Redundancy is a commonly employed strategy to enhance the reliability of systems involving human factors. However, this paper shows that a system with higher level of redundancy can be less reliable due to free-rider effects.

Lian Cheng

English

Research Papers in Economics

Redundancy and Reliability
Lian Cheng
Institute of Quantitative & Technical Economics, Chinese Academy of Social Sciences
11 July 2005
Abstract
Redundancy is a commonly employed strategy to enhance the reliability of systems
involving human factors. However, this paper shows that a system with higher level of
redundancy can be less reliable due to free-rider e⁄ects.
Keywords: redundancy; reliability; free-rider e⁄ect
JEL classi￿cation: D23; D82
1. Introduction
Redundancy is a commonly employed strategy to enhance the reliability of sys-
tems involving human factors. For example, in an organization we often see two or
more sta⁄s are assigned to a key operation, although anyone of them can complete
the task independently. And whenever people show concern for the reliability of
the operation, a natural reaction is to send more sta⁄s there. However, as some
authors argued (e.g., Sagan, 2004), higher level of redundancy may actually lead
to less reliability. This article builds a formal model to support this view. The
intuition behind is that the reliability of a redundant system depends on two el-
ements: the number of the parallel subunits and the possibility that one of them
fail to work. Unfortunately, in a redundant system involving human factors the
measure to increase the former often also increases the later because of the free-
rider e⁄ect. Whether increasing the level of redundancy enhances the reliability
depends on which e⁄ect dominates.Redundancy and Reliability 2
2. The Model
Consider a team whose task is to guard a system against possible errors. Each
team member monitors the system independently. An unfound error will cause a
crash of the system. And if that happens, each team member will be punished.
The team has n members and the object function of the ith team member is
￿0￿1￿2￿￿￿￿i￿￿￿￿n￿1￿nB + c(￿i) (1)
, where ￿0 is the possibility that there is an error in the system and ￿i is the
possibility that the ith member does not ￿nd the error (individual fail possibility).
B is the punishment for a system crash. c(￿i) is the cost of the ith member
associated with his e⁄ort devoted to system monitoring (individual monitoring
cost) and
c0(￿i) < 0; c00(￿i) > 0; c(1) = 0; c(0) = +1
.
Using the ￿rst order condition for minimizing 1
￿0￿1￿2￿￿￿￿i￿1￿i+1￿￿￿￿n￿1￿nB + c0(￿i) = 0 (2)
, we get a unique symmetric equilibrium where every team member chooses an
optimal individual fail possibility ￿
￿:
￿0￿
￿n￿1B + c0(￿
￿) = 0 (3)
. Taking the derivative of ￿
￿ with respect to n yields the relationship between theRedundancy and Reliability 3
individual fail possibility and the number of team members
d￿
￿
dn
=
ln￿
￿
c00(￿￿)
c0(￿￿) ￿ (n ￿ 1)￿
￿￿1 (4)
. Since ￿
￿ 2 (0;1);
d￿
￿
dn > 0 and we have
Lemma 1 The individual fail possibility increases with the number of team mem-
bers.
Lemma 1 implies a free-rider e⁄ect: as more people join the team, the risk of
system crash on the current e⁄ort level deceases, so each team member reduces
his e⁄ort and relies more on other members to detect the error.
Now we turn to the impact of increasing the number of team member on the
possibility of a system crash
d(￿0￿
￿n)
dn . Using equation 3, we can write the derivative
as
d[￿
￿c
0(￿
￿)=B]
dn and then 1
B ￿
d[￿
￿c
0(￿
￿)]
d￿￿ ￿
d￿
￿
dn . As
d￿
￿
dn > 0, its sign depends only on
that of
d[￿
￿c
0(￿
￿)]
d￿￿ , thus
sign
￿
d(￿0￿
￿n)
dn
￿
= sign
￿
1 +
c00(￿
￿)
c0(￿
￿)
￿
￿
￿
(5)
. The right side of the equation is negative if and only if
c
00(￿
￿)
c0(￿￿) ￿
￿ < ￿1, or
￿
￿ ￿
c
00(￿
￿)
c0(￿￿) ￿
￿
￿
￿ ￿ > 1. So we have
Proposition 2 The possibility of a system crash decreases with the number of
team members only when the elasticity of marginal individual monitoring cost with
respect to individual fail possibility is larger than 1.
The intuition for Proposition 2 is straightforward. The addition of new team
members will unavoidably decrease the e⁄ort of every member, but if a little relax-Redundancy and Reliability 4
ation of e⁄ort can alleviate the individual monitoring cost enough, this negative
e⁄ect will not be too large to keep the original level of system reliability.
3. Conclusion
In this paper we show that a system with higher level of redundancy can be
less reliable due to free-rider e⁄ects. This result implies that increasing the level
of redundancy may be not an e¢ cient strategy to ensure reliability, even without
the consideration of budget constraint.
4. References
Sagan, S. D., 2004. "The problem of redundancy problem: will more nuclear
security forces produce more nuclear security?" Risk Analysis. 24, 935-946.

The problem of redundancy problem: will more nuclear security forces produce more nuclear security?&quot; Risk Analysis.

Redundancy and Reliability

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