SEGUNDA PARTE

the set o f states in class 1 ifd < 2 q * -l, and states 2^ and 2^ if d>2q^-l.

Proof:

The only difference between the cost o f and A^ is in the transition

1 2

between classes 2 and 3. Since T2 3>T32 (as q * > —), C always has a lower cost.

This leaves two candidates for minimum cost k-tree, A^ and C^. The cost o f A^ is less than the cost o f if r2i<ri2. Hence class 1 has the lowest cost k-tree if

(l4-d)(l-q*) < (l-d)q* => d < 2q *-l.

If the inequality is reversed then class 2 has the minimum cost k-tree. QED. The long-run equilibria are illustrated in figure 2.6. Hence the long-run equilibria are the set o f states where everyone plays S2, the risk-dominant strategy

d=2q*-l 0.5 Key A' : 1 <--- 2<r 2< 1 ^ 3 Figure 2.6 Minimum cost j-trees.

increases with the degree o f risk-dominance. If d is above this critical value then class 2 has the minimum cost k-tree. The long-run equilibria are the two states where the two islands play different equilibria. In fact, from the symmetry o f the cost structure, states 2^ and 2® will each have a probability o f one half in the limit- distribution. It is easy to see why higher values o f d upset the risk-dominant equilibrium. In class 2, the island playing the payoff-dominant equilibrium becomes more populated as d increases because in equilibrium it is full to capacity. The transition to class 1 therefore becomes more difficult.

By the same token the island playing the risk-dominant equilibrium in class 2 becomes smaller as d increases and so easier to convert. Hence the transition from class 2 to class 3 where both islands play the payoff-dominant equilibrium becomes easier. However, we never observe class 3 as the long-run equilibrium. The reason for this is that although the cost o f class 3 is decreasing, the cost of class 2 is also decreasing and is always less. Consider the minimum costs o f the

tr a n sitio n s 1 ^ 2 and 3 — > 2 . A s d in c r e a se s th e sm a lle st p o s s ib le s iz e o f an islan d ,

( l - d ) N , fa lls. In c la s s e s 1 and 3 th e tra n sitio n to a sta te w h e r e o n e isla n d h as a

p o p u la tio n s iz e o f ( ( l - d ) N ) h a s z e r o c o s t . T h e tra n sitio n to c la s s 2 th en o n ly

req u ires e n o u g h m u ta tio n s o n th e sm a ll isla n d . T h e m in im u m c o s t th e r e fo r e falls

as d in c r e a se s.

2.2.2 Three islands.

W e n o w e x te n d th e m o d e l to th e c a s e o f 3 isla n d s. A s s u m e th a t th e g lo b a l

p o p u la tio n is n o w 3 N . W e o n ly c o n s id e r th e c a s e s w h e r e th e re is a p o s itiv e

p o p u la tio n o n e a c h isla n d . H e n c e th e c a p a c ity c o n str a in t is ( l + d ) N w h er e

0 < d < Y • A ll o th er d e ta ils o f th e m o d e l are th e sa m e . T h e state s p a c e is n o w

>S' = { ( — —— --- ,71, e (0,1,...,71,),7l2 E ( 0 , 1 , . . . , Tl^), 71, « 2 3 A - 7 1 , -7 Î2

7I3 G ( 0 ,1 ,...,3 A -7 1 , - 7I2 ), A (1 - 2üf) < 71,, 7%2 < A ( l + Û?)}.

w h e r e 7 1* is th e n u m b er p la y in g str a te g y Si o n islan d i an d n. is th e n u m b er o f

a g e n ts o n is la n d i. D e n o te th e sta te o f th e s y s t e m b y s = ( q i,q2,q3,n i,n2)G S w h e r e qi

is th e p ro p o rtio n o f isla n d i p la y in g Si.

A s b e fo r e th e d y n a m ic s g iv e rise to a M a r k o v p r o c e s s , P^, o n sta te sp a c e

S . F r o m an y initial c o n d itio n th e s y s te m w ill m o v e to o n e o f th e recu rren t

c o m m u n ic a tio n c la s s e s w h ic h are c h a r a c te r ised latter. A n e le m e n t o f th e p ertu rb ed

M a r k o v p r o c e s s is n o w

3 N

3 N - k k=l

From propositions 2.1 and 2.2 we know that to find the long-run equilibria w e need to find the recurrent communication class that has the lowest cost k-tree. To do this we need to characterise the recurrent communication classes and the costs o f moving between them.

Recurrent communication classes.

One recurrent communication class is the set o f all states where qi=q2=q3=0. The basin o f attraction is {(qi,q2 ,qs): q i^ q * , q2^ q*, q s^ q*} since

best replies will lead both islands to the risk-dominant equilibrium. In this class the system will move between states where qi=q2=q3=0 and ni,n2e (N (l-2 d ), N (l+ d ))

since agents move with a positive probability when they are indifferent and n\

must lie in this range due to the capacity constraint.

N ow consider any initial condition with q i> q * , q2<q* and qs<q*. Best

replies will move the system towards qi =1, q2=0, q3=0. This will result in movement into island 1, since the higher payoff equilibrium is being played there. The system will eventually move to the class (l,0,0,N (l+ d ),n2) where n2G (N (l-2d), N(l4-d)). Similarly the set o f states with q i< q * , q2^q* and q3<q*

form the basin o f attraction o f the class (0,l,0,nj,N (l-i-d)) where ni G (N (l-2d ), N(l-kd)) and the set o f states with q i< q * , q2<q* and q3>q* form

the basin o f attraction o f the class (0,0,l,ni,n2) where ni G (N (l-2d ), N(l-i-d)) and ni4-n2=N(2-d). Hence there are three classes where one island plays the efficient equilibrium and the other two play the risk-dominant one.

There are also three classes where two islands play the efficient equilibrium and one plays the risk-dominant one. The basin o f attraction o f the

class where the first tw o islands play the efficient equilibrium and the third island plays the risk-dominant one is, with q i> q * , q2^q* and q3<q*. Similarly the set

o f states with q i< q * , qz^q* and qs>q* form the basin o f attraction o f the class (0 ,l,l,N (l-2 d ),N (l+ d )) and the set o f states with q i> q * , qz<q* and qs>q* form the basin o f attraction o f the class (l,0 ,l,N (l+ d ),N (l-2 d )).

The final possibility is for all three populations to play the payoff-dominant equilibrium. The basin o f attraction is { ( q i, qz, qs): q i^ q * , qz^q*, q s^ q * }, and the recurrent communication class is the set o f all states with qi =qz=q3= l and

n i,n z e(N (l-d ), N (l+ d )). The eight recurrent communication classes are illustrated in figure 2.7.