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19
SURVIVAL MODELS OF SOME POLITICAL PROCESSES
S. Esa (Kurdistan Regional Government, Council of Ministers, Erbil, IQ)
B. Dimitrov (Kettering University, Flint, Michigan, USA)
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e-mail: sahib [email protected] [email protected]
ABSTRACT
We extend the Probabilistic ideas from stochastic processes (queuing theory and reliability) on creation of some realistic models for studying several governing political formations, and find their survival characteristics. These models were presented at the Sixth and Seventh International Conferences on Mathematical Models in Reliability (Moscow 2009, and Beijing 2011). Our focus is on a "democracy" model, where the times of survival (existence at the political scene, duration of stay in leading coalition, governing survivability, life time distribution, longevity, etc.) can be derived from the model. Markovian models of spending time in certain sets of states are explored, and some discussion on statistical properties and evaluations are presented. We are confident that other political schemes also can be modeled using appropriate probabilistic tools.
1 INSTRUCTIONS
Modeling politics based on certain scientific concepts and its inclusion into quantitative models is a challenging task. The authors of this work did an extensive review of some successful attempts in political modeling by mathematical means such as: Cioffi-Revilla (2009) recent model of a political system; the recent textbook of Clark et al. (2008) on modeling of preferences in political science; the Taylor's book (1995) with ideas for discrete and deductive mathematical approaches in international conflict resolution; Doyle nonlinear methods (2000) to describe and solve existing political questions; Monroe's (1997) evaluations of the current state of empirical political theory and guidelines to future developments in political science; the Ordeshook (1986), Morrow (1994), Hafer (2007) approaches based on the use of games theory in modeling political processes, and some non mathematical ideas as these, presented in J. March (1994), Then we started working on an approach based on construction of specific probabilistic analysis of components that form political processes inn conditions of an open democracy. In our opinion, formal political theory seeks to develop formal, mathematical models of political, demographic, and economic processes. The above mentioned authors in their books and articles, make an attempt to integrate the modern developments of the applied mathematical theories (games, decision making, multiple, interactive decisions) with models of demand and supply of public goods, and social-choice theory, part of what the political structures are considered.
The articles of Esa and Dimitrov (2009, 2011) are an illustration of how probability tools can be used to model basic components in the big political games. Reliability and service system theories provide a good assembly of approaches in analysis of various sides of the products (risks, and costs inclusive) and characteristics in the political activities. Encouraged by the welcome reception of these presentations on behalf of the experts in applied statistical modeling at the MMR' (2009 and 2011) Forums, we continue working in the same direction. We found more sites our approach may reveal. In the present work we use the results of these simple models to illustrate various important additional characteristics of the political subjects. In our opinion, appropriate models can be made to study
totalitarian schemes, monarchy, parliamentary kingdoms, even some non existing, but virtually possible political structures. And this is our reason to consider probability models in politics a challenging area of applied probability and statistics.
2 CHARACTERISTICS OF THE DEMOCRATIC POLITICAL PARTY MODEL
First we use the results from the model of a political party, considered as a formation within a finite population of active individuals N. The existence (formation) of a party requires certain minimum of members, say M+1. Each free individual may decide to join a party at any time, as well as a member can quit the party at any time. Simultaneously, there exist a numerous pool of other parties which operate in a similar way, and their members are not allowed to join (or switch to) another party.
After discussing this mechanism and dynamics, Esa and Dimitrov (2009) introduced the following mathematical model of the political life in a country:
There is a population of N individuals (citizens of a country). These are r parties in the society. Each party is considered as a service system of N available seats (servers) in each. The rate of inputs from each free individual towards party (service system) j is Xj . j=1,... ,r. At system j the individual spends some random time Sj, and goes free of politics (back to its source). A simple Markovian model in the case of exponentially distributed service times Sj of parameters ft shows the stationary probabilities Pj that an individual is free (then we use j=0), or is member of the party j=1,2,.. ,,r are given by the expressions
f
P =
A
A,
Y
1 + + ■■■ +
v M1 Mr y
P = -
Mj v
A f A O "
1+A+■..
Ml Mr y
, 7=1,2,...,r.
(1)
A multinomial model describes the entire spectrum of the party's life in the country, with N independent active free individuals. The coordinates of the random vector X =(X0, X1, ..., Xr) represent the number of individuals members of each party X0 + X1 +... + Xr = N . They are distributed according to the multinomial law
P(Xo=ko, X1=k1, Xr=kr) =
M
P0k°Pk1...Prkr, k0 + k +... + kr = N.
k 0*1!... kr !
(2)
Hence, the chance of the jth party to exist (at a minimum M+1 members required for this purpose) is having Binomial probability P(j M+1)=1 - B(M N, Pj) with the P, given by (1), and B(k; N, p) is notation for c.d.f. of Binomial distribution witn parameters N and p, and k its argument. The probability generating function of this distribution (2) is given by the expression
P(z0, *!,..., Zr) = E(ZoXozX1 ... z^ ) = (Poz0 + P1Z1 + . + Przr )N , |Zj|<1. (3)
It allows particular calculation of various average characteristics, correlations etc.
The average number of members of a party j is E(Xj)=NPj, and its variation equals V(Xj)=NPj(1 - Py).
The correlation coefficients between the counts Xi and Xj of the parties labeled as i and j are given by the expressions
Pj = Corr (X, Xt ) = -.
P
1-P
1
P
1-P
, ifj, i,j=0,1, . , r.
The correlation is always negative, since parties compete for one and the same pool of potential members. Interesting quantities are the correlation coefficients of the parties i=1,...,r with the
"party" of the free individuals, for which subscript j is zero. Their absolute values may be used as a kind of rating indices for the parties in the country. The more the value, the higher the rating is. Interesting observation here is, that the correlation coefficients between parties (as well as the rating) do no depend on the total population size N.
Probability for party j to be dissolved must be considered under condition that party is active, i.e. given that Xj>Mj+1. This is the probability that someone of its members will decide to quit the platform when its members are at the critical number Mj+1. Thus this probability equals
f N } M +1 NM +1 (Mi + 1)iAt di = P{X. (t + At) = M | X. (t) > M +1} = PMj+1(1 - P.)NMj + J 1
VM. + 1J 1 v " 1 -B(M.;N,P.j)
Probability for party (platform) j not to exist (to be in a "sleep" state) equals to the measure of the chance for the model to spend in one of the states Xj=0, 1, ..., Mj, and therefore, is given by the expression
P(Xj<Mj) = B(Mj; N, P.)
with the P., determined by (1).
Further dynamic analysis will allow to determine the duration of existence of a party. Let us note that when we look at the Markovian process of the changes in the platform (party's) states, we may imaginary consider another, absorbing Markov chain whose absorbing states are these when X=0, 1, Mj, and the states Xj=MJ+1, Mj+2, ..., N are transient. Then, the duration of the existence of party (platform) j on the political scene will be equal to the time the above described Markovian process spends in the sets of its transient states.
Denote by Bk k ...k the average time the process spends at the transition set of states if it
starts at a state given by the left hand side of equation 2. Taking into account the infinitesimal intensities of the process interstate transitions, and the respective average spending times immediately before a transition and the respective sojourn times in the same state and in the transient sets after the transition, we arrive to the following system of equations for the sojourn times Bk k ,,k , valid for kj >M+1, for all j = 1, ..., r:
r N B 1
B "V k ii ko,k1,'",kt-1,—,kr |__[__,
nk0,k1,-,kr - Zj Zj r r ^ r r ^
ki =Mi +1 ko2 A, k,I ko2 A, k,|
l=1 l=1 l=1 l=1
r N B
+ 2 2 (ko + 1)A ^1,k1,-,kiykr . (4)
i=1 k =M 1 ko2 A +2k11
1=1 1=1
On the boundary layers where some, or several ki = Mi +1 the equations (4) are still valid with Bk, ki,„k -1 ... k = o . Also is true that Bk ki,„k = o if kj < Mj, for any j = 1, ., r. System (4) always
has a solution since the chain is absorbing. This solution can be found by the method of inverse matrices.
The expected life time of party j is then given by the expression
N f N ^ r N
Bj =2 k Pk (1 - Pj )N-kj 2 2 P(Xo = ko, X1 = k1,..., Xr = kr)Bko,k1,..,kr (5)
kj =Mj +1 kj J i=o kj = Mj +1
i * j i * j
The use of the introduced parameters in respective statistical data may allow practical estimation of these parameters and give the answer to various interesting statistical questions.
3 CHARACTERISTICS OF THE ELECTION MODEL
The party model of the previous section was used in [5] to create respective election model. It is assumed the following configurations before the elections. There are C coalitions registered for the peoples vote. A coalition may consist of one ore several parties. The "Zero" party is made by those who are not members of any party. They vote with probability p, or not vote with probability q= 1-p. A vote goes to coalition Cj with probability Qj proportional to the intensities to join the parties with indices i1,...,iC. from this coalition, i.e.
A, + ••• + A,
Qj = A+A-V ■ j'=1 C (6)
A + A2 + ••• + Ar
Party members vote for the coalition where their party belongs. Under some additional assumptions it is found that the random vector (T0,Ti,... TC) of votes given to the coalitions in the elections has multinomial distribution with probability generating function of the voting results (T0,Tj,...TC) given by the expression
T(z0, *„..., Zc ) = ((1 - p)Pz0 + (pPoQj + Pj + • P,' )zj t, (7)
Here Pi, Qj are given by expressions in (1) and (6), and T0 is the number of those who do not vote. Hence, the random number of voters for coalition Cj has Binomial distribution with parameters N and aj = pP0Qj + p +-----+ PiC . Roughly speaking, the number of votes for a coalition equals to
the sum of its party members and the votes of non-party people who may vote for this coalition.
Knowledge of the distribution of the random variables Tj allows calculation of various interesting explicit and/or average characteristics related to the specific electoral laws. For instance: The average number of votes for coalition Tj equals Naj .
The probability that coalition Tj does not survive the requirements to pass the minimum y% percentage barrier is given by the expression
J V
100'
Here V=pE(X0) is the expected number of voters in the elections, X0 is the number of members of "the zero party" with the marginal distribution as given by equation (2), and B(.) is notation for the Binomial c.d.f. The complement to 1 of the probability in (8) is the probability for this coalition to survive the elections. Since here the work is mostly with binomially distributed r.v.'s, all the conventional approximations to the Binomial distribution (Poisson, normal) are legitimate tools to simplify the calculations. We omit these details.
The provability that coalition Cj is the winner in the elections will be determined from the requirement
T} = max^,..., Tc ). (9)
When take into account (7) we easily find that the probability (9) to be fulfilled equals
P[T] < j V /100) = B(--; N, a}) . (8)
N
N!
fi = I nkT^T![(1 - P)P-]k0 a'' -aCC. (10)
kj > k!,,"., kc K0!K'! KC !
k0+kj +-----+kC = N
It is intuitively clear that the coalition with the highest value of the probability aj is the expected
winner. However, (10) allows to evaluate probabilities for any other coalition to win.
One last remark here is that the results in votes are negatively correlated random variables. The correlation matrix among the coordinates of the random vector (T0,T],.TC) is given by the
entries as shown in the case for the counts Xi and Xj of the parties in the previous section, where instead the probabilities Pj the quantities aj must be used.
In many cases, the votes from the losing coalitions are distributed proportionally between the winning coalitions, according to the numbers of actual votes approved for the winners. These potential extra votes may increase the elected members from a coalition. Respective conditional distributions are also available based on the described here model. This will be subject of another study.
4 CHARACTERISTICS OF THE GOVERNING MODEL
The process of formation of governing coalition between the winners in the election has a complex structure. Here we enter in the complexity of issues well described in chapters 3 and 4 by J. March (1994). It is a challenging task to make mathematical models based on these descriptions of decision making. Usually, the coalition with major sits in the National Assembly (NA) takes responsibility to form a governing coalition. It negotiates with groups with smaller sits in the NA, until gets more than K supporters among the sits in the NA. Then the government is ready to be formed. This government survives as long as it is supported by at least K members in the NA. Assuming that the government is supported by G coalitions, with Mj, ...,MG representatives in the NA, we modeled the government as a system with subsystems connected in series, with variable number of functioning components in each subsystem (the coalitions in the government). With the notation Fj(t) for the distribution function of the time of random duration that a member of the NA from coalition j keeps his/her loyalty to the governmental formation (and decisions assumed independent between the members of the NA), we derive in [5] that the government survives time of duration t is given by the rule
G Mj fM X X ,, .
P{L > t} = n2 j 1(1 - Fj (t)JFMj- (t). (11)
j=1 i=Kj V i J
Here K is a number showing the minimum number of members required for a coalition to exist as an entity in the NA, and L is the life time of this government.
If the next elections are scheduled after expiration of time of duration T, the average life time of this government will be presented by the quantity
T
IT = E(L | T) = JP{L > t} dt (12)
o
However, this is politics, and the life is dynamically changed. There are times when the parties in opposition call for non-confidentiality vote against this particular government. The results from the vote are modeled by making use of the models from sections 2 and 3. Such votes are binary (Pro or Con where the votes "abstain" are actually in favor of no confidence). Each coalition may have its own probability pj for a member to vote "Pro". The random number Nj of "pro"-votes in each coalition Cj is binomial of parameters (Yj, pj), where Yj is the random number of surviving supporters in the jth governing coalition. Therefore, the chance to survive a non-confidentiality vote at time t has probability
M■ fM,\ _ f i }
P(N1 + ■■■ Nc > K, L > t) = 2 2 2 •1|(1 -Fj (t))'FM-i(t)
i
k
pk (1 - p. )'-kj (13)
j=1 k +-kc >K i=Kj V J V 1 J
If the calls from opposition for non-confidentiality vote for the government form a point process of certain kind, the survival probability of the government will decrease proportionally to the product of probabilities to survive each non- confidentiality vote. If just one no confidence vote is supposed to be induced with a uniform distribution within the assumed interval between elections, the
probability, say a, for surviving it equals to the integral of the expression on the right hand side of (3). Then the expected life of this government will be evaluated by the expression
ju(a) = a—T +1 (1 - a)^T = —T (1 + a).
Complications under other assumptions are evident, but not worthless to be discussed.
5 CONCLUSIONS
Political processes offer interesting area of applications of various mathematical modeling approaches and theories. We discuss probability models by keeping close to the processes of formation of political units and activities to their natural components. The uncertain elements are naturally included into specific probabilistic relationships. The obtained analytical results produce promising particular characteristics, and offer a lot of field for discussions, statistical considerations and interesting applications.
6 PREFERENCES
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