On the characteristic function construction technique in differential ...

On the characteristic function construction technique in differential games with prescribed and random duration? Ekaterina V. Gromova1 and Ekaterina V. Marova2 1

Krasovskii Institute of Mathematics and Mechanics (IMM UB RAS), Yekaterinburg, Russia; St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, St.Petersburg, Russia; E-mail: [email protected] 2 St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, St.Petersburg, Russia; E-mail: [email protected]

Abstract. This paper focuses on different approaches for calculating characteristic functions in cooperative differential games with prescribed and random duration. We construct α−, δ− and ζ−characteristic functions and examine their properties in the differential game of pollution control. Additionally, we introduce a new η− characteristic function. Keywords: differential games, prescribed duration, random duration, characteristic function, environmental resource management, pollution control.

1.

Introduction

In cooperative differential games the problem of the calculation of a characteristic function (c.f.) plays an important role since the characteristic function allows to obtain optimal (cooperative) solution of the game. The review and analysis of different methods for construction of c.f. for cooperative games with so-called negative externalities (see, e.g., [2]) in static formulation was presented in [18]. Whereas in a dynamic formulation the construction of α-, δ- characteristic functions (see, correspondingly, [16, 11]) was analyzed in [4] where a new approach to constructing ζ-characteristic function was introduced (see also [12] for the first reference in Russian). This paper presents different techniques for characteristic function construction in cooperative differential game with prescribed [16] and random duration as well. To illustrate our results we consider the dynamic game-theoretical problem of pollution control [3] which belongs to the class of games with negative externalities. 2.

Different techniques of the characteristic function construction

Let Ki (·) and ui ∈ Ui ⊆ compRl be the payoff functions and controls in a classical cooperative differential game of n players with prescribed duration [16]. Assume that all standard restrictions [7] on the parameters, controls and trajectory function are satisfied. To define the cooperative game we have to construct the characteristic ?

The construction of Nash equilibria by first author was supported by the project 17– 11–01093 from Russian Science Foundation

function V (S, ·) for every coalition S ⊆ N in the game. The c.f. in a cooperative game is a mapping from the set of all possible coalitions: V (·) : 2N → R,

V (∅) = 0.

Note that the value of the c.f. for the grand coalition N equals to V (N, ·). The value V (S) is interpreted traditionally as the power of the coalition S. The important property of a c.f. is superadditivity: V (S1 ∪ S2 ) ≥ V (S1 ) + V (S2 ),

∀S1 , S2 ⊆ N, S1 ∩ S2 = ∅.

(1)

The use of superadditive characteristic functions in solving various problems in the field of cooperative game theory in static and dynamic setting provides a number of advantages [4]. There are several approaches to the construction of characteristic functions (see [10 18, 11, 4]). In this paper we shall focus on the α-, δ-, and ζ-characteristic functions and then introduce the new one named η−characteristic function. 2.1.

α-characteristic function

A classical way to define the c.f. is to use the lower value of the zero-sum game between the coalition S, acting as the first (maximizing) player and the coalition N \S, acting as the second (minimizing) player. This approach had been introduced in [10] and now is called α-characteristic function. We have

V α (S, ·) =

 0, S = {∅},  P    max min K (u , . . . , u , ·), S ⊂ N, i 1 n u , u , i i∈S

j j∈N \S

i∈S

 n P    max Ki (u1 , . . . , un , ·), u1 ,...un i=1

(2)

S = N,

where max min is the lower value of the zero-sum game between the coalition S ui , uj , i∈S j∈N \S P and N \ S with the payoff function Ki (u1 , . . . , un , ·). It was proved in [16] that i∈S

in general the α-characteristic function is superadditive. 2.2.

δ-characteristic function

δ-characteristic function, [11], of a coalition S is constructed in two steps. First, we calculate the Nash equilibrium strategies for all players. Second, we fix (freeze) those strategies for Pplayers from N \ S while the players from S seek to maximize their joint payoff Ki (u1 , . . . , un , ·). i∈S

δ

V (S, ·) =

 0,       

S = {∅}, max

P

ui , i∈S

i∈S

E Ki (uS , uN N \S , ·), S ⊂ N,

E  uj =uN , j∈N \S j    n P    max Ki (u1 , . . . , un , ·),

u1 ,...un i=1

(3) S = N.

In general, a δ-characteristic function is a non-superadditive function (see examples in [6]).

2.3. ζ-characteristic function One of the novel approaches is to use a ζ-characteristic function, [4]. This c.f. of coalition S is computed in two stages: first, we find optimal controls maximizing the total payoff of the players; next, the cooperative optimal strategies are used by the players from the coalition S while the left-out players from N \ S use the strategies minimizing the total payoff of the players from S. We have  0, S = {∅},    P    min Ki (u∗S , uN \S , ·), S ⊂ N,    i∈S u ∈U , j∈N \S, j j  ui =u∗ V ζ (S, ·) = (4) i , i∈S  n P   max  Ki (u1 , . . . , un , ·), S = N,    u1 ,...,un , i=1    ui ∈Ui , i∈N

∗

{u∗i }i∈N

where u = is the profile of strategies for which the maximal value of payoff function is achieved for all players, u∗S = {u∗i }i∈S . The c.f. defined in this way is in general superadditive (see [4]). Note, that for the case of ζ-characteristic function players from N \ S have active reaction while players from S just use the same strategies u∗i , i ∈ S as they were use in the case of total payoff maximization. 3.

Game-theoretical model of pollution control with prescribed duration

Consider a game-theoretic model of pollution control based on the models published in [1], see also [3]. There are 3 players (companies, countries) that participate in the game, N = {1, 2, 3}. Each player has an industrial production site. It is assumed that the production is proportional to the pollution ui . Thus, the strategy of a player is to choose the amount of pollutions emitted to the atmosphere, ui ∈ [0; bi ]. In this example the solution will be considered in the class of open-loop strategies ui (t) and Pontryagin’s maximum principle (PMP) is applied [17]. The dynamics of the total amount of pollution x(t) is described by following equation 3 X x(t) ˙ = ui (t). i=1

The instantaneous payoff of i-th player is defined as: 1 i ∈ N. R(ui (t)) = bi ui (t) − u2i (t), 2 Each player has to bear expenses due to the pollution removal. Hence the instantaneous payoff (utility) of the i-th player is equal to R(ui (t)) − di x(t), di ≥ 0. The payoff of the i-th player is thus defined as Z T   1  Ki (x0 , T − t0 , u1 , u2 , u3 ) = bi − ui ui − di x dt. 2 t0 We will assume additional regularity constraints to be satisfied: ∀i ∈ N bi ≥ 3 P DN (T − t0 ), where DN = di . i=1

3.1. Construction of characteristic functions To find the profile of optimal strategies we have to solve the maximization problem max

u1 ,u2 ,u3

3 Z X i=1

T

t0



 1  bi − ui ui − di x dt. 2

The Hamiltonian is H(x, u, ψ) =

3  X i=1

3 3 X X 1  di x + ψ bi − ui ui − ui , 2 i=1 i=1

its first order partial derivatives w.r.t. ui ’s are ∂H (x, u, ψ) = bi − ui + ψ, ∂ui

i = 1, 3.

The Hessian matrix is negative definite hence we conclude that Hamiltonian H is concave w.r.t. ui . ∂2H (x, u, ψ) = −1 < 0, i = 1, 3. ∂u2i The adjoint equations and the related transversality conditions are  3 P  dψ di , dt = i=1  ψ(T ) = 0. We get the optimal control u∗ (t) = (b1 − DN (T − t),

b2 − DN (T − t),

b3 − DN (T − t)).

(5)

Given the initial conditions (t, x) the value of c.f. for the grand coalition N can be written as V α (N, x(t), T − t) = V δ (N, x(t), T − t) = V ζ (N, x(t), T − t) = (6) ˜N (T − t) − 1 DN BN (T − t)2 + 1 D2 (T − t)3 , = −DN (T − t)x(t) + 21 B 2 2 N where BN =

3 P i=1

˜N = bi , B

3 P i=1

b2i .

Now we have to calculate the value of the characteristic function for coalitions of one and two players. Let us carry out some preliminary calculations. For the case of δ−c.f. we have to find the Nash equilibrium. To find Nash equilibrium strategies for this game one has to maximize the payoff for each player i by the control ui in assumption that another players use fixed NE strategies Z T   1  max bi − ui ui − di x dt, i = 1, 2, 3. ui 2 t0 We apply the PMP to this maximization problem. The players’ Hamiltonians are

3  X 1  Hi (x, u, ψ) = bi − ui ui − di x + ψ ui , 2 i=1

i = 1, 3,

and their first order partial derivatives w.r.t. ui ’s are ∂Hi (x, u, ψ) = bi − ui + ψ, ∂ui

i = 1, 3.

The Hessian matrices are negative definite hence we conclude that Hamiltonians Hi are concave w.r.t. ui . ∂ 2 Hi (x, u, ψ) = −1 < 0, ∂u2i

i = 1, 3.

The adjoint equations and the related transversality conditions are  dψ dt = di , ψ(T ) = 0. We get the Nash equilibrium strategies uN E (t) = (b1 − d1 (T − t),

b2 − d2 (T − t),

b3 − d3 (T − t)).

(7)

According (3) we have to solve the maximization problem, taking into account (7). We have: T

Z max



t0

ui ,uj E uk =uN k

 1  1  bi − ui ui + (bj − uj uj − (di + dj )x dt. 2 2

Following PMP we get uSi = bi − (di + dj)(T − t),

uSj = bj − (di + dj)(T − t).

(8)

For the case of α−, ζ− c.f. (2), (4) we will have to solve the minimization problem Z

T

min uk



t0

 1  1  bi − ui ui + (bj − uj uj − (di + dj )x dt. 2 2

By using PMP we get uk = bk .

(9)

According to the definition of the α-c.f. (2), to construct it we have to solve the maximization problem with (9): Z

T

max ui ,uj uk =bk

t0



 1  1  bi − ui ui + (bj − uj uj − (di + dj )x dt. 2 2

Applying PMP in the same way we get uSi = bi − (di + dj)(T − t),

uSj = bj − (di + dj)(T − t).

(10)

Given the initial conditions (t, x) one can calculate the value of three above mentioned characteristic functions for coalitions of one and two players.

Considering (2), (9), (10) we obtain the value of a α−characteristic function: 1 1 1 V α ({i}, x(t), T −t) = −di (T −t)x(t)+ b2i (T −t)− BN di (T −t)2 + d2i (T −t)3 , (11) 2 2 6 V α ({i, j}, x(t), T − t) = −(di + dj )(T − t)x(t) + 21 (b2i + b2j )(T − t)− (12) − 21 BN (di + dj )(T − t)2 + 13 (di + dj )2 (T − t)3 . We get a δ−characteristic function from (3), (7), (8): V δ ({i}, x(t), T − t) = −di (T − t)x(t) + 21 b2i (T − t) − 12 BN di (T − t)2 + (13) + 61 di (2DN − di )(T − t)3 , V δ ({i, j}, x(t), T − t) = −(di + dj )(T − t)x(t) + 12 (b2i + b2j )(T − t)− − 21 BN (di + dj )(T − t)2 + 13 (dk (di + dj ) + (di + dj )2 )(T − t)3 . (14) Finally, for a ζ−characteristic function from (4), (5), (9) we obtain: V ζ ({i}, x(t), T − t) = −di (T − t)x(t) + 12 b2i (T − t) − 21 BN di (T − t)2 − (15) − 61 DN (DN − 2di )(T − t)3 , V ζ ({i, j}, x(t), T − t) = −(di + dj )(T − t)x(t) + 21 (b2i + b2j )(T − t)− (16) − 12 BN (di + dj )(T − t)2 − 31 DN (DN − 2(di + dj ))(T − t)3 . 3.2.

Superadditivity of the c.f.

Check that α−characteristic function (2) is superadditive (1). From (6), (11), (12) we have V α (N ) − V α ({k}) − V α ({i, j}) =

2 dk (di + dj )(T − t)3 ≥ 0, 3

2 di dj (T − t)3 ≥ 0, 3 since t ∈ [t0 , T ]. We conclude that the constructed function is superadditive which agrees with the previously obtained results. At the next step we check if δ−characteristic function (3) is superadditive. From (6), (13), (14) we have V α ({i, j}) − V α ({i}) − V α ({j}) =

V δ (N ) − V δ ({k}) − V δ ({i, j}) =

1 2 (D + d2k )(T − t)3 ≥ 0, 6 N

V δ ({i, j}) − V δ ({i}) − V δ ({j}) =

1 2 (d + d2j )(T − t)3 ≥ 0, 6 i

since t ∈ [t0 , T ]. This proves the superadditivity of the constructed δ− c.f. Finally we may check if ζ−characteristic function is also superadditive in this game. From (6), (15), (16) we have V ζ (N ) − V ζ ({k}) − V ζ ({i, j}) =

1 DN (di + dj + 2dk )(T − t)3 ≥ 0, 3

1 DN (di + dj )(T − t)3 ≥ 0, 3 since t ∈ [t0 , T ]. This concludes that ζ−characteristic function is also superadditive (as must be in general). V ζ ({i, j}) − V ζ ({i}) − V ζ ({j}) =

3.3.

Comparison of characteristic functions

Let us investigate how the constructed c.f.s relate to each other. Obviously, we have V δ (N ) = V α (N ), also from (11), (13) and (12), (14) we get 1 V δ ({i}) = V α ({i}) + di (dj + dk )(T − t)3 , 3 1 V δ ({i, j}) = V α ({i, j}) + dk (di + dj )(T − t)3 . 3 Thus, V δ (·) ≥ V α (·).

(17)

Furthermore, from (11), (15) and (12), (16) we have V α (N ) = V ζ (N ), 1 V α ({i}) = V ζ ({i}) + (dj + dk )2 (T − t)3 , 6 1 V α ({i, j}) = V ζ ({i, j}) + d2k (T − t)3 . 3 Thus, V α (·) ≥ V ζ (·).

(18)

Finally, (17) and (18) imply that V δ (·) ≥ V α (·) ≥ V ζ (·). 4.

Game-theoretical model of pollution control with random duration

To make the model from Sec. 3 more realistic we examine the game-theoretic model of pollution control with random duration [13, 14, 15, 9, 19] (T − t0 ), where T is a random variable with exponential distribution function F (t), t ∈ [t0 , Tf ]. The strategy of each player is to choose the amount of pollution emitted to the atmosphere, ui ∈ [0; bi ]. Let us consider case of N = {1, 2, 3}. The game starts from initial state x0 at the time t0 .

The dynamics of the total amount of pollution x(t) is described by 3 X

x(t) ˙ =

ui (t),

x(t0 ) = x0 .

i=1

The expectation of the payoff of players i = 1, 2, 3 are calculated as     Z T  1 Ki (x0 , t0 , Tf , u1 , u2 , u3 ) = E bi − ui (τ ) ui (τ ) − di x(τ ) dτ . 2 t0 We assume that di ≥ 0, ∀i = 1, 2, 3 and there are additional constraints called the 3 P di . regularity constraints: bi ≥ DλN , where DN = i=1

According to [19, 8, 5] the payoff of each player i ∈ N can be represented as Z ∞    1 Ki (x0 , t0 , Tf , u1 , u2 , u3 ) = bi − ui (τ ) ui (τ ) − di x(τ ) e−λ(τ −t0 ) dτ = 2 t0 Z ∞    1 = eλt0 bi − ui (τ ) ui (τ ) − di x(τ ) e−λτ dτ. 2 t0 We consider this game in cooperative form and construct the c.f. by three described above methods. 4.1. Construction of characteristic functions We start with solving the maximization problem in order to find the profile of optimal strategies. max

3 X

u1 ,u2 ,u3

e

λt0

Z

∞



t0

i=1

 1  bi − ui ui − di x e−λt dt. 2

The Hamiltonian is H(x, u, ψ) =

3  X i=1

3  X 1  ui , bi − ui ui − di x e−λt + ψ 2 i=1

its first order partial derivatives w.r.t. ui ’s are ∂H (x, u, ψ) = (bi − ui )e−λt + ψ, ∂ui

i = 1, 3.

The Hessian matrix is negative definite hence we conclude thet Hamiltonian H is concave w.r.t. ui . ∂2H (x, u, ψ) = −e−λt < 0, ∂u2i

i = 1, 3.

The adjoint equations and the related transversality conditions are  3   dψ = P d e−λt , dt

i

i=1

  lim ψ(t) = 0. t→∞

We have optimal controls in the following form  DN DN DN  u∗ (t) = b1 − . , b2 − , b3 − λ λ λ

(19)

Given the initial conditions (t, x), we compute the c.f. for the grand coalition N from (19). V α (N, x(t), T − t) = V δ (N, x(t), T − t) = V ζ (N, x(t), T − t) = = where BN =

1 λ3



3 P i=1

˜N B 2

(20)

 2 λ2 − BN DN λ + 1 12 DN − DN λ2 x ,

3 ˜ N = P b2 . bi , B i i=1

Next, we construct the characteristic function for coalitions of one and two players. First, we find the Nash equilibrium to construct δ−c.f. Z ∞    1 bi − ui (τ ) ui (τ ) − di x(τ ) e−λτ dτ. max ui 2 t0 According to the PMP the Hamiltonian has the form Hi (x, u, ψ) =



3  X 1  bi − ui ui − di x e−λt + ψ ui , 2 i=1

i = 1, 3

and its first order partial derivatives w.r.t. ui ’s are ∂Hi (x, u, ψ) = (bi − ui )e−λt + ψ, ∂ui

i = 1, 3.

The Hamiltonians Hi are concave w.r.t. ui , because the respective Hessian matrices are negative definite. ∂ 2 Hi (x, u, ψ) = −e−λt < 0, ∂u2i

i = 1, 3.

The adjoint equations and the related transversality conditions are ( dψ −λt , dt = di e lim ψ(t) = 0. t→∞

We get the Nash equilibrium strategies  d1 uN E (t) = b1 − , λ

b2 −

d2 , λ

b3 −

d3  . λ

Now we solve the maximization problem while considering (21). Z

T

max ui ,uj E uk =uN k

t0



 1  1  bi − ui ui + (bj − uj uj − (di + dj )x dt. 2 2

(21)

Following PMP we get the controls in form of di + dj di + dj , uSj = bj − . (22) λ λ For the case of α−, ζ−c.f. (2), (4) we will have to solve the minimization problem Z ∞    1  1  λt0 min e bi − ui ui + bj − uj uj − (di + dj )x e−λt dt. uk 2 2 t0 uSi = bi −

From the PMP we get uk = bk .

(23)

According to the definition of α-c.f. (2), to calculate it we have to solve the maximization problem having in mind (23): Z ∞   1  1  bi − ui ui + (bj − uj uj − (di + dj )x e−λt dt. max 2 2 t0 ui ,uj uk =bk

Applying the PMP to this problem we get di + dj di + dj , uSj = bj − . (24) λ λ Given the initial conditions (t, x), we calculate the value of three described above characteristic functions for one- and two-players coalitions. With (2), (23), (24) we calculate the α−c.f.  1 1 2 2 1 V α ({i}, x(t), T − t) = 3 bi λ − BN di λ + d2i − di λ2 x(t) , (25) λ 2 2 uSi = bi −

V α ({i, j}, x(t), T − t) =

1 λ3



1 2 2 (bi

+ b2j )λ2 − BN (di + dj )λ + (di + dj )2 − (26) 2



−(di + dj )λ x(t) . According (3), (21), (22) the δ−characteristic function could be calculated as  1 1 2 2 1 V δ ({i}, x(t), T − t) = 3 bi λ − BN di λ + (dj + dk )di + d2i − di λ2 x(t) , (27) λ 2 2 V δ ({i, j}, x(t), T − t) =

1 λ3



1 2 2 (bi

+ b2j )λ2 − BN (di + dj )λ + dk (di + dj )+ (28)

 +(di + dj ) − (di + dj )λ x(t) . 2

2

For the ζ−characteristic function from (4), (19), (23) we obtain  1 1 2 2 1 2 V ζ ({i}, x(t), T − t) = 3 bi λ − BN di λ − DN + di DN − di λ2 x(t) , λ 2 2  V ζ ({i, j}, x(t), T − t) = λ13 12 (b2i + b2j )λ2 − BN (di + dj )λ−

(29)

(30) 2 −DN

 + 2DN (di + dj ) − (di + dj )λ x(t) . 2

4.2.

Superadditivity of c.f.

It is easy to check the superadditivity of α−characteristic function by (20), (25), (26).

V α (N ) − V α ({k}) − V α ({i, j}) =

 1  2 2 (d + d ) + 2d + 6d (d + d ) ≥ 0, i j k i j k 2λ3

V α ({i, j}) − V α ({i}) − V α ({j}) =

 1  2 di + d2j + 4di dj ≥ 0, 3 2λ

since t ∈ [t0 , T ]. Additionally we could prove that δ−characteristic function is superadditive using (20), (27), (28). V δ (N ) − V δ ({k}) − V δ ({i, j}) =

 1  (di + dJ )2 + 2d2k + 2dk (di + dj ) ≥ 0, 3 2λ  1  2 di + d2j ≥ 0, 3 2λ

V δ ({i, j}) − V δ ({i}) − V δ ({j}) =

since t ∈ [t0 , T ], Also ζ−characteristic function is superadditive too (20), (29), (30). V ζ (N ) − V ζ ({k}) − V ζ ({i, j}) =

  1 DN di + dj + 2dk ≥ 0, 3 λ

V ζ ({i, j}) − V ζ ({i}) − V ζ ({j}) =

  1 D d + d ≥ 0, N i j λ3

since t ∈ [t0 , T ]. 4.3.

Comparison of characteristic functions

Next we are going to examine the properties of the constructed c.f. with respect to each other. It is clear that V δ (N ) = V α (N ), also from (25), (27) and (26), (28) we get V δ ({i}) = V α ({i}) +

(dj + dk )di , λ3

V δ ({i, j}) = V α ({i, j}) +

dk (di + dj ) . λ3

Thus, V δ (·) ≥ V α (·). Apart from that, from (25), (29) and (26), (30) we have V α (N ) = V ζ (N ),

V α ({i}) = V ζ ({i}) +

1 (DN − di )2 , 2λ3

(31)

V α ({i, j}) = V ζ ({i, j}) +

1 2 d . λ3 k

Thus, V α (·) ≥ V ζ (·).

(32)

Inequalities (17) and (18) imply that V δ (·) ≥ V α (·) ≥ V ζ (·). 5.

New characteristic function

Developing the idea of simplification of the technique for calculating c.f.s, we introduce the new definition for the characteristic function  0, S = {∅},      P K (u∗ , uN E , ·), S ⊂ N, i S N \S V η (S, ·) = i∈S  n  P   Ki (u1 , . . . , un , ·), S = N.  max

(33)

u1 ,...,un i=1

where u∗ = {u∗i }i∈N is the profile of strategies for which the maximal value of payoff function is achieved for all players, u∗S = {u∗i }i∈S . In (33) for players from S we use (obtained earlier) strategies u∗S from optimal profile u∗ (as in ζ−c.f.) and for players from N \ S we use (obtained earlier) strateE gies uN N \S from the Nash equilibrium strategies for all players (as in δ− c.f.). This provides certain technical advantages as will be reported in a subsequent paper. 6.

Conclusion

In this paper we considered three different approaches to the calculation of characteristic function in differential games and applied all of them to differential games with prescribed and random duration. We analyzed the obtained functions and their relations. Also a new way of the characteristic function construction has been introduced. References Breton M., Zaccour G, Zahaf A. A differential game of joint implementation of environmental projects //Automatica, V. 41, no. 10, 2005, P. 1737–1749. Chander P. The gamma–cor and coalition formation // International Journal of Game Theory, V. 35, 2007, P. 539–556. Gromova E. The Shapley value as a sustainable cooperative solution in differential games of 3 players // Recent Advances in Game Theory and Applications, Birkhauser, Springer Int. Publishing, 2016, P. 67–89. Gromova E., Petrosyan L. On an approach to constructing a characteristic function in cooperative differential games // Autom Remote Control, V. 78, Issue 9, 2017, P. 1680–1692, https://doi.org/10.1134/S0005117917090120 Gromova E., Tur A. On the form of integral payoff in differential games with random duration // XXVI International Conference on Information, Communication and Automation Technologies (ICAT), IEEE, 2017.

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