A technique to determine closed-loop Nash equilibria of n-player differential games is developed when their dynamic state-control system is composed of decoupled ODEs. In particular, the theory of Lie point symmetries is exploited to achieve first integrals of such systems.

A. Palestini

English

Research Papers in Economics

 
 
UNIVERSITA' DEGLI STUDI DI BOLOGNA 
DIPARTIMENTO DI SCIENZE ECONOMICHE 
 
 
 
On Lie point symmetries in differential games 
 
 
 
Arsen Palestini 
 
 
 
 
 
Quaderni - Working Papers DSE N° 698 
 
 On Lie point symmetries in dierential games
Arsen Palestini 
Department of Economics, University of Bologna
Strada Maggiore 45, 40125 Bologna, Italy
palestini@math.uni.it
Abstract
A technique to determine closed-loop Nash equilibria of n-player
dierential games is developed when their dynamic state-control sys-
tem is composed of decoupled ODEs. In particular, the theory of
Lie point symmetries is exploited to achieve rst integrals of such
systems.
JEL Classication: C72, C73
Keywords: dierential game, closed-loop Nash equilibrium, Lie point
symmetry, innitesimal generator, rst integral
1 Introduction
The present paper introduces a procedure to help the calculation of the
optimal trajectories of some dierential games in closed form. The analytic
tools I will utilize are the Lie point symmetries, which rarely have been used
in any game theory framework.
Dierential game theory, together with the related applications in man-
agement science, economics, engineering and a number of further elds, has
been developing very much in the last three decades. A recent survey on
the fundamental instruments in use and the foremost models in literature is
due to Jrgensen and G. Zaccour (2007). For an exhaustive overview on this
The author is thankful to Davide Barbieri and Davide Dragone for their very useful
comments and suggestions. The usual disclaimer applies.
1subject, from both theoretical and applicative standpoints, see also Dockner
et al. (2000).
Among the several equilibrium concepts usually investigated, most au-
thors prefer to linger on open-loop and feedback Nash trajectories, whereby
the determination of exact closed-loop Nash solutions seldom appears in
literature.
Since closed-loop equilibria are feedback equilibria depending on the ini-
tial conditions of the associated Cauchy problem, sometimes the determina-
tion of the former follows from the computation of the latter (see for instance
Fershtman and Kamien, 1987, or Yeung, 1989). In some other setups, par-
ticular numerical techniques are developed to show the nonuniqueness for
closed-loop Nash solutions (e.g. Mehlmann and Willing, 1983) in particular
classes of subgame perfect dierential games (Reinganum, 1982).
Alternatively, the investigation of the closed-loop information structure
helps the evaluation of the feedback eects along the equilibrium paths of
the game (Cellini and Lambertini, 2005).
Several denitions of these special solutions are given (see for example
Yeung and Petrosyan, 2006, or Sethi and Thompson, 2000) but a complete
treatise on this subject seems to be missing. A partial motivation for this
lack of analysis might stem from the fact that feedback equilibria, generally
much more complicated to deal with, can be seen as an extension of the
closed-loop equilibria, since they do not depend on the initial conditions of
the dynamic constraint of the problem at hand.
To the best of my knowledge, no attempt has been made to embed Lie
point symmetries and the related innitesimal generators in a dierential
game theory framework.
In this paper I will suggest a technique to carry out the closed form
integration of the dynamic state-control systems of a class of simultaneous
dierential games. In the next section I will outline the setup of the games
under consideration and state the conditions to achieve a state-control dy-
namic system of decoupled ODEs. In section 3 I will recall some very pre-
liminary concepts on Lie point symmetries and collect some results. Section
4 features a complete example to which all the above results apply. Section
5 concludes and introduces the possible future developments.
22 The basic setup
Consider an n-player innite horizon dierential game   in which the i-th
agent, endowed with a prot function i(x;u;t) and manoeuvring her unique
strategic variable ui, aims to maximize the following functional objective:
Ji 
Z 1
t0
e
 iti(x(t);u(t);t)dt (2.1)
s.t.: (
_ xs(t) = gs(xs(t);us(t);t)
xs(t0) = xs0
(2.2)
s = 1;:::;n, where:
 xs(t) 2 Xs  R, s = 1;:::;n are the state variables and all Xs are
open subsets of R;
 ui(t) 2 Ui  R, i = 1;:::;n are the control variables of the n players
and Ui are all open subsets of R;
 u(t) = (u1(t);:::;un(t)) 2 U1:::Un and x(t) = (x1(t);:::;xn(t)) 2
X1  :::  Xn respectively are vectors of control and state variables;
 1;:::;n 2 C2(X1  :::  Xn  U1  :::  Un  [t0;1));
 gs 2 C2(Xs  Us  [t0;1)), s = 1;:::;n;
 i is the intertemporal discount rate for the i-th player;
 the game is played simultaneously.
If we call is(t) the costate variable associated by the i-th player to the
s-th state, the current-value Hamiltonian function of the game   will read
as follows (from now on, most of the time arguments will be omitted for
brevity):
Hi() = i(x;u) + iigi(xi;ui) +
X
i6=s
isgs(xs;us):
Denition 2.1. A decision rule u
i(x; t) 2 Ui is a closed-loop strategy
if it is continuous in t and uniformly Lipschitz in x for each t.
3Denition 2.2. A vector u = (u
1;:::;u
n) 2 U1  :::Un of closed-loop
strategies is a closed-loop Nash equilibrium if
Ji(u

1;:::;u

n)  Ji(u

1;:::;u

i 1;ui;u

i+1;:::;u

n); i = 1;:::;n (2.3)
holds for all closed-loop strategies ui.
The determination of an equilibrium information structure generally re-
quires the application of Pontryagin's Maximum Principle. The crucial prop-
erty of   to be exploited is that the i-th dynamic constraint depends only
on the related i-th state and on the i-th player's strategic variable, thus
implying:
@gs
@ui
=
@gs
@xi
= 0 (2.4)
for all s 6= i, which entails the following rst order conditions (FOCs):
@Hi
@ui
=
@i
@ui
+ ii
@gi
@ui
= 0; (2.5)
for all i = 1;:::;n.
If we call ~ u(x) = (~ u1(x);:::; ~ un(x)) the vector of control variables sat-
isfying (2.5), then the associated adjoint variable dynamic system results
in:
_ ii =

i  
@gi
@xi

ii  
@i
@xi
; (2.6)
_ is = iis  
@i
@xs
 
X
s6=i
@i
@us
@~ us
@xs
; (2.7)
i;s = 1;:::;n, s 6= i.
The related transversality conditions read as:
lim
t!+1e
 itis(t) = 0: (2.8)
Since no costate variable is, where i 6= s, appears in (2.5), the n2 n adjoint
equations (2.7) are not relevant for the determination of the closed-loop Nash
equilibrium of  .
42.1 The dynamic state-control structure
After deriving (2.5) with respect to time, under suitable regularity hypothe-
ses, and subsequently substituting in (2.6), we obtain a state-control dy-
namic system of 2n equations, possibly but rarely integrable in closed form
with standard methods. I will focus on the conditions under which the above
transformation of variables is feasible.
Proposition 2.1. If along every equilibrium trajectory of   the following
conditions hold for every i;j = 1;:::;n, i 6= j:
1.
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
6= 0;
2.
@2i
@ui@uj
= 0,
then the dynamic state-control system of   is formed by (2.2) and by the
following ODEs: 0
B
@
_ u1
. . .
_ un
1
C
A = A
 1 
0
B
@
h1(u;x)
. . .
hn(u;x)
1
C
A; (2.9)
where A 1 = (bij) is a diagonal n  n matrix with the following entries:
bii =
0
B
B
@
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
1
C
C
A
 1
;
and where
0
B
@
h1()
. . .
hn()
1
C
A =
0
B B B B
@

1  
@g1
@x1

@1
@u1 +
@1
@x1
@g1
@u1  
Pn
j=1
@21
@u1@xj  gj +
@1
@u1
@g1
@u1
@2g1
@u1@x1  g1
. . . 
n  
@gn
@xn

@n
@un + @n
@xn
@gn
@un  
Pn
j=1
@2n
@un@xj  gj +
@n
@un
@gn
@un
@2gn
@un@xn  gn
1
C C C C
A
:
(2.10)
5Proof. The derivation of (2.5) yields:
n X
j=1

@2i
@ui@uj
_ uj +
@2i
@ui@xj
_ xj

+ _ ii
@gi
@ui
+ ii

@2gi
@u2
i
_ ui +
@2gi
@ui@xi
_ xi

= 0;
(2.11)
then, by using (2.6) and (2.5) we obtain:
n X
j=1

@2i
@ui@uj
_ uj +
@2i
@ui@xj
 gj

 

i  
@gi
@xi

@i
@ui
+
@i
@xi
@gi
@ui

+ (2.12)
 
@i
@ui
@gi
@ui

@2gi
@u2
i
_ ui +
@2gi
@ui@xi
 gi

= 0; (2.13)
relations that we can express in the following matrix form:
A
0
B
@
_ u1
. . .
_ un
1
C
A =
0
B B B
B
@

1  
@g1
@x1

@1
@u1 +
@1
@x1
@g1
@u1  
Pn
j=1
@21
@u1@xj  gj +
@1
@u1
@g1
@u1
@2g1
@u1@x1  g1
. . . 
n  
@gn
@xn

@n
@un + @n
@xn
@gn
@un  
Pn
j=1
@2n
@un@xj  gj +
@n
@un
@gn
@un
@2gn
@un@xn  gn
1
C C C
C
A
;
(2.14)
where A = (aij) is the n  n matrix whose coecients are:
aii =
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
; aij =
@2i
@ui@uj
: (2.15)
If the two hypotheses hold, A is nonsingular, hence (2.9) and the kinematic
equations (2.2) make up a 2n-variable state-control dynamic system.
Corollary 2.1. If for every i = 1;:::;n we have:
@
@xs
2
6 6
4
0
B B
@
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
1
C C
A
 13
7 7
5 =
@
@us
2
6 6
4
0
B B
@
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
1
C C
A
 13
7 7
5 = 0
(2.16)
for all s 6= i, then (2.9) and (2.2) form a dynamic system composed of
decoupled ODEs.
6In the following, I will discuss the cases in which the hypotheses of Corol-
lary 2.1 hold, i.e. on the games whose associated system results in:
(
_ xi = gi(xi;ui)
_ ui = hi(xi;ui)
(2.17)
forming a Cauchy problem with the initial condition of (2.2) and with the
appropriate transversality conditions implied by (2.8). A system of 2n de-
coupled ODEs can be thought of as a set of n planar systems. By the time
elimination method, every pair of ODEs locally provides an expression of
the closed-loop trajectory u
i(x
i). Lie point symmetries, to which the next
Section is devoted, constitute quite a powerful tool for the possible closed
form integration of the i-th pair of (2.17).
3 Preliminaries about Lie point symmetries
An exhaustive overview of the current developments and applications of Lie
point symmetries can be found in Hydon (2000) and in Starrett (2007). In
particular, I will rely on a notation as similar to the one used by Hydon as
possible to ease the reading of the upcoming facts.
Denition 3.1. Given a one-parameter planar Lie symmetry
S(x;u) = (^ x(x;u;); ^ u(x;u;));
the tangent vector to the orbit at the point (x;u) is the vector
((x;u); (x;u)) =

d^ x
d j=0
;
d^ u
d j=0

:
When an ODE admits a one-parameter Lie group of symmetries, the partial
dierential operator
X = (x;u)
@
@x
+ (x;u)
@
@u
is called the intesimal generator of the Lie group.
Denition 3.2. Given the rst order ODE
du
dx
=
h(x;u)
g(x;u)
:= !(x;u); (3.1)
we call the reduced characteristic the function
Q(x;u) := (x;u)   !(x;u)(x;u):
7A solution curve of (3.1) is invariant under a given nontrivial Lie group
if and only if Q(x;f(x)) = 0. In general, one of the standard procedures is
based on the determination of an integrand factor to compute a rst integral
of (3.1). We can summarize the fundamental results in the following:
Proposition 3.1. If the vector eld ((x;u); (x;u)) satises the lin-
earized symmetry condition:
@
@x
+

@
@u
 
@
@x

!  
@
@u
!
2 = 
@!
@x
+ 
@!
@u
(3.2)
then Z
du   !(x;u)dx
Q(x;u)
= C (3.3)
is the general solution of (3.1).
Proof. See Hydon (pp. 31-37).
Consider a vector eld ((x);(u)), where the i-th component only de-
pends on the i-th variable. The condition (3.2) can be simplied as shown
by the following results:
Proposition 3.2. The vector eld ((x);(u)) veries the linearized sym-
metry condition if and only if the following equality holds:
! =
@(h)
@x
+ 
@h
@u
@(g)
@u
+ 
@g
@x
: (3.4)
Proof. If  depends only on x and  depends only on u, then (3.2) reduces
to: 
@
@u
 
@
@x

! = 
@!
@x
+ 
@!
@u
; (3.5)
and by denition of !(x;u), after deriving we obtain:

@
@u
 
@
@x

hg = 

g
@h
@x
  h
@g
@x

+ 

g
@h
@u
  h
@g
@u

;
which upon collecting the terms becomes:
h

@(g)
@u
+ 
@g
@x

= g

@(h)
@x
+ 
@h
@u

;
from which (3.4) follows.
8Proposition 3.3. If the vector eld ((x);(u)) is such that 0(x) = 0(u),
then it satises (3.2) if and only if
! =

@h
@x
+ 
@h
@u

@g
@x
+ 
@g
@u
: (3.6)
Proof. The left-hand side of (3.5) vanishes so that through some algebra we
obtain:
g
@h
@x
+ g
@h
@u
= h
@g
@x
+ h
@g
@u
;
which entails the identity (3.6).
Note that all vector elds of the kind ((x);(u)) = (ax+b;au+c) verify
both previous Propositions, hence they represent a very useful ansatz for the
resolution of (2.17), as pointed out by an example in the next Section.
I chose to investigate a dierential game which is endowed with a struc-
ture slightly dierent from the usual ones, characterized by a polynomial
functional objective with cubic and 4 degree terms.
4 An example
Example 4.1. Consider a 2-player game, with agents i and j, in the setup
that we xed in Section 2. Call ui and uj the control variables, and xi
and xj the respective states. The i-th agent seeks to maximize the following
functional objective w.r.t. her strategic variable:
Ji 
Z 1
0
e
 it(u
2
ixi + x
3
i   x
2
ju
2
j)dt
subject to: (
_ xi = ui
xi(0) = xi0;
i = 1;2: For simplicity, the state and control sets are X1 = X2 = U1 = U2 =
R.
Since the relevant rst order derivatives are:
@i
@ui
= 2uixi;
@gi
@ui
= 1;
@i
@xi
= u
2
i + 3x
2
i;
@gi
@xi
= 0;
9then we have:
@2i
@u2
i
 
@i
@ui
@gi
@ui

@2gi
@u2
i
= 2xi;
vanishing if and only if ii  0, that would make the problem collapse into
a static game and not yield an optimal trajectory.
@2i
@ui@uj
= 0;
so all the hypotheses of Proposition 2.1 and of Corollary 2.1 hold. The
system (2.9) amounts to:

_ u1
_ u2

=
 1
2x1 0
0 1
2x2

21u1x1 + 3x2
1   u2
1
22u2x2 + 3x2
2   u2
2

: (4.1)
The related state-control dynamic system reads as follows:
8
> > > > > > <
> > > > > > :
_ x1 = u1
_ u1 = 1u1 +
3x1
2
 
u2
1
2x1
_ x2 = u2
_ u2 = 2u2 +
3x2
2
 
u2
2
2x2
: (4.2)
(4.2) is composed of decoupled equations.
Therefore, it is not restrictive to consider any of the two pairs of equa-
tions:
dui
dxi
= i +
3xi
2ui
 
ui
2xi
:= !i(xi;ui): (4.3)
The vector eld ((xi);(ui)) = (xi;ui) veries the hypothesis of Proposition
3.3; since (3.6) holds, as shown in the following:
xi

3
2
+
u2
i
2x2
i

+ ui

i  
ui
xi

ui
= !i(xi;ui);
then a rst integral of (4.3) turns out to be:
Z
2xiuidu + (u2
i   2ixiui   3x2
i)dx
3xiu2
i   2ix2
iui   3x3
i
= C;
10i.e., for i = 1;2;
ln
3
q
3xiu2
i   2ixiui   3x3
i +
i
3
p
2
i + 9
 (4.4)
ln
2
4
 
3xi + (i  
p
2
i + 9)ui
3xi + (i +
p
2
i + 9)ui
!2  
3ui   (i +
p
2
i + 9)xi
3ui   (i  
p
2
i + 9)xi
!3
5 = Ci;
(4.5)
where Ci is a real constant depending on the i-th initial state and control
variables.
5 Concluding remarks
In this paper I proposed some ideas to embed the Lie point symmetry theory
in the problem of the determination of exact solutions of the optimal state-
control dynamic system generated by a dierential game.
I showed that under certain assumptions, it is possible to nd an in-
nitesimal generator of the Lie group leading to an integrand factor for the
ODEs of the game.
In my view, this topic deserves further investigation and future research.
In particular, the possible next lines of research should concern two main
aspects.
First of all, it would be very interesting to check whether such a technique
can be applied to some of the economic models I recalled in the Introduction.
The example in Section 4 shows the possibility of constructing an innitesi-
mal generator that does not depend on the intertemporal discount rates. It
would be helpful to x the conditions for an innitesimal generator to be
found for every choice of the parameters usually contained in such models
(spillover, reservation price, depreciation rate, and so on).
Moreover, it is undoubtedly worth exploring the connections between
the closed-loop and the feedback information structures. There could be
some relationships between the Lie symmetry leading to the integration
of the state-control dynamic system and the characteristics of the optimal
value function solving the Hamilton-Jacobi-Bellman-Isaacs equation. In my
opinion, Lie point symmetries might own a hidden potential which has not
been exploited all the way thus far.
11References
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12

On Lie point symmetries in differential games

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