0 and Nind = N of independent legislators. The efficiency of the Parliament with no independent legislators (Nind = 0)strictly depends, for a given Legislature L, on the position of the centers of the two Parties over the Cipolla diagram, but also on their size (in terms of percentage of members) and on the radius r of their circle of tolerance (representing the degree of freedom within the Party). We assign 60% of legislators to P1 (so N1 = 300) and the remaining 40% to P2 (so N2 = 200). We consider also a sequence of Na = 1000 acts of Parliament proposed each time by a randomly chosen legislator (this implies that each member of Parliament will submit, in average, two proposals).

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In Fig.5 we show the simulation results obtained for a set of NL = 100 Legislatures, each one with a different position of P1 and P2 over the Cipolla diagram. A small radius r = 0.1, equal for both the Parties, has been chosen in this case. For each Legislature Lh the correspondent values of Nacc(Lh), Y(Lh) and Eff(Lh) have been plotted in three distinct panels (from top to bottom). At the end of the simulation the average values AV(Nacc), AV(Y) and AV(Eff) have been calculated and reported in the panels as a dashed line (with the respective numerical value on the right side). It is easy to understand that, following the voting rules, all the acts of Parliament proposed by legislators belonging to the majority Party P1 will be accepted (due to the Party discipline), while the proposals of the minority Party P2 will be accepted only if they fall in the acceptance window of P1. For these reasons we see that (top panel) the number of accepted proposals Nacc(Lh) oscillates from 60% and 100% (with AV (Nacc) ≈70%), while (middle panel) the value Y(Lh) oscillates between −1 and 1 thus producing an almost null average value AV(Y) = 0.05 (this because the values y(P1), corresponding to the y coordinate of the center of P1 circle, result uniformly distributed along the y-axis when one averages over the entire set of 100 Legislatures). Consequently, also the product between Nacc(Lh) and Y(Lh) oscillates around zero and (bottom panel) the global efficiency of the Parliament AV(Eff) = 0.57 is quite small (notice that, following the definition, the range of variation of Eff(Lh) is [−100,100]). This means that, mainly due to the Party discipline, a Parliament without legislators free from the influence of Parties results to be not very efficient, as probably happens in reality (see Pluchino et al. 2011 for more details). At first sight, the situation in which only independent legislators (Nind = N) seat in the Parliament seems completely different. In this case no Parties are considered and the 500 points corresponding to the N=500 members of the Parliament are uniformly distributed over the Cipolla diagram (as in Fig.3, but without P1 and P2) and each of them votes independently one from each other, without any kind of Party discipline. As a consequence, due to the voting rules, for a given proposal an(x,y), only about 50% of legislators with coordinates y(lj) < y(an) will vote in favour. But such a number will be clearly lower than N/2

unless y(an)≈1: in fact, only in this latter case the probability that the proposal an would be accepted will be greater than zero. Therefore we expect that only a small number of proposals will be passed during a Legislature Lh. On the other hand, we also expect a very high value of Y(Lh), since all the passed proposals will have y(an)≈1. In Fig.6 we show analogous simulations as in Fig.5, but with 100% of independent legislators (Nind=500). It is clear that, averaging over 100 Legislatures, the previous predictions are confirmed: in fact, as expected, we find a very small value for AV(Nacc) ≈2% (top panel) and

Fig.5

Fig.6

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a very high value for the social welfare AV(Y) = 0.893 (middle panel: here, the few points with Y(Lh) = 0 correspond to Legislatures with Nacc = 0%). Therefore, it follows that, as in the case with 0% independent legislators, the efficiency of a Parliament with only independent (randomly selected) members will be quite small again: in fact, as visible in the bottom panel, the product Eff(Lh)= Nacc(Lh)×Y(Lh) will stay near zero for all the Legislatures, thus giving a small average global efficiency AV(Eff)=1.78. This means that doing away entirely with parties in parliament is not beneficial to society. Once these two limiting cases have been explored, it is interesting to see how the global efficiency of a Parliament with two Parties (or, more in general, two political coalitions) is affected by an intermediate number Nind of independent members, increasing from 0 to N=500. On the basis of the results shown in the last two figures, increasing the number Nind of independent legislators from 0 to N in a Parliament with N = 500 members, and distributing the remaining legislators (N − Nind) into the two Parties P1 and P2 with percentages, respectively, of 60% and 40% and radius r=0.1, it is quite straightforward to predict (and simulations confirm this, as we will see in the next figure) that: (i) the average number of accepted proposals AV(Nacc) will decrease from ≈80% to ≈2%, while, on the other hand, (ii) the average value of the social welfare AV(Y) will increase from ≈0 to ≈0.9. But it is interesting and absolutely not trivial to explore what will be the correspondent behaviour of the product of these two quantities, i.e the behaviour of the global efficiency AV(Eff) of the Parliament, emerging from the interplay between the accepted proposals and the social welfare provided. Actually, we saw that AV(Eff) results to be quite small in both the limiting cases with 0 and N independent legislators (see Figs. 5 and 6, bottom panels), so it is reasonable to ask what happens in the intermediate region 0 < Nind < N. In Fig.7 we answer this question by plotting the global efficiency AV(Eff) against an increasing number of independent legislators. Each curve corresponds to a different increasing size of the circles of tolerance, whose radius is equal for the two Parties. Each point of a given curve represents, as usual, an average over 100 Legislatures, each one with 1000 proposals. In the insets, the correspondent behaviour of the percentage of accepted proposals AV(Nacc) and of the average social welfare AV(Y) are also reported as a function of the number independent members. It clearly appears that the efficiency, albeit with a fluctuating behaviour, rapidly increases with the number of independent legislators until it reaches a peaked maximum for N*ind=140,, then it smoothly decreases towards the known small limiting value obtained for Nind=N. This means that the introduction of a percentage of about 30% of independent legislators, out of the influence of any Party (for example randomly selected from a given list of candidates), improves the efficiency of the Parliament itself. It is also interesting to observe that the maximum value decreases when the radius increases from 0.1 to 0.6, implying that the constructive role of independent legislators is sensitive to the degree of freedom existing in the Parties: the more authoritarian Parties are (r = 0.1 or r = 0.2), the more the role of independent legislators becomes crucial. When, on the contrary, Parties are more libertarian (r = 0.4 or r = 0.6), the efficiency curve becomes smoother, even if a little dependence on the number of independents still exists.

Fig.7

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We also simulated other configurations of the Parliament, changing the size of the two Parties. In Fig.8, for example, we assigned percentages of 51% and 49% to the majority and minority Party respectively, and the result is that the peak in efficiency occurs much earlier than before, with only N*ind=20 independent legislators. Such an effect is quite reasonable because it suggests that, if the two competing Parties have a similar size, even a small number of independent members in the Parliament, playing a role of balance, can fairly improve

its global efficiency. On the other hand, when one Party is quite bigger that the other one, like in the simulations shown in Fig.7, the number of independent legislators required to enhance the efficiency of the Parliament increases. This trend is confirmed in Fig.9, where a Parliament with the majority Party very much bigger (80%) than the minority one (20%) has been considered: as expected, after a very oscillating initial behaviour, the (smoother) peak shifts on the right with respect to the previous figures and the maximum efficiency is obtained with the introduction of N*ind=280 independent legislators. Finally, also in the last two figures the efficiency depends on the radius of the circles of tolerance of the two Parties, and in general decreases increasing it. Notice that in all the cases, while the values of the AV(Eff) for Nind=0 can be different for the four curves with different radius, moving towards Nind=N all the curves tend to coincide, since the dynamics (and therefore the efficiency) becomes independent of the radius of the Parties.

It is now interesting to report the optimal number of independent legislators (i.e. the value N*ind corresponding to the peak in efficiency observed in the last three figures) as function of the size of the majority Party P1, expressed as an increasing percentage p% of the N members of the Parliament. Of course, for a given N*ind and a given percentage p%, Parties P1 and P2 will have, respectively, (N – N*ind )·p%/100 and (N – N*ind )·(100-p%)/100 members. Since the position of the peak value

N*ind does not change significantly increasing the radius of the circle of tolerance up to r≈0.5, we consider only the case r=0.1. The results of the simulations for our standard Parliament with N=500 members are shown in Fig.10. As expected from the behaviour observed in the previous figures, N*ind progressively increases with the size of the majority Party, from N*ind=20 – corresponding to the case p%=51 – to N*ind=280 – corresponding to the case p%=80 –. Beside each point, obtained averaging - as usual - over 100 Legislatures (each one with 1000 proposals), the corresponding value of the maximum global efficiency AV(Eff)[N*ind] is reported. It appears to oscillate around 15, a value that seems to be independent of N*ind since, for any p%, it derives from the product of an average AV(Nacc)=38% of accepted proposals and an average social welfare AV(Y)=0.4, values that evidently realize the best compromise between the two quantities in order to obtain the maximum efficiency in presence of both Parties and independent Legislators.

Fig.8

Fig.9

Fig.10

Fig.6

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But if, on one hand, it is absolutely not trivial to predict these last quantities since, given p% and Nind, they are very sensitive to all the features which affect the voting process, on the other hand, in our previous paper, a simple empirical argument has been advanced in order to find a sort of “efficiency golden rule”, i.e. an analytical formula designed to determine the optimal number of legislators N*ind as function of p%, i.e. able to fit the simulation points reported in Fig.10. The formula is the following (see Pluchino et al. 2011 for more details): We verified that it is independent of the total number of members of the Parliament, of the number of Legislatures and of the number of proposals for each Legislature. Being also independent of the radius of the circles of tolerance of the Parties, we argue that this ”golden rule” could be likely applied in other social situations where the introduction of independent voters among two groups competing in some voting process may enhance the collective gain. Thinking of a practical application for a real Parliament, the knowledge of the ”efficiency golden rule” would allow to fix the optimal number of legislators to be selected at random, picking up from a given list of candidates (i.e. ordinary citizens fitting ordinary requirements), after that regular elections have established the relative proportion of the two Parties or coalitions. In the next part of this paper, we extend this version of our 2D model of Parliament to a case study in which more than two Parties are present, i.e. the 2013 Italian Senate, in order to evaluate the efficiency of different possible configurations of alliances among those Parties. In this new case study, the role of the independent, randomly selected, legislators will be played by the members of the 5 Star Movement, a newly political formation established some years ago, arising from the anger against the many abuses of Italian traditional Parties. 2. The 2013 Italian Senate: a case study At the end of February 2013, the political elections in Italy offered an unusual situation: after twenty years of bipolarism (more or less effective), when the country was accustomed to a tight battle between two great political Parties, the center-right and center-left, in the new Parliament sat four political formations. In addition to the two traditional coalitions, the left one led by Democratic Party and the right one, Popolo della Libertà, led by Berlusconi, two other new significant Parties appeared: one led by Mario Monti (the former prime minister), named Scelta Civica, and another one led by Beppe Grillo, the 5 Stars Movement. In particular, it was the extraordinary success of the latter that changed the previous situation, bringing Italy to an apparent stalemate. In fact, thanks to the current electoral law (recently declared unconstitutional by the Italian Consulta), if in the Deputy Chamber the majority premium guaranteed to the center-left coalition a large majority of deputies, in the Senate, no coalition alone possessed the absolute majority of seats and, to date, no one could see clear spaces for possible alliances leading to a stable government. But was it really this result, as suggested by the common sense, the prelude to an inevitable phase of political standstill? Could a Parliament with variable majorities in the Senate be as efficient as a Parliament with a large majority in both the Houses? In this second part we will try to answer these questions going beyond common sense and analyzing the 2013 political situation of Italian Senate by using our 2D model of Parliament, extended to K=4 Parties. In this case study, we have N=319 senators distributed as follows: the center-left coalition (CSX) won 123 seats, the center-right (CDX) 117 seats, 5

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Star Movement (M5S) 54 seats, the Party led by Monti 20 seats (19 + the Monti senator for life) and the remaining 5 seats (1 Vallee d'Aosta, 1st Movement Italians Abroad and three senators nominated by the President). The simulations refer here to the two more likely scenarios at that time, assuming that some kind of government could get the vote of confidence, and take into account the performance of a full Legislature, during which a total of 1000 proposals will be advanced. The two scenarios are the following:

- Scenario "without alliances": in this scenario there are no default alliances between Parties and/or coalitions, then, after the initial vote of confidence, the Senate proceeds law after law for the whole term with variable majorities. In Fig.11 you can see two possible arrangements of CSX (in red), CDX (in blue), Monti’s party (green) and 5 remaining Senators (orange dots) on the Cipolla’s diagram. As you can see, the senators who are members of Parties or coalitions are within their "tolerance circles". The M5S senators, however, are represented by black dots distributed randomly, uniformly, on the diagram, because their origin is extremely variegate and heterogeneous. In this regard, we have also considered two possibilities: the first takes into account the fact that the M5S is characterized by particular attention to the collective benefit of ts proposals, therefore its senators are distributed only on the upper half-plane diagram (M5S in the upper half-plane, panel a); the latter, however, is more neutral and distributes the senators on the whole diagram (M5S over all the diagram, panel b).

- Scenario with "governissimo": in this alternative scenario, the two center-right and center-left coalitions join their forces together with the Monti’s party, forming a governissimo (i.e. a "big coalition" CSX + CDX + Monti) which, with its 260 seats, is well above the absolute majority of the Senate (160 seats), leaving the 5 Star Movement out of the government. In Fig.3 there are two possible arrangements of this governissimo (whose senators lie in a single circle) and of the senators M5S that, as in the previous scenario, can occupy only the upper half-plane (a) or the entire Cipolla’s diagram (b).

Fig.11

Fig.12

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The voting mechanism is always the same: senators who are members of a political Party or a coalition vote all together due to Party discipline. In particular, they always vote in favour of the proposals put forward by any other member of their own political party, while in other cases, i.e. with respect to the proposals made by others, they vote according to the centre of their own Party, i.e. the centre of their own circle of tolerance in the Cipolla’s diagram. The five senators who do not belong to any political party, always vote in a free and uncoordinated way, regardless of who propose the bill. At this point, it becomes crucial the way in which the members of the 5 stars Movement would vote. Actually, although it looks like a traditional political party, it is different not only for the selection of its members – already taken into account in the distribution of points over the Cipolla’s diagram – but also for their internal organization, which is reflected on the voting strategy.

So we imagined three possible ways in which the M5S senators could vote:

1) Party discipline: In this mode all the senators M5S vote together, as a traditional Party. The vote will always be in favour of their proposals, while the other proposals will be evaluated case by case, by comparing them with the fixed centre of gravity of their members in the Cipolla’s diagram (which will be the point with coordinates (0, 0.5) in the "M5S in the upper half-plane" case, or the point of coordinates (0.0) in the "M5S in the whole diagram" case);

2) Web Discipline: a feature of M5S is to interact with their electorate through the web, so we figured that senators can vote all together but letting the center of gravity of the movement to change law by law, according to the guidelines provided by the voters through the tool offered by web platforms such as, for example, "liquid feedback";

3) Independent: the senators M5S behave like our independent, randomly selected, senators introduced in the first part of the paper and thus vote law by law independently of each other, without a binding mandate.

Taking into account all these parameters applied to the different scenarios, we performed several sets of numerical simulations to evaluate, case by case, the efficiency of the Senate, calculated – as usual – as product of the number of accepted proposals times the collective advantage resulting from them (i.e. from the average value of the y coordinates of those proposals). Each set includes the run of 100 independent legislatures, each with a different random distributions of both senators and Parties on the Cipolla’s diagram, from which we extracted the average global efficiency with an error of about 10% (which takes into account also the possible changes in the range of tolerance of the parties). As a comparison, one should remember that, in the bipolar Parliament considered in the first part of this paper, the peak of maximum efficiency (obtained in correspondence of the optimal number of independent legislators) was always around a value of 15 (in arbitrary units). In the following we present the tables and the graphical results:

1) Results with M5S in the upper half plane (Fig.13): in this case, the "virtuous" location of M5S senators in the Cipolla’s diagram yields beneficial effects, especially in the scenario with variable majorities (without alliances) and, in particular, when the voting of 5 Star Movement follows the directives of the electorate via the web, parliament achieves an efficiency nearly twice (12.7) as larger than the other cases. The scenario carrying the formation of a governissimo CSX CDX + Monti, instead, penalizes the efficiency of the Senate, which generally tends to decrease remaining however positive thanks to the

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placement of M5S in the upper half plane. Finally, the "independent" voting mode of the M5S senators, while maintaining relatively a high efficiency (around 7), on the other hand tends to cancel the differences between the two scenarios.

2) Results with M5S over all the diagram (Fig.14): in this case we do not have the "virtuous" location of M5S in the Cipolla’s diagram any more and this has an impact on the overall efficiency in all the simulations. In particular, we note a significant difference between the results for the two scenarios (without alliances and governissimo). In fact, while the scenario with variable majorities (without alliances) continues to have high efficiency values (around 9.5), expecially in the two methods of voting for M5S senators which exclude the party discipline with a fixed center of gravity, on the other hand, the scenario of a governissimo comes out always severely penalized in terms of efficiency. We see that, in the latter scenario, the average efficiency becomes even negative in the two voting rules of M5S with center of gravity, while it remains close to zero in the most favorable case of independent vote of the M5S senators.

Fig.13

Fig.14

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Summarizing the results obtained, within the limits of the working hypotheses and the inevitable simplifications that are at the basis of our model, it is possible to draw the following conclusions:

1) The scenario "without alliances" seems to be able to ensure the greatest potential efficiency for the 2013 Italian Senate. This possibility appears at a first sight in contrast with the common perception of poor governability of a Parliament with variable majorities. However, once overcome the obstacle of an initial trust to some kind of government, this solution is the one that mostly gives back its natural function to the Parliament itself, that is discussing and improving the proposed laws without the “sword of Damocles” imposed by an abuse of the "voting trust " by the government itself (which, usually, is expression of the coalition possessing the absolute majority).

2) The role of the 5 Star Movement seems to be, in any case, decisive for the purpose of maximizing the efficiency of the Senate. In particular, it seems essential that it maintains its distance from the traditional Parties, especially regarding the voting mode of its senators. In fact, its effectiveness is closely linked to the absence of a fixed centre of gravity that collectively influence the vote: the interaction with the electorate via the web or the absence of a binding mandate for senators are both valid solutions to maintain the high efficiency of the system, whatever the location of the senators themselves in the Cipolla’s diagram. Unfortunately, after more than one year from the 2013 elections, we notice that the political forces did not follow our suggestions: actually, just after the elections, the Democratic Party was not able to obtain a vote of confidence in both the Houses, therefore two months later, at the end of April 2013, the President Napolitano gave way to the first “grand coalition” government of the Italian history. In practice, Enrico Letta became Prime Minister realizing, more or less, the scenario that we called “governissimo”. Adding to this the fact that, in the next months, the 5 Star Movement has chosen for its members a voting mode which can be considered a mixing between the “party discipline” and the “web discipline”, it is not surprising that the Italian government action in the last year (until the arrival of Renzi on February 2014) has been very inefficient, as predicted by the results of Fig.14 (with the more realistic hypothesis of M5S over the entire Cipolla’s diagram). In the third and last part of this paper, we further simplify the hypothesis of the 2D model considered up to now and propose a one dimensional version of our model of Parliament, just trying to identify the building mathematical blocks of a theoretical model of legislative bodies, still in the making.

3. A 1D model of a Parliament In ancient Greece, the cradle of democracy, governing bodies were largely selected by lot. The aim of this device was to avoid typical drawbacks of any representative institution. In modern democracies, however, the standard is choosing representatives by vote through the party system. Debate over efficiency of Parliament has therefore been centred on voting systems, on their impact on parliamentary performances and, ultimately, on the efficiency of the economic system. In Pluchino et al. (2011), as already recalled in the first part of this paper, we showed how the injection of a measure of randomness improves the efficiency of a parliamentary institution and, as a result, of the economic system itself. Although based on a mathematical model of Parliament, where legislators are represented as points in a two dimensional space, the main results there obtained have been essentially achieved through numerical simulations, since – even in its simplicity – the 2D voting dynamics was too complex to be captured analytically (also the “efficiency golden

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rule” was not derived analytically from the hypothesis, but only by empirical arguments). In this last section of the paper we further simplify our 2D model, attempting to build a 1D model of Parliament aimed at obtaining (in future works) more general results in an analytical form. We accept the standard distinction between political and economic institution (North, 1990, Acemoglu and Robinson, 2010): how Parliament is organized is a political institution, while what Parliament usually produces is economic institutions. This line of research therefore is designed to evaluate the efficiency of a political institution, i.e. its ability to produce, in its lifetime, efficient bundles of economic institutions. As is clear from the first two parts, the analysis confronts two alternative ways of electing a Parliament: the party system and random selection. In the Party system representatives (legislators) are voted from a list set up by Parties. With random selection, representatives are randomly drawn from a population somehow defined. In what follows we show that any injection of random selection in the formation of Parliaments is likely to increase the efficiency of that political institution. Randomness increases efficiency because it helps avoid bad proposals, as such proposals never get enough votes to be passed. A number of political representatives is drawn from a population. They are assumed to be normally distributed along a [-1, +1] x-axis, where x represents the minimum net welfare contribution they are prepared to accept and vote for as representatives (in other word, the x-axis in this 1D model plays the same role of the y-axis in the 2D model). This is the most likely distribution of representatives as only a few will set their limit very high or very low. This means that only a few will be prepared to accept virtually anything; similarly only a few will be extremely selective in accepting proposals. Most representatives will set their limit close to an average, which conventionally is set to zero. In such a house of representatives, proposals are discussed and adopted provided they receive half (plus one) of the votes. Proposals are drawn randomly in a sequence from a population of proposals that may (i) or may not (ii) be distributed as the representatives, along the same [-1, +1] x-axis, where x represents the net contribution to welfare of that proposal. In all cases representatives choose proposals according to their threshold; in case (i) proposals are consistent with representatives’ preferences; in case (ii) proposals stem from a different selection mechanism, like a government. At any given time interval, representing the duration of Parliament, only those proposals receiving the majority of votes will be adopted. The contribution to welfare is determined by the sum of the contribution of those proposals capable of commanding a majority. Case (i) Proposals resemble perfectly the distribution of representatives: each representative will put forward a proposal which is, at least, as good as its lower limit of acceptance. If this is the case, such a Parliament is expected to produce an overall contribution to welfare equal to (given the majority rule only the proposals falling within the interval [0, 1] will be accepted)

𝑃 𝑥 𝑥𝑑𝑥 > 0!

!

Case (ii) Proposals may stem from sources different than the various representatives. It is very often the government which puts proposals before Parliament. We can distinguish the case of a benevolent government, that is a government whose distribution of proposals

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is shifted to the right, from the case of a malevolent government (ii), a government whose distribution of proposals is shifted to the left. The overall contribution to welfare will look like the previous formula, but the distribution P(x) has a higher average in the first case, and a lower average in the second. In the case of a benevolent government, therefore, a Parliament based on sortition will act as a magnifier of government benevolence, as a majority exists for most government proposals. In the case of a malevolent government, a Parliament based on sortition will act as a limiting factor of government malevolence, as a majority exists for a very few government proposals. Let us move now to a Parliament with Parties. Two Parties or coalitions are assumed to exist (a party holding the majority of votes - 60% - and an opposition party - 40%). Party members are elected to Parliament according to some electoral system: Parties may represent all sections of society. Once in Parliament they act as a single individual: their preferences collapse into the preference of the Party leader who is assumed to be placed at the centre of the Party. In this case, too, proposals may be generated either from within the Party representatives or from outside, i.e. the government. In case they are generated within the Party, it is as if they are all generated by the Party leader. Hence all proposals coming from the Party holding a majority will be accepted. The majority will also vote for the proposals of the opposition party provided they meet their threshold. Since the Party holding a majority may represent both the good or bad sections of society the average expected distribution of the Party representatives will look like that of independent (randomly selected) representatives; furthermore, since proposals are in this case internally generated the two distributions, that of representatives and that of proposals, will look the same. Since all proposals are voted through Party discipline the likely welfare contribution is the following:

𝑃 𝑥 𝑥𝑑𝑥 = 0!

!!

The opposition Party will not make a big difference as, on average, it will have the same distribution as the bigger Party. Therefore, in the case of internally generated proposals, the Parliament with randomly selected representatives makes a contribution to society much higher than the contribution of a Parliament based on voted representatives through the Party system. It follows that any injection of randomness acts as a filter of bad proposals as no majority can be won for proposals lying within the [-1, 0] interval if they come from independent representatives. If proposals come from outside Parliament, like a government, no Party discipline necessarily applies. It follows that a Parliament with Parties acts like a Parliament with independent representatives if proposals come form outside Parliament. As in the previous case the government maybe either benevolent and produce proposals shifted to the right, or malevolent and produce proposals shifted to the left. In such circumstances party members will vote only proposals originating through government initiative; hence they will accept only those proposals consistent with their average preferences. It follows that once government (or any other external source of proposals) is brought in, there is no difference between the party system and random selection of members of Parliament. Hence if proposals are internally generated a Parliament made up of randomly selected representatives is more efficient than a Parliament based on voting through the Party system. If proposals are externally generated (e.g. by a government) Parliaments are equally efficient no matter how they are formed. This simple analysis shows that moving from a Parliament based entirely on the Party system, with internally generated proposals, any injection of randomly selected representatives will increase parliamentary efficiency. Similarly, any reduction of the

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percentage of internally generated proposals in favour of externally generated ones will increase parliamentary efficiency as it reduces the perverse role of the Party system. It must be stressed, however, that such conclusions are contingent on the assumptions that all the various regimes investigated are equally costly. As recalled at the beginning of this section the attempt made here is simply designed to establish a few building blocks of a model of legislative bodies to be developed much further. It goes without saying that all the various regimes cannot be equally costly. Furthermore, it cannot be taken for granted that costs in each individual regime change in the same way. An interesting development of the analysis can be envisaged therefore aimed at the determination of the optimal composition of legislative bodies. 4. Concluding remarks This paper develops a bit further some ideas already illustrated in the literature. Random selection of political representatives is increasingly being considered in the public and academic debate alike as a way to make Parliaments more representative and more efficient. We started from agent-based simulations applied to a 2D mathematical model of Parliament (first part), showing how randomly selected legislators (therefore independent form Parties) can improve parliamentary efficiency. Then we presented an application to the Italian Senate as it came out from 2013 political elections (second part), where the 5 Star Movement could play, to some extent, the role of independent legislators. Finally, in the last part, we set out to develop a 1D theoretical model along economic lines, designed to show how and under what (more general) conditions random selection of legislative bodies members can improve upon the efficiency of those bodies. This last point will be, of course, further discussed in future works. REFERENCES

Acemoglu, D. and J. Robinson (2012) Why Nations Fail, Kindle Edition

Bankes, S. et al. (2002). Making Computational Social Science Effective: Epistemology, Methodology, and Technology. Social Science Computer Review, 20(4), p.377-388.

Epstein, J. (2007). Generative Social Science: Studies in Agent-Based Computational Modeling. Princeton, NJ, United States: Princeton University Press.

Galam, S. (2008). Sociophysics: A Review of Galam Models. International Journal of Modern Physics, 19(3), p.409-440.

North, D. C. (1990) Institutions, Institutional Change and Economic Performance, Cambridge University Press

Pluchino A., C. Garofalo, A. Rapisarda, S. Spagano, M. Caserta (2011). Accidental politicians: how randomly selected legislators can improve Parliament efficiency. Physica A 390, p.3944.

For more details, articles and international press concerning our 2D model of Parliament please visit the web page: http://www.pluchino.it/parliament.html