Weighted voting systems: reliability versus rapidity - Semantic Scholar
Weighted voting systems: reliability versus rapidity Gregory Levitin Reliability Department, Planning, Development and Technology Division, Bait Amir, Israel Electric Corporation Ltd., P.O. Box 10, Haifa, 31000 Israel E-mail: [email protected] Abstract The weighted voting system (WVS) consists of n units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise, system output is 0. The system input is either 0 or 1. Every unit is characterized by probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if its output is not equal to its input. This paper shows that if the WVS consists of units that need different time to produce their outputs, the decision time of the entire system depends on the distribution of unit weights and on the value of τ. It shows also that a tradeoff exists between the system reliability and its rapidity. An algorithm that finds the system parameters maximizing its reliability under constraint imposed on the expected system decision time is suggested. Illustrative examples are presented.
Notation
τ
threshold factor, (0≤τ≤1)
n
number of units belonging to WVS
K
number of different voting results in WVS
I
WVS input (proposition to be accepted or rejected), I∈{0,1}
Pr{v} probability of event v pi
Pr{I = i}
dj(I)
decision (output) of individual unit j 1
D(I)
decision of the entire WVS
qis(j)
Pr{dj(I) = s | I = i}, i∈{0,1}, s∈{0,1,x}
Qis
Pr{D(I) = s | I = i}, i∈{0,1}, s∈{0,1,x}
R
Pr{D(I) = I}, WVS reliability
U Im (z ) U-function representing all the possible different states of a subsystem containing m fastest
voting units when system input is I w0j
"negative" weight of unit j (weight of unit j when dj(I)=0)
w1j
"positive" weight of unit j (weight of unit j when dj(I)=1)
tj
time when the decision dj(I) of voting unit j becomes available
T
expected WVS decision time
T*
maximum allowed expected decision time
Wm1
total weight of units that have decision time not greater than tm and vote for the proposition acceptance
Wm0
total weight of units that have decision time not greater than tm and vote for the proposition rejection
h Im0
probability that the proposition I is rejected at time tm
h Im1
probability that the proposition I is accepted at time tm
1(x)
unity function: 1(TRUE)=1, 1(FALSE)=0
Acronyms1 WVS weighted voting system UF
1
U(z)-function
The singular & plural forms of acronyms are always spelled the same. 2
1. Introduction Voting systems (k-out-of-n systems with multiple failure modes) are widely used in human organization systems as well as in technical decision making systems. The reliability of these systems has been studied in [1-6]. The weighted voting systems (WVS) are generalization of the voting systems. WVS are intensively studied in recent years [7-12]. The applications of WVS can be found in imprecise data handling [13], safety monitoring and self-testing [14], multi-channel signal processing [15], pattern recognition and target detection [7], etc. A WVS makes a decision about propositions based on the decisions of n statistically independent individual units of which it consists (for example, in target detecting system speed detectors and heat radiation detectors provide the system with their individual decisions without communicating among themselves). Each proposition is a priori right or wrong but this information is available for the units in implicit form. Therefore the units are subject to the following three errors: 1. Acceptance of a proposition that should be rejected (fault of being too optimistic), 2. Rejection of a proposition that should be accepted (fault of being too pessimistic), 3. Abstaining from voting (fault of being unavailable or indecisive). This can be modeled by considering system input I being either 1 (proposition to be accepted) or 0 (proposition to be rejected) which is supplied to each unit. Each unit j produces its decision (unit output) dj(I) which can be 1, 0 or x (in the case of abstention). Inequality dj(I)≠I means that the decision made by the unit is wrong. The listed above errors can be expressed as 1. dj(0)=1 (unit fails stuck-at-1), 2. dj(1)=0 (unit fails stuck-at-0), 3. dj(I)=x (unit fails stuck-at-x). Accordingly, reliability of each unit j can be characterized by probabilities of these errors: q01(j) for the first one, q10(j) for the second one, q1x(j) and q0x(j) for the third one (note that stuck-at-x probabilities can be different for inputs I=0 and I=1). 3
In this paper we consider an asymmetric WVS (first suggested in [9]) which is able to take advantage of knowledge about statistical asymmetry of voting units (asymmetric probabilities of making correct decisions with respect to the input I) and therefore has greater reliability than the symmetric WVS. In such system each voting unit j has two weights that express its relative importance in the WVS: "negative" weight w0j which is assigned to the unit when it votes for the proposition rejection and "positive" weight w1j which is assigned to the unit when it votes for the proposition acceptance. To make a decision about proposition acceptance, the system incorporates all the unit decisions into a unanimous system output D (Fig. 1). The proposition is rejected by the WVS (D(I)=0) if the total weight of units voting for its acceptance is less than a pre-specified fraction τ of total weight of not abstaining units (τ is usually referred to as threshold factor). The WVS abstains (D(I)=x) if all of its voting units abstain. D(I)
τ w01 w02 w03 w04 w05 w06 w 1 w12 w13 w14 w15 w16 1
d1(I) unit 1
d2(I) unit 2
d3(I)
d4(I)
unit 3
d5(I)
unit 4
d6(I) unit 5
unit 6
I
Fig. 1. Example of asymmetric WVS with n=6.
The system fails if D(I)≠I. The entire WVS reliability can be defined as R=Pr{D(I)=I}. One can see that the system reliability is a function of reliabilities of units it consists of. The reliability characteristics of WVS units as well as the propositions probability distribution can be elicited from historical statistics. In technical systems probabilities of different kinds of errors can be obtained for each unit with high precision by intensive testing. The entire WVS reliability also depends on the unit weights and the threshold. While the units' reliabilities usually can not be changed when the
4
WVS is built, the weights and the threshold can be chosen in such a way that maximizes the entire WVS reliability: (w01,w11,…, w0n,w1n,τ)=arg{R(w01,w11,…, w0n,w1n,τ)→max}.
(1)
The algorithm for WVS optimization according to this formulation is presented in Ref. [9]. The previous studies of WVS don't address the aspect of the system rapidity in making decisions. The first paper that considers the decision time factor is by Xie and Pham [11]. In this paper it is assumed that the probabilities of making mistakes decreases with the time for each voting unit but the cost of decision making by the entire system increases with the time. An algorithm for finding the optimal stopping time (the time when all the units are obliged to vote) is suggested. This model is relevant for human organizations in which the requirement of a limited time to complete the report (to make the decision) affects the results of the experts' evaluations together with their expertise. In many technical systems the time when the output (decision) of each voting unit is available is predetermined. For example, the decision time of a chemical analyzer is determined by the time of a chemical reaction. The decision time of a target detection radar system is determined by the time of the radio signal return and by the time of signal processing by the electronic subsystem. In both these cases the variation of the decision times is usually negligible small. On the contrary, the decision time of the entire WVS composed from voting units with different constant decision times can vary. As it was mentioned in [11], the system does not need to wait for decisions of slow voting units, as long, as the system can make a correct decision with reliability higher than a pre-specified level. Moreover, in some cases the decisions of the slow voting units do not affect the decision of the entire system since this decision becomes evident after the fast units have voted. This happens when the total weight of units voting for the proposition acceptance or rejection is enough to guarantee the system decision independently of the decisions of the units that have not voted yet. In such situations the voting process can be terminated without waiting for slow units' decisions and the WVS decision can be made in a shorter time. 5
The number of combinations of unit decisions that allow the entire system decision to be obtained before the outputs of all of the units become available depends on the unit weights distribution and on the threshold value. By increasing the weights of the fastest units one makes the WVS more decisive in the initial stage of voting and therefore reduces the mean system decision time by the price of making it less reliable. In applications where the WVS should make many decisions in a limited time the expected system decision time is considered to be a measure of its performance. Since the units' weights and the threshold affect both the WVS's reliability and its expected decision time, the problem of the optimal system turning can be formulated as follows: find the voting units' weights and the threshold that maximize the system reliability R while providing the expected decision time T not greater than a pre-specified value T*: (w01,w11,…, w0n,w1n,τ) = arg{R(w01,w11,…, w0n,w1n,τ) → max| T(w01,w11,…, w0n,w1n,τ) ≤ T*}.(2) This paper presents an algorithm for solving this optimization problem. Section 2 of the paper presents the voting model. Section 3 describes an algorithm for reliability and performance evaluation. Section 4 is devoted to optimization technique.
2. The weighted voting model Consider the WVS consisting of n voting units characterized by error probabilities q01(j), q10(j), q1x(j) , q0x(j) and decision times tj. The WVS incorporates all the unit outputs into a unanimous system output D using the following threshold based decision rule:
⎧ ⎪1, if ⎪ ⎪ D( I ) = ⎨0, if ⎪ ⎪ x, if ⎪⎩
∑
w1j d j ( I ) ≥ τ ∑ [ w1j d j ( I ) + w0j (1 − d j ( I ))],
∑
w1j d j ( I ) < τ ∑ [ w1j d j ( I ) + w0j (1 − d j ( I ))],
∑
wj = 0
d j ( I )≠ x
d j ( I )≠ x d j ( I )≠ x
d j ( I )≠ x
d j ( I )≠ x
∑
wj ≠ 0
∑
wj ≠ 0
d j ( I )≠ x
d j ( I )≠ x
(3)
Let us order the units such that tj ≤ tj+1 for 1 ≤ j ≤ n-1 and define the total weight of WVS units that have the decision times not greater than tm and support the proposition I as Wm1: 6
m
Wm1 = ∑ w1j 1(d j ( I ) = 1)
(4)
j =1
and the total weight of WVS units that have the decision times not greater than tm and reject the proposition as Wm0: m
Wm0 = ∑ w1j 1(d j ( I ) = 0) .
(5)
j =1
The decision rule (3) can now be rewritten as follows:
⎧1, if W n1 ≥ τ (W n1 + W n0 ), W n1 + W n0 ≠ 0 ⎪⎪ D( I ) = ⎨0, if W n1 < τ (W n1 + W n0 ), W n1 + W n0 ≠ 0 ⎪ 1 0 ⎪⎩ x, if W n + W n = 0.
(6)
Following this expression the condition that D(I)=0 can be rewritten as Wn1 < τ( Wn1+ Wn0)
(7)
(1-τ)Wn1-τ Wn0
Notation
τ
threshold factor, (0≤τ≤1)
n
number of units belonging to WVS
K
number of different voting results in WVS
I
WVS input (proposition to be accepted or rejected), I∈{0,1}
Pr{v} probability of event v pi
Pr{I = i}
dj(I)
decision (output) of individual unit j 1
D(I)
decision of the entire WVS
qis(j)
Pr{dj(I) = s | I = i}, i∈{0,1}, s∈{0,1,x}
Qis
Pr{D(I) = s | I = i}, i∈{0,1}, s∈{0,1,x}
R
Pr{D(I) = I}, WVS reliability
U Im (z ) U-function representing all the possible different states of a subsystem containing m fastest
voting units when system input is I w0j
"negative" weight of unit j (weight of unit j when dj(I)=0)
w1j
"positive" weight of unit j (weight of unit j when dj(I)=1)
tj
time when the decision dj(I) of voting unit j becomes available
T
expected WVS decision time
T*
maximum allowed expected decision time
Wm1
total weight of units that have decision time not greater than tm and vote for the proposition acceptance
Wm0
total weight of units that have decision time not greater than tm and vote for the proposition rejection
h Im0
probability that the proposition I is rejected at time tm
h Im1
probability that the proposition I is accepted at time tm
1(x)
unity function: 1(TRUE)=1, 1(FALSE)=0
Acronyms1 WVS weighted voting system UF
1
U(z)-function
The singular & plural forms of acronyms are always spelled the same. 2
1. Introduction Voting systems (k-out-of-n systems with multiple failure modes) are widely used in human organization systems as well as in technical decision making systems. The reliability of these systems has been studied in [1-6]. The weighted voting systems (WVS) are generalization of the voting systems. WVS are intensively studied in recent years [7-12]. The applications of WVS can be found in imprecise data handling [13], safety monitoring and self-testing [14], multi-channel signal processing [15], pattern recognition and target detection [7], etc. A WVS makes a decision about propositions based on the decisions of n statistically independent individual units of which it consists (for example, in target detecting system speed detectors and heat radiation detectors provide the system with their individual decisions without communicating among themselves). Each proposition is a priori right or wrong but this information is available for the units in implicit form. Therefore the units are subject to the following three errors: 1. Acceptance of a proposition that should be rejected (fault of being too optimistic), 2. Rejection of a proposition that should be accepted (fault of being too pessimistic), 3. Abstaining from voting (fault of being unavailable or indecisive). This can be modeled by considering system input I being either 1 (proposition to be accepted) or 0 (proposition to be rejected) which is supplied to each unit. Each unit j produces its decision (unit output) dj(I) which can be 1, 0 or x (in the case of abstention). Inequality dj(I)≠I means that the decision made by the unit is wrong. The listed above errors can be expressed as 1. dj(0)=1 (unit fails stuck-at-1), 2. dj(1)=0 (unit fails stuck-at-0), 3. dj(I)=x (unit fails stuck-at-x). Accordingly, reliability of each unit j can be characterized by probabilities of these errors: q01(j) for the first one, q10(j) for the second one, q1x(j) and q0x(j) for the third one (note that stuck-at-x probabilities can be different for inputs I=0 and I=1). 3
In this paper we consider an asymmetric WVS (first suggested in [9]) which is able to take advantage of knowledge about statistical asymmetry of voting units (asymmetric probabilities of making correct decisions with respect to the input I) and therefore has greater reliability than the symmetric WVS. In such system each voting unit j has two weights that express its relative importance in the WVS: "negative" weight w0j which is assigned to the unit when it votes for the proposition rejection and "positive" weight w1j which is assigned to the unit when it votes for the proposition acceptance. To make a decision about proposition acceptance, the system incorporates all the unit decisions into a unanimous system output D (Fig. 1). The proposition is rejected by the WVS (D(I)=0) if the total weight of units voting for its acceptance is less than a pre-specified fraction τ of total weight of not abstaining units (τ is usually referred to as threshold factor). The WVS abstains (D(I)=x) if all of its voting units abstain. D(I)
τ w01 w02 w03 w04 w05 w06 w 1 w12 w13 w14 w15 w16 1
d1(I) unit 1
d2(I) unit 2
d3(I)
d4(I)
unit 3
d5(I)
unit 4
d6(I) unit 5
unit 6
I
Fig. 1. Example of asymmetric WVS with n=6.
The system fails if D(I)≠I. The entire WVS reliability can be defined as R=Pr{D(I)=I}. One can see that the system reliability is a function of reliabilities of units it consists of. The reliability characteristics of WVS units as well as the propositions probability distribution can be elicited from historical statistics. In technical systems probabilities of different kinds of errors can be obtained for each unit with high precision by intensive testing. The entire WVS reliability also depends on the unit weights and the threshold. While the units' reliabilities usually can not be changed when the
4
WVS is built, the weights and the threshold can be chosen in such a way that maximizes the entire WVS reliability: (w01,w11,…, w0n,w1n,τ)=arg{R(w01,w11,…, w0n,w1n,τ)→max}.
(1)
The algorithm for WVS optimization according to this formulation is presented in Ref. [9]. The previous studies of WVS don't address the aspect of the system rapidity in making decisions. The first paper that considers the decision time factor is by Xie and Pham [11]. In this paper it is assumed that the probabilities of making mistakes decreases with the time for each voting unit but the cost of decision making by the entire system increases with the time. An algorithm for finding the optimal stopping time (the time when all the units are obliged to vote) is suggested. This model is relevant for human organizations in which the requirement of a limited time to complete the report (to make the decision) affects the results of the experts' evaluations together with their expertise. In many technical systems the time when the output (decision) of each voting unit is available is predetermined. For example, the decision time of a chemical analyzer is determined by the time of a chemical reaction. The decision time of a target detection radar system is determined by the time of the radio signal return and by the time of signal processing by the electronic subsystem. In both these cases the variation of the decision times is usually negligible small. On the contrary, the decision time of the entire WVS composed from voting units with different constant decision times can vary. As it was mentioned in [11], the system does not need to wait for decisions of slow voting units, as long, as the system can make a correct decision with reliability higher than a pre-specified level. Moreover, in some cases the decisions of the slow voting units do not affect the decision of the entire system since this decision becomes evident after the fast units have voted. This happens when the total weight of units voting for the proposition acceptance or rejection is enough to guarantee the system decision independently of the decisions of the units that have not voted yet. In such situations the voting process can be terminated without waiting for slow units' decisions and the WVS decision can be made in a shorter time. 5
The number of combinations of unit decisions that allow the entire system decision to be obtained before the outputs of all of the units become available depends on the unit weights distribution and on the threshold value. By increasing the weights of the fastest units one makes the WVS more decisive in the initial stage of voting and therefore reduces the mean system decision time by the price of making it less reliable. In applications where the WVS should make many decisions in a limited time the expected system decision time is considered to be a measure of its performance. Since the units' weights and the threshold affect both the WVS's reliability and its expected decision time, the problem of the optimal system turning can be formulated as follows: find the voting units' weights and the threshold that maximize the system reliability R while providing the expected decision time T not greater than a pre-specified value T*: (w01,w11,…, w0n,w1n,τ) = arg{R(w01,w11,…, w0n,w1n,τ) → max| T(w01,w11,…, w0n,w1n,τ) ≤ T*}.(2) This paper presents an algorithm for solving this optimization problem. Section 2 of the paper presents the voting model. Section 3 describes an algorithm for reliability and performance evaluation. Section 4 is devoted to optimization technique.
2. The weighted voting model Consider the WVS consisting of n voting units characterized by error probabilities q01(j), q10(j), q1x(j) , q0x(j) and decision times tj. The WVS incorporates all the unit outputs into a unanimous system output D using the following threshold based decision rule:
⎧ ⎪1, if ⎪ ⎪ D( I ) = ⎨0, if ⎪ ⎪ x, if ⎪⎩
∑
w1j d j ( I ) ≥ τ ∑ [ w1j d j ( I ) + w0j (1 − d j ( I ))],
∑
w1j d j ( I ) < τ ∑ [ w1j d j ( I ) + w0j (1 − d j ( I ))],
∑
wj = 0
d j ( I )≠ x
d j ( I )≠ x d j ( I )≠ x
d j ( I )≠ x
d j ( I )≠ x
∑
wj ≠ 0
∑
wj ≠ 0
d j ( I )≠ x
d j ( I )≠ x
(3)
Let us order the units such that tj ≤ tj+1 for 1 ≤ j ≤ n-1 and define the total weight of WVS units that have the decision times not greater than tm and support the proposition I as Wm1: 6
m
Wm1 = ∑ w1j 1(d j ( I ) = 1)
(4)
j =1
and the total weight of WVS units that have the decision times not greater than tm and reject the proposition as Wm0: m
Wm0 = ∑ w1j 1(d j ( I ) = 0) .
(5)
j =1
The decision rule (3) can now be rewritten as follows:
⎧1, if W n1 ≥ τ (W n1 + W n0 ), W n1 + W n0 ≠ 0 ⎪⎪ D( I ) = ⎨0, if W n1 < τ (W n1 + W n0 ), W n1 + W n0 ≠ 0 ⎪ 1 0 ⎪⎩ x, if W n + W n = 0.
(6)
Following this expression the condition that D(I)=0 can be rewritten as Wn1 < τ( Wn1+ Wn0)
(7)
(1-τ)Wn1-τ Wn0
