Van Damme (Oct 87) claims that renegotiation and stability may be mutually inconsistent concepts. This note shows that this claim may be incorrect and proposes a way to restore the compatibility of these two concepts that, in general, do not apply in the same context.Renegotiation-proof equilibrium; Pareto perfect equilibrium; Stable set of equilibria;

Ferreira, José Luis

English

Research Papers in Economics

Working Paper 93-41  Departamento de Economia 
Economics Series  16  Universidad Carlos III de  Madrid 
December  1993  Calle Madrid,  126 
28903 Getafe (Spain) 
Fax (341) 624-9875 
ON THE POSSIBILITY OF STABLE RENEGOTIATION.￿ 
A NOTE￿ 
Jose Luis  Ferreira'￿ 
Abstract  _ 
Van  Damme  (Oct  87)  claims  that  renegotiation  and  stability  may  be  mutually  inconsistent 
concepts.  This  note  shows that  this  claim may  be incorrect and proposes a way  to  restore the 
compatibility of these two concepts that,  in general, do not apply in the same context. 
Key  Words￿ 
Renegotiation-proof equilibrium, Pareto perfect equilibrium, stable set of equilibria.￿ 
•  Departamento  de  Economia,  Universidad  Carlos  III  de  Madrid.  Correspondence  to: 
Departamento de Economia, Universidad Carlos III de Madrid, Calle Madrid, 126,28903 Getafe 
(Madrid), Spain. 1.￿  INTRODUCTION 
This note discusses the relation between the concepts of renegotiation-proof and stable 
equilibria. In particular, the question about existence of stable renegotiation is addressed. This 
discussion is motivated by an example with which van Damme claims that the answer to this 
question is  negative.  The example is a fInite horizon game of two players, the defInition of 
renegotiation-proof he uses  is  the Pareto perfect equilibrium (PPE, widely accepted for this 
class  of games)  and  the  definition of stability  is  the  one  given by  Kohlberg  and  Mertens 
(1986)  (see  appendix).  The concept of Pareto perfect equilibrium was  first  introduced  by 
Farrell and Maskin (1989) and by Bernheim and Ray  (1989). Here we present the definition 
using the notation in van Damme (Oct 87): 
DEFINITION 1  Let  G  be a normal form game and let  G(T)  be the T-fold repetition of 
G.  Then  s  is  a PPE of  G(1)  if it  is a Nash equilibrium (NE) of  G  that is  not (strictly) 
Pareto dominated by another equilibrium. Inductively,  s  is a PPE of  G(t+ 1)  if 
(1)  s￿  is  a Nash equilibrium of  G(t+ 1) 
(2)  s￿  continues with a PPE of  G(t)  after each period 1 history, and 
(3)  there does not exist  s'  satisfying (1) and (2) that strictly Pareto dominates  s. 
Now we can present van Damme's example: consider the bimatrix game  G  in figure  1, 
and  let  G(2)  be the 2-fold repetition of  G. 
2 FIGURE 1￿ 
Write  A=(a,a),  B=(b,t3)  and  C=(c,'Y)  and consider the following strategy pair  s* 
in  G(2): 
rB  after A￿ 
(*)  at  t=l  play  A  at  t=2  play  {￿ 
l C  otherwise￿ 
Clearly  s*  satisfies definition 1.  This strategy results in a payoff of 14 for each player. 
Van-Damme  shows  that  all  PPEa  of  G(2)  yield  the  same  path  as  s*,  and  that  this 
equilibrium component cannot be  stable.  The intuitive reason is  as  follows: 
"Player 1 is  'guaranteed' the payoff 14 if he does not deviate from  (A,B). If 
he, however, deviates to  b  at  t= 1, then player 2 should conclude that 1 will 
not play  c  as  b  followed by  c  yields player 1 at most  12 and this is  less 
than what this player gets in equilibrium. Furthermore,  1 will not play  a  in 
the second round as this is dominated by  b.  Hence, player 2 should conclude 
that  1 will  again play  b  after a deviation, but then he should  respond with 
t3  (his best response) and 1 will indeed gain from deviating. The equilibrium 
is not self-enforcing, Le  it  is  not stable". (van Damrne (Qct 87)) 
In another place, van Damrne notes that 
3 "stability is a pure non cooperative concept, hence, it requires that all aspects 
relevant to the situation be modeled by the rules of  the game. This is important 
as it will turn out that stable equilibria depend crucially on modeling 'details'. 
Hence, stability is inconsistent with 'small words' arguments and if  one cannot 
model all  details, then stability should not be used as  the  solution concept." 
(van Danune (Aug 87». 
However, in van Danune (Oct 87), he confronts the notion of stability with a definition 
of renegotiation-proof equilibrium, the Pareto perfect equilibrium (PPE), which, in his own 
words  "seems especially justified if players have the opportunity to conununicate during the 
game".  In order  to  be  able  to  compare  both concepts  in  the  same  context,  van  Danune 
appeals to Schelling's (1960) principle of tacit bargaining to conclude that "if the requirement 
(of PPE) is really compelling, then ... players should accept the same concept also in the case 
in  which  no  such  conununication  is  possible.  (Especially  in  case  there  is  a  unique 
renegotiation-proof equilibrium, as  in our example)." 
In this  note  I will  try  to  show that the  incompatibility between the  two concepts may 
disappear once the following two claims are realized: 
CLAIM 1.  The principle of tacit bargaining does not apply in general when conununication 
may take place. 
CLAIM 2.  In the particular cases when the principle seems to be applicable, the PPE may 
not be the  "good"  definition of a renegotiation-proof equilibrium if one considers stability 
4 issues. 
2.  DISCUSSION OF CLAIM 1 
If  a definition of equilibrium is  to be  interpreted as  a theory of how players would 
behave, it is clear that any suggested strategy profile that does not conform to that definition 
will  not be obeyed:  players will  find  some  incentive to  deviate.  This is  true for the  Nash 
equilibrium and all  its  refinements.  For instance,  if players believe in the theory of perfect 
Nash equilibria, some player will unilaterally deviate from a non-perfect Nash equilibrium 
because he or she will be afraid of "trembling hand" moves by other players. However, the 
same cannot be said about the PPE if the game is  to be played without communication. To 
see this, consider the game in figure 2: 
FIGURE 2 
Only  (m,c)  and  (b,r)  are PPE.  Ifplayers believe in the theory of PPE, they will choose 
among these two pairs of strategies,  but if they cannot communicate with each other,  (t,l) 
cannot be discarded  as  an equilibrium.  Even if they  are  fervent believers  of the  PPE and 
know which are the PPEa of the game, they have no way to move from  (t,l)  if this strategy 
is  somehow proposed first:  they need communication to go to either  (m.c)  or  (b,r).  The 
knowledge of the theory of PPE is useful in general only if communication exists (that was 
5 precisely the  motivation for  renegotiation-proof equilibria).  Another examples of this  kind 
may  occur  when  coalitions  are  pennitted  to  deviate  (e.g.  in  the  Coalition-proof  Nash 
equilibrium or in the  strong Nash equilibrium concepts):  it may  happen that a coalition is 
necessary  to deviate from a "bad"  equilibrium, but if several coalitions are possible, again 
communication  is  necessary  to  decide  which  one  will  be  fonned.  For  very  particular 
examples,  however  it  is  true that communication is  not  necessary  to  apply  the  PPE.  For 
example,  in  figure  3,  the  knowledge  and  acceptance  of the  theory  of PPE make  players 
simultaneously move from  (t,l)  to  (b,r). This occurs, of course because there is only one 
place to  go according to  the theory. 
FIGURE 3 
Therefore,  in general,  we  need communication to  apply  the  definition of PPE as  a 
theory that predicts players' behavior, but then, following van Damme, stability in the sense 
of Kohlberg and Mertens is  not a relevant concept and we do  not have any  incompatibility: 
stability simply does not apply. 
I  will  finish  this  section with  two  more  examples  to  show  how  stability  issues  (and 
forward  induction arguments) may be  ruled out when communication may take place.  The 
first  one  is  the  example  given  in van Damme  (Qct  87),  it will  also be  analyzed  without 
communication (as  van Damme did) in the discussion of claim 2 in the following section. 
6 
...._---_. --_._-----._-----------_._----_._--------Let  us  consider,  then,  van  Damme's  example:  if the  PPE  (s"')  is  proposed  and 
communication can take place, then we find that player 1 will not deviate: if he does deviate 
at the first stage (as  in the discussion for stability in van Damme), at the beginning of the 
second stage he will find that player 2 will recall the first agreement (s"') which implies that 
C  will follow,  since it is  an equilibrium. Of course,  player 1 may defend his new proposal 
B  based on his last deviation, but since  B  is now viewed as a deviation from  C,  it has no 
chance to survive because not all players involved (read player 2) find it attractive. 
As a last example consider the game of battle of sexes in figure 4. 
FIGURE 4 
If player 2 has the opportunity to  "bum a dollar" before playing, it is well known that there 
is only one stable equilibrium in which player 2 does not bum the dollar and  (t,l)  follows. 
With communication,  however,  (b,r)  is  still  plausible:  if  (b,r)  is  decided and  player 2 
bums the dollar to show that he  will play aggressively afterwards  (to induce  (t,I», at the 
beginning of the second stage he will hear from player 1 something like:  "Ok, you burnt a 
dollar,  so  what?  we  planned to play now  (b,r)  and we  shall do that way  since  it is  an 
equilibrium; your deviation is  worthless so you better follow the equilibrium path. " 
3.  DISCUSSION OF CLAIM 2 
7 For  the  cases  in  which  the  principle  of tacit  bargaining  is  applicable  and  no 
communication is necessary to have a PPE as a result (e.g. when only one such equilibrium 
exists as  in the game in figure 3 above),  it may well be the case that a PPE is not stable. An 
illustration of this  is  van Damme's example,  but one  can argue  that if that game  is  to  be 
played without communication and  if players believe in the theories of both Renegotiation-
proof and stable equilibria, then it is not clear why at the beginning they will choose  s,  the 
Pareto  optimal  within  the  subgame  perfect  equilibria.  They  are  now  aware  of forward 
induction  arguments  and  know  that  a deviation  will  occur,  therefore,  the  PPE is  not  the 
"good"  renegotiation-proof equilibrium in this case.  Players should better concentrate their 
attention on the set of strategy profiles that are  "optimal within the set of stable equilibria". 
The formal definition of this "Pareto Stable equilibrium", using van Damme's notation, is as 
follows: 
DEFINITION 2.  s  is a Pareto stable equilibrium (PSE) of G(1)  if and only if it is a stable 
equilibrium (SE) of  G  that is not strictly dominated by any other equilibrium. Inductively, 
s  is  a  PSE  of  G(t+ 1)  iff 
(1)  s  is a  SE  of  G(t+1) 
(2)  s  continues with a  PSE  of  G(t)  after each period 1 stage history and 
(3)  there does not exist  s'  satisfying  (1)  and  (2)  that strictly Pareto dominates 
s. 
8 REMARK  1.  SE is the definition of stable equilibrium by  Kohlberg and Mertens. 
REMARK 2.  Since  SEa  always exist when the sets of strategies are compact, the existence 
of  PSEa  is  immediate for those games with a compact set of outcomes in SEa. 
REMARK 3.  In van Darnme's example,  it is easy  to check that the only two  PSEa  are: 
(i)  play  (m,c)  in the fIrst period and  (b,l)  in the second after any history and 
(ii)  play  (b,l)  in the first period and  (m,c)  in the second after any  history. 
5.  FINAL COMMENTS 
Pareto perfectness is a definition that is very generally applicable.  Stability, however, 
is  only meaningful in a very special kind of situations.  To make the point simpler, consider 
van Damrne's condition for stability (in the sense of forward induction) in two-player games: 
"A  solution concept  S  is  consistent with forward  induction on the class  of 
generic 2-person games  if  pE S  for  any  path  p  for  which  there exists a 
player  i  who by  unilaterally deviating from  p  can enforce that a sUbgame 
is reached for which exactly one solution (according to  S)  yields this player 
more than  p  does  and  for which all  other solutions  yield this  player less". 
(van Darnme (Aug 87». 
It is clear that, for this definition to be applicable (in the sense that it restricts the set 
of solutions), one needs a coincidence to happen (" .. , exactly one solution ... "). Furthermore, 
the  cases  in which both PPE and  SE are  applicable concepts  are even more  restricted:  in 
9 addition, one needs the coincidence that makes the principle of tacit bargaining adequate (as 
in the discussion of claim 2). As a result, only in a very particular set of games both concepts 
can be contrasted. It has been shown that, in this case, the definition of PPE can be modified 
to be a "reasonable" renegotiation-proof equilibrium concept when players accept the theory 
of stability  (they  interpret  deviations  from  a  Nash  equilibrium,  not  as  mistakes,  but  as 
something to be rationalized). 
10￿ APPENDIX 
Kohlberg and Mertens (1986): 
"We will  say  that a set of equilibria is  stable  in a game  G  if it  is  minimal with 
respect to the following property: 
PROPERTY (8):  8  is a closed set of Nash Equilibria of G  satisfying: for any  E > 0  there 
exists some  00>0  such that for any completely mixed strategy vector  "Y1, ...  ,"Yn  (n players) 
and for any  01, ... ,On'  (O<Oj<oo),  the perturbed game where every strategy  s  of player 
i  is  replaced by  (1-0j)s +Oi"Yi  has an equilibrium €-closed to  8. 
11￿ REFERENCES 
Bernheim, B.D. and D. Ray (1989):  "Collective Dynamic Consistency in Repeated Games", 
Games and Economic Behavior,  1,  295-326. 
Farrell, J.  and E. Maskin (1989): "Renegotiation in Repeated Games", Games and Economic 
Behavior,  1,  327-360. 
Kohlberg,  E.  and  J.  L.  Mertens  (1986):  "On  the  Strategic  Stability  of  Equilibria"; 
Econometrica 54,  1003-1037. 
Schelling, T.  C.  (1960):  "The Strategy of Conflict"; Harvard Univ.  Press. 
van Damme, E. (August 1987):  "Stable Equilibria and Forward Induction"; Discussion Paper 
No.  A-128, SFB 303  Universitat Bonn. 
van Damme,  E.  (October  1987):  "The  Impossibility of Stable  Renegotiation";  Discussion 
Paper No.  A-136,  SFB 303  Universitat Bonn. 
12￿ Player  2 
Q  P  7  . 
(1  10,13  -5,0  -5,0 
Player  1  b  11,-5  4,1  0,0 
c  0,-5  0,0  1,4 
Figure  1 
Player  2 
1  c  r 
2  0  0 
t 
2  0  0 
Player  1  I!l 
0  3  0 
0  3  0 
b 
0  0  3 
0  0  3 
Figure  2 
13 Player  2 
1  r 
2  0 
t 
2  0 
Player  1  .-
0 3 
b 
0 3 
Figure  3 
Player  2 
1 r 
3 0 
t  1  0 
Player  1 
0 1 
b 
0 3 
Figure  4 
14 

On the possibility of stable renegotiation. A note.

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On the possibility of stable renegotiation. A note.

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