R-core Implementation
Chen-Ying Huang
Tomas Sjöström
April, 2002
AbstractWe argue that a new solution concept by Huang and Sjöström (2002), called the r-core, extends the concept of core to games in partition function forms naturally. Moreover, the r-core can be implemented by a modi…ed Perry and Reny (1994) game in a straightforward manner. We show that: 1) every stationary subgame perfect Nash equilibrium (SSPNE) outcome must be in the r-core; 2) with the additional assumption of superadditivity, every r-core can be implemented as an SSPNE.
1 INTRODUCTION
The concept of core is one of the most important cooperative solutions because of its natural appeal. Competition underlying core is characterized by the un-hampered ability with which players can freely sign some binding agreement. Thus an allocation is in the core if no blocking coalition will form to object.
Because of its very intuitive appeal, two extensions seem relatively possible and important. The …rst regards how to extend the core concept to situations where externalities prevail across coalitions. The classical starting point of the core, namely, the characteristic function, simply assumes away this important interaction. A natural and parsimonious starting point is the partition function where externalities are incorporated. Accordingly, solution concepts like the ®-, ¯- or °- cores are de…ned. However, they are often criticized for either being too pessimistic or arbitrary.
The second important extension concerns how to non-cooperatively imple-ment the core. This task seems imminent if the concept is to stand soundly. Kalai, Postlewaite and Roberts (1979) is one of the earliest attempts. However, they show that the set of the strong Nash equilibria coincides with the core. Since the idea of strong equilibria assumes the formation of pro…table blocking coalitions, whether it truly provides a non-cooperative foundation of the core is doubtable. Other contributions include Chatterjee et. al.’ (1990) which consid-ers an extension of Rubinstein’s bargaining game. They show only some points in the core are implementable. Moldovanu and Winter (1991), on the other hand, implements the core via a class of games. In our view, the implementa-tion of the core is not well settled until Perry and Reny (1994). They provide a non-cooperative game where the moves and counter-moves closely mimic the spirit of competition behind the core, that is, players have unhampered abil-ity to sign binding agreement. They show that for a superadditive game: 1) any Stationary Subgame Perfect Nash Equilibrium (SSPNE) is in the core; 2) with the additional assumption of totally balancedness, an allocation can be supported as an SSPNE outcome if and only if it is in the core.
Since we argue that the …rst extension hasn’t been quite settled while the second extension has been well done by Perry and Reny (1994), a qualifying …rst extension must satisfy two properties. First, it must extend the core concept to partition function forms. Second, since Perry and Reny (1994) successfully implements the core in superadditive games in characteristic function form, a modi…ed Perry and Reny (1994) game must also implements the extension at least in superadditive games in partition function form.
This is exactly the purpose of this paper. We shall argue that a new solution concept by Huang and Sjöström (2002), called the r-core,1 satis…es these two
properties straightforwardly. First, the r-core extends the core to partition function forms naturally. Suppose a set of players, denoted N, is contemplating a possible ”core” allocation. In order to know whether this allocation is immune to coalitional blocking, every coalition S µ N has to know its value. However
the value of S is not de…ned in partition function forms. To solve this, the r-core postulates the following. Since S is calculating its value (which is what it can achieve independently on its own), it must predict what will happen to NnS once S has formed. Because the norm of the society is the core, members in S simply believe that players in NnS will play according to the core. This is typically not the end of the story. To predict what the core of NnS is, every coalition T µ N nS has to know its value. The r-core assumes that players in Nn(S [ T) play according to the core and so on. This naturally yields a recursive de…nition of the core where the core solution concept is used to de…ne coalitional values at every recursive step. Second, the r-core, as just roughly de…ned, can be implemented by a modi…ed Perry and Reny (1994) game in a straightforward manner. We will show that: 1) every SSPNE outcome must be in the r-core; 2) with the additional assumption of superadditivity, every r-core can be implemented as an SSPNE. Moreover, the construction of the proof is a straightforward extension of that in Perry and Reny (1994). In our view, this lends extra credit to the r-core being a natural extension of the core because if we believe that Perry and Reny well captures the non-cooperative spirit of the core and its slight modi…cation implements a new solution concept quite straightforwardly, then this new solution concept must have preserved some nice spirit of the core to make its implementation so easily.
(maybe we should mention EBA here)
The rest of the paper is organized as follows. In the next section, we provide an overview of the r-core. For more details, please refer to Huang and Sjöström (2002). The third section lays out the modi…ed Perry and Reny (1994) game informally. The fourth section formally de…ned the game. We prove two main theorems in the …fth section. In the six section we conclude by brie‡y discussing relaxing the assumption of superadditivity.
2 OVERVIEW OF THE R-CORE
Consider a situation in which the underlying opportunities can be summarized by a transferable utility game in partition function form, (P; N), where N = f1; 2; :::; ng is the set of players and P is the partition function. Precisely, for every partition on the player set N, PN, for any coalition S in PN, P (S j
PN) 2 R++ is the value that S can achieve when players partition themselves
according to PN.2 Restricting our attention to the partition function form is the
…rst natural step to allow for externalities across players since the value of S, unlike in the characteristic function form, crucially depends on what coalitional structure other players N nS form into.
A payo¤ vector x ´ (xi)i2N is called feasible under the partition PN if for
every coalition S in PN,Pi2Sxi = P (S j PN). Let C(S j S; PNnS)denote the
r-core for coalition S given others N nS partition into PNnS. As in the core,
in order to determine the r-core, we …rst need to determine the value of each sub coalition T µ S. Let V (T j S; PNnS)denote the value of each sub coalition
T µ S when we are considering the r-core for coalition S given PNnS. We can
now recursively de…ne the r-core.
For any i 2 N, and any coalition structure PNnfigfor the players in N nfig,
the value of fig is clearly what it can achieve when other players partition into PNnfig. Thus
V (fig j fig; PNnfig) ´ P (fig j fig; PNnfig). (1)
The r-core for the coalition fig given others Nnfig partition into PNnfigis the
set of feasible payo¤ vectors such that coalition fig is getting at least its value. Formally,
C(fig j fig; PNnfig) ´ fx 2 Rn: xis feasible under the partition (fig; PNnfig(2))
and xi ¸ V (fig j fig; PNnfig)g:
2Alternatively, we can assume P (S j P
N) 2 R+ or even allow P (S j PN)to take negative
Now make the induction hypothesis that we have de…ned C(S j S; PNnS)for
all S such that 1 jSj k ¡ 1 where jSj stands for the cardinality of S, and all partitions PNnSof NnS.
Consider S such that jSj = k. The value of S is clearly what it can achieve when other players partition into PNnS. Thus
V (S j S; PNnS) ´ P (S j S; PNnS). (3)
For any non-empty proper sub coalition T ½ S, since players in NnS have partitioned into PNnS, to determine its value, we need to know what coalition
structure players in SnT will form. This is predicted by the r-core for coalition SnT given other players partition into (T; PNnS), C(SnT j SnT; T; PNnS). Thus
V (T j S; PNnS) ´ minfx(T ) : x 2 C(SnT j SnT; T; PNnS)g. (4)
We take the minimum because there might be multiple coalitional structures that players in SnT could form since as long as every sub coalition in SnT is getting at least its value, the concept of the core makes no further selection. Taking the minimum gives us the least value that coalition T can get when SnT is playing according to the r-core.
We can now de…ne the r-core for S given others form PNnSsince according
to the concept of the core, every coalition has to get at least its value and their values have been de…ned by (3) and (4). Hence
C(S j S; PNnS) ´ fx 2 Rn: there exists some partition PS of S such that:(5)
1) x is feasible under the coalition structure (PS,PNnS);
2) x(S) ¸ V (S j S; PNnS);
3) x(T ) ¸ V (T j S; PNnS)for any non-empty proper sub coalition T ½ Sg:
We have now de…ned C(S j S; PNnS) for S such that jSj = k given any
partition PNnS. Continuing in this fashion, we can de…ne C(S j S; PNnS)for
larger and larger S. The …nal step occurs when S = N. Although the method in the …nal step is the same as in the preceding steps, it may be helpful to describe
this last step explicitly. First, V (N j N)3 is what it can achieve. Therefore
V (N j N) ´ P (N j N):
For any non-empty proper sub coalition T ½ N, to determine its value, we need to know what coalition structure players in NnT will form. This is predicted by the r-core for coalition NnT given others have formed T, C(NnT j NnT; T). Thus
V (T j N) ´ minfx(T ) : x 2 C(NnT j NnT; T )g.
Then, C(N j N) is the set of payo¤s where every coalition is getting at least its value. Therefore,
C(N j N) ´ fx 2 Rn: there exists some partition PN of N such that:
1) x is feasible under the coalition structure PN;
2) x(N) ¸ V (N j N);
3) x(T ) ¸ V (T j N ) for any non-empty proper sub coalition T ½ Ng: We sometimes call C(N j N) the r-core because it is what we ultimately care about. The r-core for smaller coalitions are derived in order to recursively derive the r-core for N. Besides, in the process of de…ning the r-core for S given PNnS in (5), we incidentally get some coalition structures where S could
possibly form. The set of these coalition structures of S in C(S j S; PNnS)will
be denoted by P(S j S; PNnS).
Notice that for the r-core of N to exist, it must exist for every S given any PNnS. If at any step of the recursive de…nition, the r-core for some S
given some PNnS does not exist, then the r-core for any larger coalition is not
de…ned because presumably we need to use C(S j S; PNnS)to predict what S
will do. This makes the r-core more di¢cult to exist than the ®- and ¯- cores.
3V (N j N) should be written V (N j N; P
N nN)to confo rm to (3). But PN nN is the trivial
partitioning of the empty set since N nN = ;; so we will suppress PN nN here and in the
However, when it exists, it is a subset of the ®- and ¯- cores.4 More importantly,
rightly because of its recursive nature, it has some perfection built in already. Recall that Theorem 2 in Perry and Reny (1994) requires the game to be totally balanced. The property of totally balanced is necessary because for an SSPNE to exist, on the o¤-equilibrium path when some coalitions have left, an equilibrium among the remaining players has to exist. However, this equilibrium naturally corresponds to the core among the remaining players. Thus the game has to be totally balanced. On the other hand, in a parallel Theorem of the r-core, we do not need to impose some extra property a la totally balanced. This is because by the recursive nature of the r-core, when it exists, the r-core for any S given any PNnS exists. Thus on the o¤-equilibrium path when players NnS have left
according to PNnS, the remaining players of S can simply follow the r-core for
Sgiven PNnS. This shall become clear when we construct equilibria to support the r-core.
3 THE EXTENSIVE FORM GAME
In this section we will lay out the extensive form game informally. This section follows Section 2 of Perry and Reny (1994) closely. We choose to follow their notations as closely as possible so that the results could be directly comparable. The game starts at t = 0 and time is continuous. At any point of time, a player can either: 1) make a proposal; 2) accept the current proposal; 3) stay quiet or 4) leave.5
A proposal ((wi)i2S; S)by any player who hasn’t left consists of a division
rule and a coalition S. A division rule is simply a way to distribute S’s value where wi 2 R+ represents i’s share of the value of S. Thus, Pi2Swi = 1and wi¸ 0 for all i 2 S. Notice that in Perry and Reny, a player proposes a payo¤ allocation instead of a division rule. Proposing a payo¤ allocation does not make
4Refer to Huang and Sjöström (2002 ) for details.
5Except that at the very beginning of the game when t = 0, players can only choose either
sense here because unless all other players N nS have left and consumed, players in S might not have any idea about what coalitional structure that others will form. By externalities across coalitions, the coalitional structure determines the value of S, which in turn determines the possible payo¤ allocations. Proposing a division rule avoids this di¢culty because it speci…es a way to distribute the value of S for all possible coalitional structures that might form. When a proposal is made, it is e¤ective as long as no new proposal is made. Upon a new proposal is made, the previous proposal is no longer e¤ective. To avoid the simultaneous proposals of distinct proposals, when this happens, no new proposal is e¤ective. So there is at most one e¤ective proposal at any point of time.
When an e¤ective proposal ((wi)i2S; S)is accepted by all members in S, it
becomes binding. In this case, S is a binding coalition. If any player in S chooses to leave, then all players in S leave at the same time. At any point in time, there might be multiple binding proposals among players who haven’t left. If a new proposal contains any player in any binding coalition, then it must contain all players in that binding coalition. This re‡ects the idea that annulment of a binding coalition has to be approved by every member in it. To avoid the problem where a player is involved in two di¤erent binding proposals, at the time when a new binding proposal, containing some binding coalitions, becomes binding, the old corresponding binding proposals are annulled. Hence, at any point in time, all binding coalitions are disjoint. Players consume according to their binding proposals only when all have left. Notice that when all players leave, the coalitional structure is uniquely de…ned. Players simply divide the value of the coalition to which they belong according to the binding proposals they have signed. If some player never leaves the game, all players get ¡1.6
6One can relax this rather strong assumption. For instance, if some binding coalitions have
left while others remain in the game forever, it might be argued that although the remaining players might get the worst p ossible payo¤s, any leaving coalition should at least get its value in the coalitional structure where all the leaving coalitions have formed and the remaining players form into the worst p ossible coalitional structure for this coalition in consideration. Allowing this do es not change the result.
Finally, we assume that for every time t and every history up to t, there exists an " > 0 and two open intervals (t ¡ "; t) and (t; t + ") in which every strategy of every player asks him to be quiet. Perry and Reny (1994) explain the role that this assumption plays in their Example 1.
4 THE MODEL
We now introduce the notations, the formal game rules and the equilibrium concept. As in the previous section, we choose to follow the notations in Perry and Reny (1994) as closely as possible so that readers familiar with that paper can understand this paper more easily. We also try to make the paper as self-contained as possible so that readers, who haven’t read Perry and Reny (1994) before, will not have problems in understanding this paper. We follow Perry and Reny (1994) to divide this section to four sub sections to deal with histories, payo¤s, strategies and the equilibrium concept respectively.
4.1 HISTORIES
A proposal, as stated in the previous section, is a division rule and a coalition, ((wi)i2S; S). Thus, the set of feasible proposals, denoted by P , is
P ´ f((wi)i2S; S) : S µ N;
X
i2Swi= 1and wi¸ 0 for all i 2 Sg:
Denote a the choice to accept the current e¤ective proposal, q the choice to be quiet and l the choice to leave. A history for player i up to time t > 0, is a function hi such that
hi : [0; t) ! P [ fa; q; lg: Since players can only leave once and for all, h¡1
i (l)is either empty or singleton.
We follow Perry and Reny (1994) to assume that h¡1
i (P [fag) is a …nite set. At
t = 0, since nothing has happened, hi(0) = ;. For convenience, denote a history
up to time t by the n-tuple of functions h ´ (h1; h2; :::; hn). Let H(t) denote
the set of all histories up to time t and H ´ [1
We need to introduce more notations.
Let p(h) denote the current e¤ective proposal according to h. To make it well-de…ned, if according to h, either no proposal has been made, multiple distinct proposals are simultaneously made, the most recently e¤ective proposal has become binding or some member in a binding coalition which is also involved in the most recently e¤ective proposal has exercised to leave, then p(h) ´ ;.7
Let ¿(h) for h 2 H(t) denote the amount of time that has passed up to time tsince p(h) was proposed. Whenever p(h) = ;, ¿(h) measures the time that has passed since the previous e¤ective proposal becomes binding. When there is never any e¤ective proposal, ¿(h) measures the time that has passed since time 0.
Let N(h) µ N denote the set of players who have not left and A(h) µ N(h) the set of players who have accepted p(h). Whenever p(h) = ;, A(h) ´ ;.
Player i is said to have accepted the current e¤ective proposal p(h) for h 2 H(t)if p(h) is made at time t < t and hi(t0) = afor some t02 (t; t). If everyone
involved in p(h) has accepted it, then p(h) is binding. The coalition associated with p(h) is called a binding coalition.
Let ¦(h) denote the set of binding proposals among the players in N(h). Since there exists externalities across coalitions, we also need to keep track of those binding coalitions that have left. Let C(h) denote coalitional structure that has formed by the players in NnN(h) according to h. Notice that Perry and Reny (1994) do not need to keep track of C(h) since in characteristic function form the remaining players’ values are not a¤ected by the coalitional structure those leaving players form. On the other hand in partition function form, the coalitional structure those leaving players form a¤ects the values of the
remain-7W hen some member in a binding coalition which is also involved in the most recently
e¤ective proposal has exercised to leave, we need to reset p(h) to an empty set to avoid the following from happening. Suppose players 1, 2 and 3 remain and the current proposal pertains to them. Suppose player 1 and 2 have accepted the current proposal and 3 hasn’t. Supp ose coa lition {3} is binding. If 3 exercises to leave, then by the rules in Perry and Reny, if p(h) is not reset to an empty set, players 1 and 2 cannot do anything further and they have to stay in the game forever.
ing players a great deal. Finally, let hjt denote the restriction of h 2 H(s) to [0; t)where s ¸ t.
4.2 PAYOFFS
If at some point of time, all players have left so that N(h) = ;, that must imply a partition PN on the player set N has formed. In this case, every binding
coalition distributes its value according to its binding division rule. For instance, if S 2 PNand its binding division rule is (wi)i2S, then player i 2 S simply gets
wiP (S j PN). If N(h) 6= ; where h 2 H(1), implying someone never leaves the
game, then everyone gets ¡1. The assumption on payo¤s re‡ects the idea that if a proposal ((wi)i2S; S)becomes binding and any player i 2 S leaves, then
in contrast to Perry and Reny (1994) he cannot consume immediately since a coalition’s value crucially depends on the …nal coalitional structure PN that all
players form. All he can guarantee by leaving is that coalition S will be one member of the …nal coalitional structure PN.
4.3 STRATEGIES
A strategy for any player is a function which maps every possible history to an action. Hence, a strategy for any player i 2 N , denoted by fi is:
fi: H ! P [ fa; q; lg:
Denote the n-tuple of strategies by f ´ (f1; f2; :::; fn). There are several
restric-tions on strategies.
(S0) For h 2 H(0), fi(h) 2 P [ fqg, since at the very beginning of the game,
players can only make an proposal or be quiet.
(S1) If i 2 N (h) has accepted the current e¤ective proposal p(h), then fi(h) =
q. That is, before the current e¤ective proposal becomes binding, an accepting player can only be quiet. If i 2 NnN(h), then fi(h) = q. Thus,
(S2) If p(h) = ((wi)i2S; S) and fi(h) = ((wi)i2S0; S0), then either S \ S0 = ;
or S µ S0. Moreover, S0 µ N(h). In words, if a new proposal contains
some players in the current e¤ective proposal, it has to include all of them. Quite naturally, any new proposal can only contain players who haven’t left.
(S3) If a player i is not a member of any binding coalition which hasn’t left, then fi(h) 6= l. That is, a player can only leave when it belongs to a
binding coalition which hasn’t left.
(S4) For all i and t > 0 and for all h 2 H(t) and t 2 [0; t), there exists an " > 0such that fi(hj¿) = q for all ¿ ¸ 0 and ¿ 2 (t ¡ "; t + ")nftg. This
assumption serves to make sure that players always have enough time to respond.8
Lastly, denote player i’s payo¤ induced by the strategy tuple f after h by ui(fjh). Let Fi denote the set of strategies for i which satis…es (S0) to (S4).
4.4 THE EQUILIBRIUM CONCEPT
The equilibrium concept in this paper is the stationary subgame perfect Nash equilibrium (SSPNE). Thus, a strategy pro…le bf ´ ( bf1; bf2; :::; cfn) is an SSPNE
if:
(E1) Perfection: For all i 2 N, h 2 H and fi 2 Fi,
ui( bfjh) ¸ ui((f1; bf2; :::; cfn)jh):
(E2) Stationarity: For h; h02 H, if
(p(h); ¿(h); N(h); A(h); ¦(h); C(h)) = (p(h0); ¿(h0); N(h0); A(h0); ¦(h0); C(h0));
then bf(h) = bf(h0). Notice that we have one more state variable C(h)
because of externalities across coalitions.
8Interested readers can refer to Example 1 and the last paragraph on page 806 in Perry
5 THE THEOREMS
We will prove two parallel theorems to Theorems 1 and 2 in Perry and Reny (1994). The …rst theorem will show that every SSPNE outcome of the extensive form game we set up is in the r-core of (P; N). The second theorem needs some quali…cation. We will show that for superadditive game, every r-core of (P; N) can be supported as an SSPNE outcome. The restriction of superadditivity makes the exposition a lot easier and yet it is more than necessary. We will discuss some way to relax it.
Notice however that the proof of the …rst theorem in this paper is more com-plicated than that of Theorem 1 in Perry and Reny (1994). This is because for Perry and Reny (1994), the value of each coalition is given by the characteristic function. In the r-core, the value of each coalition has to be derived recursively from the partition function instead. Rightly because of so, we …rst need to show that if there exists an SSPNE of the extensive form game, then the value of each coalition is well de…ned. Only until then can we talk about the existence of a r-core which corresponds to this SSPNE outcome.
Proposition 1 Suppose an SSPNE bf exists.
(1) For any S µ N and any partition PNnS on NnS, C(S j S; PNnS) 6= ;.
(2) Let x denote the equilibrium outcome induced by bf where players in NnS have left according to PNnS (i.e., in the subgame where the histories are
(p(h); ¿(h); N(h); A(h); ¦(h); C(h)) = (;; 0; S; ;; ;; PNnS), there must exist a
y 2 C(S j S; PNnS) such that
xi= yi for all i 2 S.
(3) The coalitional structure induced by bf must be (PS; PNnS) where PS 2
P(S j S; PNnS).
Proof. We will proceed by induction.
jSj = 1: For any S = fig and any PNnfig on Nnfig, by (2), C(fig j
fig; PNnfig) 6= ; since by de…nition, it consists of every payo¤ vector where i gets
In the subgame where all players in N nfig have left and formed PNnfig,
apparently fig must …rst propose ((1); fig) and then leave according to bfi. For
otherwise, either i is staying in the game forever and getting ¡1 or he is getting strictly less than P (fig j fig; PNnfig). Neither can be an equilibrium strategy.
Parts (2) and (3) follow immediately.
Suppose by induction that for any jSj k ¡ 1 < n and any PNnS on NnS,
parts (1), (2) and (3) hold.
jSj = k: For any such S and any partition PNnS on N nS, to show part
(1), we …rst need to make sure that V (S j S; PNnS) and V (T j S; PNnS) for
any non-empty proper sub coalition T ½ S are well de…ned. By (3), V (S j S; PNnS) is certainly well de…ned since it is simply P (S j S; PNnS). By (4),
V (T j S; PNnS)is also well de…ned because C(SnT j SnT; T; PNnS) 6= ; by the
induction hypothesis since jSnTj k ¡ 1.
Suppose by contradiction that C(S j S; PNnS) = ;. Let t be the …rst time
that all players in NnS have left according to PNnS and let h 2 H(t0) where
t0 > tand (p(h); ¿(h); N(h); A(h); ¦(h); C(h)) = (;; 0; S; ;; ;; P
NnS). Consider
the continuing equilibrium outcome x induced by bf. Since C(S j S; PNnS) = ;, there must exist a coalition S0µ S such that x(S0) < V (S0j S; P
NnS). Let yi = xi+ V (S0j S; P NnS) ¡ x(S0) jS0j for all i 2 S0; and de…ne wi= yi V (S0j S; P NnS) for alli 2 S 0:
Without loss of generalities, let S0= f1; 2; :::; s0g where jS0j = s0. Consider
at any time t00> t0, any history h 2 H(t00)such that i) p(h) = ((w
i)i2S0; S0), ii)
N(h) = S, iii) A(h) = f1; 2; :::; s0¡ 1g, iv) ¦(h) = ;, v) C(h) = P
NnS and vi)
b
fi(h) = q for all i 2 S.
Claim 2 According to bf, player s0will accept ((w
i)i2S0; S0) and thus ((wi)i2S0; S0)
will become binding.
Proof of Claim. Notice that player s0can accept the current e¤ective
accord-ing to the induction hypothesis, the coalitional structure will be (PSnS0; S0; PNnS)
where PSnS0 2 P(SnS0j SnS0; S0; PNnS). By (4)
V (S0j S; PNnS) P (S0j SnS0; S0; PNnS); because when S0 is calculating its value V (S0 j S; P
NnS), it expects the worst
possible coalitional structure and (SnS0; S0; P
NnS) is not necessarily the worst
possible one. Therefore player s0gets
ws0P(S0j SnS0; S0; PNnS) ¸ ys0 > xs0.
Suppose to the contrary that either no new proposal is made and player s0never
accepts, or some new proposal ((wi)i2S00; S00) is made before player s0 accepts
((wi)i2S0; S0) according to bf. In the …rst possibility, player s0 gets ¡1 which
is impossible. In the second possibility, in the continuing equilibrium, s0 must
get more than ws0P (S0 j SnS0; S0; PNnS) > xs0. By stationarity this means
whenever the history h0yields the states
(p(h0); ¿(h0); N(h0); A(h0); ¦(h0); C(h0)) = (((wi)i2S0; S0); 0; S; ;; ;; PNnS);
player s0 gets strictly more than xs0. But then player s0 could have proposed
((wi)i2S0; S0) at time close enough to t0. This is in contradiction to x being a
continuing equilibrium outcome.
Next consider at any time t00 > t0, any history h 2 H(t00) such that i)
p(h) = ((wi)i2S0; S0), ii) N(h) = S, iii) A(h) = f1; 2; :::; s0¡ 2g, iv) ¦(h) = ;,
v) C(h) = PNnS and vi) bfi(h) = qfor all i 2 S.
Claim 3 According to bf, players s0¡ 1 and s0 will accept ((w
i)i2S0; S0) and
((wi)i2S0; S0) will become binding.
Proof of Claim. Notice that player s0¡1 can accept the proposal to render
the states of the previous claim. Thus he can guarantee himself the payo¤ of ws0¡1P (S0j SnS0; S0; PNnS) ¸ ys0¡1 > xs0¡1.
As in the previous claim, suppose to the contrary either no new proposal is made and player s0¡ 1 never accepts or a new proposal is made before both players
s0¡ 1 and s0 accept. In the …rst possibility, player s0¡ 1 gets ¡1, which is
impossible. In the second possibility, by an analogous argument, when the new proposal is made, player s0¡ 1 must get strictly more than x
s0¡1. But he could
have made that proposal at time close enough to t0.
Proceeding step by step, we can …nally establish that at any time t00 > t0,
any history h 2 H(t00) such that i) p(h) = ((wi)
i2S0; S0), ii) N(h) = S, iii)
A(h) = f1g, iv) ¦(h) = ;, v) C(h) = PNnS and vi) bfi(h) = qfor all i 2 S,
according to bf, players 2 to s0 will accept ((wi)
i2S0; S0)and ((wi)i2S0; S0) will
become binding.
This contradicts that x is an equilibrium outcome induced by bf because player 1 can always propose ((wi)i2S0; S0) and subsequently accept it at time
close enough to t0. In this deviation, he gets at least y
1 > x1. Thus C(S j
S; PNnS) 6= ; and part (1) is proved.
For part (2), if x is the equilibrium outcome induced by bf where players in NnS have left according to PNnS (i.e., in the subgame where the histories are
(p(h); ¿(h); N(h); A(h); ¦(h); C(h)) = (;; 0; S; ;; ;; PNnS), and there does not
exist a y 2 C(S j S; PNnS)such that
xi= yi for all i 2 S,
then there must exist a coalition S0µ S such that x(S0) < V (S0j S; P
NnS). We
can now run over the same argument all over again to get a contradiction. Part (3) follows from Part (2) immediately.
An immediate application of Proposition 1 is summarized in the following theorem to make it directly comparable to Theorem 1 in Perry and Reny (1994). Theorem 4 Every SSPNE outcome is in the r-core of (P; N). More formally, suppose an SSPNE bf exists and induces an equilibrium outcome x. Then x 2 C(N j N ). Moreover, the coalitional structure induced by bf belongs to the r-core structure P(N j N).
A parallel theorem to Theorem 2 in Perry and Reny (1994) can be made easiest with the assumption of superadditivity. We …rst state the assumption, prove the theorem and …nally come back to discuss the assumption after the proof.
A game in partition function form (P; N) is superadditive if for any two non-empty disjoint coalitions S and T and any coalitional structure PNn(S[T) on
the remaining players,
P (S j S; T; PNn(S[T )) + P (T j S; T; PNn(S[T )) < P (S [ T j S [ T; PNn(S[T )):
(6) For example, the symmetric Bertrand competition with di¤erentiated commodi-ties studied in Deneckere and Davidson (1985) satisfy (6). Supperadditivity greatly simpli…es the derivation of the r-core as summarized in the following lemma.
Lemma 5 If a game (P; N) is superadditive, then for any S and PNnS, P(S j
S; PNnS) = fSg provided C(S j S; PNnS) 6= ;.
Proof. If x 2 C(S j S; PNnS) 6= ;, then x(S) ¸ V (S j S; PNnS). Since
V (S j S; PNnS) = P (S j S; PNnS), by superadditivity, if S breaks up, the sum
of payo¤s across S must be strictly lower than P (S j S; PNnS). Thus players in
S must stay together in the r-core.
With the help of Lemma 5, a theorem parallel to Theorem 2 of Perry and Reny (1994) can be constructed. That is, strategies that support r-core out-comes can be provided.
For any S and PNnS, pick a r-core payo¤ vector from C(S j S; PNnS). Denote
this payo¤ vector by x(S j S; PNnS). Hence
x(S j S; PNnS) 2 C(S j S; PNnS):
This is possible because by the recursive de…nition of the r-core, if the r-core exists, then it exists for any reduced society S given PNnS. We now follow Perry
the case where the current proposal is rejected, the other for the situation where the current proposal is accepted. That is, for any history h where
¦(h) = f((w1i)i2S1; S1); ((wi2)i2S2; S2); :::; ((wim)i2Sm; Sm)g;
de…ne zi(h) = 8 > < > : wk i P j2Sk xj(N (h) j N(h); C(h)) if i 2 Skwhere k 2 f1; 2; :::; mg; xi(N(h) j N(h); C(h)) if i 2 N(h)n(S1[ S2[ ::: [ Sm):
Without loss of generality, we can assume that there exists an integer r such that the current proposal p(h) = ((wi)i2S; S) contains all the coalitions Sk
where k r. On the other hand, S is disjoint from all the coalitions Sk where
k ¸ r + 1. Thus when the current proposal gets binding, the resulting new set of binding proposals becomes
b
¦(h) = f((wi)i2S; S); ((wr+1i )i2Sr+1; Sr+1); :::; ((wim)i2Sm; Sm)g:
De…ne b zi(h) = 8 > > > > < > > > > : wiPj2Sxj(N (h) j N(h); C(h)) if i 2 S; wk i P j2Sk xj(N (h) j N(h); C(h)) if i 2 Sk where k 2 fr + 1; :::; mg xi(N(h) j N (h); C(h)) if i 2 N(h)n(S [ Sr+1[ ::: [ Sm):
Intuitively, zi(h) (bzi(h)) will be player i’s payo¤ in the continuing equilibrium
if the current proposal gets rejected (accepted).
We …rst provide the equilibrium strategies and then show that they indeed constitute an SSPNE. The proof is essentially a re-write of that of Theorem 2 in Perry and Reny (1994). This is due to the assumption of superadditivity. This similarity hints that the r-core is a very natural extension of the core, at least for superadditive games.
(i) if ¿(h) is not a positive integer, then fi(h) = q;
(ii) if ¿(h) is a positive integer:
(a) if p(h) = ((wi)i2S; S), and bzj(h) ¸ zj(h) for all j 2 SnA(h), then
fi(h) = 8 < : a; if i 2 SnA(h); q; otherwise; (b) otherwise fi(h) = 8 > > > > > > > < > > > > > > > : l; if i 2 N(h)nA(h) and ¦(h) = f(¢; N(h))g; (( P zj(h) l2N(h) xl(N(h)jN(h);C(h)))j2N(h); N(h)), if i 2 N(h)nA(h) and ¦(h) 6= f(¢; N(h))g; q; if i 2 A(h);
where ¦(h) = f(¢; N (h))g means the binding coalition is N(h)9and ¦(h) 6=
f(¢; N(h))g means the binding coalition is not N(h).
These strategies depend only on the state variable. On the equilibrium path, all players propose ((Pxi(NjN)
j 2N
xj(NjN))i2N; N)at time 1, accept at time 2, and leave at
time 3. The equilibrium outcome is x(N j N) where every i 2 N gets xi(N j N).
For any history h, in the continuing subgame, the equilibrium outcome is where every i 2 N(h) gets xi(N(h) j N(h); C(h)).
We …rst prove two lemmas analogous to Lemmas 1 and 2 in Perry and Reny (1994).
Lemma 6 For any h 2 H, if p(h) = ((wi)i2S; S) and
¦(h) = f((w1
i)i2S1; S1); ((w2i)i2S2; S2); :::; ((wmi )i2Sm; Sm)g, then
(a) zi(h) ¸ wik
P
j2Sk
xj(N (h)nSk j N(h)nSk; Sk; C(h)) if i 2 Sk where k 2
f1; 2; :::; mg.
9So that there exists a division rule (w
(b) P i2Sk zi(h) = P i2Sk xi(N(h) j N (h); C(h)) for all k 2 f1; 2; :::; mg. (c) P i2S zi(h) = P i2Sbzi (h) = P i2S xi(N (h) j N(h); C(h)). Proof. (a) By de…nition zi(h) = wik P j 2Sk xj(N(h) j N (h); C(h)). Since x(N(h) j
N (h); C (h))belongs to the r-core for N(h) given C(h), X
j2Sk
xj(N(h) j N(h); C(h)) ¸ V (Skj N(h); C(h))
= P (Skj Sk; N(h)nSk; C(h))
where the equality follows because by Lemma 5, when Sk breaks o¤,
the remaining players in N(h)nSk stay together. Since x(N(h)nSk j
N (h)nSk; Sk; C(h))belongs to the r-core for N(h)nSk given Skand C(h)
and by Lemma 5 players in N(h)nSk stay together,
X j2Sk xj(N(h)nSkj N(h)nSk; Sk; C (h)) = P (Skj Sk; N(h)nSk; C (h)): Hence zi(h) ¸ wik X j2Sk xj(N(h)nSkj N(h)nSk; Sk; C(h)): (b) By de…nition X i2Sk zi(h) = X i2Sk wki(X j2Sk xj(N(h) j N (h); C(h))) = X i2Sk xi(N(h) j N(h); C(h)); since P i2Sk wk i = 1.
(c) By de…nition X i2S zi(h) = X k2f1;:::;rg (X i2Sk zi(h)) + X i2Sn(S1[:::[Sr) xi(N(h) j N(h); C(h)) = X i2S xi(N(h) j N(h); C(h));
where the second equality follows because of (b). By de…nition X i2S bzi(h) = X i2S wi( X j 2S xj(N(h) j N(h); C(h))) = X i2S xi(N(h) j N(h); C(h)):
Lemma 7 For any t ¸ 0, h 2 H(t) and A(h) = ;, the outcome generated by the equilibrium strategies (f1; :::; fn) after h is where player i gets zi(h) for all
i 2 N(h).
Proof. Let t1be the smallest time at least as large as t such that t1¡ ¿(h)
is a positive integer. According to the equilibrium strategies, all players are quiet between [t; t1).10 Denote the history generated by h12 H(t1). There are
four possible cases depending on what players will do at t1 according to the
equilibrium strategies.
(1) If fi(h1) = l, then it must be the case that ¦(h1) = f(¢; N(h1))g. This
im-plies there exists a division rule (wi)i2N(h1)such that ¦(h1) = f((wi)i2N(h1); N(h1))g.
Thus player i gets wi P
j2N(h1)
xj(N(h1) j N(h1); C(h1)) for all i 2 N(h) in the continuing equilibrium outcome. However, since everyone is quiet between [t; t1), ¦(h) = ¦(h1)and N(h) = N(h1). By de…nition
zi(h) = wi X j2N(h) xj(N(h) j N(h); C(h)) = wi X j2N(h1) xj(N(h1) j N(h1); C(h1));
where the second equality follows because N(h) = N(h1). Hence player i
gets zi(h).
10Note that if t
(2) If fi(h1) = a for some i 2 N(h1)nA(h1)and p(h1) = ((wi)i2N(h1); N(h1)),
since everyone is quiet between [t; t1) and thus A(h1) = A(h) = ;, then
it implies bzi(h1) ¸ zi(h1) for all i 2 N(h1). So the current proposal
becomes binding at t1. By Lemma 6 (c) P
i2N(h1)
zi(h1) = P i2N(h1)
b zi(h1),
hence bzi(h1) = zi(h1) for all i 2 N(h1). Moreover, N(h1) = N(h)and
zi(h1) = zi(h)for all i 2 N(h) because everyone is quiet between [t; t1). By the equilibrium strategies, everyone is quiet between (t1; t1+ 1). Let
t2 = t1+1and denote the history generated by h22 H(t2). Since everyone
is quiet between (t1; t2), N(h1) = N(h2)and C(h1) = C(h2). At t2, since
¦(h2) = f(¢; N(h2))g, so all players leave. Thus player i 2 N(h2)gets
wi X j2N(h2) xj(N(h2) j N(h2); C(h2)) = wi X j2N(h1) xj(N(h1) j N(h1); C(h1)) = bzi(h1) = zi(h1) = zi(h): (3) If fi(h1) = (( P zj(h1) l2N (h1 ) xl(N(h1)jN(h1);C(h1)))j 2N(h1); N(h1)), then everyone is
quiet between (t1; t1+ 1). Let t2= t1+1and denote the history generated
by h22 H(t2). By de…nition for all i 2 N(h1),
b zi(h2) = P zi(h1) j2N(h1) xj(N(h1) j N(h1); C(h1)) X l2N(h1) xl(N(h2) j N(h2); C(h2)) = zi(h1) = zi(h2):
The second equality follows because no one leaves between [t1; t2), so
N (h1) = N(h2)and C(h1) = C(h2). The third equality follows because
no new proposal binds between [t1; t2), so zi(h1) = zi(h2). Thus all
play-ers in N(h1)accept the proposal at time t2. According to the equilibrium
strategies, everyone is quiet between (t2; t2+ 1). Let t3= t2+1and denote
the history generated by h32 H(t3). Since no one leaves between [t2; t3),
player i 2 N(h3)gets zi(h1) P j2N(h1) xj(N(h1) j N(h1); C(h1)) X l2N(h3) xl(N(h3) j N(h3); C (h3)) = zi(h1) = zi(h):
The …rst equality follows because no one leaves between [t1; t3). The second equality follows because nothing happens between [t; t1).
(4) If fi(h1) = a for some i 2 SnA(h1)and p(h1) = ((wi)i2S; S)where S 6=
N (h1), since everyone is quiet between [t; t1)and thus A(h1) = A(h) = ;,
then it implies bzi(h1) ¸ zi(h1) for all i 2 S. So the current proposal
becomes binding at t1. By Lemma 6 (c) P
i2S
zi(h1) = P i2Sbzi
(h1), hence
b
zi(h1) = zi(h1)for all i 2 S. Denote ¦(h1) = f((w1i)i2S1; S1); ((wi2)i2S2; S2); :::; ((wim)i2Sm; Sm)g
and without loss of generality, assume that S contains all the coalitions Skwhere k rand is disjoint from all the coalitions Skwhere k ¸ r + 1.
Then for all i 2 N(h1)n(S [ Sr+1[ ::: [ Sm),
b
zi(h1) = zi(h1) = xi(N(h1) j N(h1); C (h1));
and for all i 2 Skwhere k ¸ r + 1,
b
zi(h1) = zi(h1) = wki
X
j2Sk
xj(N(h1) j N(h1); C(h1)):
Thus bzi(h1) = zi(h1) for all i 2 N(h1). Note that all players are quiet
between (t1; t1+ 1). Let t2= t1+ 1and denote the history generated by h22 H(t2). Since no one leaves between [t1; t2), so N(h1) = N(h2). Since S 6= N(h2), at time t2, for all i 2 N(h2) fi(h2) = (( P zj(h2)
l2N(h2)
xl(N(h2)jN(h2);C(h2)))j2N(h2); N(h2)).
Everyone is quiet between (t2; t2+1). Let t3= t2+1and denote the history
generated by h32 H(t3). By de…nition for all i 2 N(h2),
b zi(h3) = zi(h2) P j2N(h2) xj(N(h2) j N(h2); C(h2)) X l2N(h2) xl(N(h3) j N(h3); C(h3)) = zi(h2) = zi(h3):
The second equality follows because no one leaves between [t2; t3), so
N (h2) = N(h3)and C(h2) = C(h3). The third equality follows because
no new o¤ers binds between [t2; t3). Thus all players in N(h2)accept the
proposal at time t3. Everyone is quiet between (t3; t3+ 1). Let t4= t3+ 1
and denote the history generated by h42 H(t4).
Since no one leaves between [t3; t4), so N(h3) = N(h4) and C(h3) =
C (h4). At t4, since ¦(h4) = f(¢; N(h4))g, so all players leave. Thus player
i 2 N (h4)gets zi(h2) P j 2N(h2) xj(N(h2) j N(h2); C(h2)) X l2N(h4) xl(N(h4) j N(h4); C(h4)) = zi(h2):
However zi(h2) = bzi(h1)because at time t1, p(h1)binds. Combining with
b
zi(h1) = zi(h1), thus zi(h2) = zi(h1) = zi(h). The last equality follows
because nothing happens between [t; t1).11 Thus player i 2 N(h) gets
zi(h)in the continuing equilibrium.
We can now prove a theorem parallel to Theorem 2 in Perry and Reny. Theorem 8 If (P; N) is superadditive, then any element of its r-core can be supported as an SSPNE outcome by (f1; :::; fn).
Proof. The proof is almost the same as that in Perry and Reny (1994) except with some minor modi…cations.
For any t ¸ 0, h 2 H(t) and i 2 N (h)nA(h), if fj(h) = l for some j 2
N(h)nfig, then according to the equilibrium strategies, it must be the case that ¦(h) = f(¢; N(h))g. Hence player i has to leave anyway. So fi(h) = l is
clearly optimal. Thus we only need to show that for any t ¸ 0, h 2 H(t) and i 2 N(h)nA(h), if fj(h) 6= l for all j 2 N(h)nfig, then using the equilibrium
strategy fi given all others are using their corresponding equilibrium strategies
f¡i is optimal for player i.
11Hence N (h) = N(h 1).
Consider another strategy f0
i for player i. Notice that fi0and f¡igenerate a
unique continuation path h0subsequent to h. Since i gets at least z
i(h) > ¡1
by following fi, he does no better if he never leaves according to h0. Thus
suppose i leaves at time t0 ¸ t with the proposal ((w0
j)j2S0; S0). Notice that
since others are following the equilibrium strategies f¡i, everyone must leave the game ultimately. The coalitional structure thus formed, denoted by P0
N,
must satisfy:
S 2 P0
N for all S 2 C(h) and S02 PN0 :
That is, it must respect the coalitions that have left. Thus player i obtains the payo¤ of w0
iP (S0j PN0 ). Since the proposal ((wj0)j2S0; S0)must be binding
before the coalition S0can leave, thus ((w0
j)j2S0; S0) 2 ¦(h0jt0). By Lemma 6 zi(h0jt0) ¸ wi0 X j 2S0 xj(N(h0jt0)nS0j N(h0jt0)nS0; S0; C(h0jt0)) = wi0P (S0j PN0 ):
The equality follows because the coalitional structure formed is unique, thus P0
N = fN (h0jt0)nS0g [ fS0g [ C(h0jt0). Moreover, the vector x(N (h0jt0)nS0 j
N(h0jt0)nS0; S0; C(h0jt0))is in the core C(N(h0jt0)nS0j N(h0jt0)nS0; S0; C(h0jt0)),
hence the sum of payo¤s for players in S0is simply P j2S0
xj(N(h0jt0)nS0j N(h0jt0)nS0; S0; C (h0jt0)) =
P (S0j P0 N).
We follow Perry and Reny (1994) to consider three exhaustive cases. Case A: A(h) = ;.
Case B: p(h) = ((wj)j2T; T ) and either bzi(h) zi(h) or bzj(h) < zj(h) for
some j 2 TnA(h).
Case C: p(h) = ((wj)j 2T; T ), bzi(h) > zi(h) and bzj(h) ¸ zj(h) for all j 2
T nA(h).
Notice that case B covers the instances where i =2 T since then bzi(h) = zi(h).
Case A covers the instances where p(h) = ;. As in Perry and Reny (1994), the argument applying to cases A and B have a common component, thus we treat them together until it is necessary to separate them.
Cases A or B: By Lemma 7, in case A, player i will get zi(h) by using the
equilibrium strategy fi. In case B, by the equilibrium strategies, either player
idoes not belong to T , he is the only player who hasn’t accepted the current proposal p(h) and will accept it or at least a player will reject the proposal by making a new proposal involving all the players N(h), in any case, player i gets zi(h). Thus suppose to the contrary that by following fi0, player i made a
pro…table deviation. Thus w0
iP (S0j PN0 ) > zi(h).
Since zi(h0jt0) ¸ w0iP (S0j PN0 ) > zi(h), it must be the case that t0> t. This
is because h0jt = h. Hence, let
t¤= inf fbt2 [t; t0] j z
i(h0jbt) > zi(h)g.
It follows that zi(h) ¸ zi(h0jt¤). To see this, note if t¤= t, then since h0jt = h,
it is certainly true. If t¤ > tand suppose to the contrary that z
i(h) < zi(h0jt¤),
then by (S4), there exists an " > 0 small enough so that t¤¡ " > t and nothing
happens between [t¤¡ "; t¤). Hence z
i(h0jt¤¡"2) = zi(h0jt¤) > zi(h). But then
t¤ is not the in…mum. Hence zi(h) ¸ zi(h0jt¤). This implies t¤ 6= t0 because
zi(h0jt0) > zi(h). Thus t0> t¤.
Because zi(h) ¸ zi(h0jt¤)and t¤is the in…mum, there must exist a sequence
of positive numbers f"ng where limn!1"n= 0 and zi(h0jt¤+ "n) > zi(h) ¸
zi(h0jt¤)for every "n. By (S4), there must exist an n¤large enough such that
nothing happens between (t¤; t¤+"
n¤). Thus something must happen at time t¤
for otherwise it cannot be the case that zi(h0jt¤+ "n¤) > zi(h0jt¤). Thus either
the current proposal p(h0jt¤) which contains player i becomes binding at t¤ or
someone leaves at t¤. The latter cannot happen because according to f0 i, player
ileaves at t0> t¤. Players other than i cannot leave at t¤ either for otherwise
since they are playing according to the equilibrium strategies f¡i, when one
leaves, all must leave, contradicting player i leaves at t0 > t¤. Therefore, the
current proposal p(h0jt¤) becomes binding at t¤. Let p(h0jt¤) = ((w¤
j)j 2S; S).
Notice that i 2 S.
Since the current proposal p(h0jt¤)becomes binding at t¤ and nothing
hap-pens between (t¤; t¤+ "
zi(h0jt¤+ "n¤) > zi(h0jt¤), it follows bzi(h0jt¤) > zi(h0jt¤). By part (c) of Lemma
6 P
j2S
zj(h0jt¤) = P j2Sb
zj(h0jt¤). Because i is in S, this implies there exists a
player k in S such that bzk(h0jt¤) < zk(h0jt¤). We now separate the discussion
for cases A and B.
Case A: Since p(h0jt¤) becomes binding at t¤ and at time t no one has
accepted any proposal because A(h) = ;, thus player k must accept p(h0jt¤)at
some point of time tk 2 [t; t¤]. At that time, since k is playing according to
the equilibrium strategy, so bzk(h0jtk) ¸ zk(h0jtk). Note that from ¦(h0jtk) =
¦(h0jt¤), N(h0jtk) = N(h0jt¤)and C(h0jtk) = C(h0jt¤) since p(h0jtk) = p(h0jt¤).
Hence bzk(h0jtk) = bzk(h0jt¤)and zk(h0jtk) = zk(h0jt¤). This implies bzk(h0jt¤) ¸
zk(h0jt¤), yielding a contradiction. Hence there is no pro…table deviation for
case A.
Case B: Note that p(h) = ((wj)j2T; T ) will not bind. This is because if
there exists some j 6= i where j 2 TnA(h) such that bzj(h) < zj(h), then he
will not accept the proposal and will make another proposal pertaining to N(h) at the next integer time if no one has done so. If there exists no j 6= i where j 2 T nA(h) such that bzj(h) < zj(h), then either bzi(h) = zi(h)or bzi(h) < zi(h).
When bzi(h) = zi(h), according to the equilibrium strategies, all j 6= i where
j 2 T nA(h) will accept the proposal at the next integer time. Thus if player i accepts as well, the proposal will bind. However, once it binds, say at time t00,
then p(h0jt00) = ;. Since we have shown in Case A that no pro…table deviation
is possible, player i’s optimal strategy is to follow the equilibrium strategy fi
from t00 on. This implies i’s payo¤ will be bz
i(h) = zi(h) by using fi0. Since
bzi(h) = w0iP (S0j PN0 ). This is in contradiction to w0iP (S0j P0N) > zi(h). When
bzi(h) < zi(h), there are two possibilities. Either there exists an j 6= i where
j 2 T nA(h) or fig = T nA(h). In the …rst situation when there exists an j 6= i where j 2 TnA(h), then according to player j’s equilibrium strategy, he will not accept the proposal and will make another proposal pertaining to N (h) if no on has done so. In the second situation where fig = TnA(h), player i will not accept the proposal. For if he did, say at time t000, then p(h0jt000) = ;. Again,
since we have shown in Case A that no pro…table deviation is possible from t000
on, player i gets bzi(h) < zi(h), yielding a contradiction. Therefore in all possible
situations, p(h) = ((wj)j2T; T )will not bind.
Since p(h0jt¤)becomes binding at t¤ and p(h) = ((w
j)j2T; T )will not bind,
p(h0jt¤)must be proposed at time t or later but before t¤. Hence player k must
accept p(h0jt¤)at some point of time t
k 2 [t; t¤]. Now apply exactly the same
logic in case A to get a contradiction. Thus there is no pro…table deviation for case B.
Case C: We will show that player i has no pro…table deviation. If player i plays according to the equilibrium strategy fi, since all others are also playing
the equilibrium strategies, his payo¤ is bzi(h) > zi(h). If instead player i deviates
to another strategy f0
i, there are two possibilities.
In the …rst possibility p(h) becomes binding. This implies player i accepts p(h) at some time t00. Since all other players accept p(h) by the equilibrium
strategies at the next integer time, say t000, this implies p(h) becomes binding at
maxft00; t000g. Therefore p(h0j maxft00; t000g) = ;. By the argument in case A, it
is optimal for player i to follow the equilibrium strategy from time maxft00; t000g
on. Hence player i’s payo¤ from using f0
i is at most bzi(h).
In the second possibility p(h) does not become binding. This implies either player i makes another proposal at some time t00 or i leaves before
accept-ing. In the …rst situation, since others are playing according to the equilibrium strategies, if the next integer time arrives before than or at t00, all others
ac-cept p(h) at the next integer time exac-cept player i. If the next integer time arrives after t00, this new proposal is made before anyone has accepted it. Both
imply p(h0jt00) = ;. By the argument in case A, it is optimal for player i to
follow the equilibrium strategy from time t00 on. Hence player i’s payo¤ from
using f0
i is at most zi(h). In the second situation where i leaves before
ac-cepting, it must be the case that player i is in a binding coalition Sk. After
Sk leaves, all players still play according to the equilibrium strategies. Thus
Hence player i’s payo¤ is wk i
P
j2Sk
xj(N(h)nSk j N(h)nSk; Sk; C(h)) zi(h)
by part (a) of Lemma 6.
Thus there is no pro…table deviation for case C.
6 Conclusion
There are some di¢culties in extending the proof of Theorem 8 to cases where superadditivity does not hold. For example, suppose the r-core structure of N should consist of three sub coalitions S1, S2 and S3. To implement a
r-core payo¤ consistent with this coalitional structure, without loss of generality, suppose S1 has to form …rst, S2 second and S3 last.
When S1forms, the relevant partition function is no longer (P; N). Instead,
we should now treat S1 as a composite player and consider the new partition
function where S1is treated as a player. This implies some sort of consistency is
needed. For instance, a player in S2may not want to deviate originally but now
since players in S1 cannot break apart, he may get a new incentive to deviate.
Imposing some consistency on the partition functions presumably could rule this kind of deviation out.
Moreover, once S1has formed, it can leave …rst. When this happens,
accord-ing to 4, we expect the r-core for NnS1 to occur in the continuing equilibrium.
In this situation, we naturally want the r-core structure for N nS1to consist of
S2and S3because we have to make sure that S1does not have the incentive to
deviate to leave so fast to induce another continuing equilibrium.
How to solve di¢culties of these kinds is worth further consideration if not impossible. We leave this for future research.
