Abstract In cooperative dynamic games with non-transferable payoffs, the players&apos; agreed-upon cooperative actions would determine the resulting payoff that each player receives. This article develops a mechanism for the derivation of individual player&apos;s payoff functions in cooperative stochastic dynamic games with nontransferable payoffs. This is the first time that individual player&apos;s payoff functions are characterized in an analytically derivable form in such a framework. An illustrative example is provided. Mathematics Subject Classifications: 91A12, 91A2

David W K Yeung

CiteSeerX

 
International Journal of Algebra, Vol. 7, 2013, no. 13, 597 - 606 
HIKARI Ltd,  www.m-hikari.com 
http://dx.doi.org/10.12988/ija.2013.3775 
 
 
 
Nontransferable Individual Payoffs in Cooperative  
 
Stochastic Dynamic Games 
 
 
David W. K. Yeung 
 
SRS Consortium for Cooperative Dynamic Games  
Hong Kong Shue Yan University 
and 
Center for Game Theory, St Petersburg State University 
Petrodivrets, 198904, St Petersburg, Russia 
 
   Copyright © David W. K. Yeung. This is an open access article distributed under the Creative 
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in 
any medium, provided the original work is properly cited. 
 
 
Abstract 
 
In cooperative dynamic games with non-transferable payoffs, the players’ 
agreed-upon cooperative actions would determine the resulting payoff that each 
player receives. This article develops a mechanism for the derivation of individual 
player’s payoff functions in cooperative stochastic dynamic games with 
nontransferable payoffs. This is the first time that individual player’s payoff 
functions are characterized in an analytically derivable form in such a framework. 
An illustrative example is provided. 
 
Mathematics Subject Classifications: 91A12, 91A25 
 
Keywords: Cooperative stochastic dynamic games, nontransferable payoffs, 
stochastic difference equations 
 
 
1 Introduction  
 
In cooperative games, players negotiate to establish an agreement on how to 
act and assign their payoffs. A necessary condition is that the agreement must 
satisfy individual rationality. To verify individual rationality, individual players’  
598                                               David W. K. Yeung 
 
 
payoff functions have to be derived. In the case when the game is dynamic and 
stochastic while payoffs are nontransferable, analytically tractable individual 
payoff functions, though difficult to be obtained, are needed for this verification 
process. Yeung (2004) provided a formulation to characterize the players’ 
individual payoffs in continuous-time cooperative stochastic differential with 
non-transferable payoffs. In this article, we develop a mechanism for the 
derivation of individual player’s payoff functions for discrete-time cooperative 
stochastic dynamic games. An illustrative example is given. 
 
 
 
2 Game Framework  
 
Consider the general −T stage −n person nonzero-sum discrete-time 
stochastic dynamic game with initial state 01x . The state space of the game is 
mRX ∈  and the state dynamics of the game is characterized by the stochastic 
difference equation: 
 k
n
kkkkkk uuuxfx θ+=+ ),,,,(
21
1 L ,     (2.1) 
for κ≡∈ },,2,1{ Tk L  and 011 xx = ,  
where imik Ru ∈  is the control vector of player i  at stage k , Xxk ∈  is the 
state, and kθ  is a set of statistically independent random variables. 
The objective of player i  is 
 
⎩
⎨
⎧
T
E θθθ ,,, 21 L ),,,,(
21
1
n
kkkk
i
k
T
k
uuuxg L∑
=
)( 1++ T
i xq
⎭
⎬
⎫
,          (2.2) 
 
for Nni ≡∈ },,2,1{ L ,  
where 
T
E θθθ ,,, 21 L  is the expectation operator with respect to the statistics of 
1θ , 2θ , Tθ,L , and )( 1+T
i xq  is the terminal payoff that player i  will received in 
stage 1+T .  
The payoffs of the players are not transferable. Let )({ xikφ , for κ∈k  and 
}Ni∈  denote a set of strategies that provides a feedback Nash equilibrium 
solution (if it exists) to the game (2.1)-(2.2), and ),( xkV i , for Kk ∈ , denote the 
value functions indicating the expected payoff to player i  over the stages from 
k  to T . The theorems characterizing a Nash of the game (2.1)-(2.2) can be 
found in standard textbooks (for instance see Theorem 13.1 in Yeung and 
Petrosyan (2012)).   
For the sake of exposition, we sidestep the issue of multiple equilibria and 
focus on solvable games in which a particular noncooperative Nash equilibrium is 
chosen by the players in the entire game.  
Nontransferable individual payoffs                                  599 
 
 
 
3 Cooperation Scheme and Individual Payoffs  
 
Now consider the case when the players agree to cooperate and enhance their 
payoffs according to an agreed-upon cooperative scheme. In the scheme, the 
players would adopt the agreed-upon cooperative strategies which would directly 
determine the payoffs of the players.  
 
3.1. Optimal Cooperative Strategies   
To obtain a group optimal outcome one has to consider the derivation of a set 
of cooperative strategies using payoff weights in which the players agree to adopt 
a vector of constant payoff weights ),,,( 21 nαααα L=  for 1
1
=∑
=
j
n
j
α . Dockner 
and Jorgensen (1984), Hamalainen et al (1986), Leitmann (1974), and Yeung and 
Petrosyan (2005) provided analysis along this line. Conditional upon an 
agreed-upon vector of weights α , the agents’ optimal cooperative strategies can 
be generated by solving the following stochastic control problem of maximizing 
their joint weighted expected payoff: 
⎩
⎨
⎧
T
E θθθ ,,, 21 L ⎭
⎬
⎫
⎥
⎦
⎤
+⎢
⎣
⎡
+
==
∑∑ )(),,,,( 121
11
T
jjn
kkkk
j
k
j
T
k
n
j
xquuuxg αα L   (3.1) 
subject to (2.1).    
An optimal solution to the problem (3.1)-(2.1) can be characterized by the 
following Theorem. 
Theorem 3.1. A set of strategies )({ )( xik
αψ , for κ∈k  and }Ni∈  provides an 
optimal solution to the problem (3.1)-(2.1)  if there exist functions ),()( xkW α , 
for Kk ∈ , such that the following recursive relations are satisfied: 
),1()( xTW +α )( 1
1
+
=
∑= Tjj
n
j
xqα ,                   (3.2) 
=),()( xkW α
⎩
⎨
⎧
kn
kkk
E
uuu
θ
,,, 21
max
L
),,,,( 21
1
n
kkkk
j
k
n
j
j uuuxg L∑
=
α  
        ]),,,,(,1[ 21)( k
n
kkkkk uuuxfkW θ
α +++ L
⎭
⎬
⎫
  
=
⎩
⎨
⎧
k
Eθ ∑
=
n
j
n
kkk
j
k
j xxxxg
1
)(2)(1)( )](,),(),(,[ ααα ψψψα L  
            ]))(,),(),(,(,1[ )(2)(1)()( k
n
kkkk xxxxfkW θψψψ
αααα +++ L
⎭
⎬
⎫
, (3.3) 
Proof.  The results in (3.2)-(3.3) comes directly from the stochastic optimal 
control techniques (See Basar and Olsder (1999) and Yeung and Petrosyan 
(2012)).               ■ 
 
600                                               David W. K. Yeung 
 
 
Substituting the optimal control )({ )( xik
αψ , for κ∈k  and }Ni∈  into the 
state dynamics (2.1), one can obtain the dynamics of the cooperative trajectory as: 
kk
n
kkkkkkkk xxxxfx θψψψ
ααα +=+ ))(,),(),(,(
)(2)(1)(
1 L ,   (3.4) 
 We use )(αkx
)(α
kX∈  to denote the value of the state at stage k  generated by 
(3.4), where )(αkX  is the set of realizable values of 
)(α
kx . The term ),(
)( xkW α  
gives the expected weighted cooperative payoff over the stages from k  to T  if 
xxk =
)(α )(α
kX∈  is realized at stage κ∈k . 
 
3.2. Individual Payoff under Cooperation 
Given that all players are adopting the cooperative strategies in Section 3.1 
the expected payoff of player i  under cooperation can be obtained as: 
=),()( xtW iα
⎩
⎨
⎧
+ Ttt
E θθθ ,,, 1 L )](,),(),(,[
)()()(2)()(1)()( ααααααα ψψψ k
n
kkkkkk
i
k
T
tk
xxxxg L∑
=
 
                
)( )( 1
α
++ T
i xq xxt =
⎭
⎬
⎫
,   for Ni∈  and κ∈t .  (3.5) 
To allow the derivation of the functions ),()( KtW iα  in a more direct way 
we establish a discrete-time analog of Yeung’s (2004) characterization of the 
players’ individual expected payoffs under cooperation in the following Theorem. 
 
Theorem 3.2.  
The expected payoff of player i  can be characterized as  
),1()( xTW i +α )( 1+= T
i xq ,                    
=),()( xtW iα
⎩
⎨
⎧
t
Eθ )](,),(),(,[
)(2)(1)( xxxxg nttk
i
t
ααα ψψψ L  
  ]))(,),(),(,(,1[ )(2)(1)()( t
n
tttt
i xxxxftW θψψψ αααα +++ L
⎭
⎬
⎫
,  
for Ni∈  and κ∈t .             (3.6) 
Proof.  Note that ),()( xtW iα  in (3.5) can be expressed as: 
=),()( xtW iα
⎩
⎨
⎧
+ Ttt
E θθθ ,,, 1 L )](,),(),(,[
)(2)(1)( xxxxg nttt
i
t
ααα ψψψ L  
 )](,),(),(,[ )()()(2)()(1)()(
1
ααααααα ψψψ k
n
kkkkkk
i
k
T
tk
xxxxg L∑
+=
+ )( )( 1
α
++ T
i xq xxt =
)(α
⎭
⎬
⎫
.(3.7) 
Invoking (3.5), we have: 
 
=+ + ),1(
)(
1
)( αα
t
i xtW  
 
Nontransferable individual payoffs                                  601 
 
 
 
⎩
⎨
⎧
++ Ttt
E θθθ ,,, 21 L )](,),(),(,[
)()()(2)()(1)()(
1
ααααααα ψψψ k
n
kkkkkk
i
k
T
tk
xxxxg L∑
+=
)( )( 1
α
++ T
i xq
⎭
⎬
⎫
,    
 (3.8) 
Using (3.7) and (3.8), we have 
 
=),()( xtW iα
⎩
⎨
⎧
t
Eθ )](,),(),(,[
)(2)(1)( xxxxg nttk
i
t
ααα ψψψ L  
        ]))(,),(),(,(,1[ )(2)(1)()( t
n
tttt
i xxxxftW θψψψ αααα +++ L
⎭
⎬
⎫
,  
for Ni∈  and κ∈t .                                            (3.9)  
Hence Theorem 3.2 follows.                                         ■ 
For individual rationality to be maintained at the outset, it is required that: 
),1( 01
)( xW iα ≥ ),1( 01xV
i , for Ni∈ .    (3.10) 
For individual rationality to be maintained throughout all the stages κ∈k , it is 
required that: 
),( )()( αα k
i xkW ≥ ),( )(αk
i xkV , for Ni∈  and κ∈k .    (3.11) 
If there exists an agreed-upon set of solution weights α  that satisfies (3.11) 
the cooperative solution satisfies both individual rationality and group optimality 
throughout the cooperative duration. 
 
 
4. An Illustration 
 
We consider a non-transferable payoff version of the game in Yeung and 
Petrosyan (2010). Consider two economies which can extract a renewable 
resource. The planning horizon for resource extraction begins at stage 1 and ends 
at stage 3 for these two extractors. Let iku  denote the quantity of resource 
extracted by extractor i  at stage k , for }2,1{∈i . Let iU  be the set of 
admissible extraction, and +⊂∈ RXxk  the size of the resource stock at stage k . 
In particular, we have iU ∈ R+ and kkk xuu ≤+
21 . The social plus private 
extraction cost for extractor i  at stage k  is k
i
ki xuc /)(
2 . The benefit of a unit 
of resource to economy i  is iP . 
The growth dynamics of the resource is governed by the stochastic 
difference equation: 
 =+1kx ∑
=
−−+
2
1j
j
kkk ubxax kθ+ ,  for }3,2,1{∈k  and 
0
1 xx = ,  (4.1) 
where kθ  is a random variable with non-negative range },,{
321
kkk θθθ  and 
corresponding probabilities },,{ 321 kkk λλλ . 
602                                               David W. K. Yeung 
 
 
 The objective of extractor }2,1{∈i  is to maximize the present value of the 
expected payoff: 
 
⎪⎩
⎪
⎨
⎧
321 θθθ
E ⎥
⎦
⎤
⎢
⎣
⎡
−∑
=
2i
3
1
)( k
k
ii
ki
k
u
x
cuP
1
1
1 −
⎟
⎠
⎞
⎜
⎝
⎛
+
k
r ⎭
⎬
⎫ ,   for }2,1{∈i ,        (4.2) 
subject to (4.1). 
 The payoffs are nontransferable. Now consider the case when the extractors 
agree to use the weight ),( 21 ααα =  to maximize their expected weighted payoff   
 
⎪⎩
⎪
⎨
⎧
321 θθθ
E ∑
=
2
1j
jα ⎥
⎦
⎤
⎢
⎣
⎡
−∑
=
2j
3
1
)( k
k
jj
kj
k
u
x
c
uP
1
1
1 −
⎟
⎠
⎞
⎜
⎝
⎛
+
k
r ⎭
⎬
⎫
   (4.3) 
subject to (4.1). 
 Invoking Theorem 3.1, one can characterize the optimal controls in the 
stochastic dynamic programming problem (4.1) and (4.3). In particular, a set of 
control strategies )({ )( xik
αψ , for }3,2,1{∈k  and }}2,1{∈i  provides an optimal 
solution to the problem (4.1) and (4.3) if there exist functions :),()( xkW α RR → , 
for }3,2,1{∈k , such that the following recursive relations are satisfied: 
=),()( xkW α
⎩
⎨
⎧
21 ,
max
kk uu
⎥
⎦
⎤
⎢
⎣
⎡
−∑
=
2j
2
1
)( k
jj
kj
j
j u
x
c
uPα
1
1
1 −
⎟
⎠
⎞
⎜
⎝
⎛
+
k
r
 
 ∑
=
+
3
1y
y
kλ ,1[
)( +kW α ]
2
1
y
k
j
j
kubxax θ+−−+ ∑
= ⎭
⎬
⎫
, for }3,2,1{∈k .  
0),1()( =+ xTW α .           (4.4) 
Performing the indicated maximization in (4.4) yields the optimal cooperative 
strategies: 
=)()( xik
αψ  
       iP⎜⎜
⎝
⎛ ∑
=
−
3
1
1
y
y
ki λα
,1[)(
1
+
=
kW
kx
α ])(
2
1
)( y
k
j
j
k xbxax θψ
α +−−+ ∑
= i
k
c
xr
2
)1( 1 ⎟⎟
⎠
⎞
+ − ,  
for }2,1{∈i  and }3,2,1{∈k .                                      (4.5) 
 
Proposition 4.1.    The value function 
=),()( xkW α ][ )()( αα kk CxA + , for }3,2,1{∈k ,      (4.6) 
where )(αkA  and 
)(α
kC , for }3,2,1{∈k , are constants given in (A.4), (A.9) and 
(A.12) in Appendix A. 
Proof.   See Appendix A.            ■ 
 Using (4.5) and Proposition 4.1, the optimal cooperative strategies of the 
agents can be expressed as: 
=)()( xik
αψ
i
i
k
k
i c
xrAP
2
])1([
1)(
1
α
α −
+ +− ,     for }2,1{∈i  and }3,2,1{∈k .   (4.7) 
 
Nontransferable individual payoffs                                  603 
 
 
Invoking Theorem 3.2 we can characterize the value function ),()( xkW iα  which 
indicates the expected payoff to player i  under cooperation by: 
=),()( xkW iα iP⎢
⎣
⎡
i
i
k
k
i c
xrAP
2
])1([
1)(
1
α
α −
+ +− ⎥
⎦
⎤+
−−
−
+
i
i
k
k
i c
xrAP
4
])1([ 2
1)(
1
α
α 1
1
1 −
⎟
⎠
⎞
⎜
⎝
⎛
+
k
r
 
 ∑
=
+
3
1y
y
kλ ,1
)( +⎜⎜
⎝
⎛
kW iα ∑
=
−−+
2
1j
bxax
j
j
k
k
j c
xrAP
2
])1([
1)(
1
α
α −
+ +− ⎟⎟
⎠
⎞
+ ykθ , 
for }3,2,1{∈k , 
0),4()( =xW iα .      (4.8) 
Proposition 4.2.    The value function 
=),()( xkW iα ][ )()( ik
i
k CxA
αα + , for }3,2,1{∈k  and }2,1[∈i ,   (4.9) 
where ikA
)(α  and ikC
)(α , for }3,2,1{∈k  and }2,1[∈i , are constants given in 
(B.2), (B.3) and (B.4) of Appendix B. 
Proof.  See Appendix B.             ■ 
 
 
5. Concluding Remarks 
 
This article develops a mechanism for the derivation of individual player’s 
payoff functions in cooperative stochastic dynamic games with nontransferable 
payoffs. The analysis can be readily applied to cooperative dynamic games by 
removing the stochastic elements. Further applications of the results in 
discrete-time dynamic games are expected.  
 
 
Appendix A: Proof of Proposition 4.1.  
 
Consider first the last stage, that is stage 3. Invoking that 
=),3()( xW α ][ )(3
)(
3
αα CxA +  from Proposition 4.1 and 0),4()( =xW α , the 
conditions in equation (4.4) become 
=),3()( xW α ][ )(3
)(
3
αα CxA + =
⎩
⎨
⎧
2
3
1
3 ,
max
uu ⎥⎦
⎤
⎢
⎣
⎡
−∑
=
2j
33
2
1
)(u
x
c
uP jjj
j
jα
2
1
1
⎟
⎠
⎞
⎜
⎝
⎛
+ r ⎭
⎬
⎫
.   (A.1) 
Performing the indicated maximization in (A.1) yields the optimal cooperative 
strategies in stage 3 as: 
  =)()(3 x
iαψ
i
i
c
xP
2
,    for }2,1{∈i .          (A.2) 
Substituting (A.2) into (A.1) yields: 
][ )(3
)(
3
αα CxA +
2
1
1
⎟
⎠
⎞
⎜
⎝
⎛
+
=
r
x
c
P
j
j
j
j
4
)( 22
1
∑
=
α ,   for }2,1{∈i ,   (A.3) 
604                                               David W. K. Yeung 
 
 
where =)(3
αA
2
1
1
⎟
⎠
⎞
⎜
⎝
⎛
+ r j
j
j
j
c
P
4
)( 22
1
∑
=
α  and 0)(3 =
αC .           (A.4) 
Now we proceed to stage 2, the conditions in equation (4.4) become  
=),2()( xW α ][ )(2
)(
2
αα CxA + =
⎩
⎨
⎧
2
2
1
2 ,
max
uu ⎥⎦
⎤
⎢
⎣
⎡
−∑
=
2j
22
2
1
)(u
x
c
uP jjj
j
jα ⎟
⎠
⎞
⎜
⎝
⎛
+ r1
1  
 ∑
=
+
3
1
2
y
yλ ,3[)(αW ]2
2
1
2
y
j
jubxax θ+−−+ ∑
= ⎭
⎬
⎫
.      (A.5) 
Invoking (A.3), the condition in (A.5) can be expressed as: 
][ )(2
)(
2
αα CxA + =
⎩
⎨
⎧
2
2
1
2 ,
max
uu ⎥⎦
⎤
⎢
⎣
⎡
−∑
=
2j
22
2
1
)(u
x
c
uP jjj
j
jα ⎟
⎠
⎞
⎜
⎝
⎛
+ r1
1  
 ∑
=
+
3
1
2
y
yλ [)(3
αA ]2
2
1
2
y
j
jubxax θ+−−+ ∑
= ⎭
⎬
⎫
.     (A.6) 
Performing the indicated maximization in (A.6) yields the optimal cooperative 
strategies in stage 2 as: 
=)()(2 x
iαψ ])1([ )(3
α
α
ArP ii
+
−
ic
x
2
,    for }2,1{∈i .           (A.7) 
Substituting (A.7) into (A.6) yields:  
][ )(2
)(
2
αα CxA + ,                           (A.8) 
where   
=)(2
αA ∑
=
+
−
2
1
)(
3 ]
)1([
j
jj
j ArP α
α
α
j
j
j
c
ArP
4
/)1( )(3 α
α++
 
)(
3
αA+ )1( b− ])1([ )(3
2
1
α
α
ArP jj
j
+
−−∑
= jc
A
2
)(
3
α
, and 
=)(2
αC aA ⎜⎜
⎝
⎛)(
3
α
⎟
⎟
⎠
⎞
+∑
=
y
y
y
2
3
1
2θλ .       (A.9) 
Finally, we proceed to the first stage, using (A.8) the conditions in equation 
(4.4) become 
][ )(1
)(
1
αα CxA + =
⎩
⎨
⎧
2
1
1
1 ,
max
uu ⎥⎦
⎤
⎢
⎣
⎡
−∑
=
2j
11
2
1
)(u
x
c
uP jjj
j
jα  
 ∑
=
+
3
1
1
y
yλ [)(2
αA⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
++−−+ ∑
=
)(
21
2
1
1 ]
αθ Cubxax y
j
j
⎭
⎬
⎫
.   (A.10) 
Performing the indicated maximization in (A.10) yields the optimal cooperative 
strategies in stage 1 and upon substitution of the optimal strategies into (A.10) 
yields: 
][ )(1
)(
1
αα CxA + ,                           (A.11) 
where   
Nontransferable individual payoffs                                  605 
 
 
 
)(
1
αA =∑
=
−
2
1
)(
2 )(
j
jj
j AP
α
α
α
j
j
c
AP
4
)(
2
α+ )(
2
αA+ )1( b− ∑
=
−
2
1j
)(
)(
2
jj
AP
α
α
−
jc
A
2
)(
2
α
, and 
=)(1
αC aA ⎜⎜
⎝
⎛)(
2
α
⎟
⎟
⎠
⎞
+∑
=
y
y
y
1
3
1
1θλ
)(
2
αC+ .               (A.12) 
Hence Proposition 4.1 follows.                                   Q.E.D. 
 
 
 
Appendix B: Proof of Proposition 4.2.  
 
From (4.8) we have: 
=),()( xkW iα iP⎢
⎣
⎡
i
i
k
k
i c
xrAP
2
])1([
1)(
1
α
α −
+ +− ⎥
⎦
⎤+
−−
−
+
i
i
k
k
i c
xrAP
4
])1([ 2
1)(
1
α
α 1
1
1 −
⎟
⎠
⎞
⎜
⎝
⎛
+
k
r
 
  ∑
=
+
3
1y
y
kλ ,1
)( +⎜⎜
⎝
⎛
kW iα ∑
=
−−+
2
1j
bxax
j
j
k
k
j c
xrAP
2
])1([
1)(
1
α
α −
+ +− ⎟⎟
⎠
⎞
+ ykθ ,  
for }3,2,1{∈k .  
0),1()( =+ xTW iα .    (B.1) 
 
Following the analysis in the proof of Proposition 4.1 in Appendix A one can 
readily obtain: 
 
=iA )(3
α
2
1
1
⎟
⎠
⎞
⎜
⎝
⎛
+ r i
i
c
P
4
)( 2  and 0)(3 =
iC α ;                        (B.2) 
=iA )(2
α ])1([ )(3
α
α
ArP ii
+
−
i
i
i
c
ArP
4
/)1( )(3 α
α++
 
iA )(3
α+ )1( b− ])1([ )(3
2
1
α
α
ArP jj
j
+
−−∑
= j
i
c
A
2
)(
3
α
, and 
=iC )(2
α aA i⎜⎜
⎝
⎛)(
3
α
⎟
⎟
⎠
⎞
+∑
=
y
y
y
2
3
1
2θλ ;           (B.3) 
 iA )(1
α = )(
)(
2
ii
AP
α
α
−
i
i
c
AP
4
)(
2
α+ iA )(2
α+ )1( b− ∑
=
−
2
1j
)(
)(
2
jj
AP
α
α
−
j
i
c
A
2
)(
2
α
, and 
=iC )(1
α aA i⎜⎜
⎝
⎛)(
2
α
⎟
⎟
⎠
⎞
+∑
=
y
y
y
1
3
1
1θλ
iC )(2
α+ .               (B.4)  
Q.E.D. 
 
606                                               David W. K. Yeung 
 
 
 
References 
 
[1] T. Basar and Olsder, G.J., Dynamic Noncooperative Game Theory, SIAM, 
Philadelphia, 1999. 
[2] E. Dockner and S. Jørgensen, Cooperative and Non-cooperative Differential 
Games Solutions to an Investment and Pricing Problem, The Journal of the 
International Operational Research Society, 35(1984), 731-739. 
[3] R. Hamalainen, A. Haurie, and V. Kaitala, Equilibria and Threats in a 
Fishery Management Game, Optimal Control Applications and Methods, 
6(1986), 315-333. 
[4] G. Leitmann, Cooperative and Non-Cooperative Many Players Differential 
Games, Springer-Verlag, New York, 1974. 
[5] D.W.K. Yeung, Nontransferable Individual Payoff Functions under 
Stochastic Dynamic Cooperation, International Game Theory Review, 
6(2004), 281-289. 
[6] D.W.K. Yeung and L.A. Petrosyan, Subgame Consistent Solution of a 
Cooperative Stochastic Differential Game with Nontransferable Payoffs,  
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cooperative Stochastic Dynamic Games. Journal of Optimization Theory and 
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Optimization: An Advanced Cooperative Dynamic Game Analysis, 
Birkhäuser, Boston, 2012.  
 
Received: August 1, 2013 


Nontransferable Individual Payoffs in Cooperative Stochastic Dynamic Games

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Nontransferable Individual Payoffs in Cooperative Stochastic Dynamic Games

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