JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 55, No, 2, NOVEMBER 1987
Identification of Classes of Differential Games
for Which the Open Loop Is a
Degenerate Feedback Nash Equilibrium 1
C. FERSHTMAN 2
Communicated by G. Leitmann
Abstract.
In general, it is clear that open-loop Nash equilibrium and
feedback Nash equilibrium do not coincide. In this paper, we study the
structure of differential games and develop a technique using which we
can identify classes of games for which the open-loop Nash equilibrium
is a degenerate feedback equilibrium. This technique clarifies the
relationship between the assumptions made on the structure of the game
and the resultant equilibrium.
Key Words.
Differential games, open-loop Nash equilibria, feedback
Nash equilibria.
L Introduction
In general, it is clear that open-loop Nash equilibrium and feedback
Nash equilibria do not coincide. However, Clemhout and Wan (Ref. t),
Leitmann and Schmitendorf (Ref. 2), and more recently Reinganum (Ref.
3) and Kamien and Muller (Ref. 4) identified and examined different classes
of games for which the Nash equilibrium in feedback strategies is degenerate
such that it depends on time only and not on the state variables. Clemhout
and Wan studied the class of trilinear games, and Leitmann and Schmitendoff studied dynamic advertising competition among firms. In both of these
studies, the degeneration occurs because the feedback form does not depend
on the state variables. Reinganum studied a class of games which has a
linear quadratic structure in the strategies, but is exponential in the state
variables. For this structure, the feedback form depends on the state variables. However, by using the value-function approach, Reinganum identified
The author would like to thank E. Dockner, A. Mehlmann, and an anonymous referee for
helpful comments.
2 Lecturer, Economics Department, Hebrew University, Jerusalem, Israel.
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feedback Nash equilibrium strategies that depend only on time. The uniqueness of this feedback equilibrium was studied by Mehlmann and Willing
(Ref. 5), who examined the same class of games. Kamien and Muller studied
a game in which the two players have identical payoff functions which
they jointly maximize. Under this structure, the cross-effect term in the
Pontryagin-type necessary conditions disappears, which implies that the
open-loop and the closedqoop Nash equilibria coincide.
The classes of games in which the open-loop Nash equilibria is a special
case of the feedback equilibrium is of special interest, since in such games
the open-loop Nash equilibrium is subgame perfect (Selten, Ref. 6).
Moreover, as the paper will indicate, an open-loop Nash equilibrium is
subgame perfect if and only if it is a degenerate feedback equilibrium.
The objective of this paper is to study the structure of differential games
in order to identify the classes of games in which the open-loop Nash
equilibrium is subgame perfect (a special case of the feedback equilibrium).
In order to make the suggested technique useful, it is based on analyzing
the structure of the game itself, rather than finding the feedback equilibrium
strategies and showing that they are functions of the time only. We demonstrate this technique by identifying several classes of games for which the
open-loop equilibrium is a degenerate feedback equilibrium. The paper,
however, does not try to p~omote the use of such particular classes of games
or to encourage researchers to gain tractability by squeezing their problems
into molds that will satisfy the conditions presented in this paper. The
purpose is to contribute to our understanding of differential games as an
appropriate framework for discussing dynamic interaction by investigating
the relationship between the assumptions that are made on the structure of
the interaction and the resultant equilibrium.
2. Strategy Spaces, Nash Equilibrium, and Set of Possible Initial Conditions
Consider an n-person nonzero-sum differential game in which:
(i)
n is the numer of players, indexed by i;
(ii) t is the calendar time, which belongs to the time interval [0, T];
(iii) u, is the control variable of player i, which belongs to the set of
admissible control U ~C_Rti;
(iv) x(t) = ( x l ( t ) , . . . , xm(t)) is a vector of state variables that evolves
on the time interval [0, T] according to the kinematic equation:
2c=f(x, u, t),
where f < C2;
x(O) =xo,
(1)
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(v)
219
the payoff to player i is given by the form
J~=
gi(x, u, t) dt,
(2)
i = 1 . . . . , n,
0
where gi ~ C 2, for every" i.
Definition 2.1. An open-loop strategy is a time path u~(x0, t) such
that, given the initial state variable Xo, it assigns for every t a control in
U i. The set of all possible open-loop strategies is denoted as q~, and
T=~I,...,~'.
In most cases in the literature, when only the open loop is considered,
the parameter xo is suppressed. Since in this work we wish to compare
different types of strategies, xo appears in the definition in order to indicate
that, for every initial state variable, a different time path of the control is
chosen.
Definition 2.2. A closed-loop strategy is a decision rule u~(xo, x, t)
such that it is continuous in t and uniformly Lipschitz in x for each t. The
set of all possible closed-loop strategies is denoted as (b;, and • = c) 1, . . . , qb'.
Definition 2.3. A feedback strategy is a decision rule u~(x, t) such that
it is continuous in t and uniformly Lipschitz in x for each t. The set of all
possible feedback strategies is denoted as Ai, and 5 = 5 1 , . . . , A"
The definitions of open.loop and closed-loop Nash equilibrium follow
immediately, such that open-loop (closed-loop) Nash equilibrium is an
n-tuple of open-loop (closed-loop) strategies ( u * , . . . , u*) satisfying the
following condition:
Ji(u*~,.. . ,
u*)>-Ji(u*,., . , u*,, u,, u % , . . . , u , )*,
i=l,...,n,
(3)
for every possible ui ~ q~i in the open-loop case and ui ~ ~i in the closed-loop
case.
In defining the open-loop Nash equilibrium, notice that an n-tuple of
time paths constitute an equilibrium for a game that starts at a particular
(to, Xo), but they do not necessarily constitute an equilibrium for a game
that starts at a different (to, xo). This implies that the open-loop Nash
equilibrium is not subgame perfect (Selten, Ref. 6). Similarly, notice that
the same conclusion holds for the closed-loop equilibrium. An n-tuple of
decision rules can constitute an equilibrium for a game that starts at a
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particular (to, xo), but they do not necessarily constitute an equilibrium for
a game that starts at a different (to, Xo).
In defining the feedback equilibrium, we require that the n-tuple of
feedback strategies ( u * , . . . , u*) satisfies the Nash condition (3) for every
possible (to, x0). Thus, the feedback equilibrium is an n-tuple of decision
rules that constitute an equilibrium for every possible initial condition and
can be found by solving the Hamiltonian-Jacobi-Bellman set of partial
differential equations (see Starr and Ho, Ref. 7). Since the feedback equilibrium strategies constitute an equilibrium for every possible subgame [i.e.,
(to, xo)], we can conclude that the feedback Nash equilibrium is subgame
perfect.
Lemma 2.1. The open-loop Nash equilibrium is a special case of the
closed-loop Nash equilibrium.
Proof. Since the set of all possible decision rules ui(xo, x, t) contains
also all the open-loop strategies ui(xo, t), we can conclude that • is a subset
o f ~ . Let ( u * , . . . , u*) be an open-loop Nash equilibrium, arid let us consider
the game that starts at (to, x0). Since ~Cqb,
u* ~ d) i,
for every i.
In order to prove that u* is also a closed-loop Nash equilibrium, one
needs to show that u* satisfies the Nash condition (3) for every possible
closed-loop strategy. Since u* is an open-loop equilibrium, condition (3)
holds for all u~~ xtt( So, it remains to be shown that condition (3) holds
also for all members in ~ \ ~ . For every ui ~ qbi\~ i, let ~(t) be the trajectory
generated by the kinematic equation (1), the closed-loop strategy ui, and
the open-loop equilibrium strategies u* for j ¢ i. Now, let us represent the
closed-loop strategy u~ by an open-loop strategy a (xo, t) which is generated
by the path £(t) and the strategy ui such that
ol(Xo, t) = ui(Xo, :~( t), t).
For such a, we have
J'(u*, ...,
u%,
u,, u*+,, . . . , u * ) =
J ' ( U l*, . . .
, u/~_l,/~E, u / ~ + l , . . . , Un~).
Since (Ul*,..., u*) is an open loop equilibrium, Ineq. (3) holds for all
members of ~ , and in particular for c~. Thus,
Ji(u*,...,u*)>-Ji(u*,...,u*_l,u~,u*+l,...,u*),
forevery u~ ~ ~ (
(4)
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22t
Remark 2.1. The above result was discussed previously in the
literature. See Basar and Olsder (Ref. 8) and Sandell (Ref. 9).
Definition 2.4.
For convenience, let us define a differential game as
G = ( n , J , U , f ( . ) , Yo), such that
n = number of players;
J = n-tuple vector of payoffs as defined in Eq. (2);
U = ( U l, .. ., Un), where U i c R li, is the set of admissible controls
for player i;
f ( . ) = kinematic equation (1);
Yo = set of possible initial conditions (to, xo).
The above definition deviates from the standard definition by including
the possibility of restricting the set of possible initial conditions. When the
set of initial conditions is not restricted and can include all possible (to, Xo),
the game will be denoted as (n, J, U,f(. ), UR). Since the meaning of a
subgame in a differential game setting is a game that starts at a particular
(to, Xo), the restriction on the possible set of initial conditions is a restriction
on the set of all possible subgames. A subgame can start only at points
included in Yo. By restricting the set of possible initial conditions to Y0,
we actually assume that, if the game arrives ~,t/x ~ Yo, the players do not
have at this state the opportunity to reconsider their strategies, which implies
that a subgame cannot start at ~ ~ Y0.
Proposition 2.1. Consider the game (n,J, U , f ( . ) , UR). In such a
game, there exists an open-loop Nash equilibrium which is a special case
of the feedback Nash equilibrium if and only if there exists an n-tuple of
time paths of the control (u*(t) . . . . , u*(t)) that constitutes an open-loop
Nash equilibrium for all possible initial conditions.
Proof. Every time path u~(t) belongs to the set of all possible feedback
strategies. Now, in order to prove that ( u * , . . . , u*) constitutes a feedback
equilibrium, it is sufficient to show that the Nash condition (3) is satisfied.
This can be done by repeating the technique used in Lemma 2.1.
The other direction is straightforward. If there is a degenerate feedback
equilibrium with strategies that depend only on t, then these strategies
constitute an open-loop equilibrium for every possible initial condition. []
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Corollary 2.1. The open-loop equilibrium is not a subgame perfect
equilibrium, unless it is also a degenerate feedback equilibrium.
3. Classes of Equivalence and Structure of Games
Previous works [see Clemhout and Wan (Ref. 1), Leitmann and
Schmitendorf (Ref. 2), Reinganum (Ref. 3), Kamien and Muller (Ref. 4),
Dockner, Feichtinger, and Jorgensen (Ref. 10), and Jorgensen (gef. 11)]
identify and examine several classes of games in which the open-loop and
the feedback equilibria coincide.
Their method of proof is to solve the feedback equilibrium and to
prove that the equilibrium strategies degenerate so as to depend only on t.
Since the feedback equilibrium is, in most cases, difficult (if not impossible)
to compute, this method is not useful in the general case. In this section,
we will develop a method by which we can examine the structure of a game
and find out if, for this specific structure, the open-loop Nash equilibrium
is a special case of the feedback equilibrium.
Definition 3.1. Two games, G1=(n, J1, U~,f~(.),Y1) and G~=
(n, J~, U2,f2(" ), Y2) are equivalent if every n-tuple of strategies that constitutes a Nash equilibrium for G1 constitutes also a Nash equilibrium
for G2.
The definition of equivalence can be made for open-loop, closed-loop,
or feedback strategies. In this work, we use equivalence classes with respect
to open-loop strategies. Therefore, for convenience; from now on we wilt
say that two games are equivalent when they are equivalent with respect to
opendoop strategies.
An example for the above definition can be constructed by taking a
game G~ and changing the payoff function such that .~'= aJ i, for every
i = 1. . . . , n, where a is some positive constant. Examining now the games
Gl=(n,J, U,f('), Yo) and
G2=(n,J, U , f ( - ) , Yo),
we can claim that these two games are equivalent. This claim does not need
a proof. The only change that was done is similar to changing the currency
in which the payoffs are made. Thus, any n-tuple of open-loop strategies
that constitutes an open-loop Nash equilibrium for G~ constitutes also an
equilibrium for G2.
Proposition 3.1. Consider the game G = (n, J, U,f('), UR). For such
a game, open-loop Nash equilibrium, if it exists, is also a feedback Nash
equilibrium , if and only if there exists an initial condition y c UR such that,
for every r/c UR, the game G , = (n, J, U , f ( . ), ~) is equivalent to the game
(n,J, U,f('),y).
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Proof. The proof is straightforward from Proposition 2.1 and the
definition of equivalence.
[]
Remark 3.1. Note that Proposition 3.1 indicates also that, if an openloop Nash equilibrium is a degenerate feedback Nash equilibrium, the
equilibrium strategies do not depend on the initial conditions [see also
Kamien and Muller (Ref. 4)].
Given a specific structure of a game, Proposition 3.1 provides us with
a tool by which we can find whether, for the game under discussion, the
open-loop Nash equilibrium, if it exists, is also a feedback equilibrium.
Alternatively, we can say that this tool can be used in order to find whether
the open-loop Nash equilibrium is subgame perfect. In order to demonstrate
the usefulness of this technique, we discuss a few examples. Some of these
examples are known in the literature, and our discussion just simplifies the
proofs and points out the characteristics of the structure that guarantee that
the open-loop equilibrium is a degenerate feedback equilibrium. Other
examples, which will be specified below, identify new classes of games for
which the above result holds.
Example 3.1. Games that Have Exponential Structure in the State Variables. Reinganum (Ref. 3) identified and examined a class of differential
games that have a linear quadratic structure in the strategies and exponential
structure in the state variable. For this class of games, Reinganum proved
that feedback and open-loop equilibria coincide. More recently, Jorgensen
(Ref. 11) proved that the same property holds for a class of games that
have exponential structure in both the control and the state variables. In
this subsection, we will generalize these results by examining a class of
games that have exponential structure in the state variables, but without
requiring linear-quadratic or exponential structure in the control variables.
We will show that, for this class of games, the open-loop Nash equilibrium
is also a feedback equilibrium.
Let G be a game in which there are n players, indexed by i = 1. . . . , n.
The payoff functions are assumed to be
ji=
fo
gi(u, t) exp(-Aix) dt,
(5)
where A i = ( A I , . . . , I ~) is a row vector of scalars, u~ ~ U ~C N ~, and g~(u, t) e
R ~, for every u and t. The state variables evolve according to
=f(u, t),
and the initial condition Xo can take any value in R ~.
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Lemma 3.1. For the above game, if there exists a n open-loop Nash
equilibrium for the game starting at Xo = 0, then these open-loop strategies
constitute also a feedback equilibrium for the game (n, J, U , f ( . ), UR).
ProoL For every initial state xo, the payoff function of this game can
be written as
J~= exp(-A~xo)
g~(u, t) exp[-A~(x -Xo)] dt,
i = 1 . . . . , n.
(6)
Let
A,(xo) = e x p ( - h 'xo).
For every initial state Xo, Ai(xo) is a constant that multiplies the payoff
function. Thus, if we define
.~ -- ji / a~( xo),
i -~ 1 , . . . , n,
then, for the above structure, GI = (n, J, U , f ( . ), Xo) is equivalent to G2 =
(n, J, U , f ( . ), Xo), for every Xo.
Let us define a state variable y e ~ " as
y(t) = x(t)-xo.
Since xo is fixed, it is clear that
3~= f ( u , t),
y(0) = x o - x o = 0 ,
and
J ' = f0 T g~(u, t) exp(-Aiy) dt.
Using y as a state variable defines the game G3 = (n, J, U, f ( - ), 0), which is
identical and therefore equivalent to G2. Now, since multiplication of the
payoff function by a constant leaves the game in the same equivalence class,
we define G4 = (n, J, U , f ( . ), 0), where J = JA(xo). Clearly, G4 is equivalent
to G3. Thus, we can conclude that G1 and (34 are equivalent.
Now, we can complete our proof by using Proposition 3.1, which claims
that, if for every Xo the game (n, J, U , f ( . ) , Xo) is equivalent to the game
(n, J, U , f ( . ) , 0), then the open-loop Nash equilibrium is also a feedback
equilibrium.
[]
Analyzing the above game structure, we can see that the semigroup
property 3 of the exponential function and the separability in x and u lead
to the above result. However, if we assume that the payoff function is
continuous in x, then the only continuous function that satisfies the semigroup property is the exponential function [see Rudin (Ref. 12, p. 178)].
3The function f(-) satisfies the semigroup property iff(x+y)=f(x).f(y).
JOTA: VOL. 55, NO. 2, NOVEMBER 1987
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An example of using the above game structure for studying economic
problems can be found in Reinganum (Ref. 13), in which a problem of
dynamic games of innovation is discussed.
Example 3.2. Games with Linear Structure in the State Variable. Let
G be a game in which there are n players, each striving to maximize the
payoff function
J'=
f0
g,(x) dt,
i = 1 , . . . , n,
(7)
where gi(. ) is a linear functional such that
gi( °~X + BY) : a g i ( x ) + flgi(Y).
The state variable x evolves according to
2 =f(u, t),
and the initial conditions Xo can take any value in R". Using the same
technique as in Lemma 3.1, we can prove that, for the above class of games,
open-loop Nash equilibrium, if it exists, is also a feedback equilibrium.
Let G~o b e the above game that starts at xo. Here, g~(.) is a linear
functional; therefore,
g i ( x ) -----gi( x -- XO) +
gi( xo).
Since gi(xo) is constant, we can define
.~ = J,
-
gi(Xo) T,
and the games (n, J, U, f ( . ), Xo) and (n, J, U, f ( . ), Xo) are equivalent. Now,
we can define a state variable y ( t ) = - x ( t ) - x o , which implies that
~=f(u,t)
and
y(0)=0.
Following the same steps as in Lemma 3.1, we can prove that G~ois equivalent
to Go, the game that starts as x = 0. Using Proposition 3.1, we can conclude
that, for the above game structure, the open-loop Nash equilibrium is
subgame perfect.
Example 3.3.
Let G be a differential game in which the payoffs are
given by
ji=
mi(t)[gli(u, t)(1 -x)+g21(t)x] dt,
i= 1 , . . . , n,
(8)
and the evolution of the state variable is governed by
2 = h(u, t)(1 - x ) ,
x(0) =Xo.
(9)
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This setting was use'd in the literature to discuss various one-player
problems [see, for example, Kamien and Schwartz (Ref. 14)]. In such a
setting, the state variable x denotes the cumulative probability that an event
occurred before the time t; g1~(u, t) is the payoff to player i if the event
did not occur; and g2~(t) is the payoff to player i if the event occurred prior
to time t.
Lemma 3.2. For the differential game defined by (8) and (9), an
open-loop equilibrium, if it exists, is also a feedback equilibrium.
Proof. We will prove the above lemma twice in order to demonstrate
the various techniques that can be used in proving such a claim.
Method A. Consider the transformation
(1 - x ) = (1 - y ) ( 1 -Xo).
Substituting this transformation into (9) implies the
~=h(u,t)(1-y),
which is identical in its structure to (9). Substituting x in (8) implies that
J'=
m~(t)[gli(u, t ) ( 1 - y ) ( 1 - X o ) + g 2 i ( t ) ( y + x o - x o y ) ] dt.
(10)
Rearranging (10) yields
J~ = (1 - xo)]+ A~(xo),
where
A~(xo) = xo
m~(t)g2~( t) dt,
and
ji= f0 T mi(t)[gl~(U, t)(1 -y)+g2i(t)y] dt.
Since a linear transformation of the payoff function leaves the game in the
same equivalence class, we can conclude that, for every Xo, the game
(n, J, U,f(. ), Xo) is equivalent to the game (n, J, U,f(. ), 1). Now, it remains
to apply Proposition 3.1.
Method B. Alternatively, we can prove that the above game is
equivalent to the game presented in Example 3.1 [see also Mehlmann and
Willing (Ref. 5)]. Using the state transformation
y = -In(1 - x),
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the problem (8)-(9) can be reformulated as
ji=
[mi(t)(gl~(u, t ) - g 2 i ( t ) e x p ( - y ) + g 2 i ( t ) m ~ ( t ) ] dr,
= h(u, t).
(8')
(9')
The game defined by (8')-(9') is identical to the class of games discussed
in Example 3.1.
[]
Example 3.4. Let G be a game in which there are n players. The
payoff functions are given by
ji=
fo
gi(u, t)x~ dt,
i= l,. .., n,
(11)
where x, a ~ R 1 and the state variable evolves according to the kinematic
equation
2 = f ( u , x, t) = m(u, t)x.
(12)
Lemma 3.3. For the above class of games, an open-loop Nash equilibrium, if it exists, is also a feedback equilibrium.
Proof.
Given a specific xo, the payoff functions can be written as
t'T
j i = xo~ j
g~(u, t ) x ~ ( t ) / x ~ dt,
i= 1 , . . . , n.
(13)
o
Let us define
J /xo,
i=l,...,n.
Since, under multiplication of the payoff function, the game remains in the
same equivalence class, we can conclude that, for every given Xo, the games
G1 = (n, J, U,f(" ), Xo) and G2 = (n, J, U , f ( . ), Xo) are equivalent. Now, let
us define a state variable
y(t) = x(t)/Xo.
Since x(0) = x0 is fixed, it is clear that y(0) = 1 and
)) = 2/Xo = m(u, t)X/Xo= re(u, t)y,
)~=
fo
g~(u, t)y ~ at.
(14)
(15)
Using y as a state variable, we define the game G3 = (n, .~ U , f ( . ), 1) which
is identical to and therefore equivalent to G2.
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Since multiplication of the payoff function by a constant leaves the
game in the same equivalence class, G4 = (n, J, U,f(. ), 1), where J = Jxo,
is equivalent to G3. Thus, we can conclude that G~ and Ga are equivalent.
Since, for every initial condition x0, the games (n, J, U,f('),Xo) and
(n, J, U , f ( . ) , 1) are equivalent, the conditions of Proposition 3.1 are
satisfied.
[]
Remark 3.2. Note that, by using the state transformation y - - x ~, the
game defined by (11)-(12) can be transformed into a game which is linear
with respect to the state variables.
The above class of games can be extended to a larger class if we
consider the payoff functions
J~=
g,l(u, t)g,2(x) at,
(16)
where gi2 satisfies the condition that, for every x, y ~ R 1,
g~2(xy) =-g~2(x)g,2(y).
Note that, although we discuss this structure for state variables in R 1, if we
let gi2(" ) be a function from R m to R 1, we can apply the same technique to
state variables in R m.
Example 3.5. Examining the Necessary and Sufficient Conditions for the
Open-Loop Nash Equilibrium. For the above classes of games, we examine
the equivalence problem by discussing directly the structure of the payoff
function and the kinematic equation without presenting the necessary and
sufficient conditions for equilibrium, However, investigating the Pontryagintype necessary and sufficient conditions for open-loop equilibrium can be
useful in proving the required equivalence relationship. It was already noted
in the literature [see Leitmann and Schmitendorf (Ref. 2), Mehlmann and
Willing (Ref. 5), Dockner, Feichtinger, and Jorgensen (Ref. 10)] that, when
the necessary and sufficient conditions for open-loop Nash equilibrium are
independent of the state variables, the open-loop Nash equilibrium is a
degenerate feedback equilibrium. In what .follows, we use the technique
suggested by Proposition 3.1 to reprove this result.
Lemma 3.4. Consider a differential game G. If the Pontryagin-type
necessary conditions for open-loop Nash equilibrium do not depend on
the state' variables, then open-loop Nash equilibrium, if it exists, is a
degenerate feedback Nash equilibrium. 4
4 As Jorgensen (Ref. 15) pointed out, under the assumptions of Lemma 3.4, the necessary
conditions for open-loop Nash equilibrium are also sufficient.
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Proof. Using Proposition 3.1, it is sufficient to prove that, for every
different initial condition Xo and Yo, the games G1 = (n, J, U , f ( . ), xo) and
G2 = (n, J, U , f ( . ) , Yo) are equivalent. Let ( u * , . . . , u~*) be an n-tuple of
open-loop strategies that constitute an open-loop Nash equilibrium for Gr.
Thus, ( u * , . . . , u*) satisfies the necessary and sufficient conditions for
open-loop Nash equilibrium for the game G~. Since these conditions do
not depend on the state variable, the necessary and sufficient conditions of
open-loop equilibrium for G2 are identical to those of G1. Thus, ( u ~ , . . . , u*)
constitutes also an equilibrium for the game G2. Thus, G~ and G2 are
equivalent.
[]
Example 3.6. Leitmann and Schmitendorf (Ref. 2) studied the advertising competition among firms. Under their assumption, the firms' sales
rate, denoted by x(t), evolves in the following way:
Yc~=gi(u,)-k~(u;)x~-a~x,,
i = 1. . . . . n;
(17)
here, u~ is the advertising expenditure rate of firm i; g~ is concave and
describes the effect of the firm's own advertising on its sales; ki is positive
and increasing and describes the effect of competitive advertising. The firms'
payoff functions are
ji =
exp(-rit)(qixi 2 ui) dt,
i = 1 , . . . , n,
(18)
where r~ is the ith firm discount rate and q~ is the price minus cost per unit,
which is assumed to be constant. The necessary and sufficient conditions
that the open-loop Nash equilibrium has to satisfy are
-exp(-r~t)+A~g'i(u~) =0,
i = 1 , . . . , n,
(19)
,~ = -exp(-r~t)q~+ A~k~(uj)+ A,a,,
i= 1 , . . . , n,
(20)
and the transversality conditions A~(T)= 0, for every i. Since the above
conditions do not depend on the state variable x, the condition of Lemma
3.4 is met and, as Leitmann and Schmitendorf indicated, in this setting, the
open-loop Nash equilibrium is also a degenerate feedback equilibrium.
4, Concluding Comments
In this paper, we examine the structure of differential games and specify
the conditions under which an open-loop equilibrium is also a feedback
230
JOTA: VOL. 55, NO. 2, NOVEMBER 1987
equilibrium. We use these conditions to identify classes of games for which
this property exists.
The paper does not try to promote the use of such specific structures.
What characterizes the games, in which this property exists, is the irrelevance
of the initial conditions. The players' actions do not depend on the initial
conditions. Thus, although the problem is formulated in a dynamic
framework, some of the dynamic aspects are missing. Thus, in investigating
a dynamic interaction among agents using a differential game framework,
one should be very careful in choosing the players' objective function and
in the assumptions that are made on the evolution of the state variables.
I m p r o p e r setting can diminish the dynamic aspects of the problem.
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