The probability of the correct majority made decision

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Abstract


Aim: the probability of correctness of the collegial decision, which is made by a majority vote of some collective (board), consisting of an odd number of members is investigated, if the probability of correctness of the individual decision of each member of Board is known.

Мaterials and methods: Bernoulli scheme, asymptotic representation, estimation via geometric series, power series expansion, the formula of Wallis, a power scale of averages, average of Kolmogorov.

Result: it is established, that if for each member of the board the probability of correctness of the individual decision is more than ½, then with an unlimited increase in the number of members of the Board the probability of correctness of the collegial decision tends to 1. The asymptotic representation and a number of bilateral estimates characterizing the speed of this aspiration are obtained. For heterogeneous Board (that is a Board, whose members make the right individual decision with different probability) introduced the concept of collegial average as an average characteristics, which can replaced the individual probability of each member of the board with the preservation of the probability of a collegial decision. The existence and uniqueness of the collegial average are proved.

We derive a collegial inequality showing that the collegial average of some a set of numbers is not less than the geometric average of the same numbers with the equality takes place in the case and only if all the numbers are equal to each other. The collegial inequality serves as an analogue and complement to known set of inequalities establishing a connection between different averages (for example, Cauchy inequality for arithmetic average and geometric average).

Conclusion: thus, the results of the study fully meet the aim of determining the probability of correctness of collegial decision taken by a majority of votes under the assumptions. As a result we obtain an asymptotic representation and bilateral estimates characterizing the rate of striving for the correct solution. For a heterogeneous board, the existence uniqueness of the concept of collegial average as an average characteristics is introduced and strictly proved, which can be replaced by an individual probability of each with preserving the probability of correctness of the collegial decision. It is established that the collegial average is not less than the geometric average. Possible applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors of various transport systems, including high-speed transport systems on magnetic suspension.


INTRODUCTION

With the aim to increase the reliability of responsible decisions, the group decision making is used. For instance, significant calculations are tasked to several individuals, and the result is considered to be reliable if it is the same at all the individuals. Another example is obtaining experiment findings. In case of discrepancies in measuring sensors’ weak signals values, the preference is given to the majority’s data. This approach rests on the hypothesis that the decision that is made by the majority vote of some group (group of experts) is more correct than that of each expert’s individually. The major aim of this work is to verify this hypothesis and assess quantitatively the collectiveness effect.

MAIN NOTIONS AND TASK SETTING

Let one assume that the group consists of (2n+1) experts (for short - (2n+1) -group) and is characterised by vector , where  – probability that k-th expert takes the right decision. Then, by the law of total probability [1], the probability that the majority vote results in the right decision, is determined by the expression

 

ПX=k=0ni1,...ik1,...,  2n+11xi1...1xikx1...x2n+1/xi1...xik. (1)

 

In the expression (1) the k-th number of the outer sum is the probability of the incorrect decision by k experts, and the sum is the probability that the incorrect decision has been made by less than half of the experts. If the fraction in (1) loses its literal sense as a result of some components equaling 0, it should be considered equal to multiplication of all components except for .

If X = (p,..., p), let the group be called homogeneous (a good one, if p > 0,5, and a bad one if p < 0,5). For this type of group, the expression (1) turns into  the Bernoulli scheme [2]: ПX=P2n+1=k=0nC2n+1kp2n+1kqk, where q =1– p. Further on, it is more convenient to deal with the probability of mistake Q2n+1=1–P2n+1, i.e. the probability that the group makes the incorrect decision. It is obvious,

 

Q2n+1=k=0nC2n+1kpkq2n+1k. (2)

HOMOGENEOUS GROUP

Two-sided assessment of probability of mistake

Let the group be considered to be a good one. If the multiplication q(pq)n is put outside the brackets, from (2) Q2n+1=qpqn k=0nC2n+1kq/pnk. is obtained. With p > 0,5 it follows that q/pnq/pnk1. This implies that qpqnq/pnk=0nC2n+1kQ2n+1qpqnk=0nC2n+1k. Considering that  k=0nC2n+1k=0,5k=02n+1C2n+1k=22n [3], and making other obvious simplifications, we obtain

q4q2nQ2n+1q4pqn (3)

 

Let us introduce the parameter α = 2p –1. It is obvious that p =0,5(1+α), q = 0,5(1–α), whereby for a good group  and for a bad one  Expressing p and q through a, the inequality (3) can be converted to

 

q1α2nQ2n+1q1α2n. (4)

 

For a good group 0 ≤ (1–α)2 < (1–α2) < 1, means that the probability of mistake is assessed from both sides by infinitely decreasing geometric progressions. In particular, with unlimited growth in the number of experts, the probability of mistake tends towards 0.

Recurrence relations, decrease of probability of mistake

(2n+1)-group is a combination of (2n–1)-group and a pair of experts. Let be the probability that (2n–1)-group have made the correct (incorrect) decision dominating by one vote. (2n+1)-group makes the incorrect decision in the following cases:

1) (2n–1)-group have voted incorrectly by a margin of more than one vote; the probability of this equals Q2n-1q1;

2) (2n–1)-group have voted incorrectly by a margin of one vote, whereas a pair of experts has at least one that voted incorrectly; the probability of this event equals q1(1–p2);

3) (2n–1)-group have voted correctly by a margin of one vote, and a pair of experts has two that have voted incorrectly; the probability of this event equals p1q2.

Adding these probabilities together and considering that p1=C2n1npnqn1, q1=C2n1n1pn1qn, p=0,51+αq=0,5(1α) and C2n1n=C2n1n1=0,5C2nn, we can obtain

 

Q2n+1Q2n1=α/2C2nn1α2/4n. (5)

 

For a good group (0< α ≤ 1) the right side (5) is negative, wherefrom it can be seen that the sequence Q2n+1 is a decreasing one, i.e. adding an additional pair of experts results in the decrease of probability of mistake.

Economical calculation formula

Having recorded Q2n+1 as Q1+ (Q3 Q1) +...+ (Q2n+1 Q2n-1), putting subtractions (Q2k+1 Q2k–1) from (5) and considering that Q1= q = 0,5(1–α), we can obtain for Q2n+1 the following representation:

Q2n+1=1/2α/2k=0nC2kk1α2/4  k=1/2α/2k=0nAkxk, (6)

where x=1α2, Ak=C2kk/4k.

 

Let us also consider that, in accordance with [4] Ak=C2kk/4k=2k1!!/2k!!.

As n increases by 1, there is an additional of one term added to the addition in (6), and the already existing terms remain unchanged. In (2) as n increases, all the terms are changed. Therefore, the formula (6) is more economical for numerical calculations.

Asymptotic representation of the probability of mistake at n→∞

Considering the last representation for Ak, k=0nAkxk is a partial sum of binominal series with the facto r–1/2 [5]. Then k=0nAkxk=1x1/2=1/α, wherefrom k=0nAkxk=1/αk=n+1Akxk. Putting this expression to (6) and putting the first member of the addition outside the brackets, we can obtain the following representation:

 

Q2n+1=α/2An+1xn+1k=0Ak+n+1/An+1  xk. (7)

Let us prove that at n →∞ the sum in (7) tends towards k=0xk, i.e. we will assess the subtraction of these sums. According to Wallis’ product [6]

 

1/πn+1/2  <An  <1/πn. (8)

 

From this above, we can easily obtain: Ak+n/An  >n/k+n+1/2, wherefrom we have 1Ak+n/An<1n/k+n+1/2. The right side is less than the value  consequently, 0 < 1– Ak+n+1/An+1< (k+1/2)/2(n+1). This means that

 

0<k=0xkk=0Ak+n+1/An+1xk=k=01Ak+n+1/An+1xk<k=0k+1/2/2n+1xk=1/2n+1k=0kxk+1/2k=0xk. (9)

 

where k=0xk=1/1x (the sum of geometric progression [7]); k=0kxk=x/1x2 [8]; wherefrom, considering x=1α2, we have: k=0xk=1/α2, k=0kxk=1α2/α4. Adding this to (9) and expressing k=0Ak+n+1/An+1xk in the resultant inequality , we can obtain:

1/α211/2n+11/α21/2<k=0Ak+n+1/An+1xk<1/α2. (10)

The sum in (10) differs from Q2n+1 by the multiplier (α/2) An+1 xn+1 (see. (7)). Multiplying (10) by this multiplier and assessing An+1 from both sides by virtue of (8), we can obtain the following two-sided assessment for Q2n+1:

 

1α2n+1/2απn+1/2  11/2n+11/α21/2<Q2n+1<1α2n+1/2απn. (11)

 

It is obvious that the second multiplier in the left part is 1+О(1/n). Therefore, the following asymptotic representation follows from (11):

 

Q2n+1=1α2n+1/2απn1+О1/n. (12)

 

Let us compare the assessments (4) and (11). It is seen that the second one is asymptotically more precise, i.e. for each fixed α > 0 there exists n, starting from which the range of change Q2n+1, in (11) is more narrow than the range in (4). However, the assessment (4) has the advantage that it is even along α, whereas in (11) the range expands without limits at α→ 0. Therefore, the relevance of this or that assessment for certain calculations depends on numerical values of input data.

COLLECTIVE AVERAGE

Let us return to (2n+1)-group of the general form, which is characterised by vector X= (x1,..., x2n+1). Let us select such a homogeneous (2n+1)-group that has the same probability of the correct decision just the first one does. The value p of this homogeneous group (the probability of the right individual decision of its member) can be interpreted as an average value for a set of probabilities x1,..., x2n+1. Let them be referred to as a collective average of the values x1,..., x2n+1. In accordance with this definition, the collective average p of values x1,..., x2n+1 is a root of the expression

 

ПX=P2n+1p, (13)

lying in the interval 0 ≤ p ≤ 1, where P(X) is determined by formula (1), P2n+1p=Пp,p...p=k=0nC2n+1kp2n+1k1pk.

Existence and uniqueness of collective average

Let us study the monotony of Π(X) by each variable, e.g. x2n+1. Let us say that 2n-group of experts with numbers 1, …, 2n have voted by a margin of k, if the right votes have been given by k more experts than incorrect ones. The margin acquires odd values from от –2n to 2n. With k ≠ 0 the decision of (2n+1)-group does not depend on the decision of (2n+1) expert, and with  it coincides with it. It means that Π(X) =R2n+...+ R2+R0 x2n+1, where Rk – the probability of margin k. The probabilities R2n,..., R0 do not depend on x2n+1, and R0 ≥0, therefore Π(X) is either strictly increasing function (at R0 ≥ 0), or a constant one (at R0 = 0). The latter takes place when over half of the members of 2n-group vote unanimously, and therefore their votes cannot be divided equally. In other words, among x1,..., x2n there are more n components equalling 0, or more n, equalling 1. In the first case Π(X) =0, in the second Π(X) =1.

Thus, since P2n+1(p) = Π(p, p,...p), then P2n+1(p) strictly increases at the interval (0; 1), where it is different from 0 and 1. But P2n+1(0) = 0, P2n+1(1) =1. Thereby, P2n+1(x) strictly increases by [0; 1] from 0 to 1. It is uninterrupted, therefore the equation (13) has the sole solution [9].

Note: if in the set (x1,..., x2n+1) there are zeros or ones, for a collective average one of the properties of the Kolmogorov mean may be breached [10]: replacement of values of any subset in the set (x1,..., x2n+1) for the average value of this subset does not change the average value of the whole set. Therefore, for the collective average, generally speaking, the universal representation by Kolmogorov does not take place [10, 11] in the form , where φ is some strictly monotonic function.

COLLECTIVE INEQUALITY  

Further on, it will be proved that the collective average is more than or equal to the geometric mean:

 

pg=x1...xm1/m, где m=2n+1. (14)

 

Let us start with special cases g = 0 и g = 1. In both cases, the inequality (14) is obvious, therefore it may be further assumed that 0 < g < 1. This implies that all x1,..., x2n+1 are different from 0 and not all of them are equal to 1.

Due to strict increase of the function P2n+1 (14)  P2n+1(p) ≥ P2n+1(g). But P2n+1(p) =P(X) (13) and P2n+1(g) =P(G), where G=(g,...,g). Thus, (14)  Π(X) ≥ Π(G). And this inequality, considering exclusion of special cases, is equipotent to the statement  , which is exactly what needs proving.

 Let g0;1, and A is a set of vectors, the ones, that, and x1∙...∙xm= gm. In this set, the function Π(X) reaches the lowest value at X=G, where G=(g,..., g).

Note: due to equality of x1∙...∙xm= gm it is sufficient to prove that all components of the vector, delivering the minimum of П(X), are equal between each other.

Let us proceed to the proof. Expressing xm from the condition x1∙...∙xm= gm and putting it to Π(X), we acquire rational and fractional function of the variables x1,..., xm–1, an uninterrupted one since its denominator x1∙...∙xm–1 differs from 0 due to 0 < g < 1. And for all . Thus, the function is determined at the compactum [12], meaning it reaches the minimal value [13]. Consequently, the vector constituting the minimum Π(X), exists. Now, let us prove that it cannot have irregular components.

Let us consider the vector XA, that has more than n components equalling 1. For this Π(X)=1, whereas Π(G) < 1 (because g < 1). It means that the vector, which constitutes the minimum of the function Π(X), has no more than n components equalling 1. Let X be such a vector, and let not all of its components be equal to each other. Without limiting the generality, it can be assumed that x1x2. Let us prove that then the value of Π(X) is not the lowest at A.

Since all the components are different from 0, the fraction in (1) can be perceived literally and the multiplication of all components x1...x2n+1=gm. can be put outside the brackets. Now, ПX=gmk=0ni1,...ik1,...,mti11...tik1, where tj=1/xj, and the term of the outer sum, corresponding to k=0, equals 1, and the vector T = = (t1,..., tm) possesses the following properties:

1) t1t2;

2) among t1,..., tm no more than n units equal 1;

3) the vector T delivers the function ΘT=k=0ni1,...ik1,...,  mti11...tik1 the minimum at the set of vectors, which have tj1, and t1...tm=rm>1, where r=1/g.

Let us be view Θ (T) as the function of t1, t2 at constancy of the other variables. Separating the terms and co-multipliers, containing variables t1, t2, expressed by Θ (T) one can convert them to

 

ΘT=C2t1t2+C1C2t1+t2+C0, (15)

 

where coefficients C1, C2, C0 are not dependent on t1, t2, and

 

C1=k=0n=1i1,...ik3,...,  mti11...tik1, C2=k=0n2i1,...ik3,...,  mti11...tik1. (16)

 

For further work, the signs of coefficients in (16) are significant at t1·t2 and t1+ t2. In the sum for C2 all the terms are non-negative, with one of them being equal to 1. It means, C2 > 0. Let us prove that C1C2 > 0. In accordance with (16) the outer sum for C1 differs from the outer sum for C2 on additional term with the number k = n –1. Therefore,  as the sum of non-negative terms. Let us assume that this sum equals 0. Then all the variables are equal to 0, i.e. any multiplication of , where , is equal to 0. Thus among the subtractions and  there are no more than n–2 that are different from 0, and it means there are at least 2n –1– (n–2) = n+1 equalling 0. Or, which is the same, among the components t3,...,tm there are n+1, equalling 1. It is especially true for a full set of components t1, ..., tm. However, this contradicts the above-mentioned property 2) of the vector T. So, C1C2 > 0.

Now, let us change t1 and t2 leaving the rest of the variables constant. Then the multiplication t1t2 = r m/(t3∙...∙tm) will also be constant, meaning in the expression (16) only the second term will be changed, which equals (with the precision of up to the positive multiplier) t1+c/t1, where c = r m/(t3∙...∙ tm). The sum t1+c/t1 acquires the minimal value at  [14]. But then as well, i.e. . Meanwhile, as to the assumption,, it means we have found the vector different from the initial one, at which the function  acquires the lesser value. This contradicts the assumption that the initial vector delivers the function Θ the minimum.

Returning to the initial variables x1,...,tm, we obtain the statement (*). Indeed, the vector that has not all components equal to each other, is not a vector which delivers the function Π(X) minimum. But this vector does exist, and it means that this is a vector having different components. And this is nothing more like the vector G. Thus the collective inequality has been proved.

Let us add that due to the well-known properties of the power scale of averages [15], it additionally follows from this inequality that the collective average is no less than any power mean with non-positive value.

The notions of the collective average and collective inequality introduced here provide simplified lower bound assessment of the quality of decisions by the non-homogeneous group. The probability of its correct decision is no less than that of the homogeneous group comprising the same number of experts, where the probability of the correct individual decision by one expert equals the geometric mean of similar probabilities in the initial non-homogeneous group.

CONCLUSION

The expressions (6, 11, 12 и 14–16) fully resolve the issue stated to determine the probability of the right majority made decision, with the assumptions accepted. The asymptotic representation and two-sided assessment, which, characterise the speed of tending towards the right decision.

For a non-homogeneous group, the existence and uniqueness of the concept of collective average as an averaged characteristic were introduced and firmly proved, which can be used to replace an individual probability of each group member, whereby preserving the probability of correctness of the collective decision

Potential applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors in various transport systems, including high-speed transport systems on magnetic suspension [16-17].

Konstantin E. Voevodskii

St. Petersburg State University

Email: kv5832@mail.ru
ORCID iD: 0000-0002-0519-5527
SPIN-code: 2579-7541

Russian Federation, 7/9, Universitetskaya embankment, Saint-Petersburg, 199034

Ph.D., assistant professor

Vladimir M. Strepetov

Emperor Alexander I St. Petersburg State Transport University

Author for correspondence.
Email: strepetovvm@mail.ru
ORCID iD: 0000-0002-4072-4519
SPIN-code: 4649-2141

Russian Federation, 9 Moskovsky ave., St. Petersburg, 190031

Ph.D., assistant professor, Department "Theoretical Foundations of Electrical Engineering"

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