It is well known that, in general, Nash equilibria in

open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate in the sense that it depends on time (t) only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies

Reinganum, Jennifer F.

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
A CLASS OF DIFFERENTIAL GAMES WHERE THE CLOSED-LOOP 
.AND OPEN-LOOP NASH EQUILIBRIA COINCIDE 
Jennifer F. Reinganum 
E'::D:-,·o�, .:;8 R::SEARCH LIBRARll:..:�=��_:"Jr;,1:::;\;T OF ECONOMICSii,\ /t.i:::s,-;·"( o::- rillNNt.SOTA
'"ej,'1\lUTf OF 
�.... �� � � � 0 
3 1g . � 
.... 
. 
� l:f 
� � 
{,,..,, �� SlfALL tAfi-�� 
SOCIAL SCIENCE WORKING PAPER 333 
July 1980 
A CLASS OF DIFFERENTIAL GAMES WHERE THE 
CLOSED-LOOP AND OPEN-LOOP NASH EQUILIBRIA COINCIDE 
Jennifer F. Reinganum* 
ABSTRACT 
It is well known that, in general, Nash equilibria in 
open-loop strategies do not coincide with those in closed-loop 
strategies. This note identifies a class of differential games 
in which the Nash equilibrium in closed-loop strategies is degenerate 
in the sense that it depends on time (t) only. Consequently, the 
closed-loop equilibrium is also an equilibrium in open-loop strategies. 
*Assistant Professor of Economics, California Institute of Technology,
Pasadena, California 91125
2 
1. INTRODUCTION
It is well known that, in general, Nash equilibria in 
open-loop strategies do not coincide with those in closed-loop 
strategies. Notable exceptions are reported in [l], [2]. This note 
identifies another class of differential games in which the Nash 
equilibrium in closed-loop strategies is degenerate in the sense 
that it depends on time (t) only. Consequently, the closed-loop 
equilibrium is also an equilibrium in open-loop strategies. 
2. FORMULATION 
game. 
Consider the following n-person nonzero-sum differential 
2.1. n is the number of players, indexed by i. 
> 2. 2. m is the number of states, indexed by j; m < n. 
3 
2.3. t denotes calendar time; t ranges over [t0,T]; t0 E [O,T].
2. 4. z(t) � (z1(t), . . .  ,zm(t))', a column vector of state
variables. 
2.5. u(t,z) = (u1 (t,z), ••. ,un(t,z))', a column vector of
(closed-loop) strategies; strategy ui(t,z) is player i's strategy.
2. 6. Player i's strategy is assumed to belong to the admissible 
set 
ui {ui(t,z):ui(t,z) is measurable in t for each
fixed z and continuous in z for each fixed t; 
lui(t,z)I _'.:. k�(t)(l + lzl) for (t,z) E [t0,TJ
f T i x IRm where k0(t)dt < "'; andto 
lu/t,z) - ui(t,
--;)I _'.:. ki(t)lz - --;I if lzl < R,
1;1 _'.:. R, t0 < t < T}.
2.7. The state variables evolve according to the kinematic 
equations 
where A(t) 
z = A(t)u(t,z), z(t0) = z0, ( 1) 
[a�(t)] is an m x n matrix; aj (t) is the j-th row1 
vector and ai(t) is the i-th column vector.
are assumed continuous on [t0,TJ.
The functions a�(t)
1 
Remark 2. 1. We will try, in general, to use a superscript to 
indicate row vectors; subscripts for column vectors. Vectors with 
no sub- or superscripts will be column vectors. 
Remark 2.2. Because of (2.6), there exists a unique solution �(t) 
4 
to the system (1) on [t0,T] for any n-tuple of admissible strategies
[3]. 
Suppose that player i's payoff functional depends on 
t, �. and u in the following way: 
. f T . 2.8. J1(u;�) = [c(t) 'u + u'B(t)u]exp(-A1�)dt
to 
where Ai = (A�, •. . ,A!) is a row vector of scalars, c(t) = (c1(t), • . .  , 
cn(t))' is a column vector of continuous functions and B(t) =
[b�(t)] is an n x n matrix of continuous functions, with bi(t)
the i-th row vector and bk(t) the k-th column vector. We assume
that bi(t) < 0 Vi  and [B(t) + B(t)'] is nonsingular V t  with
D(t) = [B(t) + B(t)'J-1•
Remark 2. 3. This problem has a linear-quadratic structure in the stra-
tegies, but is exponential in the state variables. It will be apparent 
that Theorem 3. 1 below is true for other specifications of the payoffs 
and kinematic equations as well, so long as exponential state dependence 
is retained (though existence of equilibrium may be more difficult to 
prove since one would need to establish an instantaneous fixed point 
property at each t) . 
5 
3. RESULTS
Definition 3.1. A strategy n-tuple (u!·····u�) is a Nash equilibrium 
in pure closed-loop strategies on [O,T] x lRm if, for each
(t0,z0) E [O,T] x lR
m: v i
(a) ut E Ui; and
(b) Ji(u!, . . .  ,u�;<r*) � Ji(ut, . . .  ,ut_1,u1, ut+l' · · · ,u�;<P)
V ui E Ui.
Definition 3. 2 .  Let u(i) = (uf•···•ut_1,u1,ut+l'"""'u�) for
convenience of notation. 
Theorem 3. 1. For T sufficiently near O, there exists a Nash 
equilibrium in closed-loop strategies for the game described in 
Section 2. Furthermore, the equilibrium depends only on t .  
Proof: 
Define the value functions 
vi(to,zo) - sup ui Eui 
J
T . 
[c(t)'u(i) + u(i)'B(t)u(i)]exp(-A1z)dt 
to
subject to z = A(t)u(i), z(t0) = z0,
where u�(t,z), k r �is taken as given by player i. 
This definition requires that V i 
Vi(T,z(T)) o. (2) 
First we remark that, to be a Nash equilibrium, u 
must satisfy the necessary conditions below [4]: V i  
vi+ t max uiEUi 
{ViA(t)u(i)z 
+ [c(t)'u(i) + u(i)'B(t)u(i)]exp(-Aiz)}
at each point of differentiability of Vi(t,z), where Viz
0 
(Vi , • • .  ,Vi ) is a row vector of partial derivatives of i'szl zm 
value function . 
6 
(3) 
Performing the indicated maximization yields the candidates 
for a Nash equilibrium: V i 
i i . V a.(t) + {c.(t) + [b (t) + b.(t)']u*}exp(-A1z)z i i i 0. (4) 
Since b�(t) < O, these are also sufficient to guarantee
i 
a maximum in (3). 
Define yi - i i Vzai(t)exp(A z) + ci(t). Then we can rewrite
(4) as 
yi + [bi(t) + b.(t)']u*
i 
or, more compactly, 
y + [B(t) + B(t)'Ju* 
o, i 1, ... ,n (5) 
0, (6) 
7 
where y (y1, • • •  ,yn)' is a column vector.
The candidates for a Nash equilibrium (in feedback form) 
are 
u* = -D(t)y. (7) 
Substituting the candidates (7) into the system (3) 
and recalling (2) gives the following system of partial differential 
equations with terminal conditions (2): V i  
i i ; Vt - VzA(t)D(t)y + [-c(t)'D(t)y + y'D(t)'B(t)D(t)y]exp(-A z) o. (8)
Now we note that if we have in hand a system of value 
functions V* which are c1 and solve (8)-(2), then by the verification
theorem of [S], the strategies (7) form a Nash equilibrium in 
closed-loop pure strategies. 
Claim: Vi*(t,z) = h. (t)exp(-Aiz), i = 1 , 2, . . •  , n  is such 
i 
a system of value functions , where h(t) = (h1(t), • . .  ,hn(t))' is the
unique, continuously differentiable solution of: Vi 
hi
and 
-h.AiA(t)D(t)y(t) + c(t)'D(t)y(t) - y(t)'D(t)'B(t)D(t)y(t)
i 
hi(T) 0 
(9) 
(10) 
i where yi(t) = -hiA ai(t) + ci(t).
To see this , substitute i i V� and v; into (8), note
8 
i i that yi = V� ai(t)exp(A z) + ci(t) 
i -hiA ai(t) + ci(t), and cancel
i exp(-A .z) from each term.
The solution h(t) through h(T) = 0 exists (at least 
near T) and is unique because the r.h. s. of (9) is continuous 
in t and c1 in h.
Since 
u*(t,z) -D(t)y(t) 
D(t) 
1 h1(t)A a1(t) - c1(t)
n • h (t)A a (t) - c (t)n n n 
u* is continuous in t (and independent of z), so ut E Ui and
(uf·····u�) provides the Nash equilibrium as claimed. QED.
Theorem 3. 2. If the functions a{(t) , b!(t) and ci(t) are C00, then
the Nash equilibrium of Theorem 3.1 is unique. 
Proof: 
If (8)-(2) has a unique solution, then the equilibrium 
is unique. But if all the coefficient functions are C
00, then
the system (8)-(2) is C00 in (t , z,Vt ,Vz) and has a unique C
00 solution
V*(t ,z) near t = T [6]. 
4. EXAMPLES 
The following application to research and development is 
reported in greater detail and generality in [7) , [8) • 
Suppose two identical firms are engaged in a race for an 
invention. However, there is some uncertainty regarding the 
feasibility of the invention. In particular, firm i's probability 
9 
of success by time t depends upon zi(t), i's stock of accumulated
knowledge. This probability of success is assumed to be exponential: 
Fi(t) = 1 - exp(-Azi(t)). Player i's knowledge stock may be
incremented at the rate zi = u.(t,z) (> 0) by incurring costs ofl. -
$(1/2)u� . If player i wins the race, then i receives a patent
with current value Pert (constant present value). The game ends
as soon as one player succeeds or at T, whichever occurs earlier. 
Discounting at rate r, player i's payoff can be written as follows: 
Ji(u;z) f T -rt rt 2 
0
e [Pe Aul - (l/2)u1Jexp{-A(z1+z2)}dt
subject to z. = uj, z.(O) J J 
0, u. > 0, j = 1,2.
J -
That is, player i receives Pert at t if no one has yet
succeeded (this occurs with probability (1 F1) (1 - F2)
exp{-A(z1+z2)}) and if i succeeds at t (the conditional density of
success is Aui); costs are incurred only so long as no one has
succeeded. 
The analogs of equations (2), (3), (7) and (8) are given 
below. 
i Vt +
{ i i -rt 2 }max V u. + V u. + [PAui - e (l/2)ui)exp{-A(z1+z2)} EU z. l. z. J ui i l. J 
u. 
l. 
i rt rt vz.e exp{A(z1+z2)} + PAe ;
l. 
i i 2 rt } i j rt { } v + (l/2)(V ) e exp{A(z1+z2) + V V e exp A(z1+z2) t zi zj zj 
+ (Vj +Vi )PAert + (l/2)P2A2ertexp{-A(z1+z2)}Zj Zi 
Vi(T,z(T)) 0.
O; 
We know by Theorems 3.1-3.2 that the unique solution 
to (13)-(14) is of the form V*i(t,z) hi(t)exp{-A(z1+z2)}. 
Symmetry suggests that h1(t) = h2(t) - h(t). Substituting V�,
V* and V* into (13)-(14) implies that h(t) must solvezl z2 
h( t) 
h E -A2ert(l/2){3h2 - 4Ph + P2}, h(T) = 0.
The ordinary differential equation (15) has solution 
{ 2 rT rt } P[l - exp PA (e - e )/r )
{ 2 rT rt } [l - 3exp PA (e - e )/r ) 
Thus the Nash equilibrium strategies are: for t E [O,T), 
u* i 
2PAert . 
2 rt rT ' l. 3 - exp{PA (e - e )/r} 
1,2. 
10 
= O; 
(11) 
(12) 
(13) 
(14) 
(15) 
11 
Another example concerns the noncooperative exploitation 
of a common property resource of unknown size. That is, suppose 
n adjacent property-owners discover a shared pool of oil. Although 
i does not know the size of the reserve, S, i believes that S is 
distributed exponentially: 
Pri{S 2_ x} 1 - e-
).ix
Let X(t) be cumulative extraction to date. Then (to i) 
the probability that the pool is not exhausted by t is 
-AiX(t) 
Pr{S � X(t)} = e . The rate of extraction (strategy) of
player i is denoted by u (t,X) and the aggregate rate of extraction 
n 1 
is X = � uj(t,X).j=l 
If the players sell their resource in a market with 
linear demand curve 
P(t,Euj) 
j 
a(t) - b(t)�uj(t,X), 
J 
each spends $ci(ui) on extraction, and each discounts profits at
the rate r, then player i's payoff becomes 
Ji(u;X) J
r"'
e-A
iX(t)e-rt{ [a(t) - b(t)2: uj Jui - Ci ( ui) }dt
where X 
0 J 
�u /t ,X), X(O) 
J 
0. 
12 
If ci(ui) is nonnegative and quadratic, and a(t), b(t)
are positive and continuous, then by Theorem 3.1 there exists a 
Nash equilibrium in closed-loop strategies which is independent of 
x. 
13 
5. REFERENCES
1. Clemhout, S. and Wan, H. Y. , Jr. , "A Class of Trilinear
Differential Games," Journal of Optimization Theory and Applications,
Vol. 14, No. 4, pp . 419-424, 1974. 
2. Leitmann, G. and Schmitendorf, W. , "Profit Maximization
Through Advertising: A Nonzero Sum Differential Game Approach,"
IEEE Transactions on Automatic Control, Vol. AC-2l, pp. 645-650, 
August 1978. 
3. Freidman, Avner, Differential Games, John Wiley and Sons, New
York, 1971. 
4. Starr, A. W. and Ho, Y. C., "Nonzero Sum Differential Games,"
Journal of Optimization Theory and Applications, Vol 3, No. 3, 
pp. 184-208, 1969. 
5. Stalford, H. and Leitmann, G. , "Sufficiency Conditions for
Nash Equilibrium in N-Person Differential Games," Topics in
Differential Games, Edited by Austin Blaquiere, North-Holland
Publishing Company, New York, New York, 1973. 
6. Bernstein, D. L., Existence Theorems in Partial Differential
Equations, Princeton University Press, Princeton, New Jersey,
1950. 
7. Reinganum, J. F. , "Dynamic Games of Innovation," California
Institute of Technology, Social Science Working Paper #287, 
July 1979. 
8. 
------ • "A Dynamic Game of R & D: Patent Protection and
Competitive Behavior," California Institute of Technology, 
Social Science Working Paper #289 , August 1979. 
14 


English

A Class of Differential Games Where the Closed-Loop and Open-Loop Nash Equilibria Coincide

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
A CLASS OF DIFFERENTIAL GAMES WHERE THE CLOSED-LOOP 
.AND OPEN-LOOP NASH EQUILIBRIA COINCIDE 
Jennifer F. Reinganum 
E'::D:-,·o�, .:;8 R::SEARCH LIBRARll:..:�=��_:"Jr;,1:::;\;T OF ECONOMICSii,\ /t.i:::s,-;·"( o::- rillNNt.SOTA
'"ej,'1\lUTf OF 
�.... �� 
� � � 0 
3 1g . � 
.... 
. 
� l:f 
� � 
{,,..,, �� SlfALL tAfi-�� 
SOCIAL SCIENCE WORKING PAPER 333 
July 1980 
A CLASS OF DIFFERENTIAL GAMES WHERE THE 
CLOSED-LOOP AND OPEN-LOOP NASH EQUILIBRIA COINCIDE 
Jennifer F. Reinganum* 
ABSTRACT 
It is well known that, in general, Nash equilibria in 
open-loop strategies do not coincide with those in closed-loop 
strategies. This note identifies a class of differential games 
in which the Nash equilibrium in closed-loop strategies is degenerate 
in the sense that it depends on time (t) only. Consequently, the 
closed-loop equilibrium is also an equilibrium in open-loop strategies. 
*Assistant Professor of Economics, California Institute of Technology,
Pasadena, California 91125
2 
1. INTRODUCTION
It is well known that, in general, Nash equilibria in 
open-loop strategies do not coincide with those in closed-loop 
strategies. Notable exceptions are reported in [l], [2]. This note 
identifies another class of differential games in which the Nash 
equilibrium in closed-loop strategies is degenerate in the sense 
that it depends on time (t) only. Consequently, the closed-loop 
equilibrium is also an equilibrium in open-loop strategies. 
2. FORMULATION 
game. 
Consider the following n-person nonzero-sum differential 
2.1. n is the number of players, indexed by i. 
> 2. 2. m is the number of states, indexed by j; m < n. 
3 
2.3. t denotes calendar time; t ranges over [t0,T]; t0 E [O,T].
2. 4. z(t) � (z1(t), . . .  ,zm(t))', a column vector of state
variables. 
2.5. u(t,z) = (u1 (t,z), ••. ,un(t,z))', a column vector of
(closed-loop) strategies; strategy ui(t,z) is player i's strategy.
2. 6. Player i's strategy is assumed to belong to the admissible 
set 
ui {ui(t,z):ui(t,z) is measurable in t for each
fixed z and continuous in z for each fixed t; 
lui(t,z)I _'.:. k�(t)(l + lzl) for (t,z) E [t0,TJf T i x IRm where k0(t)dt < "'; andto 
lu/t,z) - ui(t,
--;)I _'.:. ki(t)lz - --;I if lzl < R,
1;1 _'.:. R, t0 < t < T}.
2.7. The state variables evolve according to the kinematic 
equations 
where A(t) 
z = A(t)u(t,z), z(t0) = z0, ( 1) 
[a�(t)] is an m x n matrix; aj (t) is the j-th row1 
vector and ai(t) is the i-th column vector.
are assumed continuous on [t0,TJ.
The functions a�(t)
1 
Remark 2. 1. We will try, in general, to use a superscript to 
indicate row vectors; subscripts for column vectors. Vectors with 
no sub- or superscripts will be column vectors. 
Remark 2.2. Because of (2.6), there exists a unique solution �(t) 
4 
to the system (1) on [t0,T] for any n-tuple of admissible strategies
[3]. 
Suppose that player i's payoff functional depends on 
t, �. and u in the following way: 
. f T . 2.8. J1(u;�) = [c(t) 'u + u'B(t)u]exp(-A1�)dt
to 
where Ai = (A�, •. . ,A!) is a row vector of scalars, c(t) = (c1(t), • . .  , 
cn(t))' is a column vector of continuous functions and B(t) =
[b�(t)] is an n x n matrix of continuous functions, with bi(t)
the i-th row vector and bk(t) the k-th column vector. We assume
that bi(t) < 0 Vi  and [B(t) + B(t)'] is nonsingular V t  with
D(t) = [B(t) + B(t)'J-1•
Remark 2. 3. This problem has a linear-quadratic structure in the stra-
tegies, but is exponential in the state variables. It will be apparent 
that Theorem 3. 1 below is true for other specifications of the payoffs 
and kinematic equations as well, so long as exponential state dependence 
is retained (though existence of equilibrium may be more difficult to 
prove since one would need to establish an instantaneous fixed point 
property at each t) . 
5 
3. RESULTS
Definition 3.1. A strategy n-tuple (u!·····u�) is a Nash equilibrium 
in pure closed-loop strategies on [O,T] x lRm if, for each
(t0,z0) E [O,T] x lR
m: v i
(a) ut E Ui; and
(b) Ji(u!, . . .  ,u�;<r*) � Ji(ut, . . .  ,ut_1,u1, ut+l' · · · ,u�;<P)
V ui E Ui.
Definition 3. 2 .  Let u(i) = (uf•···•ut_1,u1,ut+l'"""'u�) for
convenience of notation. 
Theorem 3. 1. For T sufficiently near O, there exists a Nash 
equilibrium in closed-loop strategies for the game described in 
Section 2. Furthermore, the equilibrium depends only on t .  
Proof: 
Define the value functions 
vi(to,zo) - sup ui Eui 
JT . [c(t)'u(i) + u(i)'B(t)u(i)]exp(-A1z)dt to
subject to z = A(t)u(i), z(t0) = z0,
where u�(t,z), k r �is taken as given by player i. 
This definition requires that V i 
Vi(T,z(T)) o. (2) 
First we remark that, to be a Nash equilibrium, u 
must satisfy the necessary conditions below [4]: V i  
vi+ t max uiEUi 
{ViA(t)u(i)z 
+ [c(t)'u(i) + u(i)'B(t)u(i)]exp(-Aiz)}
at each point of differentiability of Vi(t,z), where Viz
0 
(Vi , • • .  ,Vi ) is a row vector of partial derivatives of i'szl zm 
value function . 
6 
(3) 
Performing the indicated maximization yields the candidates 
for a Nash equilibrium: V i 
i i . V a.(t) + {c.(t) + [b (t) + b.(t)']u*}exp(-A1z)z i i i 0. (4) 
Since b�(t) < O, these are also sufficient to guarantee
i 
a maximum in (3). 
Define yi - i i Vzai(t)exp(A z) + ci(t). Then we can rewrite
(4) as 
yi + [bi(t) + b.(t)']u*
i 
or, more compactly, 
y + [B(t) + B(t)'Ju* 
o, i 1, ... ,n (5) 
0, (6) 
7 
where y (y1, • • •  ,yn)' is a column vector.
The candidates for a Nash equilibrium (in feedback form) 
are 
u* = -D(t)y. (7) 
Substituting the candidates (7) into the system (3) 
and recalling (2) gives the following system of partial differential 
equations with terminal conditions (2): V i  
i i ; Vt - VzA(t)D(t)y + [-c(t)'D(t)y + y'D(t)'B(t)D(t)y]exp(-A z) o. (8)
Now we note that if we have in hand a system of value 
functions V* which are c1 and solve (8)-(2), then by the verification
theorem of [S], the strategies (7) form a Nash equilibrium in 
closed-loop pure strategies. 
Claim: Vi*(t,z) = h. (t)exp(-Aiz), i = 1 , 2, . . •  , n  is such 
i 
a system of value functions , where h(t) = (h1(t), • . .  ,hn(t))' is the
unique, continuously differentiable solution of: Vi 
hi
and 
-h.AiA(t)D(t)y(t) + c(t)'D(t)y(t) - y(t)'D(t)'B(t)D(t)y(t)
i 
hi(T) 0 
(9) 
(10) 
i where yi(t) = -hiA ai(t) + ci(t).
To see this , substitute i i V� and v; into (8), note
8 
i i that yi = V� ai(t)exp(A z) + ci(t) 
i -hiA ai(t) + ci(t), and cancel
i exp(-A .z) from each term.
The solution h(t) through h(T) = 0 exists (at least 
near T) and is unique because the r.h. s. of (9) is continuous 
in t and c1 in h.
Since 
u*(t,z) -D(t)y(t) 
D(t) 
1 h1(t)A a1(t) - c1(t)
n • h (t)A a (t) - c (t)n n n 
u* is continuous in t (and independent of z), so ut E Ui and
(uf·····u�) provides the Nash equilibrium as claimed. QED.
Theorem 3. 2. If the functions a{(t) , b!(t) and ci(t) are C00, then
the Nash equilibrium of Theorem 3.1 is unique. 
Proof: 
If (8)-(2) has a unique solution, then the equilibrium 
is unique. But if all the coefficient functions are C00, then
the system (8)-(2) is C00 in (t , z,Vt ,Vz) and has a unique C
00 solution
V*(t ,z) near t = T [6]. 
4. EXAMPLES 
The following application to research and development is 
reported in greater detail and generality in [7) , [8) • 
Suppose two identical firms are engaged in a race for an 
invention. However, there is some uncertainty regarding the 
feasibility of the invention. In particular, firm i's probability 
9 
of success by time t depends upon zi(t), i's stock of accumulated
knowledge. This probability of success is assumed to be exponential: 
Fi(t) = 1 - exp(-Azi(t)). Player i's knowledge stock may be
incremented at the rate zi = u.(t,z) (> 0) by incurring costs ofl. -
$(1/2)u� . If player i wins the race, then i receives a patent
with current value Pert (constant present value). The game ends
as soon as one player succeeds or at T, whichever occurs earlier. 
Discounting at rate r, player i's payoff can be written as follows: 
Ji(u;z) f T -rt rt 2 
0
e [Pe Aul - (l/2)u1Jexp{-A(z1+z2)}dt
subject to z. = uj, z.(O) J J 
0, u. > 0, j = 1,2.J -
That is, player i receives Pert at t if no one has yet
succeeded (this occurs with probability (1 F1) (1 - F2)
exp{-A(z1+z2)}) and if i succeeds at t (the conditional density of
success is Aui); costs are incurred only so long as no one has
succeeded. 
The analogs of equations (2), (3), (7) and (8) are given 
below. 
i Vt +
{ i i -rt 2 
}
max V u. + V u. + [PAui - e (l/2)ui)exp{-A(z1+z2)} EU z. l. z. J ui i l. J 
u. 
l. 
i rt rt vz.e exp{A(z1+z2)} + PAe ;
l. 
i i 2 rt } i j rt { } v + (l/2)(V ) e exp{A(z1+z2) + V V e exp A(z1+z2) t zi zj zj 
+ (Vj +Vi )PAert + (l/2)P2A2ertexp{-A(z1+z2)}Zj Zi 
Vi(T,z(T)) 0.
O; 
We know by Theorems 3.1-3.2 that the unique solution 
to (13)-(14) is of the form V*i(t,z) hi(t)exp{-A(z1+z2)}. 
Symmetry suggests that h1(t) = h2(t) - h(t). Substituting V�,
V* and V* into (13)-(14) implies that h(t) must solvezl z2 
h( t) 
h E -A2ert(l/2){3h2 - 4Ph + P2}, h(T) = 0.
The ordinary differential equation (15) has solution 
{ 2 rT rt } P[l - exp PA (e - e )/r )
{ 2 rT rt } [l - 3exp PA (e - e )/r ) 
Thus the Nash equilibrium strategies are: for t E [O,T), 
u* i 
2PAert . 
2 rt rT ' l. 3 - exp{PA (e - e )/r} 
1,2. 
10 
= O; 
(11) 
(12) 
(13) 
(14) 
(15) 
11 
Another example concerns the noncooperative exploitation 
of a common property resource of unknown size. That is, suppose 
n adjacent property-owners discover a shared pool of oil. Although 
i does not know the size of the reserve, S, i believes that S is 
distributed exponentially: 
Pri{S 2_ x} 1 - e-
).ix
Let X(t) be cumulative extraction to date. Then (to i) 
the probability that the pool is not exhausted by t is 
-AiX(t) Pr{S � X(t)} = e . The rate of extraction (strategy) of
player i is denoted by u (t,X) and the aggregate rate of extraction 
n 1 
is X = � uj(t,X).j=l 
If the players sell their resource in a market with 
linear demand curve 
P(t,Euj) 
j 
a(t) - b(t)�uj(t,X), 
J 
each spends $ci(ui) on extraction, and each discounts profits at
the rate r, then player i's payoff becomes 
Ji(u;X) J
r"'
e-A
iX(t)e-rt{ [a(t) - b(t)2: uj Jui - Ci ( ui) }dt
where X 
0 J 
�u /t ,X), X(O) 
J 
0. 
12 
If ci(ui) is nonnegative and quadratic, and a(t), b(t)
are positive and continuous, then by Theorem 3.1 there exists a 
Nash equilibrium in closed-loop strategies which is independent of 
x. 
13 
5. REFERENCES
1. Clemhout, S. and Wan, H. Y. , Jr. , "A Class of Trilinear
Differential Games," Journal of Optimization Theory and Applications,
Vol. 14, No. 4, pp . 419-424, 1974. 
2. Leitmann, G. and Schmitendorf, W. , "Profit Maximization
Through Advertising: A Nonzero Sum Differential Game Approach,"
IEEE Transactions on Automatic Control, Vol. AC-2l, pp. 645-650, 
August 1978. 
3. Freidman, Avner, Differential Games, John Wiley and Sons, New
York, 1971. 
4. Starr, A. W. and Ho, Y. C., "Nonzero Sum Differential Games,"
Journal of Optimization Theory and Applications, Vol 3, No. 3, 
pp. 184-208, 1969. 
5. Stalford, H. and Leitmann, G. , "Sufficiency Conditions for
Nash Equilibrium in N-Person Differential Games," Topics in
Differential Games, Edited by Austin Blaquiere, North-Holland
Publishing Company, New York, New York, 1973. 
6. Bernstein, D. L., Existence Theorems in Partial Differential
Equations, Princeton University Press, Princeton, New Jersey,
1950. 
7. Reinganum, J. F. , "Dynamic Games of Innovation," California
Institute of Technology, Social Science Working Paper #287, 
July 1979. 
8. 
------ • "A Dynamic Game of R & D: Patent Protection and
Competitive Behavior," California Institute of Technology, 
Social Science Working Paper #289 , August 1979. 
14 


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A Class of Differential Games Where the Closed-Loop and Open-Loop Nash Equilibria Coincide

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