equilibria and stability in a model of political and social con⁄ict. · 2019-12-03 · konrad kai...

Equilibria andStability in AModel of

Political andSocialConict.

FaustoCavalli*,Mario Gilli*and AhmadNaimzada*

Equilibria and Stability in A Model of Politicaland Social Conict.The Symmetric Case

Fausto Cavalli*, Mario Gilli* and Ahmad Naimzada*

*Department of Economics, University of Milan-Bicocca.

February 2015VERY PRELIMINARY AND INCOMPLETE




The Model ofPolitical andSocial Conict

Restrictionson Utilitiesand ConictTechnology

Specic Casesof Models ofPolitical andSocial ConictCase 1: lineartechnology,linear costs andno externalitiesCase 1-A: zeroxed costsCase 1-A withConvex orConcave CostFunctionCase 1-B: nonzero xed costsCase 2: ConictTechnology WithExternalityCase 2-A: noxed costsCase 2-B: xedcostsCase 3:PolarizationWith ExternalityCase 3-A: noxed costs

The Limits ofthe Model

Interpretationof Results

Conclusion

The structure of the presentation

1 Introduction:

1 stylized facts2 the general research program3 specic questions addressed in this paper4 reference literature5 the contribution of the paper

2 The model

3 The cases

4 Interpretation of the results

5 Conclusion









Conclusion

Introduction - 1Stylized facts

Real conicts (wars, political, social, interpersonal) arecharacterized by

1 sudden surge2 huge diversity in evolution through time and in outcomes.

Why?

Two possible explanations:

1 structural change in the environment , comparativestatics, which "requires" uniqueness of equilibria

2 dynamics out of equilibrium , stability, which "requires"multiplicity of equilibria to be interesting









Conclusion

Introduction - 2Research Program - 1

GENERAL PROBLEM: Why actual conicts present so manydi¤erent situations and evolution through time?Can we provide an explanation of the two commoncharacteristics of actual conicts, i.e. that

1 very often they came as a surprise

2 their evolution through time is very heterogenous,sometimes they quickly disappear sometimes theydegenerate in catastrophe









Conclusion

Introduction - 3Research Program - 2

SPECIFIC PROBLEM:What are the e¤ects of di¤erent

1 conict technologies

2 costs functions

3 agents goals on

1 Set of equilibria (unique or multiple)2 comparative statics3 dynamics out of equilibria (stability and cycles)

In a perfectly symmetric game with two players.









Conclusion

Introduction - 4Specic questions addressed in this paper

The main focus of this work is to explore the joint work of

1 conict technologies2 costs3 polarization

on the

1 relation between this model and standard conict models

2 existence of asymmetric and multiple equilibria

3 evolution of conicts out of equilibrium according to thebest reply dynamics

4 characterization of basins of attraction.









Conclusion

Introduction - 5Reference literature

Corchon Luis C. 2007. The theory of contests: a survey.Review of Economic Design,

Garnkel, Michelle R. and Stergios Skaperdas 2007.Economics of Conict: An Overview.

Konrad Kai A. 2009. Strategy and Dynamics in Contests.

Chowdhury Subhasish M. and Roman M. Sheremeta2011a. A generalized Tullock contest. Public Choice, 147:413-420.

Chowdhury Subhasish M. and Roman M. Sheremeta2011b. Multiple equilibria in Tullock contests. EconomicLetters, 112: 216-219.

Szidarovszky, F. and Koji O., 1997. On the existence anduniqueness of pure Nash equilibrium in rent-seeking.Games and Economic Behavior.









Conclusion

Introduction - 6The contribution of the paper - 1

1 we qualify the characteristics of our model and itsproperties with respect to the literature on contests

2 For the case of simple conict technology withoutexternalities and

1 linear/concave/convex cost function with or without xedcosts

we provide a characterization of

i. equilibrium

ii. comparative statics

iii. stability.









Conclusion

Introduction - 7The contribution of the paper - 2

3. For the case of simple technology with externalities and

1 linear cost function with or without xed costs

we provide a qualitative characterization of

i. equilibriumii. comparative staticsiii. stability4. For the case of simple conict technology withoutexternalities and linear cost function and

1 polarizion with externalities

we provide a characterization of

i. equilibriumii. comparative staticsiii. stability.









Conclusion

The Model - 1

Denition

A Model of Political and Social Conict is dened by thefollowing element:

1 two agents, part 1 and 2, denoted by

i , j 2 f1, 2g ;

2 the agentse¤orts:

xi 2 Xi R+, i 2 f1, 2g ;

3 the agentse¤ectiveness of e¤orts:

Si (xi , xj ) : Xi Xj ! R+, i 2 f1, 2g ;









Conclusion

The Model - 2

Denition

4. the agentsgoals:

gi : Xi Xj !gi , gi

R, i 2 f1, 2g ;

θ (xi , xj ) := g2 (x2, x1) g1 (x1, x2) > 0is a measure of social and political polarization;

5. the outcome function describes the result of the conict asa function of both agentse¤ortse¤ectiveness

ζ : R+ R+ ! [g1, g2] ;

as ζ is near g2, we have that 2 is prevailing, while if ζ isnear g1 it states the superiority of 1;









Conclusion

The Model - 3

Denition

6. the conict technology connecting agentse¤ectiveness ofthe e¤orts Si (xi , xj ) to the probability of getting anoutcome:

P : R+ R+ ! ∆ ([g1, g2]) ,

7. the agentsutility function evaluating the (possiblyrandom) outcomes:

Ui : [g1, g2]! R+, i 2 f1, 2g ;

8. the agentscost function

Ci (xi , xj ) : Xi Xj ! R+, i 2 f1, 2g .









Conclusion

Observations on the Model

Remark

1 If the set of possible players e¤ort Xi is bounded, thenwlg we can normalize the intensity of the e¤ort xi 2 [0, 1].

2 If we assume that the agentsgoals gi do not depend onthe playerse¤orts, then we put our model within the classof rent seeking models, while in the models of productionand conict the value of the goals is endogenouslydetermined by the agentschoices

3 the separation between utility and cost functions is acommon restriction in the literature of economics ofinformation.









Conclusion

The Model - 4

Consider the standard denition of Contest, e.g. Corchon 2007:

Denition

A Contest is dened by the following element:

1 A nite set of agents, the contenders:

N = f1, ..., ng ;

2 A set of possible actions taken by the agents before a prizeis allocated:

ai 2 Ai 8i 2 N;3 A prize that may depend on agentsactions:

Vi : A1 ... An ! R 8i 2 N;

let denote the set of i 0s possible prizes as Ωi ;









Conclusion

The Model - 5

Denition

4. A contest success function relating agentsactions to theprobability of obtaining the prize:

pi : A1 ... An ! ∆ (Ωi ) 8i 2 N;

5. A utility function evaluating each agent prize:

Ui : Ωi ! R 8i 2 N;

6. A cost function relating agentsactions to the cost of theactions:

Ci : A1 ... An ! R 8i 2 N.









Conclusion

Remark

It is immediate to see that a Model of Political and SocialConict is a particular specic Contest.

Remark

A peculiar di¤erence between contests and MPSC is that in ourmodel outcomes and conict technology are objective, whilesubjectivity, i.e. agentsdependence, regard the e¤ectiveness ofe¤ort and the preferences, a characteristic that we believebetter match real conicts, helping interpretation andunderstanding.









Conclusion

AgentsUtilities

Hypothesis

The agentsutility function is linearly decreasing in the distancedi between its goal and the outcome:

d1 (x1, x2) = ζ (Si (xi , xj ) ,Sj (xj , xi )) g1 (x1, x2) ,

d2 (x2, x1) = g2 (x2, x1) ζ (Si (xi , xj ) ,Sj (xj , xi ))

Then wlg we normalize the linear utility function, so that

Ui (xi , xj ) = di (xi , xj ) .

Remark

This functional form is useful to focus on the role ofpolarization.









Conclusion

Outcomes - 1

Hypothesis

the outcome of the conict is either deterministic and equal to

ζ (S1,S2) =S1 (x1, x2) g1 (x1, x2)

S1 + S2+S2 (x2, x1) g1 (x1, x2)

S1 + S2,

with the following conict technology

P (z jS1 (x1, x2) ,S2 (x2, x1)) =1 if z = S1g1

S1+S2+ S2g2

S1+S20 otherwise.









Conclusion

Outcomes - 2

Hypothesis

or random

ζ (S1 (x1, x2) ,S2 (x2, x1)) 2 fg1 (x1, x2) , g2 (x2, x1)g

with the following conict technology

P (z jS1,S2) =

8><>:S1(x1,x2)

S1(x1,x2)+S2(x2,x1)if z = g1 (x1, x2)

S2(x2,x1)S1(x1,x2)+S2(x2,x1)

if z = g2 (x2, x1)0 otherwise.









Conclusion

Outcomes - 3

Remark

1 risk neutrality means that these two possibilities arestrategically equivalent

2 the functional form of the weights or of the conicttechnology when the outcome is random means that weare restricting ourself to the class of "Ratio SuccessFunctions"









Conclusion

Costs Functions

In production theory cost functions are derived fromtechnology and from a "reduced form" dynamics, i.e. thedistinction between short and long run

Fixed costs are a reduced form of the impact of pastdecisions on today decisions only if this past e¤ects areredeemable

Their application to conict theory requires furtherconsiderations.









Conclusion

In conicts costs should be included not only the directcosts of exerting e¤ort, including opportunity costs, butalso this "reduced form" dynamics

If we restrict ourself to direct conicts costs, then all xedcosts are sunk, they can not be redeemed since usuallythere is no market for conicts e¤orts

Conicts xed costs might be used to model a "reducedform" of continuation payo¤, which means that xed costsmight be negative when conicting is seen as a way ofestablishing future positive reputation.









Conclusion

The reduced form of our MPSC

Proposition

Hypotheses 1 to 7 imply

E [ζ (S1 (x1, x2) ,S2 (x2, x1))] =

=S2 (x2, x1)

S1 (x1, x2) + S2 (x2, x1)θ(x1, x2) + g1(x1, x2) =

= S1 (x1, x2)S1 (x1, x2) + S2 (x2, x1)

θ(x1, x2) + g2(x2, x1),

hence

πi (xi , xj ) = Sj (xj , xi )

S1 (x1, x2) + S2 (x2, x2)θ (x1, x2) Ci (xi , xj ) .









Conclusion

MPSC and contest

Remark

Hence, under these assumptions, a MPSC is equivalent to acontest where

1 the contest success function is

Sj (xj , xi )S1 (x1, x2) + S2 (x2, x1)

2 the prize isθ (x1, x2)

which means that from the point of view of the contesttheory we are dealing with a very peculiar setting. Inparticular, note that our contest success function isdi¤erent from the general technology used by Szidarovszkyand Okuguchi 1997 which might look very similar.









Conclusion

Case 1: linear technology, linear costs and noexternalities

Hypothesis

1 Each agents e¤ectivity function is linear in the intensity ofthe e¤ort:

S(xi , xj ) = βxi + 1,

2 Each agents cost function is linear in the intensity of thee¤ort :

c1 = 0, c2 = c , c3 2 f0,Kg .3 Polarization is constant

θ(xi , xj ) = θ > 0.









Conclusion

The Best Replies in case 1 & zero xed costs









Conclusion

The Equilibrium in case 1 & zero xed costs

Proposition

EXISTENCE OF NASH EQUILIBRIA: When conicttechnologies and costs are linear without externalities and thereare no xed costs, then there exists a unique symmetric purestrategy Nash equilibrium such that

xNE1 , xNE2

=

8><>:(0, 0) if β

4c 20, 1θ

θ4c

1β ,

θ4c

1β

if β

4c 2 1

θ ,1

θ4c

(1, 1) if β4c 2

1θ4c ,∞









Conclusion

Comparative statics in case 1 & zero xed costs - 1

Proposition

COMPARATIVE STATIC: When conict technologies andcosts are linear without externalities and there are no xedcosts, then

1 an increment in the marginal cost of conicts c reducesthe intensity of conict in equilibrium,

2 an increase of the marginal productivity of the e¤ort β onits e¤ectivity increases the intensity of conict inequilibrium

3 similarly, an increase in polarization θ increases theintensity of conict in equilibrium.









Conclusion

Comparative statics in case 1 & zero xed costs - 2

Proposition

COMPARATIVE STATIC: When conict technologies andcosts are linear without externalities and there are no xedcosts, then

1 an increase in the marginal cost of conict c implies thatthe interval of β such that there is zero conict isincreasing, however has an ambiguous e¤ect on the otherintervals, since

4cθ ,

4cθ4c

is increasing but the condition

θ 4c is stronger while 4c

θ ,+∞and

4cθ4c ,+∞

are

both shrinking;

2 on the other hand an increase in polarization θ has thecertain e¤ect of shrinking the interval of β such that thereis zero conict and increasing all the other intervals.









Conclusion

A Global View of the Equilibria in case 1 & zeroxed costs









Conclusion

Evolution of conict in case 1 & zero xed costs

Proposition

STABILITY: When conict technologies and costs are linearwithout externalities and there are no xed costs, then all thethree possible equilibria are fully stable wrt to the best replydynamics, independently of β, c , θ.









Conclusion

Equilibrium & Evolution with convex or concavecosts & zero xed costs

Proposition

EXISTENCE OF NASH EQUILIBRIA: When conicttechnologies are linear without externalities, costs are convex orconcave and there are no xed costs, then there exists a uniquesymmetric pure strategy Nash equilibrium such that either

xNE1 , xNE2= (0, 0)

or xNE1 , xNE2

2 (0, 1) (0, 1)

or xNE1 , xNE2

= (1, 1) .









Conclusion

Proposition

STABILITY: When conict technologies are linear withoutexternalities, costs are convex or concave and there are no xedcosts, the possible equilibria are fully stable with respect to thebest reply dynamics, independently of β, c , θ.









Conclusion

The Equilibrium with xed costs

Proposition

When conict technologies and costs functions are linearwithout externalities and there are xed costs, then

1 when the xed costs are small, i.e.K 2

h0, βθ2

βθ+4c +cβ

34 θi, then they are negligible and we

get the equilibria of case 1-A;

2 when the xed costs are intermediate, i.e.K 2

hβθ2

βθ+4c +cβ

34 θ,Z (β, θ, c)

i, then there is no pure

strategy equilibrium, hence the outcomes are random;

3 when the xed costs are huge, i.e. K 2 [Z (β, θ, c) ,∞] ,then in equilibrium there is no conict;

4 when the xed costs are negative, then in equilibriumeither they are irrelevant or they induce maximum conict









Conclusion

The Best Replies with linear costs & negligiblexed costs









Conclusion

The Best Replies with linear costs & intermediatexed costs









Conclusion

The Best Replies with linear costs & huge xedcosts









Conclusion

Comparative statics with linear costs & xed costs

Proposition

When conict technologies and costs functions are linearwithout externalities and there are xed costs, then

1 an increase in the marginal productivity of e¤ort β has anon monotonic e¤ect on the region of negligible xedcosts;

2 an increase in the marginal cost of conict c has a nonmonotonic e¤ect on the region of negligible xed costs;

3 an increase in polarization θ has a non monotonic e¤ecton the region of negligible xed costs.









Conclusion

A Global View of the Equilibria with linear costs &xed costs









Conclusion

Evolution of conict with linear costs & xed costs

Proposition

When conict technologies and costs functions are linearwithout externalities and there are xed costs, then all thepossible pure strategy equilibria are fully stable wrt to the bestreply dynamics, independently of K , β, c , θ.However, in the case of non existence of pure strategyequilibria there exists synchronous cycles of period three andasynchronous cycles of period six.









Conclusion

Basins of attractions and cycles with linear costs,xed costs and no pure strategy equilibria









Conclusion

CASE 2: Conict technology With Externality

Hypothesis

1 Each agents e¤ectivity function is

Si (xi , xj ) = βxi (1 αxj ) + 1,

where α 2 [0, 1] is a measure of the capacity of theantagonists in reducing the e¤ectivity of the other agentse¤ort;

2 Each agents cost function is linearci1 = 0, ci2 = c , ci3 2 f0,Kg ,

3 Polarization is constant θ(xi , xj ) = θ > 0.









Conclusion

The Best Replies in case 2 with zero xed costs









Conclusion

The Equilibrium in case 2 with zero xed costs

Proposition

When conict technologies and costs functions are linear withexternalities and there are no xed costs, then there exists aunique symmetric pure strategy Nash equilibrium such that

xNE1 , xNE2

=

8<: (0, 0) β4c 2

0, 1θ

, α 2

0, 12

(x, x) 2 (0, 1) (0, 1) otherwise(1, 1) α 2

12 , 1

This proposition shows that from a qualitative point of view,the externalities on the conict technology matters only if theexternality parameter is big enough, i.e. α 2

12 , 1.









Conclusion

Comparative statics in case 2 with zero xed costs

Proposition

When conict technologies and costs functions are linear withexternalities and there are no xed costs, then

1 an increment in the marginal cost of conicts c reducesthe intensity of conict in equilibrium;

2 an increase of the marginal productivity of the e¤ort β onits e¤ectivity increases the intensity of conict inequilibrium;

3 an increase in polarization θ increases the intensity ofconict in equilibrium;

4 an increase in α, increases the likelihood of getting anequilibrium with maximum conict intensity.









Conclusion

Evolution of conict in case 2 with zero xed costs

Proposition

When conict technologies and costs functions are linear withexternalities and there are no xed costs, then the possibleequilibria are fully stable with respect to the best replydynamics, independently of α, β, c , θ.









Conclusion

The Equilibrium in case 2 with xed costs

Proposition

When conict technologies and costs functions are linear withexternalities and there are xed costs, then the set of purestrategy Nash equilibria might be empty; when it is not emptythen:

xNE1 (K ) , xNE2 (K )

=

8>><>>:(0, 0) K big enoughxNE1 , xNE2

K intermediate

xNE1 , xNE2& (1, 1)

K negative8xj 1

2α

Hence, the evolution of conict in case 2-B with negative xedcosts is interesting.









Conclusion

Evolution of conict in case 2 with xed costs

Proposition

hen conict technologies and costs functions are linear withexternalities and there are xed costs,

1 if the xed costs are positive, then all the possible purestrategy equilibria are fully stable, independently ofK , β, c , θ

2 if xed cost are negative 8xj 12α , then the pure strategy

equilibria are locally stable and there is the possibility of atwo period cycle, where the players alternate combininglow with high conict.









Conclusion

Best replies and basin of attraction in case 2 withxed negative costs

In case 2 we have multiple equilibria that can be Pareto rankedand cycles alternating asymmetric intensity of conict. Thefollowing picture reports this case.









Conclusion

CASE 3: Polarization With Externality

Hypothesis

1 Each agents e¤ectivity function is linearS(xi , xj ) = βxi + 1,

2 Each agents cost function is linearc1 = 0, c2 = c , c3 2 f0,Kg .

3 Polarization is increasing in both agentse¤orts

θ(xi , xj ) = θ + δxi + δxj .









Conclusion

The Best Replies in case 2 with zero xed costs









Conclusion

The Equilibrium in case 3 with zero xed costs

Proposition

When conict technologies and costs are linear withoutexternalities, there are no xed costs, and there are externalitiesin polarizations, then there exists a unique symmetric purestrategy Nash equilibrium such that

xNEi =

8>>>>>><>>>>>>:

0 if β 2h0, 32c

2+16δc+2δ(8+δ)θ

i(βθ2δ)8c+

p(βθ2δ)216δc

8βc

if θ 8c &β 2

h32c2+16δc+2δ

(8+δ)θ; 2δ+8c

θ8c

i1

if θ 8c orβ 2

2δ+8cθ8c ;∞









Conclusion

A Global View of the Equilibria in case 3 & zeroxed costs









Conclusion

The Limits of the Model

Two players

Specic functional forms for

1 utility2 conict technology3 costs functions









Conclusion

The Interpretation of Previous Results - 1

1 Existence, uniqueness and stability are robust to thespecication of the conict technology and of the costsfunctions without xed costs

hence in this case

the interpretation of the stylized facts relies oncomparative statics i.e. on the transition from anequilibrium to another as changes in

marginal productivity of e¤ortmarginal cost of e¤ortpolarization

interpretation of stylized facts as a consequence ofstructural changes.









Conclusion

The Interpretation of Previous Results - 2

2. Non existence, multiplicity and interesting dynamicsrequires the specication of peculiar xed costs functions

hence

the interpretation of the stylized facts as dynamictransitions from a region to another relies on the role ofcosts functions even with slightly complex conicttechnologies.









Conclusion

Conclusion

Our results are partially disappointing since we wouldprefer to interpret the stylized facts in terms of dynamictransitions

however

the model stress the relevance of costs functions inconict models

and

provides a general framework to discuss stylized facts.

Our future work will focus on asymmetries.

equilibria and stability in a model of political and social con⁄ict. · 2019-12-03 · konrad kai...

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