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Rational Sabotage in Cooperative Production with
Heterogeneous Agents.
Carmen Beviá
y
Universitat Autònomade Barcelona-CODE
LuisC.Corchón
DepartamentodeEconomía. UniversidadCarlosIIIde Madrid
April, 2006.
Abstract: We present amodel ofcooperativeproduction in which rational agents
might carry out sabotage activities that decrease output. We provide necessary and
su¢ cient conditions for the existence of a Nash equilibrium without sabotage. It is
shown that the absence of sabotage in equilibrium depends on the interplay between
technology, relativeproductivityofagents and thedegreeofmeritocracy. In particular
we show that, ceteris paribus, meritocratic systems give more incentives to sabotage
than egalitarian systems.
Keywords: Cooperativeproduction, sharingrules, sabotage.
JEL Classi…cation Number(s): D20, D72, D78, J54.
We thank the Editor, , two referees, Francisco Alcala, Clara Eugenia Garcia, Shasikanta Nandeibam, Ines Macho-Stadler,
David Perez-Castrillo, and Clara Ponsati for their helpful comments. . The …rst t author acknowledges …nancial support from grants
SEJ2005-01581/ECON- FEDER, 2005SGR-00454 and the support of f the Barcelona Economics program (CREA). The second
author acknowledges …nancial l support from m BEC2002-02194.
y
Address for r correspondence: Carmen n Beviá. Universitat Autonoma de Barcelona. Edi…ci i B. 08193. Bellaterra, , Barcelona.
Spain. e-mail: Carmen.Bevia@uab.es s phone: : 935812190
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1. Introduction
In this paper we consider a model where rational agents have the option to carry out
sabotageactivities. Traditionally,sabotagehasbeenassociatedwithcapitalism-atleast
sincethedaysoftheLudditerevolt-and/orwithenvy(seeMui[1995]). Inthispaperwe
showthat sabotagecan ariseas arational action under cooperative production, where
outputisentirelydistributedtoworkers. Anexampleofhowsabotagemayariseinthis
framework follows.
Two people are collecting grapes. Andy collects white grapes -whose quantity is
denoted by R
1
andBethcollectsred grapes, whose quantity isdenoted byR
2
. These
grapes are transformed into wine -denoted by X  according to the production func-
tion X = (R
1
+R
2
)
1=2
. The quantity of wine allocated to each worker -C
1
and C
2
respectively- is given by theProportional Sharing Rule, i.e.
C
i
=
R
i
R
1
+R
2
(R
1
+R
2
)
1=2
; i=1;2:
Forfuturereferencewenoticethat this sharingruleis meritocratic, inthesensethatit
allocates wine dependingon relative inputs. Suppose that when an unexpected event
forces Beth toleave, R
1
=R
2
=50: Thus, X =10, C
1
=C
2
=5. Choices forAndy are
toremain faithfully devoted tocollectinggrapes, inwhich case hewould obtain,say,21
extra units or todestroythecrop assembled byBeth and pretend that somebody stole
it.
1
In the …rst case hisconsumption ofwineis
C
1
=
71
121
(121)
1=2
'6;45:
In thesecond case, Andy’sconsumption of wine is
C
1
=
50
50
(50)
1=2
'7;07:
1
Andycanexplainthathisoutputisjust50bysayingthathespenttheremainingtimechasingthe
thief.
2
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Therefore,ifAndyisrationalandonlycaresabouthiswineconsumption,hewilldestroy
Beth’scrop. Supposenowthat thesharingruleis Egalitarian, i.e.
C
i
=
(R
1
+R
2
)
1=2
2
; i=1;2:
We notice that this rule is not meritocratic at all, in the sense that it allocates wine
irrespective of relative inputs. In this case, faithful work yields toAndy C
1
=5:5 and
sabotageC
1
=3:5, i.e. sabotageis not arational action.
Whatisgoingonin thisexample? Whenan agentdecidestosabotagetherearetwo
e¤ects. On the one hand, output falls re‡ecting the decrease in the quantity of input
suppliedbyboththesaboteurand theagentwhohasbeensabotaged. Thisisbadfrom
the saboteur’s point of view because there is less to be distributed. We will call this
theProduction E¤ect. Noticethattheimportanceofthis e¤ectdependson thereturns
embodied in the production function. Since returns re‡ect the degree of congestion
amonginputs, themagnitude of the production e¤ect dependsinversely on the degree
ofcongestion. Ontheotherhand,therelativerankingofthesaboteurrises and thatis
good for him. We will call this the Distribution E¤ect. The importance of this e¤ect
dependsonhow meritocraticthesharingruleis; forinstance in theegalitarian sharing
rule this e¤ect does not exist. When the rule is meritocratic and there is congestion,
thedistribution e¤ect maydominateandsabotageisarational action,as inthecaseof
theproportional sharingruleabove.
2
The model of cooperative production is presented in Section 2. In order to make
the model tractable we make a number of simpli…cations. First we assume that the
totalquantityoflaborsupplied byeachagentis …xed. Thus, laborcan bespenton the
2
A related example may help to further understand this relationship. Suppose that a group of
athletesarerunninginanevent. Runnerscandevotetheirenergieseithertorunningortosteppingon
otherpeopleshoes,i.e.tosabotage.Onewouldexpectthattheoccurrenceofsabotagedependsonhow
e¤orts translateinto run/sabotage,howrunnersare rewarded-themoremeritocraticthereward,the
moresabotage-andthedegreeofcongestioni.e. ifrunnersformacompactpackortheyarescattered.
3
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production of an intermediate input (by exerting e¤ort) or on the destruction of the
inputsofother agents (sabotage). Thisassumption is madein ordertofocusattention
on the choice between productive and sabotage activities. It is appropriated when
length of working time is …xed exogenously by law, custom, etc. Second, we assume
that the production function and the sharing rule are symmetric in inputs. However
the model is not symmetric because agents have di¤erent productivities.
3
This paper
studies the existence and properties of Pareto E¢ cient Nash equilibria, which, under
our assumptions, istheNash equilibrium with no sabotage.
In Section 3 we present a necessary condition for sabotage not to arise in a Nash
equilibrium that says that foreach pairofagents, either therelativecapabilityofsab-
otage with respecttoproductiveactivities issmall relativetotherelativeproductivity,
or, thedegreeofmeritocracyshould bebounded. Thisbound depends on thedegreeof
congestion, therelativeproductivityofagents, therelativecapabilityofagentsbetween
sabotage and productive activities and the complementarity or substitutability among
inputs. This result is recorded as Proposition 1. We alsoshow howthis condition can
be simpli…ed becauseforagiven agent, sayi, theful…llmentofthe necessarycondition
with respect to another agent, say j, implies the ful…llment of thenecessary condition
ofi with respecttoanyagentlessproductivethan j (Proposition 2). However, the ful-
…llmentofthenecessaryconditionforsomeidoesnotimplytheful…llmentofnecessary
conditions for any other agent, see Example 1. In our Example 2 we explore the role
of the di¤erent variables in the necessary condition in the case where the production
function is of the CES type and the sharing rule is a convex combination of the pro-
portionaland theegalitarian sharingrules. Aninterestingimplication ofouranalysisis
that, for more than twoagents with di¤erent productivities, an increase in the degree
3
Agentsmustbeheterogeneousbecausewithidenticalagentsallsharingrulesyieldequaldistribution
so thequestion ofthedistributionof theoutput, does notarise. Moreover,given that theegalitarian
sharingruleyieldsincentivesnottosabotage,thissharingruleispreferrableto anyotherinthiscase.
4
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ofcomplementarityamonginputsdecreasesthelevelofmeritocracycompatiblewith no
sabotage. In the limit (perfect complementarity), the unique sharing rule compatible
with nosabotage istheegalitarian sharingrule.
In Section 4 we study the existence of aNash equilibrium with nosabotage. As it
happens withthenecessarycondition,wehavetwocases. Whenthecapabilitiesofsab-
otagewithrespecttoproductiveactivities aresmall,nosabotageisaNash equilibrium
(Proposition 3). Furthermore, when these relative capabilities are su¢ ciently small,
zerosabotageis theuniqueNash equilibrium (Proposition4). However, when theseca-
pabilitiesarenotsmallweneedanextraassumptionthatisstronger-butwiththesame
‡avor- than the necessary condition in order toguarantee that no sabotage is a Nash
equilibrium (Proposition 5). Example3shows thenecessity ofthis newassumption.
4
Finally, Section 5commentsourassumptions and suggests further research.
Let us now comment on other papers dealing with sabotage. As far as we know
Nalebu¤ and Stiglitz (1983) were the …rst to acknowledge that ”In the competitive
system.... there are... rewards from engaging in destructive activity”(id. p. 40).
5
If saboteurs are identical, we have the following contributions. In the case of a pro…t
maximizing…rm, Lazear(1989)showedthat, ifsabotageispossible,largedi¤erencesin
salariesbecome dysfunctional. In his model, agents arepaid according totheposition
achieved in acontest.
6
Auriol, Friebel and Pechlivanos, (1999) considered a Principal-
4
OurresultscontrastwithHolmstrom(1982)wheree¢ciencycannotbeachieved. Thereareseveral
di¤erences between our modeland Holmstrom’s,themost important being that in our modelagents
produceanintermediateinputthatiscontractible. AshasbeenshownbyNandeibam(2002)inamodel
wheresabotageisnotpossible,e¢ ciencymightariseinequilibriumiftherearecontractibleintermediate
inputs.Thus,ourresultsandNandeibam’sshowtheimportanceofcontractibleintermediateinputs.
5
Itoh(1991)andMacho-StadlerandPérez-Castrillo(1993)analyzethepolarcasewherecooperation
among agentsispossible.
6
Whensomeaspectsof performancearerewarded but othersarenot,other types ofdysfunctional
behaviorarepossible,seeHolmstromandMilgrom(1991). ThisliteratureissurveyedinGibbons(1998)
andPrendergast(1999).
5
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Agent model wherethe formercan not commit so shehas too¤er collectively oriented
incentive schemes in order to avoid sabotage. Konrad (2000) considered a model of
Rent-Seeking where the e¤ort of an agent reduces rival´s performance by sabotaging
her activities. He shows that, in equilibrium, sabotage disappears if the number of
agentsissu¢ cientlylarge. Heterogeneousagentshavebeenconsideredbythreedi¤erent
strandsofliterature: Experimental(Harbring,Irlenbusch,Krakël,andSelten[2004]and
Harbring and Irlenbusch [2005]), Management (Chen [2003] and Yumoto [2003]) and
Politicaleconomy(SkaperdasandGrofman[1995]). Allthesepapersidentifyconditions
underwhich some ofthe agents aremore likely toengage and/ortoreceivesabotage.
7
However,noneofthesepapersdealswiththemainfocusofourpaperwhichistoidentify
conditions on the sharingrulesuch thatsabotage doesnotarisein equilibrium.
In all the models referenced above agents competedirectly so the existenceof sab-
otage is quite natural. In amodel ofcooperativeproduction whereagentsatthesame
timecompete(the distribution e¤ect)and cooperate(the production e¤ect) things are
more involved. In our model the possibility of sabotage depends on: (1) The degree
of meritocracy (as in pro…t-maximizing …rms), (2) the relative productivity of agents
(under identical agents the relative productivity of an agent is just one divided by the
number of agents, hence the role of the later in rent-seeking models), (3) the relative
capability of agents between sabotage and productive activities, (4) the relationship
among inputs (complementarity or substitutability) and (5) the degree of congestion.
Thus,ouranalysisofthenecessaryandsu¢ cientconditionsforabsenceofsabotagepro-
duces apicturewhereall thecauses of sabotageconsidered before havearole and new
causes emerge, i.e. points (4) and (5) above which are associated with the technology
of the …rm.
7
ForinstanceSkaperdasandGrofman(1995)consideredamodelinwhichcandidatesmayfocusthe
campaignonpositiveornegativeaspects,theroleofthelatterbeingsimilartosabotage. Theyshowed
thatinequilibriumtheleading candidateislesslikelytoengageinnegativecampaigning.
6
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2. The Model
Thesetting is one of cooperative production, see, e.g., Sen (1966), Fabella (1988) and
Roemer and Silvestre (1993) for examples and applications. There are n agents. The
inputprovidedbyagentiisdenotedbyR
i
2R
+
:LetXbetotaloutput. Theproduction
functioniswrittenf(R
1
;::;R
n
)=F(
P
n
k=1
(R
k
)):Wewillassumethatf isC
1
,concave,
strictly increasing in all its arguments and   is homogeneous of degree  2 R with
1:This form generalizes the well known CES form. Total output is shared among
agentsbymeansofasharingrule,i.e. alistoffunctionss
i
:R
n
+
![0;1];i=1;::::;nsuch
that
P
n
i=1
s
i
(R
1
;::;R
n
)=1forall(R
1
;::;R
n
)2R
n
+
.Weassumethatforalli2f1;::;ng;
s
i
(R
1
;::;R
n
)=s(R
i
;
P
k6=i
R
k
); where s() is aC
1
function, non decreasing on R
i
;non
increasingin
P
k6=i
R
k
;homogeneousofdegreecero,andsuchthats(R
i
;
P
k6=i
R
k
)>0if
R
i
>0. Thehomogeneityassumption ensuresthatsharesdonotdepend on howinputs
aremeasured. An exampleofaclassofsharingrules ful…llingtheseconditionsis:
s
i
(R
1
;::;R
n
)=
R
i
P
n
k=1
R
k
+
1 
n
; 2[0;1];i =1;:::;n:
(2.1)
This family of sharing rules is parametrized by : If  = 0 we get the egalitarian
sharing rule and if  =1; we have the proportional sharing rule. The parameter  is
ameasure ofhowrelativee¤ortis valued and thusmeasuresthedegree ofmeritocracy.
Theinterested readercan …ndin Moulin (1987)andP…ngsten(1991)otherexamplesof
sharingrules ful…llingour conditions.
Agentscareonlyabouttheirownconsumption. AsweremarkedintheIntroduction,
the quantity of labor time is …xed. An agent, say i, can divide her working time,
denoted by T; between productive labor, denoted by l
P
i
and sabotage activities. Let
l
ij
be thequantity of labor allocated by i tosabotage the input ofagent j. The time
constraint reads T = l
P
i
+
P
j6=i
l
ij
: The input provided by agent i depends on her
own productive e¤ort, l
P
i
; the amount of time devoted by the remaining agents to
7
sabotaging the input of i and a parameter, 
i
> 0, re‡ecting the productivity of i:
Formally, R
i
=
i
R(l
P
i
;l
1i
;:::l
(i 1)i
;l
(i+1)i
;:::l
ni
)whereR() isaC
1
function such that
@R
@l
P
i
>0and
@R
@l
ji
<0:
Without lossof generality wewill assumethat
1
:::
n
. Given theseelements,we
describe the sabotage game as follows: foreach agent i, a strategy is thetime devoted
to sabotage activities, i.e. the vector l
i
= (l
i1
;l
i2
;l
i(i 1)
;l
i(i+1)
;l
in
): Time devoted to
productive activities is determined by the constraint l
P
i
= T  
P
j6=i
l
ij
: By l
i
we
denote the vector (l
1
;:::;l
i 1
;l
i+1
;::;l
n
): For each agent i; given a vector of strategies
(l
i
;l
i
); thepayo¤function is given by
i
(l
i
;l
i
)  s
i
(R
1
(l
i
;l
i
);:::;R
n
(l
i
;l
i
))f(R
1
(l
i
;l
i
);:::;R
n
(l
i
;l
i
)) where
R
j
(l
i
;l
i
)  
j
R(T  
X
j6=i
l
ji
;l
1j
;:::l
(j 1)j
;l
(j+1)j
;:::l
nj
); j =1;:::;n:
A Nash equilibrium of the sabotage game, denoted by NE, is a vector of strategies
(l
1
;::;l
n
)suchthat forallagent i; 
i
(l
i
;l
i
)
i
(l
0
i
;l
i
)forall l
0
i
:
We postpone until Section 4 the problem of the existence of a NE. In Section 3
belowwe concentrateon the implications ofthenecessary condition guaranteeingthat
noagenthasincentives toengageinsabotage.
3. A Necessary Condition for No Sabotage
If all working time is devoted to productive activities, (l
i
;l
i
)= (0;0): De…ne R
0
j
j
R(0;0) for agent j; and let R
0
denote the vector of inputs evaluated at the point of
zerosabotage, thatis,R
0
=(R
0
1
;:::;R
0
n
): Ifnoagenthas incentivetosabotagewhen all
other agents donot sabotage, it must be that8i;j;
@
i
(0;0)
@l
ij
0; where
@
i
(0;0)
@l
ij
= f(R
0
)(
@s
i
(R
0
)
@R
j
@R
j
(0;0)
@l
ij
@s
i
(R
0
)
@R
i
@R
i
(0;0)
@l
P
i
)
+s
i
(R
0
)(
@f(R
0
)
@R
j
@R
j
(0;0)
@l
ij
@f(R
0
)
@R
i
@R
i
(0;0)
@l
P
i
)
8
If this requirement isnotful…lled,all NE implysabotage. Let
M
ij
 
@R
j
(0;0)
@l
ij
@R
i
(0;0)
@lP
i
j
@R(0;0)
@l
ij
i
@R(0;0)
@lP
i
:
M
ij
isameasureof the relativeimpact ofchangein inputs induced byan in…nitesimal
reallocationofi’slaborfromproductivetosabotageactivitiestowardsagentj evaluated
atthepointofzerosabotage. Thus, M
ij
isameasureofthepowerofdestructionversus
production capabilities. From our assumptionsit follows thatM
ij
>0: Let
M 
@R(0;0)
@l
ij
@R(0;0)
@l
P
i
;
which isindependent ofi and j: Thus, we can write
M
ij
=
j
i
M:
By our assumptionson the production function,
@f(R
0
)
@R
j
=
@F(y
0
)
@y
0
(R
0
j
); where y
Xn
k=1
(R
k
):
Since  ishomogeneousofdegree;  
0
(R
0
j
)=
 1
j
0
(R(0;0)): Thus,
@f(R
0
)
@R
j
=
j
i
 1
@f(R
0
)
@R
i
:
Usingthede…nition ofM
ij
and dividingbyf(R
0
); the necessarycondition reads
@s
i
(R
0
)
@R
j
M
ij
@s
i
(R
0
)
@R
i
+s
i
(R
0
)
@f(R
0
)
@R
i
1
f(R0)
j
i
 1
M
ij
1)0:
Or,
@s
i
(R
0
)
@R
j
j
i
@s
i
(R
0
)
@R
i
+s
i
(R
0
)
@f(R
0
)
@R
i
1
f(R0)
j
i
M 1)0:
Letz
P
k6=i
R
k
:Becauseourassumptions on thesharingrules,
@s
i
(R
0
)
@R
j
=
@s(R
0
i
;z)
@z
;
9
and sincethesharingruleis homogenousof degreecero,
@s(R
0
i
;z)
@R
i
R
0
i
+
@s(R
0
i
;z)
@z
z=0:
Thus,
@s
i
(R
0
)
@R
j
=
@s
i
(R
0
)
@R
i
R
0
i
P
k6=i
R
0
k
=
@s
i
(R
0
)
@R
i
i
P
k6=i
k
:
Usingthisrelation, the …rst order condition can bewritten as
(
j
P
k6=i
k
M 1)
@s
i
(R
0
)
@R
i
1
s
i
(R
0
)
@f(R
0
)
@R
i
1
f(R
0
)
(
j
i
M+1):
MultiplyingbyR
0
i
;the aboveinequalityimplies that
(
j
P
k6=i
k
M 1)
@s
i
(R
0
)
@R
i
R
0
i
s
i
(R0)
@f(R
0
)
@R
i
R
0
i
f(R0)
(
j
i
M+1):
(3.1)
Nowwe havetwo cases. IfM 
P
k6=i
k
j
,theinequality (3.1) always holds. However, if
M>
P
k6=i
k
j
;
@s
i
(R
0
)
@R
i
R
0
i
s
i
(R0)
@f(R
0
)
@R
i
R
0
i
f(R0)

j
i
M+1
0
B
@
P
k6=i
k
j
P
k6=i
k
1
C
A
:
(3.2)
Next Proposition summarizesourpreviousarguments:
Proposition1. If zero sabotage is a Nash equilibrium of the sabotage game, then
either
1. The possibilities ofdestructionare small compared with therelativeproductivityof
all otheragents, i.e., M 
P
k6=i
k
=
j
;or
2. if M >
P
k6=i
k
=
j
;in the point of zero sabotage, the elasticity of the share
with respect to the input of agent i is bounded by the elasticity of the production
functionwithrespecttotheinputofagenti;multiplied byafactorthatdependsonthe
possibilitiesof destruction and therelativeproductivityof agents (see (3.2) above).
10
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