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3Nash and Stackelberg Solutions Numerical Construction in a Two-Person Nonantagonistic Linear Positional Differential
Game*
Anatolii F. Kleimenov1, Sergei I. Osipov2 and Dmitry R. Kuvshinov3
1 Inst, of Math. and Mech.,
Ural Branch of RAS,
16, S.Kovalevskaja street, Ekaterinburg, 620219, Russia,
E-mail: kleimenov@imm.uran.ru
2 Ural State University,
51, Lenin ave., Ekaterinburg, 620017, Russia,
E-mail: sergei.osipov@usu.ru
3 E-mail: evetro@2-u.ru
Abstract The paper suggests numerical methods for constructing Nash and Stackelberg solutions in a linear two-person positional differential game with terminal payoffs of players and polygonal constraints for players controls. Formalization of players’ strategies in the game is based on formalization and the results of positional antagonistic differential games theory, developed by N. N. Krasovskii and his scientific school. The game is such, that it could be reduced to a game on the plane and the problem is transformed to solving non-standard optimal control problems. For the approximation of trajectories in these problems a set of computational geometry algorithms in plane is used, including convex hull construction, union and intersection of polygons and a Minkowski sum for polygons.
Keywords: nonantagonistic differential game, Nash solution, Stackelberg solution, algorithm.
1. Introduction
Problems of computation of solutions in antagonistic and nonantagonistic differential games are important (Basar and Olsder, 1999, Krasovskii and Subbotin, 1974, Krasovskii, 1985, Kleimenov, 1993). One can note algorithms proposed for antagonistic games in Isakova et al., 1984, Vahrushev et al., 1987, as well as in other studies of the same and other authors. Comparing to this, there are distinctly less studies concerning non-antagonistic games and they usually deal with linear quadratic games. The present paper describes algorithms for Nash equilibrium solutions and Stackelberg solutions in linear differential game with geometrical constraints for players’ controls and terminal cost functionals of players.
The paper is organized as follows. Section 2 contains problem statement. Section 3 describes common method for Nash and Stackelberg solutions construction based on reduction of original problem to non-standard problems of (optimal) control. Section 4 solves Stackelberg problem approximating admissible trajectories via repetitive intersections of stable bridges and local attainability sets. Section 5 presents description of two algorithms. The first one builds a Nash solution with the
* This work was partially supported by the Russian Foundation for Basic Research, grants No. 06-01-00436, No. 09-01-00313.
help of auxiliary bimatrix game. The second one based on modification of previous algorithm and is aimed to construction of unimprovable (according to Pareto) Nash solutions. Brief description of the program implementation is given in Section 6. Results of numerical experiment for a material point motion in plane are presented in Section 7. Finally, Section 8 proposes possible study perspectives.
2. Formalization of Two-Person Nonantagonistic Positional Differential
Let dynamics of two-person nonantagonistic positional differential game (NPDG) be described by the equation
where x € Rn is a phase vector, the controls u € P C Rk and v € Q C R1 are handled by Player 1 (P1) and Player 2 (P2) respectively, and 0 is a fixed final time. Sets P and Q are convex polyhedra. Matrix functions A(t), B(t), and C(t) are continuous and have sizes n x n, n x k, and n x l respectively.
Player i chooses his control in order to maximize the cost functional
where <Xj: Rn ^ R1 are given continuous and concave functions.
Suppose, that both players have complete information about the current position (t, x(t)) of the game. The formalization of players’ strategies and of motions generated by them in NPDG is similar to the formalization introduced for antagonistic positional differential games (APDGs) in (Krasovskii and Subbotin, 1974, Krasovskii, 1985) with the exception of technical details (Kleimenov, 1993). A pure strategy (or strategy for short) of P1 is identified with a pair U ^ {u(t, x, e), ^(e)}, where u(-) is an arbitrary function depending on the position (t, x) and on a positive precision parameter e and having values in P. The function ^1: (0, to) ^ (0, to) is a continuous monotone one and satisfies the condition ^1 (e) ^ 0 if e ^ 0. The function ^1(-) has the following sense. For a fixed e the value ^1(e) is the upper bound for the step of a subdivision of the interval [to, 0] which P1 uses for forming step-by-step motion. A strategy V ^ {v(t, x, e), ^2 (e)} of P2 is defined analogously.
Motions of two types: approximated (step-by-step) ones and ideal (limit) ones are considered as motions generated by a pair of strategies of players. Approximated motion x[-,t0,x0,U,e1 ,^1,V,e2,^2] is introduced for fixed values of players’ precision parameters e1 and e2 and for fixed subdivisions 41 = {t(1)} and (2)
^2 = {tj } of the interval [t0,0] chosen by P1 and P2, respectively, under the conditions S(Ai) < pi(ei), i = 1, 2. Here S(Ai) = max(tk++1 — t^). A limit motion generated by the pair of strategies (U, V) from the initial position (t0, x0) is a continuous function x[t] = x[t,t0, x0,U, V], for which there exists a sequence of approximated motions
Game
x = A(t)x + B(t)u + C(t)v, t € [t0,0], x(t0) = x0,
(1)
I = ffi(x(0)), i = 1, 2,
(2)
uniformly converging to x[t] on [t0,0] as
k ^ to, ek ^ 0, ek ^ 0, tik ^ t0, x§ ^ x0, S(A’k) < ^i(ek).
A pair of strategies (U, V) generates a nonempty compact (in the metric of the space C[t0, 0]) set X (t0, x0, U, V) consisting of limit motions x [■, t0, x0, U, V].
Now we introduce the following definition (Kleimenov, 1993).
Definition 1. A pair of strategies (UN, VN) is called a Nash equilibrium solution (NE-solution) of the game, if for any motion x*[-] € X(t0,x0,UN, VN), any t € [t0, 0], and any strategies U and V the following inequalities hold
maxct1(x[0, t, x* [t], U, VN]) < mina1(x[0, t, x*[t], Un, VN]), xH x[-]
max a2 (x[0, t, x* [t], UN, V]) < min a2(x[0, t, x* [t], UN, VN]). x[-] x[-]
Definition 2. An NE-solution (UP, VP), which is Pareto unimprovable with respect to the values 11,12 (2) is called a P-solution.
Now let the following assumptions be fulfilled.
10. P1, called the leader, announced his strategy U* ^ {u*(t,x,e),^*(e)} ahead of time to P2.
20. P2, called the follower, in view of P1’s strategy U*, chooses his rational strategy V* from the condition
mina2(x[0,t0, x0, U*,V]) —► max,
where the minimum of the function a2 is taken over the set X(t0, x0, U*, V).
The problem of P1 is to find such a strategy US1, which ensures maximal value of his cost functional a1 (x[0]) (2) under the condition of the rationality of P2. (More detailed statement including the consideration of various variants of choice from the set of rational strategies for P2 can be found in (Kleimenov, 1993).)
Definition 3. A pair of strategies (US1, VS1), where VS1 is a rational strategy of P2 corresponding to announced strategy US1, is called an S1-solution.
S2-solution is defined analogously.
3. Auxiliary APDGs. Theorems on Structure of Solutions of NPDG
Now we consider auxiliary antagonistic positional differential games r1 and r2. Dynamics of both games is described by (1). In the game ri Player i maximizes his payoff functional ai (x(0)) (2) and Player 3 — i opposes him.
It is known (see (Krasovskii, 1985)) that both games J\ and r2 have universal saddle points
{u(i)(t, x, e), v(i) (t, x, e)}, i = 1,2 (3)
and continuous value functions
Y1(t x) Y2(^ x). (4)
The property of strategies (3) to be universal means that they are optimal not only for the fixed initial position (t0, x0) € G but also for any position (t*, x*) € G assumed to be initial one.
Now we formulate the following problems.
Problem 1. Find measurable functions u(t) and v(t), t0 < t < 0, which generate a trajectory x(t), t0 < t < 0, satisfying the inequalities
Yi(t,x(t)) < Yi(0,x(0)), t0 < t < 0, i = 1,2. (5)
Problem 2.i (i = 1,2). Find measurable functions u(t) and v(t), t0 < t < 0, which generate a trajectory x(t), t0 < t < 0, satisfying the inequality
Y3-i(t,x(t)) < Y3-i(0,x(0)), t0 < t < 0, (6)
and maximize the payoff functional ai (x(0)).
Let piecewise continuous functions u*(t) and v*(t), t0 < t < 0, generate a trajectory x*(t), t0 < t < 0, of the system (1). Consider the strategies of P1 and P2
U0 -{u°(t,x,e), ^0(e)}, V0 -{v°(t,x,e), $0(e)},
where
u0 (t x e) = f u*(t), if llx — x*(t)|
(, , ) 1 u(2) (t, x,e) , if ||x — x* (t)|| > e^(t),
f (7)
v0 (tve)=J v*(t), if |x — x*(t)ll <e^(t)
(, , ) 1 v(1) (t, x,e) , if ||x — x*(t)|| > e^(t),
for all t € [t0,0]. Functions $ (■) and positive increasing function <^(-) are chosen so that the following inequality
||x(t,t0,x0,U0,e,^1 ,V°,e,^2) — x*(t))| <e^(t), (8)
holds for e > 0, ¿(^i) < $i(e). Functions m(2)(-) and v(1)(-) are defined by (3). They can be interpreted as universal “penalty strategies” used when the partner refuses to follow the trajectory x*(-) at some moment of time t € [t0, 0]. Penalty strategies were considered in (Kononenko, 1976, Tolwinski et al., 1986).
The following results are valid (see (Kleimenov, 1993)).
Theorem 1. Let the controls u*(-) and v*(-) be solution of Problem 1. Then the pair of strategies (U0, V0) (7, 8) is an NE-solution. On the contrary, for any NE-solution there exists an equivalent solution of the same type having the form (U0, V0) (7, 8) where u*(-) and v*(-) is a solution of Problem 1.
Theorem 2. Let Assumptions 10 and 20 be fulfilled. Let the controls u*(-) and v*(-) be solution of Problem 2.i. Then the pair of strategies (U0, V0 ) (7, 8) is an Si-solution. On the contrary, for any Si-solution there exists an equivalent Sj-solution having the form (U0 ,V0) (7, 8) where u*(-) and v*(-) is a solution of Problem 2.i.
Thus, Theorems 1 and 2 establish correspondences between the sets of solutions of Problems 1 and 2.i, and the sets of NE- and Sj-solutions, respectively. These theorems determine a structure of solutions of the game. The existence theorems for NE- and Sj-solutions are corollaries of Theorems 1 and 2.
4. Stackelberg Solutions Building
The general idea for the algorithm is to search maxa ai (xa [0]), where xa[0] are node points of a grid constructed for a set of admissible trajectories final states Di. This xa [0] serves an endpoint for Si-trajectory, which is then built back in time (controls u(t), v(t) may be found simultaneously). The Di set approximation is constructed by a procedure, that builds sequences of attainability sets (step-by-step in time), repeatedly throwing out the positions that do not satisfy (6). The procedure is described in brief below. More details (some designations differ) about it could be found in (Kleimenov and Osipov, 2003).
A special construction from the theory of antagonistic positional differential games called stable bridge in pursuit-evasion game is used. The aim of the follower in this game is to drive the phase vector to Lebesgue’s set for the level function (for a chosen constant c) of his cost functional. Note, that any position at the bridge holds the inequality, but positions outside the bridge do not. Thus the bridge is used to find positions satisfying the inequality Y3-i (t, x) < c.
A set Wi'=t designates an approximation of a bridge (in the pursuit-evasion game) section in the time moment t = const. The following discrete scheme is used for building admissible trajectories pipe G' section approximation G' tk (here k runs through some time interval subdivision, k = 0 corresponds to t0 moment and k = N corresponds to 0):
ОС
i , tfc+1
G'tk © 4(B(tk)P © C(tk)Q) \WCtk+i, (9)
where G£t0 = { x0 }, ¿k = tfc+1 — tk. Operation A © B = { a + b | a € A, b € B } denotes Minkowski sum of two sets A and B.
We iterate through a sequence of c values to make up Di of corresponding sequence of D' = G' e 0 using (9) as follows:
1. We have some initial step value ¿c > 0 constrained by ¿cmin < ¿c.
2. Let Di be empty set, c = cmax = max a3_i(x).
x£R"
3. Build a pipe G' and a set D' as in (9);
4. supplement Di := Di U { (x, c) | x € D' }.
5. If ¿c > ¿cmi^then we choose next c value:
— if x0 € Wf to then a) return to the previous value c := c + ¿c, b) decrease step ¿c;
— take next value c := c — ¿c;
— repeat from item 3.
6. Quit.
One example of S-trajectories numerical computation results was presented in (Kleimenov et al., 2006). Program, used for value function calculation, is based on results (Isakova et al., 1984, Vahrushev et al., 1987). An example of a bridge and a pipe is depicted on Fig.1. It illustrates all the basic constructions, that are described in the algorithm, for one iteration with fixed c value.
5. Nash Solutions Building
The proposed algorithms use BM-procedure and modified BM-procedure, which are based on: the principle of non-decrease of player payoffs, the maximal shift in
the direction best for one and another player, and Nash equilibrium in auxiliary bimatrix games made up for each step in a subdivision of the time interval (see (Kleimenov, 1997)). The procedures implies that Player i is interested in increasing the function Yi (t, x) along a trajectory.
The BM-procedure, described in subsection 5.1, allows to approximate a NE-solution, which is a P-solution and, in the same time, it mostly satisfies both players.
The modified BM-procedure, described in subsection 5.2, allows to approximate the rest of P-solutions as well.
5.1. BM-procedure
Suppose, that a position (t*, x*) is fixed. The system is moving from the position (t*, x*) to the position (t*, x*), where t* = t* + h and h is considered to be fixed provided t* + h < 0 holds. We are going to find x* as well as u* and v* leading to that position (with u(t) = u* and v(t) = v*, t* < t < t*).
A set
G(t*; t*, x*) = { x[t*; t*, x*, u, v] | u € P, v € Q } (10)
is an approximation of local attainability set of system (1) when moving from the position given to the next time moment.
Two sets (i = 1, 2)
Wi(t*; t*,x*) = { x € | Yi(t*, x) > Yi(t*,x*) } (11)
are approximated by known procedures of building maximal stable bridges in some pursuit-evasion games (see Krasovskii and Subbotin, 1974, Isakova et al., 1984).
We analyze a set
H(t*; t*, x*) = G(t*; t*,x*) П Wi(t*; t*, x*) П W2(t*; t*,x*) (12)
in order to find wl(t* ; t*, x*) , which is a maximum point of 7*(t*, x) on it, (i = 1, 2). Note, that these maximum points may be found either exactly on the set (and hence it should be simple to get corresponding controls uio, vio leading to wl ) or in some neighborhood of x*. In the latter case one may try to maximize shift in the direction of a maximum point found. Below some details follow.
Consider vectors
Now we run through all the elements of polyhedra P and Q to find out controls u10, u20, v 10, and v20:
Then, an auxiliary bimatrix 2 x 2 game (A, B) is constructed. In this game P1 has two strategies: “to choose u10” and “to choose u20”. Similarly, P2 has two strategies: “to choose v10” and “to choose v20”. Therefore, payoff matrices of players are defined as follows:
Bimatrix game (A,B) has at least one Nash equilibrium in pure strategies. It is possible to take a Nash equilibrium as both players’ controls for semi-interval (t*, t*] (or [t0,t*] if t0 = t*). Such an algorithm of players’ controls construction generates a trajectory, which is an NE-trajectory. When (A,B) has two equilibria (1,1) and (2,2), then one is chosen after another in turn (they are being interleaved).
We take a solution of (A, B) and try to fit both players’ controls to maximize shift along the motion direction of this equilibrium, while staying inside H (this may be done similar to (13)).
This way of players’ controls generation is called BM-procedure here. To provide some Nash equilibrium in the game, controls generated by BM-procedure (giving BM-trajectory) must be paired with penalty strategies (see (7), (8)). Note, that each player watches for the trajectory and applies a penalty strategy when the other evades following BM-trajectory.
s1(t*; t*,x*) = w1 (t*; t*, x*) - x*, s2(t*; t*,x*) = w2(t*; t*, x*) - x*.
maxuGp,vGQ s1T[B(t*)u + C(t*)v] = s1T[B(t*)uio + C(t*)vw], maxuGp,vGQ s2T[B(t*)u + C(t*)v] = s2T[B(t*)u2o + C(t*)v2o],
holding the conditions x[t*;t*,x*,uio, vio] G H(t*;t*,x*), x[t*; t*,x*, u2o, v2o] G H(t*; t*, x*).
(13)
aij = Yl(t*,x[t*; t*,x*, uio, v¿o])), bij = Y2(t*,x[t*; t*, x*, uio, vjo])),
i,j = 1,2.
5.2. Modified BM-procedure
The BM-procedure introduced above can be modified by applying the additional inequality for every step (t*, x*)
Yi(t*,x*) < Yi(t*,x*) + e(t*,t*,x*), (14)
where e(-, ■, ■) is some scalar function, which may be reformulated in terms of the set H(t* ; t* , x* ). In trivial case it may be a constant, but even in simple examples constant value proved being impractical because Yi may grow with non-constant rate.
Having some e value, we may force condition (14) by replacing the original H set (12) with
H|(t*; t*,x*) = H(t*; t*, x*)\{ x € | Yi(t*, x) > Yi(t*,x*) + e(t*,t*, x*) } (15)
Modified BM-procedure with i € {1,2} and fixed e(-, ■, ■) function is BM-procedure for which the set H(t*; t*,x*) from (12) is substituted by H|(t*; t*,x*) from (15).
Then all other elements of BM-procedure may be applied to obtain a modified BM-trajectory. A set of endpoints of modified BM-trajectories is of a special interest as we want to approximate a part of the Nash trajectories endpoints set bound, which is Pareto unimprovable.
1. Let K be an empty set.
2. Let some step c > 0 be fixed, c := c0.
3. Try to build a modified BM-trajectory T, we may fail if computed H^ ^ approximation turns out to be empty at some step.
4. If we failed or endpoint of T is in K then quit.
5. Supplement K := K U {endpoint of T}.
6. Increase c := c + c0.
7. Repeat from item 3.
The procedure is to be run twice: for i = 1 and for i = 2. This way both halves of the border approximation are built. It must be mentioned that the resulting approximation quality depends dramatically on choice of function e(-, ■, ■).
An c* > c0 may exist such that elements of K corresponding to c0 < c < c* approximate endpoints of NE-trajectories, which are not P-trajectories.
6. Program Implementation
A completely new approach to program implementation was successfully applied at first to NE-solution computation and after that — to Si-solutions. Computational geometry algorithms library, developed by S. Osipov in Fortran somewhere in 80’s, became outdated, so it was temporarily substituted with a C++ wrapper library, which builds upon polygon tessellation facilities from OpenGL Utility Library (GLU). GLU functions and structures for polygonal primitives are intensively used by algorithms, mentioned herein. Examples of the next section were obtained with GLU being used for polygons processing.Advantages of GLU include straightforward API in C, which is simple to use in almost any programming environment. Many implementations exist (usually bundled with operational system), both proprietary and open source.
Despite positive results achieved by the implementation, another library (which is an open source project) was tested, as a possible future base for our algorithms. It was Computational Geometry Algorithms Library (CGAL), which goal “is to provide easy access to efficient and reliable geometric algorithms in the form of a C++ library” (see http://www.cgal.org/). While GLU is a convenient external component, CGAL provides a complex framework to expand upon and is not bounded by hardware-supported double precision arithmetics.
In the case of Si-solutions, OpenMP was adopted (discrete scheme (9) is run for
different c values in parallel). Tests on a machine with Intel Core 2 Duo processor
demonstrated twofold run-time improvement for two-threaded computations against one-threaded.
7. An Example
The following vector equation
£'= u + v C(t0) = ^ £(t0) = £0 (16)
£,u, v € R2, ||u|| < ^, ||v|| < v,
describes the motion of a material point of unit mass on the plane (£l,£2) under the action of a force F = u + v. P1 (P2), who governs the control u (v), tends to lead the material point as close as possible to the given target point a(1) (a(2)) at the moment of time 0. Then players’ cost functionals are
*i(£(0)) = -||£(0) - «(i)||, (17)
£ =(£1,6), a(i) = (a1i) ,a2i)), i = 1, 2,
where 0 is final time.
By setting y1 = £1, y2 = £]_, y3 = £2, = £2 and making the following change
of variables X1 = y1 + (0 — t)y3, X2 = y2 + (0 — t)y4, X3 = y3, X4 = y4 we get a system, which first and second equations are
x 1 = (0 — t)(u1 + V1), (18)
X2 = (0 — t)(u2 + V2).
Further, (17) can be written
ffi(x(0)) = —||x(0) — a(i)||, x =(x1,x2), i = 1, 2. (19)
Since the cost functional (19) depends on variables x1 and x2 only and the right-
hand side of (18) does not depend on other variables, one can conclude, that it is
sufficient to consider only reduced system (18) with cost functionals (19).
Then initial conditions for (18) are given by formulae
xi(t0) x0i £0i + (0 t0)£0i, i 1, 2.
It can easily be shown, that value functions in antagonistic differential games
and r2 are given by formulae (if ^ > v)
7i(i,x) = min{ 11x - a(1)|| - ^ ^ (n - J/),0},
72(i,x) = min{ 11x a(2) 11 + ^ (m -v), 0}
and universal optimal strategies (3) are given by
,x — a(i)
(t, x, e) = ( — l)*yU,
(i)
(t, x, e) = —(—1)iv
||x — a(i) ||' x — a(i)
||x — a(i) ||
One can see modified BM-procedure results (with several NE-trajectories) on Fig.2 for the following setting: ^ = v =1, t0 = 0, 0 = 2, x0 = £0 = (0.6, 0.9) (initial velocity is zero), a(1) = ( — 1,6), a(2) = (5, 5), ec(t*; t, x) = c(t* — t)(^+v)(0 — t). This variant, where ^ = v, was studied analytically in Kleimenov, 1993 (Section 1.13). Good coincidence of Fig.2 and Fig. 1.6, p. 53 of the book mentioned could be seen.
Another settings were used to demonstrate Si- and BM-trajectories. Let the following conditions be given: ^ = 1.4, v = 0.6, t0 =0, £0 = (0.5,0.5), a(1) = (4,5),
and a(2) = (3.5, —2.5). Two variants of target points were considered:
(V1) 0 = 2.5, £0 = (—0.25, —1), then x0 = (—0.125, —2), see Fig. 3 and Fig. 5;
(V2) 0 = 2.1, £0 = (—1, —1), then x0 = ( — 1.6, —1.6), see Fig. 4.
Time step for NE-trajectory is 0.001, and time step for Si-trajectories is 0.005.
The points S1 and S2 on Fig. 3 and Fig. 4 denote endpoints of S1- and S2-trajectories, respectively, while the point N denotes NE-trajectory endpoint, generated by BM-procedure. On Fig. 3 symbol “x” is used to show a point (t = 1.347), where BM-procedure switches from nonantagonistic game to antagonistic one (V2 gives no antagonistic trajectory part).
Four modified BM-trajectories are given on Fig.3 and Fig.4 as well. The same ec(-) function as for Fig.2 is used.
For V1, both players controls generating Nash trajectory are shown on Fig. 5. Note, that every 25th pair of vectors u, v is presented there.
Further, on Fig. 3 and Fig. 4 border of relevant D = (D1 0 D2)\L set is painted. Here the symbol L denotes a half-plane bounded by a line connecting points a(1) and a(2), not containing point £0. Set D contains all NE-trajectories endpoints, but, in general, there may be also points, which are not endpoints of any NE-trajectories. Sets Di were built with time step 0.005.
8. Conclusion
As it is seen today, there are at least two ways of development of the results presented. First, it seems to be possible to transparently generalize NE- and Si-solution algorithms for non-linear systems with dynamics
x(t) = F1 (t, x(t), u(t)) + F2 (t, x(t), v(t)).
Second, software development may lead to a powerful and flexible framework simplifying solution computations in a class of differential games. The last program implementation uses techniques of generic programming, which is common for modern C++ software like CGAL. This supports flexibility of its structure and simplifies future modernizations. For example, the early experience allows to suggest, that facilities supplied by the library could give the algorithms literally new dimension: polygons could be changed to polyhedrons without deep change in generic algorithms constructing the solutions.
Fig.2. Modified BM-trajectories (the case ß = v)
Fig.5. V1: Player controls generating NE-trajectory
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