NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
NEGOTIATION AND MEDIATION IN CONFLICTS I: THE ROLE OF MATHEMATICAL APPROACHES AND METHODS
Andrzej Wierzbicki November 1983
WP-83-106
Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, Austria
NEGOTIATION AND MEDIATION I N CONFLICTS
I: THE ROLE OF MATHEMATICAL APPROACHES IWD METHODS
Andrzej P. W i e r z b i c k i
I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems A n a l y s i s , Laxenburg, A u s t r i a
A b s t r a c t . There a r e many p o s s i b l e a p p r o a c h e s t o t h e a n a l y s i s of c o n f l i c t s i t u a t i o n s and ways of d e v e l o p i n g m e d i a t i o n o r n e g o t i a t i o n p r o c e d u r e s . I n g e n e r a l , we c a n d i s t i n g u i e h between so&o-political a p p r o a c h e s , p r o c e d u r e s f o r m e d i a t i o n and n e g o t i a t i o n which h a v e developed h i s t o r i c a l l y t h r o u g h p o l i t i c a l e x p e r i e n c e , and systems- analytical, m a t h e m a t i c a l a p p r o a c h e s , which r e l y on v a r i o u s b r a n c h e s of d e c i s i o n and game t h e o r y o r on c o m p u t e r i z e d s i m u l a t i o n o r gaming models. While t h e s o c i o - p o l i t i c a l a p p r o a c h e s must remain t h e b a s i s of any a n a l y s i s of c o n f l i c t s i t u a t i o n s , t h e y can u s e f u l l y be sup- plemented by more-formalized a p p r o a c h e s ; b o t h a p p r o a c h e s , when used s e p a r a t e l y , have t h e i r l i m i t a t i o n s .
T h i s p a p e r , a f t e r a s h o r t r e v i e w of e x i s t i n g s y s t e m s - a n a l y t i c a l and m a t h e m a t i c a l a p p r o a c h e s , p r o p o s e s new c o n c e p t s and methods f o r t h e m a t h e m a t i c a l a n a l y s i s of c o n f l i c t p r o c e s s e s . The proposed approach a t t e m p t s t o b r i d g e t h e gap between s i m u l a t i o n gaming models and ex- p e r i m e n t s and more f o r m a l g a m e - t h e o r e t i c a l a n a l y s i s by t a k i n g c e r t a i n i d e a s from i n t e r a c t i v e d e c i s i o n s u p p o r t e d gaming, assuming t h a t t h e p l a y e r s have m u l t i p l e o b j e c t i v e s t h a t t h e y wish t o s a t i s f i c e ( n o t o p t i m i z e ) , and i n t r o d u c i n g s p e c i a l m e d i a t i o n p r o c e d u r e s a s a n e l e m e n t of d e c i s i o n s u p p o r t d u r i n g t h e game. The m a t h e m a t i c a l c o n c e p t s u s e d a r e t h e s e l e c t i o n of s a t i s f i c i n g game e q u i l i b r i a , t h e d e f i n i t i o n of c o n s t r u c t i v e and d e s t r u c t i v e b e h a v i o r (on t h e p a r t of t h e p l a y e r s ) , and a f o r m a l i z a t i o n of c o n f l i c t e s c a l a t i o n and d e - e s c a l a t i o n p r o c e s s e s . Keywords. Game t h e o r y ; gaming; c o n f l i c t a n a l y s i s ; n e g o t i a t i o n t h e o r y ; m e d i a t i o n ; i n t e r a c t i v e d e c i s i o n s u p p o r t ; s a t i s f i c i n g .
INTRODUCTION
The s t u d y of c o n f l i c t s and t h e p o s s i b l e r o l e of n e g o t i a t i o n and m e d i a t i o n h a s a l o n g t r a d i t i o n i n t h e p o l i t i c a l and s o c i a l s c i e n c e s ( s e e , e.g
. ,
M a c h i a v e l l i , 1515).The need f o r a b e t t e r u n d e r s t a n d i n g of t h e p r o c e s s e s i n v o l v e d h a s become even more u r g e n t s i n c e t h e development of n u c l e a r weapons. A s a r e s u l t , t h e s t u d y of nego- t i a t i o n s h a s r e c e n t l y become t h e s u b j e c t o f v e r y i n t e n s i v e r e s e a r c h v i a a number of d i s c i p l i n e s . For example, a p r o j e c t a t Harvard U n i v e r s i t y b r i n g s t o g e t h e r l a w y e r s , p o l i t i c a l s c i e n t i s t s , e c o n o m i s t s , mathe- m a t i c a l game t h e o r i s t s and d e c i s i o n the- o r i s t s a s w e l l a s r e p r e s e n t a t i v e s of o t h e r d i s c i p l i n e s ( s e e , e . g . , F i s h e r and Ury, 1981; R a i f f a , 1982). We c a n d i v i d e t h e s e d i s c i p l i n a r y approaches i n t o two b r o a d c a t e g o r i e s : s o c i o - p o l i t i c a l a p p r o a c h e s and systems-ana l y t i c a t , m a t h e m t i c a l a p p r o a c h e s .
T r a d i t i o n a l l y , t h e s t u d y of n e g o t i a t i o n and m e d i a t i o n d u r i n g c o n f l i c t s h a s b e e n t h e domain of p o l i t i c a l and h i s t o r i c a l s c i e n t i s t s , d i p l o m a t s , p s y c h o l o g i s t s and s o c i o l o g i s t s . These s o c i o - p o l i t i c a l ap- proaches w i l l p r o b a b l y remain t h e b a s i s o f a l l f u t u r e a n a l y s e s o f c o n f l i c t , nego- t i a t i o n and m e d i a t i o n ; w i t h o u t a deep u n d e r s t a n d i n g of t h e u n d e r l y i n g s o c i o - p o l i t i c a l p r o c e s s e s , no f o r m a l i z e d c h a r - a c t e r i z a t i o n o f any d e c i s i o n p r o c e s s c a n e v e r b e r e l e v a n t ( s e e , e . g . , W i e r z b i c k i , 1983). T h i s does n o t mean, however, t h a t s y s t e m s - a n a l y t i c a l o r e v e n m a t h e m a t i c a l f o r m a l i z a t i o n c a n n o t i n c r e a s e o u r under- s t a n d i n g of c o n f l i c t p r o c e s s e s . As i n many o t h e r a r e a s of s c i e n t i f i c e n d e a v o r , m a t h e m a t i c a l f o r m a l i z a t i o n can h e l p , f i r s t , i n checking t h e c o n s i s t e n c y of b a s i c con- c e p t s and images. When t h e u n d e r l y i n g con- c e p t s a r e c l e a r , r e l e v a n t a n d f o c u s e d on es- s e n t i a l s , m a t h e m a t i c a l f o r m a l i z a t i o n c a n a l s o h e l p u s t o d e a l w i t h c o m p l e x i t y by producing frameworks f o r p o s s i b l e mathe- m a t i c a l models. F i n a l l y , g i v e n t h e s e
c o n c e p t s and frameworks, computerized models can be b u i l t and used f o r v a r i o u s purposes : f o r overcoming c o m p l e x i t y , f o r e d u c a t i n g and a s s i s t i n g d e c i s i o n makers i n p a r t i c u l a r a s p e c t s of a problem, a s a t o o l f o r s t i m u l a t i n g i n t e r a c t i o n between t h e p a r t i e s i n v o l v e d i n a d i s p u t e , and s o on
( s e e , e . g . , S e b e n i u s , 1981).
The s y s t e m s - a n a l y t i c a l and m a t h e m a t i c a l approaches have produced a l a r g e v a r i e t y of c o n c e p t s and t o o l s which c o u l d b e use- f u l i n s t u d y i n g c o n f l i c t , n e g o t i a t i o n s and m e d i a t i o n . One of t h e e a r l i e s t mathe- m a t i c a l d e s c r i p t i o n s of armed c o n f l i c t (a
" b a t t l e model") was produced by L a n c h e s t e r (1 916) ; s u c h b a t t l e models, developed f u r - t h e r by Richardson (1960), Rappaport (1957), and r e c e n t l y by Avenhaus and F i c h t n e r (1983), c a n a l s o s u g g e s t i n i t i a l c o n d i t i o n s t h a t would reduce t h e l i k e l i h o o d of war and could t h u s be h e l p f u l i n nego- t i a t i o n s .
S i n c e t h e e a r l y days of game t h e o r y ( s e e von Neumann and Morgenstern, 1944) i t h a s been hoped t h a t g a m e - t h e o r e t i c a l a n a l y s i s might h e l p t o i n c r e a s e o u r u n d e r s t a n d i n g of c o n f l i c t s . However, w h i l e game t h e o r y h a s f o r m a l i z e d many e s s e n t i a l c o n c e p t s i n t h i s f i e l d , i t s u n d e r l y i n g assumptione-- namely, t h a t p l a y e r s behave i n such a way a s t o maximize t h e i r own u t i l i t y function-- may s t i l l l i m i t i t s p o t e n t i a l u s e f u l n e s s . I n p a r t i c u l a r , f o r m a l models of b a r g a i n i n g
( s e e , e.g., Roth, 1979) a r e n o t n e c e s s a r i l y s u p p o r t e d by e x p e r i m e n t a l s t u d i e s of bar- g a i n i n g b e h a v i o r (Roth, 1983); t h e paradox of s h o r t - s i g h t e d r a t i o n a l b e h a v i o r exemp- l i f i e d by t h e s o - c a l l e d ' p r i s o n e r ' s dilemma', which was s o l v e d long ago by a n t h r o p o l o g i s t s s t u d y i n g p r i m i t i v e forms of
-- b a r t e r , h a s o n l y r e c e n t l y f o u n d m o r e f o r m a l r e s o l u t i o n i n e v o l u t i o n a r y game t h e o r y (Maynard-Smith, 1977; Axelrod, 1983;
H o f s t a d t e r , 1983).
The more a p p l i c a t i o n - o r i e n t e d m a t h e m a t i c a l approaches o f t e n r e l y on l e s s s o p h i s t i c a t e d b u t more r e l i a b l e t o o l s . I n p a r t i c u l a r , d e c i s i o n - t r e e a n a l y s i s i s f r e q u e n t l y used t o examine c r i s i s s i t u a t i o n s . Dynamic s i m u l a t i o n and gaming models a l s o have a wide range of a p p l i c a t i o n s ( s e e , e . g . , S t a h l , 1983). However, d e s p i t e t h e i r many e x c e l l e n t f e a t u r e s , gaming s i m u l a t i o n models and experiments do n o t g i v e s u f - f i c i e n t i n s i g h t i n t o t h e p o s s i b l e c o n f l i c t e s c a l a t i o n o r d e - e s c a l a t i o n p r o c e s s e s a r i s i n g from t h e p r o p e r t i e s of t h e game.
Although e s c a l a t i o n and d e - e s c a l a t i o n have been observed i n many gaming e x p e r i m e n t s , t h e r e a r e a s y e t no a n a l y t i c a l t o o l s t o h e l p us understand t h e s e e m p i r i c a l obser- v a t i o n s . The need f o r f u r t h e r r e s e a r c h on s u c h t o o l s i s t h e r e f o r e v i t a l .
1. While i t i s r e c o g n i z e d t h a t gaming s i m u l a t i o n models and e x p e r i m e n t s a r a a l - ready v e r y s u c c e s s f u l i n c o n f l i c t a n a l y s i s , we a t t e m p t t o e x t e n d t h e s e gaming simula- t i o n t e c h n i q u e s by p r o v i d i n g d e c i s i o n sup- p o r t based on c o n c e p t s and methods from game
t h e o r y and m u l t i o b j e c t i v e o p t i m i z a t i o n . T h i s l e a d s t o a new c l a s s of gaming tech- n i q u e s which we c a l l i n t e r a c t i v e decision- s u p p o r t e d gaming.
2. Many gaming e x p e r i m e n t s show t h a t t h e p l a y e r s do n o t t h i n k i n terms of o n e u t i l i t y f u n c t i o n , b u t t r y t o b a l a n c e s e v e r a l g o a l s o r o b j e c t i v e s . A l s o , p l a y e r s do n o t neces- s a r i l y behave as maximizers, o f t e n exhib- i t i n g s a t i s f i c i n g b e h a v i o r i n s t e a d ( s e e , e.g., Simon, 1969). Thus, t h e game- t h e o r e t i c a l c o n c e p t s and methods used i n d e c i s i o n s u p p o r t a r e m o d i f i e d t o t a k e t h i s m u l t w b j e c t i v e
and satisficing
b e h a v i o ri n t o a c c o u n t .
3. I n t e r a c t i v e d e c i s i o n - s u p p o r t e d gaming a s d e s c r i b e d above c a n be used e i t h e r a s a means of t e a c h i n g p l a y e r s a b o u t t h e p r o c e s s of c o n f l i c t e s c a L a t i o n , o r i n c o n j u n c t i o n w i t h f o r m a l i z e d m e d i a t i o n p r o c e d u r e s t o il- l u s t r a t e how conflict d e - e s c a k t i o n c a n b e achieved.
A COASTAL WATERS AND OPEN SEA
FISHING GAME
A s a n i l l u s t r a t i o n , l e t u s c o n s i d e r a f i s h i n g game developed by M. S t a l e y and C. Walters a t I I A S A . ~ F o r s i m p l i c i t y , we s h a l l l i m i t t h e problem t o o n l y two coun- t r i e s ( I and 2 ) ; t h e y can f i s h e i t h e r i n t h e i r own c o a s t a l w a t e r s o r i n t h e o p e n s e a . The f i s h b e i n g caught i s assumed t o spawn i n t h e r i v e r s of a g i v e n c o u n t r y ( l i k e sal- mon); t h u s , t h e f i s h r e t u r n from t h e open
s e a t o t h e s e r i v e r s e a c h y e a r and i t i s r e a s o n a b l e t o r e f e r t o t h e s t o c k of f i s h o r i g i n a t i n g from each c o u n t r y . Suppose t h e s t o c k s i n a g i v e n y e a r a r e xi, i = 1 , 2 ; t h e s i z e of t h e n a t i v e f l e e t f i s h i n g i n i t s own c o a s t a l w a t e r s i s c ( f i s h i n g i n t h e o t h e r
i
c o u n t r y ' s c o a s t a l w a t e r s i s p r o h i b i t e d ) ; t h e s i z e of e a c h f l e e t f i s h i n g i n t h e open s e a i s si. Then t h e c a t c h of e a c h c o u n t r y can be approximated by
where t h e f i r s t term d e s c r i b e s t h e c a t c h i n t h e open s e a , t h e second t h e c a t c h i n coas- t a l w a t e r s ,
f
d e n o t e 8 t h e v a r i a b l e s of t h e T h i s p a p e r s u g g e s t s a number of new con-c e p t s and methods f o r t h e m a t h e m a t i c a l a n a l y s i s of c o n f l i c t p r o c e s s e s . T h e b a s i c assumptions of t h i s approach a r e a s f o l l o w s :
P r i v a t e communication; a l t h o u g h many ex- perimentshavebeenperformedwiththis game, a f u l l d e s c r i p t i o n h a s n o t y e t been pub- 11s hed
.
" o t h e r " c o u n t r y , and a i s a p a r a m e t e r . L i k e t h e s t o c k , t h e c a t c h c a n b e measured d i r e c t l y i n monetary u n i t s , and s h o u l d b e compared w i t h t h e c o s t of m a i n t a i n i n g t h e f l e e t , p(ci+si). However, t h i s a n a l y s i s c o n s i d e r s o n l y s h o r t - t e r m b e n e f i t s and c o s t s . The long-term b e n e f i t s c a n b e char- a c t e r i z e d by t h e s t o c k s e x p e c t e d i n t h e f o l l o w i n g y e a r :
x r ( x exp [-a(si+s3+ci)l i , t
where r ( . ) i s a r e p r o d u c t i o n r a t e c o e f - f i c i e n t . T y p i c a l l y , t h e e x p r e s s i o n ( 2 ) i s h i g h l y n o n l i n e a r and random; i f t h e s t o c k s a r e a l r e a d y s t r o n g l y d e p l e t e d , however, we can approximate ( 2 ) by assuming
r ( . ) = r = c o n s t a n t . Taking i n t o account b o t h s h o r t - and long-term b e n e f i t s and c o s t s , t h e payoff f u n c t i o n s f o r p l a y e r s r e - p r e s e n t i n g b o t h c o u n t r i e s a r e
w i t h z e v a l u a t e d a t x = x
i i i , t
.
The ac-t u a l gaming model used i n p r a c t i c e may be c o n s i d e r a b l y more c o m p l i c a t e d t h a n t h e b a s i c framework o u t l i n e d above, i n c l u d i n g dynamic s i m u l a t i o n o v e r s e v e r a l y e a r s , t h e p o s s i b i l i t y of i n c r e a s i n g t h e s i z e of t h e f l e e t t h r o u g h i n v e s t m e n t , t h e u s e of hatch- e r i e s , e t c . A model of t h i s t y p e h a s been used i n i n t e n s i v e e x p e r i m e n t a l gaming, w i t h some i n t e r e s t i n g r e s u l t s . Three c l u s t e r s of outcomes seem t o emerge from t h e gaming e x p e r i m e n t s . One t y p e of r e s u l t ( c a l l e d c o o p e r a t i v e ) a r i s e s when t h e p l a y e r s a g r e e
t a c i t l y n o t t o f i s h t o o e x t e n s i v e l y ; t h i s r e s u l t s i n h i g h p a y o f f s qi and s t o c k s
X i. t+l a t t h e end of t h e game. The second t y p e of r e s u l t c a n b e c a l l e d dominated-- t h i s o c c u r s when one p l a y e r makes much h i g h e r c a t c h e s zi and r e c e i v e s much h i g h e r p a y o f f s q t h a n t h e o t h e r . The t h i r d t y p e
i
can be c a l l e d d e s t r u c t i v e ; i n t h i s c a s e b o t h p l a y e r s u s e v e r y l a r g e - f l e e t s , d r a s - t i c a l l y r e d u c i n g b o t h p a y o f f s and f u t u r e s t o c k s of f i s h . Moreover, t h e p l a y e r s t y p i c a l l y r e d u c e t h e s i z e of t h e f l e e t f i s h i n g i n t h e i r c o a s t a l w a t e r s ci and i n c r e a s e t h e i r sea-going f l e e t s R e s u l t s
i '
of t h e s e t h r e e t y p e s have a l s o been ob- s e r v e d i n a c t u a l f i s h i n g d i s p u t e s between n e i g h b o r i n g c o u n t r i e s .
The q u e s t i o n t h a t w e a d d r e s s h e r e i s w h e t h e r g a m e - t h e o r e t i c a l a n a l y s i s can h e l p
us t o u n d e r s t a n d gaming r e s u l t s s u c h a s t h e s e , and t o e x p l a i n t h e b e h a v i o r of p l a y e r s a s t h e c o n f l i c t d e v e l o p s . The p r e s e n t answer i s a q u a l i f i e d y e s , where t h e q u a l i f i c a t i o n a r i s e s from t h e l i m i t a t i o n s o f c l a s s i c a l t o o l s . F i g u r e 1 r e p r e s e n t s t h e image of
t h e game2 ( t h e s e t of a t t a i n a b l e p a y o f f s ) , c a l c u l a t e d making t h e s i m p l e a s s u m p t i o n s x 1 q 2 = 5
,
a = p = l,
r ( . ) = r - 3,
si+cf
[0;51.
By a c l a s s i c a l game a n a l y s i s , we can e a s i l y show t h a t ci=O i s o p t i m a l f o r b o t h p l a y e r s and t h a t t h e game h a s s e v e r a l n o n c o o p e r a t i v e
(Nash) e q u i l i b r i a (Nash, 1950, 1953) d e n o t e d i n F i g . 1 by No, N1, N2; o b v i o u s l y , N i s
0 t h e dominant Nash e q u i l i b r i u m and t h e s i n g l e P a r e t o p o i n t f o r t h e game.
Thus, how c o u l d i t happen t h a t p l a y e r s l e a v e t h e p o i n t N t o produce r e s u l t s c l u s t e r e d
0
i n r e g i o n s ( I ) , (111, 1 1 2 ) , and (111) i n F i g . l ? How can we e x p l a i n t h e p r o c e s s of c o n f l i c t e s c a l a t i o n t h a t t y p i c a l l y o c c u r s i n e x p e r i m e n t a l gaming, and may b e il- l u s t r a t e d by t h e s e q u e n c e of p o i n t s
(AO)-(A1 o r A2)-A3-(A4 o r A 5 ) 4 i n F i g . l ? 6
To e x p l a i n t h i s , we must assume t h a t t h e p l a y e r s a r e m o t i v a t e d by something more c o m p l i c a t e d t h a n s i m p l e payoff maximiza- t i o n . F i r s t , we o b s e r v e t h a t t h e p o i n t No c o r r e s p o n d s t o z e r o c a t c h e s , z1=z2=0
.
However, a p o s i t i v e c a t c h i s n e c e s s a r y t o keep up t h e f i s h s u p p l y (and t h e c a s h f l o w ) ; h e n c e , b o t h p l a y e r s must have more t h a n o n e o b j e c t i v e , k e e p i n g an e y e o n f i n a l p a y o f f s b u t c o n c e n t r a t i n g on f i s h c a t c h e s .
We c o u l d now f o l l o w t h e c l a s s i c a l approach and assume t h a t t h e p l a y e r s have u t i l i t y o r v a l u e f u n c t i o n s u l ( q l , z l ) and u 2 ( q 2 , z 2 ) t h a t r e s u l t i n an e q u i l i b r i u m , s a y , a t t h e p o i n t A However, t h e u t i l i t y f u n c t i o n
0 '
approach h a s s e v e r a l drawbacks i n c o n f l i c t a n a l y s i s . F i r s t , any i n f o r m a t i o n on u t i l i t y f u n c t i o n s h a s c r u c i a l s t r a t e g i c v a l u e and i s u s u a l l y c a r e f u l l y p r o t e c t e d ; t h e r e f o r e , we c a n n o t h o p e t h a t f u l l and p r e c i s e i n f o r m a t i o n on u t i l i t y f u n c t i o n s w i l l b e a v a i l a b l e f o r i n c o r p o r a t i o n i n t o a gaming model. Second, a p r o c e s s of c o n f l i c t e s c a l a t i o n o r d e - e s c a l a t i o n u s u a l l y i n v o l v e s m o d i f i c a t i o n s t o t h e u t i l i t y f u n c t i o n s , s i n c e e x p e r i m e n t a l l y ob- s e r v e d u t i l i t y f u n c t i o n s a r e t y p i c a l l y n o t c o n t e x t - f r e e ( s e e , e . g . , Tversky, 1972).
T h i r d , r e a l p l a y e r s u s u a l l y do n o t t h i n k i n terms of u t i l i t y maximization, and even i f a s s u r e d t h a t such i n f o r m a t i o n would b e t r e a t e d a s c o n f i d e n t i a l , would f i n d i t d i f f i c u l t t o communicate t h e i r u t i l i t y f u n c t i o n s t o a t h i r d p a r t y (computer o r m e d i a t o r ) .
A d e t a i l e d m a t h e m a t i c a l a n a l y s i s i s g i v e n i n t h e second p a r t of t h i s p a p e r :
" S a t i s f i c i n g S e l e c t i o n s of Game E q u i l i b r i a " .
T h e r e f o r e , we a d o p t i n t h i s paper a n o t h e r approach t h a t i s b o t h f l e x i b l e and prag- m a t i c and, a t t h e same time, opens new o p p o r t u n i t i e s f o r c o n f l i c t a n a l y s i s and f o r communication between p l a y e r s and model. We assume t h a t p l a y e r s e i t h e r ac-
t u a l l y t h i n k i n terms of aspiration levezs ( W i e r z b i c k l , 1982) f o r t h e i r v a r i o u s ob- j e c t i v e s , o r c a n b e t a u g h t t o do s o . Moreover, we assume t h a t t h e s e a s p i r a t i o n l e v e l s can be a d a p t i v e l y modified d u r i n g a gaming e x e r c i s e . We suppose t h a t t h e pro- c e s s of a s p i r a t i o n l e v e l f o r m a t i o n o c c u r s i n p l a y e r s ' minds, b u t t h a t p l a y e r s might b e induced (by a s s u r a n c e s o f confiden- t i a l i t y ) t o communicate t h e i r c u r r e n t as- p i r a t i o n l e v e l s t o t h e computer i n o r d e r t o r e c e i v e , i n r e t u r n , some d e c i s i o n o r m e d i a t i o n s u p p o r t .
To p u t t h e s e c o n c e p t s i n t o p r a c t i c e , we c a l c u l a t e s e t s of e q u i l i b r i a s u c h t h a t t h e outcomes a r e P a r e t o f o r t h e o b j e c t i v e s zi ' q i of e a c h p l a y e r b u t n o n c o o p e r a t i v e b e t - ween p l a y e r s . These s e t s a r e r a t h e r l a r g e ( s e e s e t s NP1, NP2 i n Fig. 2 ) , b u t e x c l u d e some outcomes ( s e t s U1, U ) t h a t a r e un-
2
s t a b l e t o t h e i n t e r e s t s and d e c i s i o n s of e i t h e r of t h e p l a y e r s ( o b s e r v e t h a t t h e s e t s of t y p i c a l outcomes shown i n F i g . 1 l i e almost completely w i t h i n NPI, NP2). I f we add t o t h e image o f t h e game t h e l i n e s of c o n s t a n t outcomes zi f o r e a c h p l a y e r , a s i n F i g . 2, we can e x p l a i n t h e p r o c e s s o f c o n f l i c t e s c a l a t i o n through t h e phe- nomenon of i n f l a t e d aspirations. By in- f l a t e d a s p i r a t i o n s we mean a s p i r a t i o n s t h a t exceed t h e l i m i t s imposed ( f o r example) by p h y s i c a l , b i o l o g i c a l , and en- v i r o n m e n t a l f a c t o r s : t h e r o l e of t h e model i s t o r e p r e s e n t t h e s e n a t u r a l l i m i t s and t o h e l p t h e p l a y e r s t o l e a r n about them.
L e t us now suppose t h a t t h e game i s p l a y e d r e p e t i t i v e l y ( r e p r e s e n t i n g f i s h i n g ac- t i v i t y o v e r many y e a r s ) and t h a t b o t h p l a y e r s i n i t i a l l y make d e c i s i o n s t h a t l e a d t o outcome Ao, corresponding t o c a t c h e s z 1 3 z 2 9 and p a y o f f s q1=q2?13
.
I f t h e p l a y e r s behave c o o p e r a t i v e l y , t h e y can s t a y c l o s e t o N =P and g a i n h i g h f u t u r e0
p a y o f f s . However, suppose p l a y e r 1 wants t o expand and d e c i d e s t o i n c r e a s e h i s c a t c h t o z1=6 ( p o i n t A1) i n o r d e r t o i n - c r e a s e h i s s h o r t - t e r m r e t u r n s and t o f i - nance new i n v e s t m e n t i n h i s f i s h i n g f l e e t . I f t h e o t h e r p l a y e r does n o t r e c i p r o c a t e . then p l a y e r 1 s u c c e e d s i n h i s g o a l s and o b t a i n s a dominant p o s i t i o n i n f u t u r e rounds. I f p l a y e r 2 r e c i p r o c a t e s , however, t h e c a t c h of t h e f i r s t p l a y e r d e c r e a s e s t o z =z % ( p o i n t Ag). However, p l a y e r 1 may
1 2
have a l r e a d y made l a r g e investment comnit- ments and must i n c r e a s e h i s c a t c h ; t h u s ,
h e might i n c r e a s e h i s f l e e t and hence h i s c a t c h s t i l l f u r t h e r ( p o i n t A ~ ) . I f t h e o t h e r p l a y e r r e c i p r o c a t e s , t h e outcome would b e t h e w o r s t Nash e q u i l i b r i u m (N2),
=z 5 5 and a l m o s t no f i s h w i t h c a t c h e s z l 25
remaining f o r t h e f o l l o w i n g y e a r . T h i s Nash e q u i l i b r i u m i s n o t v e r y r o b u s t t o m u l t i o b j e c t i v e b e h a v i o r w i t h i n f l a t e d as- p i r a t i o n s ; f u r t h e r e s c a l a t i o n can e a s i l y l e a d t o t h e a t t r i t i o n p o i n t A6
.
Although one c o u l d l e a r n much from t h e above s i m p l e example, and draw many a n a l - o g i e s (f o r example, t o t h e arms r a c e )
,
we s h a l l conclude t h e a n a l y s i s w i t h t h e ob- s e r v a t i o n t h a t t h e b e h a v i o r of p l a y e r s i n s u c h a game can b e e x p l a i n e d , f i r s t , by t h e i r m u l t i p l e o b j e c t i v e s , and second, by t h e i r i n f l a t e d a s p i r a t i o n s . I f t h e y l e a r n q u i c k l y t o r e v i s e t h e i r a s p i r a t i o n l e v e l s downwards, t h e y can s u r v i v e ; however, i f they m a i n t a i n t h e i r i n f l a t e d a s p i r a t i o n s o n l y s l i g h t l y t o o l o n g , c o n f l i c t can es- c a l a t e v e r y s p e e d i l y .T h i s l e s s o n can b e t a u g h t when p l a y i n g s i m u l a t i o n games; however, we t h e n need t o c o n c e n t r a t e t h e a t t e n t i o n o f t h e p l a y e r s on t h e i r a s p i r a t i o n s . I d e a l l y , t h e
d e c i s i o n s u p p o r t system s h o u l d produce a f u l l image o f t h e game, such a s t h a t shown i n F i g . 2. However, t h i s might r e q u i r e e x c e s s i v e computer t i m e f o r more compli- c a t e d games. F a i l i n g t h i s , t h e d e c i s i o n s u p p o r t system could compute t h e s t r a t e g i e s t h a t each p l a y e r s h o u l d a d o p t t o come c l o s e t o some e x p l i c i t l y s t a t e d a s p i r a - t i o n l e v e l s of t h e i r own, making c e r t a i n assumptions concerning t h e a s p i r a t i o n l e v e l s of t h e o t h e r p l a y e r s . I n o r d e r t o a c h i e v e t h i s , we must i n t r o d u c e some p r i n - c i p l e s f o r selecting gwne e q u i l i b r i a t h a t r e l a t e t o t h e concept of a s p i r a t i o n l e v e l s and t h u s t o some form of s a t i s f i c i n g be- h a v i o r . Although t h e s e c o n c e p t s a p p l y t o m u l t i o b j e c t i v e games, i t i s s i m p l e r t o
i n t e r p r e t them g r a p h i c a l l y f o r a s i n g l e - o b j e c t i v e game, provided i t s s e t of non- c o o p e r a t i v e e q u i l i b r i a i s l a r g e . ( S e t s of m u l t i o b j e c t i v e n o n c o o p e r a t i v e e q u i l i b r i a : a r e g e n e r a l l y l a r g e , g i v i n g t h e p l a y e r s a w i d e r c h o i c e of p o s s i b l e outcomes i n which
t h e y a r e guided by t h e i r a s p i r a t i o n l e v e l s , )
A "FISHING IN NEIGHBORS' COASTAL WATERS" GAME
We c o n s i d e r h e r e a s i m p l i f i e d example o f a game t h a t h a s o n l y h i s t o r i c a l s i g n i f i c a n c e , because most c o u n t r i e s now r e f r a i n from f i s h i n g i n each o t h e r ' s c o a s t a l w a t e r s . Two c o u n t r i e s , i = 1 , 2 , f i s h i n e a c h o t h e r ' s , r = 2 , 1 , c o a s t a l w a t e r s . Each c o u n t r y can d e c i d e how much t o t a k e from f o r e i g n w a t e r s (we d e n o t e t h i s d e c i s i o n by x ) i and what
1
r e s t r i c t i o n s t o impose o n f o r e i g n b o a t s
F i g . 1. Image of an open s e a f i s h i n g game. N , N , N a r e n o n c o o p e r a t i v e Nash e q u i l i b r i a ,
0 1 2
w i t h NO>N1>N2
.
I , 111, 112, I11 a r e a p p r o x i m a t i o n s o f t y p i c a l s e t s o f e x p e r i m e n t a l outcomes. The s e r i e s o f moves AO-(A1, A2)-A3-(A4, A )- e i t h e r N o r A6 r e p r e s e n t s5 2
a c o n f l i c t e s c a l a t i o n p r o c e s s .
F i g . 2. F u l l image of an open s e a f i s h i n g game. N , N , N a r e economic Nash e q u i l i b r i a ;
0 1 2
NP1, NP a r e s e t s of m u l t i o b j e c t i v e Nash-Pareto e q u i l i b r i a ; U1, U a r e s e t s o f
2 2
u n s t a b l e outcomes. The s e r i e s of moves (A1, A2)-A3-(A4, A )-A6 r e p r e s e n t s a 5
p r o c e s s of c o n f l i c t e s c a l a t i o n i n which b o t h p l a y e r s i n s i s t on f i s h c a t c h e s
z1=6, z2=6
.
-f i s h i n g i n t h e i r w a t e r s ( x 2 ) ; i f t h e r e - i s t r i c t i o n s a r e d i s o b e y e d , b o t h t h e o f f e n d i n g c o u n t r y and t h e e n f o r c i n g c o u n t r y i n c u r ad- d i t i o n a l c o s t s . The payoff f u n c t i o n s h a l e t h e form:
where (.)+ d e n o t e s t a k i n g t h e p o s i t i v e p a r t , aiO-ailx: r e p r e s e n t s t h e n e t g a i n from t h e c o u n t r y ' s own w a t e r s ( d i m i n i s h i n g w i t h
i
f
i n c r e a s e d f i s h i n g b y o u t s i d e r s ) , ai2x1-ai3x2 r e p r e s e n t s t h e n e t g a i n from f i s h i n g i n f o r e i g n w a t e r s ( d i m i n i s h i n g w i t h i n c r e a s e d r e s t r i c -
t i o n s ) , t h e l a s t b u t one t e r m r e p r e s e n t s t h e p e n a l t i e s f o r d i s o b e y i n g t h e r e s t r i c t i o n s of o t h e r c o u n t r i e s and t h e l a s t t e r m t h e c o s t of e n f o r c i n g i t s w n r e s t r i c t i o n s . A l l deci- s i o n v a r i a b l e s , r ' ( x : , x i ,x: ,x:)c
n4
a r e c o n s t r a i n e d byfiki
1 1 '
T h i s exampleservestoillustratethemathe-
m a t i c a l a n d computationaldifficultiesin- v o l v e d i n d e t e r m i n i n g sets of n o n c o o p e r a t i v e e q u i l i b r i a ; 3 however, t h e s e d i f f i c u l t i e s c a n beovercomeand t h e i m a g e o f thegame, t o g e t h e r w i t h t h e set of Nash e q u i l i b r i a N i s shown
q
'
i n F i g . 3 f o r aiO-2.4, ail-6, ai2-1, ai3=2, ai4=ai5=4, -1 ~ ~ " 0 . 8 , j , i = 1 , 2
.
TheNash e q u i l i b r i a i n t h i s c a s e have a r a t h e r s i m p l e i n t e r p r e t a t i o n : t h e y c o r r e s p o n d t o t h e s i t u a t i o n i n which e a c h c o u n t r y s t r i c t l y obeys t h e r e s t r i c t i o n s of o t h e r s . The Nash outcomes of thisgamearenotParetooutcomes; P a r e t o outcomes c o r r e s p o n d t o d r o p p i n g r e s t r i c t i o n s e n t i r e l y , o r , a t t h e p o i n t Po, t o t h e complete c e s s a t i o n o f f i s h i n g i n f o r e i g n w a t e r s . It i s i n t e r e s t i n g t o n o t e t h a t t h e p o i n t Po h a s finallybeenreachedthrough t h e h i s t o r i c a l development of f i s h i n g p r a c t i c e s .4
T h i s i s e q u i v a l e n t t o a min-max problem i n - v o l v i n g nondif f e r e n t i a b l e f u n c t i o n s t h a t do n o t remain convex a f t e r t h e f i r s t maxim- i z a t i o n ; s e e P a r t I I o f t h i s p a p e r .
The p o i n t P i s n o t a Nash e q u i l i b r i u m f o r 0
t h e o n e p e r i o d g a m e , b u t c a n b e s h o w n t o b e a n e v o l u t i o n a r y s t a b l e e q u i l i b r i u m f o r a repe- t i t i v e g a m e . T h u s w e m i g h t h o p e t h a t a l l ' p r i s o n e r ' s dilemmas'
,
c h a r a c t e r i z e d by t h e d i f f e r e n c e of Nash a n d P a r e t o p o i n t s , w i l l f i n a l l y b e r e s o l v e d i n a n e v o l u t i o n a r y way ( s e e a l s o H o f s t a d t e r , 1983). I n t h e c a s e o f t h e a r m s r a c e , h o w e v e r , i t i s b u t s m a l l c o n s o l a t i o n f o r u s t o hope t h a t o t h e r r a c e s i n t h e u n i v e r s e m i g h t l e a r n f r o m o u r own e v o l u t i o n a r y m i s t a k e s .However, i n t h e c o u r s e of t h i s h i s t o r i c a l p r o c e s s t h e r e have a l s o been c a s e s i n which t h e w o r s t p o i n t SD h a s b e e n r e a c h e d . T h i s p o i n t is a t t a i n e d when b o t h c o u n t r i e s d e c i d e t o f i s h a s much a s p o s s i b l e i n e a c h o t h e r ' s w a t e r s and, a t t h e same t i m e , t o impose and
t r y t o e n f o r c e extreme r e s t r i c t i o n s on any- body f i s h i n g i n t h e i r own waters--a c a s e of open f i s h i n g war. There a r e c o n c e p t s i n game t h e o r y t h a t e x p l a i n t h e development of t h i s s i t u a t i o n , a l b e i t i n a r a t h e r s i m p l i f i e d f a s h i o n .
An o l d c o n c e p t i n game t h e o r y i s t h a t of a S t a c k e l b e r g e q u i l i b r i u m ( s e e , e .g.
,
Aubin, 1979). Suppose one of t h e p l a y e r s h a s enough i n f o r m a t i o n t o compute t h e re- s p o n s e s of t h e o t h e r p l a y e r s (who wish t o maximize t h e i r own p a y o f f s ) t o any of h i s own d e c i s i o n s . I f t h e r e s p o n s e s a r e non- unique, he c a n assume, t o b e on t h e s a f e s i d e , t h a t o n l y t h o s e t h a t c o n t r i b u t e l e a s t t o h i s own p a y o f f w i l l b e chosen. These r e s p o n s e f u n c t i o n s u n i q u e l y d e t e r m i n e t h e dependence o f h i s own payoff o n h i s own d e c i s i o n s , t a k i n g i n t o a c c o u n t t h e r e s p o n s e s of o t h e r s , and h i s own p a y o f f c a n t h e n b e maximized. A p l a y e r who makes h i s d e c i - s i o n s i n s u c h a w a y i s c a l l e d t h e ( S t a c k e l b e r g ) l e a d e r ; i f o t h e r p l a y e r s respond a s pre- d i c t e d , t h e y a r e c a l l e d ( S t a c k e l b e r g ) f o l - l o w e r s ; t h e r e s u l t i n g outcome i s c a l l e d t h e S t a c k e l b e r g e q u i l i b r i u m ( t h i s i s one o f t h e Nash e q u i l i b r i a , chosen t h r o u g h t h e ( s a f e ) maximization of t h e p a y o f f o f t h e l e a d e r ) . I n t h e example c o n s i d e r e d h e r e , i f t h e f i r s t p l a y e r wants t o b e t h e l e a d e r , h e c o n c l u d e s t h a t by s e n d i n g t h e l a r g e s t pos- s i b l e f l e e t t o f i s h i n h i s o p p o n e n t ' sw a t e r s and by imposing t h e s e v e r e s t p o s s i b l e r e s t r i c t i o n s o n i n t r u d e r s i n t o h i s ovn w a t e r s , h e might f o r c e t h e o t h e r p l a y e r t o
"follow" him. I n d e e d , s i n c e b o t h e n f o r c i n g r e s t r i c t i o n s and v i o l a t i n g them a r e v e r y c o s t l y i n t h i s game, t h e second p l a y e r m i g h t maximize h i s own s h o r t - t e r m i n t e r e s t s by imposing o n l y t h e minimal r e s t r i c t i o n s com- p a t i b l e w i t h t h e f l e e t of t h e l e a d e r ( o r even d r o p p i n g r e s t r i c t i o n s a l t o g e t h e r - - b u t t h e l e a d e r c a n n o t count on t h i s ) and s e n d i n g o n l y t h e s m a l l e s t p o s s i b l e f l e e t t o t h e l e a d e r ' s w a t e r s .
T h i s i n t e r p r e t a t i o n shows, however, t h a t t h e r e a s o n i n g of t h e S t a c k e l b e r g l e a d e r i s completely u n r e a l i s t i c i f no a d d i t i o n a l l e g a l o r i n s t i t u t i o n a l c i r c u m s t a n c e s f o r c e t h e o t h e r p l a y e r t o become a f o l l o w e r . A s o v e r e i g n c o u n t r y would n o t a c c e p t t h e f o l - l o w e r ' s r o l e and would denounce a s hypo- c r i t i c a l t h e e x p l a n a t i o n s o f t h e a s p i r i n g l e a d e r t h a t t h e f o l l o w e r ' s r o l e i s l o g i c a l f r o m t h e p o i n t o f v l e w o f e c o n o m i c p a y o f f s . I n t h e example c o n s i d e r e d h e r e , t h e second p l a y e r m i g h t w e l l r e s p o n d b y r e p e a t i n g t h e a c t i o n s of t h e f i r s t - - t h i s would r e s u l t i n a So-called S t a c k e l b e r g d i s e q u i l i b r i u m ( a s i t u a t i o n i n w h i c h b o t h p l a y e r s t r y t o b e c o m e t h e l e a d e r ) and c o r r e s p o n d s t o an open f i s h i n g war i n o u r example.
F i g . 3. Image o f t h e game: q l = f l ( x ) 2 1 2 1 2 2 1 2.4-6x1+x1-2x2-4 [(x1+x2-l)++(x 1 +x 2- I ) + ]
F i g . 4. S a t i s f i c i n g game e q u i l i b r i a t h a t c o u l d b e s e l e c t e d by p l a y e r 2: SC r e p r e s e n t s a 2
c o n s t r u c t i v e s a t i s f i c i n g move; SD2 r e p r e s e n t s a h i d d e n d e s t r u c t i v e s a t i s f i c i n g move; OD r e p r e s e n t s a n o p e n l y d e s t r u c t i v e move. s(q-q) r e p r e s e n t s a maximized
2
f u n c t i o n t h a t h e l p s t o s e l e c t a c o n s t r u c t i v e s a t i s f i c i n g s t r a t e g y .
Thus, t h e concepts of S t a c k e l b e r g l e a d e r - s h i p and S t a c k e l b e r g d i s e q u i l i b r i u m e x p l a i n how open c o n f l i c t s can occur--however, t h e e x p l a n a t i o n i s n o t completely s a t i s f a c t o r y s i n c e c o n f l i c t s do n o t u s u a l l y develop t o t h i s s c a l e immediately. H i s t o r i c a l e v i d e n c e shows t h a t i f one c o u n t r y were t o send i t s f i s h i n g f l e e t t o a n o t h e r ' s w a t e r s , t h e o t h e r country would n o t n e c e s s a r i l y r e c i p - r o c a t e ; i n o r d e r t o s e c u r e i n t e r n a t i o n a l s u p p o r t , t h e i n j u r e d c o u n t r y would p r e f e r t o l i m i t t h e f i s h i n g war t o i t s own w a t e r s . We t h e r e f o r e need some a d d i t i o n a l c o n c e p t s t h a t could e x p l a i n t h e p r o c e s s e s of con- f l i c t e s c a l a t i o n and d e - e s c a l a t i o n .
SATISFICING BEHAVIOR AND SELECTION OF GAME EQUI- LIBRIA
We have a l r e a d y s e e n t h a t c o n f l i c t es- c a l a t i o n might be caused by some p l a y e r s having i n f l a t e d a s p i r a t i o n s , 1.e.. wishing t o exceed t h e l i m i t s imposed by t h e n a t u r a l ( p h y s i c a l , b i o l o g i c a l , e t c . ) c h a r a c t e r i s t i c s o f t h e game. We s h o u l d t h e r e f o r e c o n s i d e r a t y p e of decision-making b e h a v i o r r e l a t e d t o t h e concept of a s p i r a t i o n levels-- s a t i s f i c i n g b e h a v i o r . As o r i g i n a l l y in- troduced (Simon, 1959). t h i s t y p e of b e h a v i o r i s based on t h e i d e a t h a t a d e c i s i o n - maker does n o t o p t i m i z e due t o u n c e r t a i n t y
about v a r i o u s a s p e c t s of t h e problem, in- c l u d i n g t h e r e a c t i o n s of o t h e r d e c i s i o n - makers, and i s t h u s s a t i s f i e d i f h e r e a c h e s c e r t a i n adap t i v e l y formed a s p i r a t i o n l e v e l s f o r s p e c i f i c o b j e c t i v e s . T h i s does n o t ex- p l a i n , however, what a decision-maker would do i f c e r t a i n outcomes were below h i s as- p i r a t i o n l e v e l s ; we might s a f e l y assume t h a t h e would f i r s t try t o r e a c h h i s as- p i r a t i o n l e v e l s (which i s e q u i v a l e n t t o maximizing t h e outcomes i f they a r e below t h e s e l e v e l s - - s e e Kortanek and P f o u t s , 1982) and would t h e n e i t h e r r e s t a f t e r a c h i e v i n g h i s t a r g e t s , o r change h i s as- p i r a t i o n s . T h i s t y p e o f b e h a v i o r w i l l be c a l l e d strict s a t i s f i m ' n g ; a l t h o u g h o f c o n - s i d e r a b l e i n t e r e s t , S i t is s t i l l n o t s u f - f i c i e n t l y f l e x i b l e t o e x p l a i n t h e p r o c e s s of c o n f l i c t e s c a l a t i o n .
A more f l e x i b l e concept i s t h a t of q w s i - s a t i s f i c i n g b e h m ~ i o r : p l a y e r s maximize t h e i r o b j e c t i v e s , but w i t h a g r e a t e r in- t e n s i t y below t h e i r a s p i r a t i o n l e v e l s t h a n above them. Mathematically, s u c h a d i s - t i n c t i o n seems t o have no s e n s e : payoff maximization b e h a v i o r i s n o t changed by t h e i n t e n s i t y of maximization, and t h e s e t of Nash e q u i l i b r i a i s n o t changed by assuming q u a s i - s a t i s f i c i n g b e h a v i o r . However,quasi- s a t i s f i c i n g b e h a v i o r might i n f l u e n c e t h e way a p l a y e r sele-cts a Nash e q u i l i b r i u m : having a t t a i n e d h i s a s p i r a t i o n l e v e l , he might devote h i s remaining freedomof a c t i o n t o some o t h e r p u r p o s e , s u c h a s c o n s t r u c t i v e l y
See P a t t I1 of t h i s paper.
p r e v e n t i n g c o n f l i c t e s c a l a t i o n by l e t t i n g o t h e r p l a y e r s maximize t h e i r o b j e c t i v e s , o r d e s t r u c t i v e l y h u r t i n g o t h e r p l a y e r s by t r y i n g t o n e g a t i v e l y a f f e c t t h e i r o b j e c t i v e s . I n t h e example from t h e p r e v i o u s s e c t i o n , t h e s a t i s f i c i n g Nash e q u i l i b r i a f o r p l a y e r 2, who h a s some a s p i r a t i o n l e v e l
y2,
a r e a l l of t h e Nash e q u i l i b r i a above and i n - c l u d i n g t h e l i n e q 2 = t 2 ( s e e F i g . 4 ).
S e l e c t i o n of t h e p o i n t SC2 t h a t s a t i s f i e s q =q and i s a l s o good f o r t h e o t h e r p l a y e r
2 2
i s a c o m t r u c t i v e s a t i s f i c i n g s t r a t e g y ; s e l e c t i o n of t h e p o i n t OD2, which is t h e w o r s t p o s s i b l e f o r t h e o t h e r p l a y e r , i s an openly d e s t r u c t i v e ~ t r a t e g y ; ~ s e l e c t i o n of t h e p o i n t SD2 t h a t s a t i s f i e s q =p b u t i s
2 2 t h e w o r s t c h o i c e f o r t h e o t h e r p l a y e r on t h i s l i n e is a hidden d e s t r u c t i v e satis- f i c i n g s t r a t e g y . The i n t e r p r e t a t i o n of t h e d i f f e r e n c e between c o n s t r u c t i v e and hidden d e s t r u c t i v e s a t i s f i c i n g s t r a t e g i e s i n t h e example c o n s i d e r e d i s q u i t e i n t e r e s t i n g . The p a r a m e t e r s of t h e example a r e s u c h t h a t f i s h s t o c k s a r e a l r e a d y h e a v i l y d e p l e t e d and f i s h i n g i n c o a s t a l w a t e r s h u r t s t h e h o s t c o u n t r y more t h a n i t b e n e f i t s t h e f i s h i n g c o u n t r y . A c o n s t r u c t i v e satis- f i c i n g s t r a t e g y i s t h e n t o d e c r e a s e a s much a s p o s s i b l e your c a t c h i n t h e c o a s t a l w a t e r s o f o t h e r s ( b e a r i n g i n mind y o u r economic a s p i r a t i o n s ) , w h i l e imposing t h e s t r i c t e s t p o s s i b l e r e s t r i c t i o n s on o u t s i d e r s f i s h i n g i n your own w a t e r s . A hidden
d e s t r u c t i v e s t r a t e g y i s t o a c h i e v e t h e same economic a s p i r a t i o n l e v e l by f i s h i n g as much a s p o s s i b l e i n t h e c o a s t a l w a t e r s of o t h e r s and imposing o n l y such r e s t r i c t i o n s on f o r e i g n e r s f i s h i n g i n your own w a t e r s as a r e n e c e s s a r y t o a t t a i n your a s p i r a t i o n l e v e l . Each hidden d e s t r u c t i v e s t r a t e g y can b e ' r a t i o n a l i z e d ' by invoking some seemingly p l a u s i b l e argument, f o r example,
' w e b e l i e v e i n t h e freedom of f i s h i n g and r e s t r i c t i t o n l y o u t of economic n e c e s s i t y ' ; n e v e r t h e l e s s , i t s t i l l remains d e s t r u c t i v e i n t h e e y e s of t h e o t h e r p l a y e r .
A s a t i s f i c i n g game e q u i l i b r i u m c a n be s e l e c t e d u n i l a t e r a l l y when t h e a s p i r a t i o n l e v e l s of a p a r t i c u l a r p l a y e r and t h e type of a c t i o n t o be t a k e n ( c o n s t r u c t i v e , h i d d e n d e s t r u c t i v e , e t c . ) a r e known; i f t h e m u l t i p l e o b j e c t i v e s o f t h e o t h e r s i d e a r e t o b e t a k e n i n t o a c c o u n t , i t i s a l s o n e c e s s a r y t o have a t l e a s t e s t i m a t e d a s p i r a t i o n l e v e l s f o r t h e o t h e r s i d e . I n f a c t , no m a t t e r whether you want t o b e c o n s t r u c t i v e o r d e s t r u c t i v e , you must have some i d e a of t h e a s p i r a t i o n s of t h e o t h e r player-say, what economic and what e c o l o g i c a l r e s u l t s would s a t i s f y him;
o n l y when you assume ( s i m p l i s t i c a l l y ) t h a t
-
I n t h i s c a s e ( a l t h o u g h n o t n e c e s s a r i l y i n g e n e r a l ) , t h e openly d e s t r u c t i v e s t r a t e g y OD c o i n c i d e s w i t h t h e S t a c k e l b e r g
2
maximizing s t r a t e g y .
t h e o t h e r p l a y e r h a s o n l y a s i n g l e ob- j e c t i v e c a n you d i s r e g a r d h i s a s p i r a t i o n s . A s a t i s f i c i n g game e q u i l i b r i u m f o r a g i v e n m a t h e m a t i c a l model of t h e game c a n a l s o b e computed by maximizing an a p p r o p r i a t e f u n c - t i o n over t h e s e t of Nash ( o r Pareto-Nash i n t h e m u l t i o b j e c t i v e c a s e ) game equi- l i b r i a . I n t h e s i m p l e example c o n s i d e r e d h e r e , t h e c o n s t r u c t i v e s a t i s f i c i n g o p t i o n f o r p l a y e r 2 w i t h a s p i r a t i o n l e v e l
i2
canbe computed by s o l v i n g t h e f o l l o w i n g pro- lem :
1
-
maximize
-
(f 2(x)-q2)+-
a
Pwhere
~ m t f i : min 4(x,y)-0) ;
+x
-and p>>1 i s a c o e f f i c i e n t . I f w e d e n o t e
1
-
s ( q - i ) = (q2-a2)+
-
p(qiq2)+ + q1 9t h e n t h e e q u i v a l e n t problem max s ( q - a ) qaq
can b e i n t e r p r e t e d i n outcome s p a c e a s shown i n F i g . 4. We s e e t h a t i t i s neces- s a r y t o maximize a n o n d i f f e r e n t i a b l e func- t i o n o v e r a non-convex set: a l t h o u g h t h i s i s a d i f f i c u l t problem, i t i s n o t beyond t h e c a p a b i l i t i e s of modern o p t i m i z a t i o n t e c h n i q u e s ( s e e Nurminski, 1982; Demyanov, 1983).
CONFLICT ESCALATION AND DE-ESCALATION
F i g u r e 5 i l l u s t r a t e s a p r o c e s s of c o n f l i c t e s c a l a t i o n , s t r u c t u r e d u s i n g t h e concept of s e l e c t i n g s a t i s f i c i n g e q u i l i b r i a . We assume t h a t b o t h p l a y e r s have chosen s a t i s f i c i n g c o n s t r u c t i v e s t r a t e g i e s
-
SC1, SC2 a t some { 1,q2
,
b u t t h e s e as- p i r a t i o n l e v e l s a r e m u t u a l l y i n c o m p a t i b l e and t h u s t h e o v e r a l l r e s u l t of t h e s e two d e c i s i o n s , OO,
i s n o t an e q u i l i b r i u m p o i n t . Suppose t h a t p l a y e r 1 d e c i d e s , i n t h e n e x t round, t o i n f l u e n c e t h e a s p i r a - t i o n s of p l a y e r 2 downwards by choosing h i d d e n d e s t r u c t i v e SD1 ; p l a y e r 2 d o e s n o t change h i s s t r a t e g y , and t h e j o i n t outcome i s O1.
However, s i n c e p l a y e r 2 w i l l pro- b a b l y r e c o g n i z e t h e d e s t r u c t i v e c h a r a c t e r of t h e move of p l a y e r 1 , he w i l l n o t n e c e s s a r i l y respond by r e v i s i n g h i s as- p i r a t i o n s downwards; he might r a t h e r choose a n openly d e s t r u c t i v e p o l i c y OD2,
which'
See P a r t I1 o f t h i s p a p e r .l e a d s t o t h e outcome 0 i n t h e n e x t round.
2
I f p l a y e r 1 r e c i p r o c a t e s , t h e n e x t round r e s u l t s i n a n open f i s h i n g war,
o3
• BY p l a y i n g a game of t h i s t y p e s t r u c t u r e d u s i n g a s p i r a t i o n l e v e l s and t h e d i s t i n c t i o n between c o n s t r u c t i v e and d e e t r u c t i v e be- h a v i o r , p a r t i c i p a n t s c a n l e a r n much a b o u t t h e dynamics of c o n f l i c t e s c a l a t i o n . S i m i l a r t e c h n i q u e s c a n b e used t o i l l u e t - r a t e c o n f l i c t d e - e s c a l a t i o n ; however, i m - p l i c i t o r e x p l i c i t n e g o t i a t i o n s o r medi- a t i o n a r e needed f o r c o n f l i c t d e - e s c a l a t i o n , w e n i f t h i s o n l y i n v o l v e s u n i l a t e r a l d e c i - s i o n s . We u s e t h e t e r m " i m p l i c i t nego- t i a t i o n s " t o d e s c r i b e a u n i l a t e r a l s t a t e - ment from one of t h e p l a y e r s t h a t h e wants t o d e - e s c a l a t e c o n f l i c t ; h e t h e n m o d i f i e s h i s a c t i o n s i n a n t i c i p a t i o n of s i m i l a r be- h a v i o r from t h e o t h e r s i d e . I n e x p l i c i t n e g o t i a t i o n s , however, h e would d i s c u s s s u c h a c t i o n s w i t h t h e o t h e r s i d e b e f o r e a c t u a l l y implementing them. Mediation i n - v o l v e s a t h i r d p a r t y who a s s i s t s i n t h e n e g o t i a t i o n s . These well-known s i t u a t i o n s can be i l l u s t r a t e d by a gaming model used i n a d e c i s i o n - s u p p o r t mode. Even i f t h e a c t u a l a s p i r a t i o n l e v e l s of e a c h p l a y e r s h o u l d b e t r e a t e d a s c o n f i d e n t i a l by t h e d e c i s i o n - s u p p o r t system ( p r e c i s e knowledge of t h e a s p i r a t i o n s of t h e . o t h e r s i d e g i v e s a p l a y e r a s t r a t e g i c a d v a n t a g e ) , d e c i s i o n - s u p p o r t e d gaming h e l p s t h e p l a y e r s b o t h t o a d j u s t t h e i r am a s p i r a t i o n s and t o l e a r n a b o u t t h e a s p i r a t i o n s of t h e o t h e r s . A s i m p l e p r o c e s s of t h i s t y p e is i l l u s - t r a t e d i n F i g . 6. Suppose b o t h p l a y e r s have chosen c o n s t r u c t i v e s a t i s f i c i n g moves SC and SC,
b u t t h e i r a s p i r a t i o n191 2.1
l e v e l s a r e f a r from b e i n g c o m p a t i b l e and t h e r e s u l t of t h e i r d e c i s i o n s i s OO. Sup- p o s e , t h r o u g h i m p l i c i t n e g o t i a t i o n s o r under t h e i n f l u e n c e of t h e m e d i a t o r , p l a y e r
1 d e c i d e s t o r e v i s e h i s a s p i r a t i o n s down- ward and chooses a c o n s t r u c t i v e s a t i s - f i c i n g s t r a t e g y SC1,2 ; t h i s l e a d s t o O 2 i n t h e n e x t round, which i s s l i g h t l y b e t t e r f o r p l a y e r 1 and much b e t t e r f o r p l a y e r 2.
The c r u c i a l p o i n t i s w h e t h e r p l a y e r 2 w i l l r e c i p r o c a t e ; i f h e d o e s , c h o o s i n g SC
2,2 which l e a d s t o outcome O 2
,
t h e n p l a y e r 1 might b e m o t i v a t e d t o go f u r t h e r , s e l e c t i n g SC1,3 and outcome 0 ; p l a y e r 2 may t h e n3 s e l e c t SC and outcome 0
2.3 4 ' A t t h i s
p o i n t , t h e d i f f e r e n c e s between a s p i r a t i o n l e v e l s and a c t u a l outcomes a r e s o low t h a t t h e p l a y e r s can a g r e e t o a c c e p t t h e o u t - come O4 a s a n e g o t i a t e d s t a t u s quo.
An a c t i v e m e d i a t o r c a n b e h e l p f u l even when p l a y e r s w i l l n o t c o n s i d e r m u l t i l a t e r a l ac- t i o n s and proceed u n i l a t e r a l l y ; i n f a c t , a m e d i a t o r might even be n e c e s s a r y t o s t i m - u l a t e c o n f l i c t d e - e s c a l a t i o n through uni- l a t e r a l a c t i o n . I f t h e m e d i a t o r e n j o y s t h e
F i g . 5 . A c a s e of c o n f l i c t e s c a l a t i o n . S C 1 , S C represent s a t i s f i c i n g c o n s t r u c t i v e moves 2
f o r p l a y e r s 1 and 2 ; S D 1 , S D represent s a t i s f i c i n g (hidden) d e s t r u c t i v e moves 2
f o r p l a y e r s 1 and 2 ; OD1, OD represent openly d e s t r u c t i v e moves f o r players 1 and 2
2 . The c o n f l i c t e s c a l a t i o n process i s represented by ( S C 1 ' S C 2 ) 3 0 ; ( S D 1 , S C 2 ) q 1 ; ( S D 1 , O D 2 ) q 2 ; (OD1, O D 2 ) q 3
.
F i g . 6 . A c a s e of c o n f l i c t d e - e s c a l a t i o n : ( s c l , l , s c 2 , l)*o; ( s c 1 , 2 , s c 2 , 1 ) q 1 ; ( S C 1 , 2 . S C 2 , 2 ) q 2 ; ( S C 1 , 3 , S C 2 , 2 ) q 3 ; ( S C 1 , 3 , S C 2 , 3 ) q 4
.
c o n f i d e n c e of b o t h s i d e s and knows t h e i r a s p i r a t i o n l e v e l s , h e c a n m e d i a t e i n t h e choice of game e q u i l i b r i a ( F i g . 7 ) . To do t h i s , h e c o u l d u s e t h e i n f o r m a t i o n con- t a i n e d i n two p o i n t s : t h e c u r r e n t outcome of t h e game, p o i n t O,,
,
w i t h o u t c o m e l e v e l s- -
q1,q2
,
and t h e c u r r e n t a s p i r a t i o n l e v e l s- -
q l , q 2
.
A game e q u i l i b r i u m s e l e c t e d v i a m e d i a t i o n c a n be d e f i n e d , f o r example, a sa s o l u t i o n of t h e problem:
a = max emin ( f i ( x ) - i i )
/
(Qi-Gi) ( 7 )a
1-1,2where N i s d e f i n e d a s i n ( 6 ) . The v a l u e of a (measured a s t h e d i s t a n c e between t h e c u r r e n t a s p i r a t i o n l e v e l s and t h e c u r r e n t outcome) g i v e s some i d e a of how much b o t h s i d e s s h o u l d d e c r e a s e t h e i r a s p i r a t i o n l e v e l s i n o r d e r f o r t h e p l a y e r s t o uni- l a t e r a l l y s e l e c t m u t u a l l y c o n s i s t e n t game e q u i l i b r i a .
C l e a r l y , t h e p l a y e r s m i g h t do much b e t t e r by a g r e e i n g on m u l t i l a t e r a l a c t i o n ,
l e a d i n g t o outcomes i n t h e neighborhood of t h e p o i n t Po
.
However, e x p l i c i t nego- t i a t i o n s o r m e d i a t i o n a r e r e q u i r e d t o ob- t a i n agreement on m u l t i l a t e r a l a c t i o n (such a s t h e h i s t o r i c agreement on s o v e r e i g n t y o v e r c o a s t a l w a t e r s ) .MEDIATION TO OBTAIN AGREEMENT
ON MULTILATERAL AmION
Attempts t o r e a c h agreement o n m u l t i - l a t e r a l a c t i o n always s t a r t w i t h a n a s s e s s - ment of t h e n o n c o o p e r a t i v e s t a t u s quo. The s t a t u s quo i s n o t n e c e s s a r i l y a r e s u l t of c o n f l i c t d e - e s c a l a t i o n , s u c h a s t h e p o i n t O,, i n F i g . 6 ; i t c o u l d j u s t a s w e l l be t h e r e s u l t of c o n f l i c t i n g s t r a t e g i e s , s u c h a s t h e p o i n t OO
.
However, a s t a t u s quo can always be e s t a b l i s h e d i n a gaming model, i f o n l y by r e p e a t e d gaming s i m u l a t i o n s . A c t u a l n e g o t i a t i o n s o f t e n p r o c e e d w i t h o u t a m e d i a t o r , and a r e s t r u c t u r e d by t h e pro- p o s a l s o r p o s i t i o n s of t h e p a r t i e s i n - v o l v e d ; t h e s e p o s i t i o n s a r e s u c c e s s i v e l y m o d i f i e d i n t h e c o u r s e of n e g o t i a t i o n s . While t h e dynamics of s u c h n e g o t i a t i o n s a r e , i n g e n e r a l , a v e r y i n t e r e s t i n g t o p i c( s e e R a i f f a , 1982), w e s h a l l l i m i t o u r a t - t e n t i o n h e r e t o n e g o t i a t i o n s i n v o l v i n g a m e d i a t o r . By a g r e e i n g t o t h e u s e of a gaming model t h a t r e p r e s e n t s some sub- s t a n t i v e ( p h y s i c a l , b i o l o g i c a l , e t c . ) as- p e c t s of t h e system, t h e p a r t i e s i n v o l v e d a l s o i m p l i c i t l y a g r e e t o t h e model a s m e d i a t o r . As a s t a r t i n g p o i n t f o r p o s s i b l e f u r t h e r i n v e s t i g a t i o n s , we s h a l l a l s o assume t h a t t h e p a r t i e s i n v o l v e d a l s o a g r e e on some r u l e s of f a i r n e s s f o r d i v i d i n g t h e b e n e f i t s g a i n e d from m u l t i l a t e r a l a c t i o n ,
and on t h e u s e and assessment of t h e pro- p o s a l s g e n e r a t e d by t h e d e c i s i o n and m e d i a t i o n s u p p o r t system a c c o r d i n g t o t h e s e a c c e p t e d r u l e s of f a i r n e s s ( s e e F i s h e r and Ury
,
1981).There a r e many p r i n c i p l e s by which common g a i n s o r c o s t s measured i n comparable u n i t s c a n be d i v i d e d f a i r l y among t h e p a r t i c i - p a n t s : t h e r e i s t h e c l a s s i c a l p r i n c i p l e
'one d i v i d e s , t h e o t h e r c h o o s e s ' ; t h e S t e i n h a u s g e n e r a l i z a t i o n of t h i s p r i n c i p l e t o many p a r t i c i p a n t s ; and many o t h e r pro- c e d u r e s of v a r i o u s t y p e s ( s e e Young e t a l . , 1982). T h e r e a r e a l s o many ways of
d e f i n i n g c o o p e r a t i v e solutions t o s i n g l e - payoff games ( t h e c o n c e p t s of c o r e ,
n u c l e o l u s , Shapley v a l u e , e t c . - - s e e S h a p l e y , 1965; Aubin, 1979); however, t h e s e c o n c e p t s a r e n o t e a s i l y extended t o m u l t i o b j e c t i v e games. A t y p e of c o o p e r a t i v e s o l u t i o n t h a t can be r e l a t i v e l y e a s y e x t e n d e d t o m u l t i - o b j e c t i v e games was proposed f i r s t by R a i f f a and t h e n by K a l a i and Smorodinsky
(1975); w e p r e s e n t a m o d i f i c a t i o n of t h i s i d e a a s a p o s s i b l e p r i n c i p l e f o r f a i r d i v i s i o n of j o i n t g a i n s made i n a m u l t i - o b j e c t i v e game.
Assume t h a t a n o n c o o p e r a t i v e s t a t u s quo h a s been r e a c h e d i n t h e game ( f o r example, t h r o u g h s i m u l a t i o n gaming, o r noncoopera- t i v e gaming w i t h d e c i s i o n s u p p o r t )
.
Letif
d e n o t e t h e s t a t u s quo v a l u e of t h e j - t hJ
o b j e c t i v e o r outcome f o r t h e i - t h p l a y e r , and assume t h a t a l l outcomes s h o u l d . be maximized. Then t h e d e c i s i o n s u p p o r t sys- tem can compute t h e maximum p o s s i b l e i n - creaseinthevalueofeachobjective f o r e a c h p l a y e r , assuming c o o p e r a t i o n between
p l a y e r s :
i
..
f--f
-
max f x ;x
= : f J ( x ) h j,
j
f i g
jf o r a l l f , j
#
i , j l (8)These maximum c o o p e r a t i v e i n c r e a s e s Aq i a r e t a k e n a s u n i t s i n t h e a l e used t o
1
a l l o c a t e j o i n t g a i n s . According t o t h i s r u l e , t h e c o o p e r a t i v e g a i n s q ( x ) i
-
..ij j
r e s u l t i n g from some m u l t i l a t e r a l d e c i s i o n x a r e a l l o c a t e d f a i r l y i f
f o r a l l i , j (9) where
P 4
and P d e n o t e s t h e number of o b j e c t i v e s i
c o n s i d e r e d by each p l a y e r . The X! a r e J
c o r r e c t i v e s c a l i n g c o e f f i c i e n t s s p e c i f i e d by each p l a y e r a f t e r r e v i e w i n g proposed
