Houba, H.E.D.

Wen, Q.

DSpace at VU

On The Power to Hurt: Costly Conflict with Completely
Informed States 
Harold Houba Quan Wen
Vrije Universiteit and Tinbergen Institute Vanderbilt University
Abstract
Slantchev (2003, American Political Science Review, 97) studies a class of negotiation
models to explain costly conflict between two completely informed nations. In one of his
main propositions (Proposition 2.3), Slantchev provides a strategy profile to support the
so−called extremal subgame perfect equilibrium, where one nation receives its lowest
equilibrium payoff. By means of a counter example, we demonstrate the existence of an
equilibrium with one nation's payoffs below the strategy profile provided in his Proposition
2.3 (Case 2).
The authors are indebted to Gerard van der Laan and Michiel Keyzer for their support. Quan Wen is grateful to the Tinbergen
Institute and Centre for World Food Studies for their hospitality and generosity. Harold Houba conducted large part of this
research while being affiliated to the Centre for World Food Studies at the Vrije Universiteit.
Citation: Houba, Harold and Quan Wen, (2006) "On The Power to Hurt: Costly Conflict with Completely Informed States."
Economics Bulletin, Vol. 3, No. 3 pp. 1−6
Submitted: December 22, 2005.  Accepted: January 25, 2006.
URL: http://www.economicsbulletin.com/2006/volume3/EB−05C70039A.pdf
1 Introduction
In a recent article, Slantchev (2003) discusses the merits of analyzing warfare as a negotia-
tion process between two nations in the shadow of ghting. Wielding actions of warfare is
an intangible part of such processes that may inuence the nal resolution. This negotiation
process is modeled as a bargaining game with endogenous threats, also known as the nego-
tiation model pioneered by Haller and Holden (1990), Fernandez and Glazer (1991), Haller
(1991), Bolt (1995) and Houba and Bolt (2000) in the context of wage contract negotia-
tions and followed by Busch and Wen (1995) and Houba (1997) in a general context of the
disagreement game. Slantchev (2003) analyzes a class of negotiation games where there is
one disagreement outcome dominating any other disagreement outcome while the two na-
tions have di¤erent time preferences that are modelled as discount factors. When discount
factors are su¢ ciently large, this negotiation model generally admits many equilibrium out-
comes featuring costly ghting before a delayed peace treaty can be reached, despite many
restrictions to stack the model against ghting. The strategy proles supporting those
equilibria, however, critically depend upon every nations worst or extremal equilibrium.
These extremal equilibria represent a nations worst fear of what might happen if it would
unilaterally cease re and, therefore, what keeps nations trapped in continuing the warfare.
The purpose of this note is to alert the scientic community of the complications involved
with extremal SPE in the negotiation model when two nations have di¤erent time preferences.
For that purpose, we critically re-examine a claim in Slantchev (2003) about the extremal
equilibrium: By means of a simple counter example, we demonstrate that the supposedly
extremal strategy prole provided in Proposition 2.3 (Case 2) of Slantchev (2003) is not
necessarily the extremal equilibrium to the nation with a lower discount factor, no matter
how close the two discount factors are to 1. The proof of Proposition 2.3 in Slantchev (2003)
merely veries that the stated strategy prole constitutes a subgame perfect equilibrium
(SPE). The strategy prole extends the extremal SPE when the two nations have the same
discount factor, with appropriate modications to the case where the two nations di¤er in
their discount factors. Under the framework of Slantchevs model, our counter example
demonstrates a simple SPE where the nation with a lower discount factor receives less than
from what Slantchev claims to be this nation extremal SPE.
Our counter example is presented in the next section and a concluding remark is put in
a separate section.
2 The Counter Example
We now provide an example to demonstrate that the SPE provided in Part 2 of Proposition
2.3 in Slantchev (2003) fails to be the extremal SPE to the nation whose discount factor is
lower than the other nations, no matter how small both discount factors are. Consider the
warfare negotiation model with the following symmetric disagreement game:
1
1 n 2 L M R
T 1
2
; 1
2
0; 0 0; 0
M 0; 0 0; 0 0; 
B 0; 0 ; 0 0; 0
where 0   < 1
2
. As constructed, this disagreement game has one Nash equilibrium a =
(T; L) that Pareto dominates any other disagreement outcome and it is on the bargaining
frontier. In order to reduce the number of equilibrium conditions, we assume three other
Nash equilibria: (B;R), (M;R) and (B;M), of which Nash equilibrium (M;R) turns out
to be crucial. This disagreement game satises all the assumptions imposed in Slantchev
(2003).
In order to support nation 1s extremal SPE, Slantchev suggests to maximize nation
2s disagreement payo¤ by describing (T; L) in all odd periods and minimize nation 1s dis-
agreement payo¤by describing (B;R) in all even periods. Since both disagreement outcomes,
(T; L) and (B;R), are Nash equilibria in the disagreement game, a SPE can be derived based
on the innite sequence of alternating between these two disagreement outcomes. Proposi-
tion 2.3 of Slantchev (2003) identies the following two equilibria:
A. When 1  2, both nations will make acceptable proposals. Accordingly, nation 1
receives 1
2
1 2
1 12 in any odd period and
1
2
1(1 2)
1 12 in any even period.
B. When 1 < 2, only nation 2 makes an acceptable proposal. Accordingly, nation 1
receives 1
2
1
1+1
in any odd period and 1
2
1
1+1
in any even period.
It is easy to see why nation 1 behaves di¤erently in these two cases. Given the alternating
disagreement outcomes, nation 1 can secure a payo¤ of 1
2
1
1+1
in any odd period and 1
2
1
1+1
in
any even period from simply collecting its alternating disagreement payo¤s 1
2
and 0 forever,
called the no-concession strategy by Bolt (1995) in the wage negotiation model. Nation 1
will make an acceptable proposal if and only if
1
2
1  2
1  12 
1
2
1
1 + 1
, 1  2: (1)
Slantchev (2003) claims that Proposition 2.3 characterizes nation 1s extremal SPE in these
two cases.
Now we present a SPE strategy prole from which nation 1 receives strictly less than
1
2
1
1+1
in any odd period and 1
2
1
1+1
in any even period for most of 1 < 2. Consider the
following strategy prole:
C. Both nations make acceptable proposals that are calculated from the constant sequence
of disagreement outcomes (M;R) in all periods, and the two nations would play the
Nash equilibrium (M;R) in any disagreement game.
It is obvious that no nation has an incentive to deviate in the disagreement game. Unlike
the SPE of described by strategy prole B, forever collecting the disagreement payo¤s yields
nation 1 a payo¤of 0. Suppose that nation 1 proposes (x1; 1 x1) and nation 2 proposes (1 
x2; x2). After rejecting a proposal, the responding nation will receive his disagreement payo¤
2
during the current period and his equilibrium share in the following period. A responding
nation should be indi¤erent between accepting and rejecting the standing o¤er:
1  x1 = (1  2) + 2x2 and 1  x2 = 1x1;
which yield nation 1s payo¤s in an odd period and an even period as
x1 =
1  2
1  12 (1  ) and 1  x2 =
1(1  2)
1  12 (1  ) : (2)
If nation 1 makes an unacceptable proposal in an o¤er period, its payo¤will be 1 (1  x2) =
21x1 < x1. Therefore, nation 1 cannot benet from the no-concession strategy in strategy
prole C. This establishes that the strategy prole C is a SPE.
Now the issue is whether x1 and 1x1 in (2) are less than nation 1s SPE payo¤s predicted
by Proposition 2.3 of Slantchev (2003). First observe that
x1 =
1  2
1  12 (1  ) >
1
2
1  2
1  12
for all  2 [0; 1
2
). This implies that when 1  2, the SPE resulted from strategy prole A
yields nation 1 a lower payo¤ then our SPE of strategy prole C. When 1 < 2, however,
we have
x1 =
1  2
1  12 (1  ) <
1
2
1
1 + 1
; (3)
which yields
 >   1
2
  1
2
2   1
(1 + 1) (1  2) :
Obviously, 1 < 2 implies 
 < 1
2
. In other words, when 1 < 2, (3) holds for all  2
 
; 1
2

.
Moreover, it is even possible to obtain  < 0, which requires 2 > 1+212+1 which is larger than
1. This last result means that for a large set of parameter values 1 < 2, no value of  can
be found such that the SPE of strategy prole B is nation 1s worst SPE.
Figure 1 illustrates the region of 1 and 2 for which the strategy prole C yields nation 1
a lower equilibrium payo¤ than the SPE resulted from strategy prole B. Since this requires
that nation 2 is relatively more patient than nation 1, it follows that strategy prole A is
not a SPE when 1 < 2. The curved boundary of region C depends upon  and this curve
shifts downward as  increases with the forty-ve degree line at the border case  = 1
2
. So,
region C expands as  increases and covers about one quarter to half of the unit square. So,
region C is quite large.
3 Concluding Remark
The key implication of our examples is that when nations have di¤erent time preferences,
the current game theoretic literature fails to characterize the lowest equilibrium payo¤s, not
only in the warfare negotiation model of Slantchev (2003), but also in the general negotiation
model of Busch andWen (1995). Thus far, only a few have analyzed di¤erent time preferences
3
-6
1
2
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
1
1
3
1
2
 = 0
 = 14  =
1
2
C
Figure 1: Condition (3) for  = 0,  = 1
4
and  = 1
2
. The region where the strategy prole
C yields nation 1 a payo¤ lower than strategy prole B is indicated by C.
in the negotiation model (often for some restrictive class of disagreement games). The
characterization of extremal equilibrium payo¤s requires a thorough analysis of equilibrium
bounds using the method of Shaked and Sutton (1984), see Osborne and Rubinstein (1990),
Muthoo (1999) or Houba and Bolt (2002) for surveys. However, since strategy proles may
involve unacceptable proposals it also implies that the method of Shaked and Sutton (1984)
should also be adapted to allow for such deliberate delays.
It is worthwhile to point out that our nding in this note does not change the qualitative
aspect of the main message of Slantchev (2003): The existence of equilibria with wasteful
ghting and delayed peace treaties. Since these equilibrium strategies revert to each nations
worst or extremal equilibrium as the most e¤ective punishments, downward modications in
the nationsextremal equilibrium payo¤s imply a weakening of conditions under which such
equilibria can be sustained in equilibrium. We hope our example raises the awareness of the
complications involved with extremal SPE in the negotiation model when two nations have
di¤erent time preferences.
References
Bolt, W. (1995). Striking for a bargain between two completely informed agents: Com-
ment. American Economic Review 85, 13441347.
Busch, L.-A. and Q. Wen (1995). Perfect equilibria in a negotiation model. Economet-
rica 63, 545565.
Fernandez, R. and J. Glazer (1991). Striking for a bargain between two completely in-
formed agents. American Economic Review 81, 240252.
4
Haller, H. (1991). Wage bargaining as a strategic game. In R. Selten (Ed.), Game theoretic
equilibrium models III: Strategic bargaining, pp. 230241. Springer Verlag:Berlin.
Haller, H. and S. Holden (1990). A letter to the editor on wage bargaining. Journal of
Economic Theory 52, 232236.
Houba, H. (1997). The policy bargaining model. Journal of Mathematical Economics 28,
127.
Houba, H. and W. Bolt (2000). Holdouts, backdating and wage negotiations. European
Economic Review 44, 17831800.
Houba, H. andW. Bolt (2002). Credible threats in negotiations; A game theoretic approach.
Theory and Decision Library. Series C: Game Theory Mathematical Programming and
Operations Research, vol. 32. Kluwer Academic: Norwell, Mass. and Dordrecht.
Muthoo, A. (1999). Bargaining theory with applications. Cambridge: Cambridge Univer-
sity Press.
Osborne, M. and A. Rubinstein (1990). Bargaining and markets. Academic Press: Boston.
Shaked, A. and J. Sutton (1984). Involuntary unemployment as a perfect equilibrium in
a bargaining model. Econometrica 62, 13511364.
Slantchev, B. (2003). The power to hurt: costly conict with completely informed states.
American Political Science Review 97, 123133.
5


On The Power to hurt: Costly Conflict with Completely Informed States

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VU Research PortalOn The Power to hurt: Costly Conflict with Completely Informed StatesHouba, H.E.D.; Wen, Q.published inEconomics Bulletin2006Link to publication in VU Research Portalcitation for published version (APA)Houba, H. E. D., & Wen, Q. (2006). On The Power to hurt: Costly Conflict with Completely Informed States.Economics Bulletin, 3, 1-6. http://www.accessecon.com/pubs/eb/2006/volume3/EB-05C70039A.pdfGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 22. May. 2021On The Power to Hurt: Costly Conflict with CompletelyInformed States Harold Houba Quan WenVrije Universiteit and Tinbergen Institute Vanderbilt UniversityAbstractSlantchev (2003, American Political Science Review, 97) studies a class of negotiationmodels to explain costly conflict between two completely informed nations. In one of hismain propositions (Proposition 2.3), Slantchev provides a strategy profile to support theso−called extremal subgame perfect equilibrium, where one nation receives its lowestequilibrium payoff. By means of a counter example, we demonstrate the existence of anequilibrium with one nation's payoffs below the strategy profile provided in his Proposition2.3 (Case 2).The authors are indebted to Gerard van der Laan and Michiel Keyzer for their support. Quan Wen is grateful to the TinbergenInstitute and Centre for World Food Studies for their hospitality and generosity. Harold Houba conducted large part of thisresearch while being affiliated to the Centre for World Food Studies at the Vrije Universiteit.Citation: Houba, Harold and Quan Wen, (2006) "On The Power to Hurt: Costly Conflict with Completely Informed States."Economics Bulletin, Vol. 3, No. 3 pp. 1−6Submitted: December 22, 2005.  Accepted: January 25, 2006.URL: http://www.economicsbulletin.com/2006/volume3/EB−05C70039A.pdf1 IntroductionIn a recent article, Slantchev (2003) discusses the merits of analyzing warfare as a negotia-tion process between two nations in the shadow of ghting. Wielding actions of warfare isan intangible part of such processes that may inuence the nal resolution. This negotiationprocess is modeled as a bargaining game with endogenous threats, also known as the nego-tiation model pioneered by Haller and Holden (1990), Fernandez and Glazer (1991), Haller(1991), Bolt (1995) and Houba and Bolt (2000) in the context of wage contract negotia-tions and followed by Busch and Wen (1995) and Houba (1997) in a general context of thedisagreement game. Slantchev (2003) analyzes a class of negotiation games where there isone disagreement outcome dominating any other disagreement outcome while the two na-tions have di¤erent time preferences that are modelled as discount factors. When discountfactors are su¢ ciently large, this negotiation model generally admits many equilibrium out-comes featuring costly ghting before a delayed peace treaty can be reached, despite manyrestrictions to stack the model against ghting. The strategy proles supporting thoseequilibria, however, critically depend upon every nations worst or extremal equilibrium.These extremal equilibria represent a nations worst fear of what might happen if it wouldunilaterally cease re and, therefore, what keeps nations trapped in continuing the warfare.The purpose of this note is to alert the scientic community of the complications involvedwith extremal SPE in the negotiation model when two nations have di¤erent time preferences.For that purpose, we critically re-examine a claim in Slantchev (2003) about the extremalequilibrium: By means of a simple counter example, we demonstrate that the supposedlyextremal strategy prole provided in Proposition 2.3 (Case 2) of Slantchev (2003) is notnecessarily the extremal equilibrium to the nation with a lower discount factor, no matterhow close the two discount factors are to 1. The proof of Proposition 2.3 in Slantchev (2003)merely veries that the stated strategy prole constitutes a subgame perfect equilibrium(SPE). The strategy prole extends the extremal SPE when the two nations have the samediscount factor, with appropriate modications to the case where the two nations di¤er intheir discount factors. Under the framework of Slantchevs model, our counter exampledemonstrates a simple SPE where the nation with a lower discount factor receives less thanfrom what Slantchev claims to be this nation extremal SPE.Our counter example is presented in the next section and a concluding remark is put ina separate section.2 The Counter ExampleWe now provide an example to demonstrate that the SPE provided in Part 2 of Proposition2.3 in Slantchev (2003) fails to be the extremal SPE to the nation whose discount factor islower than the other nations, no matter how small both discount factors are. Consider thewarfare negotiation model with the following symmetric disagreement game:11 n 2 L M RT 12; 120; 0 0; 0M 0; 0 0; 0 0; B 0; 0 ; 0 0; 0where 0   < 12. As constructed, this disagreement game has one Nash equilibrium a =(T; L) that Pareto dominates any other disagreement outcome and it is on the bargainingfrontier. In order to reduce the number of equilibrium conditions, we assume three otherNash equilibria: (B;R), (M;R) and (B;M), of which Nash equilibrium (M;R) turns outto be crucial. This disagreement game satises all the assumptions imposed in Slantchev(2003).In order to support nation 1s extremal SPE, Slantchev suggests to maximize nation2s disagreement payo¤ by describing (T; L) in all odd periods and minimize nation 1s dis-agreement payo¤by describing (B;R) in all even periods. Since both disagreement outcomes,(T; L) and (B;R), are Nash equilibria in the disagreement game, a SPE can be derived basedon the innite sequence of alternating between these two disagreement outcomes. Proposi-tion 2.3 of Slantchev (2003) identies the following two equilibria:A. When 1  2, both nations will make acceptable proposals. Accordingly, nation 1receives 121 21 12 in any odd period and121(1 2)1 12 in any even period.B. When 1 < 2, only nation 2 makes an acceptable proposal. Accordingly, nation 1receives 1211+1in any odd period and 1211+1in any even period.It is easy to see why nation 1 behaves di¤erently in these two cases. Given the alternatingdisagreement outcomes, nation 1 can secure a payo¤ of 1211+1in any odd period and 1211+1inany even period from simply collecting its alternating disagreement payo¤s 12and 0 forever,called the no-concession strategy by Bolt (1995) in the wage negotiation model. Nation 1will make an acceptable proposal if and only if121  21  12 1211 + 1, 1  2: (1)Slantchev (2003) claims that Proposition 2.3 characterizes nation 1s extremal SPE in thesetwo cases.Now we present a SPE strategy prole from which nation 1 receives strictly less than1211+1in any odd period and 1211+1in any even period for most of 1 < 2. Consider thefollowing strategy prole:C. Both nations make acceptable proposals that are calculated from the constant sequenceof disagreement outcomes (M;R) in all periods, and the two nations would play theNash equilibrium (M;R) in any disagreement game.It is obvious that no nation has an incentive to deviate in the disagreement game. Unlikethe SPE of described by strategy prole B, forever collecting the disagreement payo¤s yieldsnation 1 a payo¤of 0. Suppose that nation 1 proposes (x1; 1 x1) and nation 2 proposes (1 x2; x2). After rejecting a proposal, the responding nation will receive his disagreement payo¤2during the current period and his equilibrium share in the following period. A respondingnation should be indi¤erent between accepting and rejecting the standing o¤er:1  x1 = (1  2) + 2x2 and 1  x2 = 1x1;which yield nation 1s payo¤s in an odd period and an even period asx1 =1  21  12(1  ) and 1  x2 =1(1  2)1  12(1  ) : (2)If nation 1 makes an unacceptable proposal in an o¤er period, its payo¤will be 1 (1  x2) =21x1 < x1. Therefore, nation 1 cannot benet from the no-concession strategy in strategyprole C. This establishes that the strategy prole C is a SPE.Now the issue is whether x1 and 1x1 in (2) are less than nation 1s SPE payo¤s predictedby Proposition 2.3 of Slantchev (2003). First observe thatx1 =1  21  12(1  ) > 121  21  12for all  2 [0; 12). This implies that when 1  2, the SPE resulted from strategy prole Ayields nation 1 a lower payo¤ then our SPE of strategy prole C. When 1 < 2, however,we havex1 =1  21  12(1  ) < 1211 + 1; (3)which yields >   12  122   1(1 + 1) (1  2):Obviously, 1 < 2 implies  < 12. In other words, when 1 < 2, (3) holds for all  2 ; 12.Moreover, it is even possible to obtain  < 0, which requires 2 > 1+212+1 which is larger than1. This last result means that for a large set of parameter values 1 < 2, no value of  canbe found such that the SPE of strategy prole B is nation 1s worst SPE.Figure 1 illustrates the region of 1 and 2 for which the strategy prole C yields nation 1a lower equilibrium payo¤ than the SPE resulted from strategy prole B. Since this requiresthat nation 2 is relatively more patient than nation 1, it follows that strategy prole A isnot a SPE when 1 < 2. The curved boundary of region C depends upon  and this curveshifts downward as  increases with the forty-ve degree line at the border case  = 12. So,region C expands as  increases and covers about one quarter to half of the unit square. So,region C is quite large.3 Concluding RemarkThe key implication of our examples is that when nations have di¤erent time preferences,the current game theoretic literature fails to characterize the lowest equilibrium payo¤s, notonly in the warfare negotiation model of Slantchev (2003), but also in the general negotiationmodel of Busch andWen (1995). Thus far, only a few have analyzed di¤erent time preferences3-6120                  111312 = 0 = 14  =12CFigure 1: Condition (3) for  = 0,  = 14and  = 12. The region where the strategy proleC yields nation 1 a payo¤ lower than strategy prole B is indicated by C.in the negotiation model (often for some restrictive class of disagreement games). Thecharacterization of extremal equilibrium payo¤s requires a thorough analysis of equilibriumbounds using the method of Shaked and Sutton (1984), see Osborne and Rubinstein (1990),Muthoo (1999) or Houba and Bolt (2002) for surveys. However, since strategy proles mayinvolve unacceptable proposals it also implies that the method of Shaked and Sutton (1984)should also be adapted to allow for such deliberate delays.It is worthwhile to point out that our nding in this note does not change the qualitativeaspect of the main message of Slantchev (2003): The existence of equilibria with wastefulghting and delayed peace treaties. Since these equilibrium strategies revert to each nationsworst or extremal equilibrium as the most e¤ective punishments, downward modications inthe nationsextremal equilibrium payo¤s imply a weakening of conditions under which suchequilibria can be sustained in equilibrium. We hope our example raises the awareness of thecomplications involved with extremal SPE in the negotiation model when two nations havedi¤erent time preferences.ReferencesBolt, W. (1995). Striking for a bargain between two completely informed agents: Com-ment. American Economic Review 85, 13441347.Busch, L.-A. and Q. Wen (1995). Perfect equilibria in a negotiation model. Economet-rica 63, 545565.Fernandez, R. and J. Glazer (1991). Striking for a bargain between two completely in-formed agents. American Economic Review 81, 240252.4Haller, H. (1991). Wage bargaining as a strategic game. In R. Selten (Ed.), Game theoreticequilibrium models III: Strategic bargaining, pp. 230241. Springer Verlag:Berlin.Haller, H. and S. Holden (1990). A letter to the editor on wage bargaining. Journal ofEconomic Theory 52, 232236.Houba, H. (1997). The policy bargaining model. Journal of Mathematical Economics 28,127.Houba, H. and W. Bolt (2000). Holdouts, backdating and wage negotiations. EuropeanEconomic Review 44, 17831800.Houba, H. andW. Bolt (2002). Credible threats in negotiations; A game theoretic approach.Theory and Decision Library. Series C: Game Theory Mathematical Programming andOperations Research, vol. 32. Kluwer Academic: Norwell, Mass. and Dordrecht.Muthoo, A. (1999). Bargaining theory with applications. Cambridge: Cambridge Univer-sity Press.Osborne, M. and A. Rubinstein (1990). Bargaining and markets. Academic Press: Boston.Shaked, A. and J. Sutton (1984). Involuntary unemployment as a perfect equilibrium ina bargaining model. Econometrica 62, 13511364.Slantchev, B. (2003). The power to hurt: costly conict with completely informed states.American Political Science Review 97, 123133.5

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