ON THE CHARACTERISTIC FUNCTION CONSTRUCTION TECHNIQUE IN DIFFERENTIAL GAMES WITH PRESCRIBED AND RANDOM DURATION Текст научной статьи по специальности «Математика»

 On the Characteristic Function Construction Technique in Differential Games with Prescribed and Random Duration*

Ekaterina V. Gromova1,2 and Ekaterina V. Marova2

1 Krasovskii Institute of Mathematics and Mechanics (IMM UB RAS), Yekaterinburg, Russia;

E-mail: e. v. gromova@spbu. ru 2 St. Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: marovaek@gmail. com

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., Chander, 2007) in static formulation was presented in (Reddy and Zaccour, 2016). Whereas in a dynamic formulation the construction of α-, δ- characteristic functions (see, correspondingly, Petrosjan and Danilov, 1982, Petrosjan and Zaccour, 2003) was analyzed in (Gromova and Petrosyan, 2017) where a new approach to constructing C-characteristic function was introduced (see also (Petrosyan and Gromova, 2014) for the first reference in Russian). This paper presents different techniques for characteristic function construction in cooperative differential game with prescribed (Petrosjan and Danilov, 1982) and random duration as well. To illustrate our results we consider the dynamic game-theoretical problem of pollution control (Gromova, 2016) which belongs to the class of games with negative externalities.

2. Different techniques of the characteristic function construction

Let Ki(·) and и; є Ui Ç compR1 be the payoff functions and controls in a classical cooperative differential game of n players with prescribed duration (Petrosjan and Danilov, 1982). Assume that all standard restrictions (Krasovskii and Subbotin, 1988) on the parameters, controls and trajectory function are satisfied. To define the cooperative game we have to construct the characteristic function V(S, ·) for

* The construction of Nash equilibria by first author was supported by the project 1711-01093 from Russian Science Foundation

every coalition S Ç 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) = 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(Sı U S2) > V(Si) + V(S2), VSi, S2 Ç N, Si П S2 = 0. (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 (Gromova and Petrosyan, 2017).

There are several approaches to the construction of characteristic functions (see Neumann and Morgenstern, 1953, Reddy and Zaccour, 2016, Petrosjan and Zaccour, 2003, Gromova and Petrosyan, 2017). In this paper we shall focus on the α-, δ-, and C-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 (Neumann and Morgenstern, 1953) and now is called α-characteristic function. We have

( 0, S = {0},

I max min Σ Ki(u1,..., un, ·), S C N,

Vа(S, ·) =i “ES j-Лies (2)

I n

I max Ki(ui,...,un, ·), S = N,

^ i=i

where max inin is the lower value of the zero-sum game between the coalition S

iES jEN\S

and N \ S with the payoff function Σ Ki(ui,..., un, ·). It was proved in (Petrosjan

ies

and Danilov, 1982) that in general the α-characteristic function is superadditive.

2.2. δ-characteristic function

δ-characteristic function, (Petrosjan and Zaccour, 2003), 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 players from N \ S while the

players from S seek to maximize their joint payoff Σ Ki(ui,..., u„, ·).

ies

Г 0, S = {0},

Vδ (S,

max Σ Ki(us, uNEs, ·), S C N,

ui, ies ies

uj =ujNE, jeN\s

n

max ^ Ki(u i,...,un, ·), S = N.

4 u1 j.. .un i= i

(3)

In general, a δ-characteristic function is a non-superadditive function (see examples in (Gromova et ah, 2017)).

2.3. ζ-characteristic function

One of the novel approaches is to use a C-characteristic function, (Gromova and Petrosyan, 2017). 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

Vζ (S, ·)

'0, S = {0},

min Σ Ki(uSs,un\s, ·), S C N,

Uj eUj ,jeN\S, ieS < Ui=u* , iGS

n

max Σ Ki(ui,...,un, ·), S = N,

U1 ,...,Un, i=1

UiEUi, i^N

(4)

where u* = {u*}ieN is the profile of strategies for which the maximal value of payoff function is achieved for all players, uS = {u*}ieS.

The c.f. defined in this way is in general superadditive (see Gromova and Petrosyan, 2017). Note, that for the case of C-characteristic function players from N\ S have active reaction while players from S just use the same strategies u*, i є 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 (Breton et ah, 2005), see also (Gromova, 2016). 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 є [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 (Pontryagin, 2018).

The dynamics of the total amount of pollution x(t) is described by following equation

3

X(t) = Σ ui(t).

i=1

The instantaneous payoff of i-th player is defined as:

R(ui(t)) = biui(t) - 2 u2(t), i є N.

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)) — dix(t),di > 0. The payoff of the i-th player is thus defined as

f t 1

Ki(x о, T — to,ui,u2,u3) = J (^(bi — 2 u^ ui — di xjdt.

We will assume additional regularity constraints to be satisfied: Vi є N bi >

3

Dn(T — to), where Dn = Σ di.

i=1

3.1. Construction of characteristic functions

To find the profile of optimal strategies we have to solve the maximization problem

3 Ç t 1

max > ( ( bi-uA ui — d,x)dt.

U1,U2 ,u3^ 2 J J

i=1J to

The Hamiltonian is

з і з з

H (x, η,ψ)=Σ (bi — 2 Ui) Ui — ^ di x + 'ФУ^/1

i= 1 i= 1 i=1

its first order partial derivatives w.r.t. ui’s are dH,

du.

■(x, u, Ф) = bi — Ui + ψ, i = 1,3.

The Hessian matrix is negative definite hence we conclude that Hamiltonian H is concave w.r.t. ui.

д% h __

„ 2 (x, u, ψ) = —1 < 0, i = 1, 3. duİ

The adjoint equations and the related transversality conditions are

з

d± = sp d·

dt ^2^ di, i= 1

Ψ(Τ )=0.

We get the optimal control

u*(t) = (b1 — Dn (T — t), b2 — Dn (T — t), Ьз — 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 a(N,x(t),T — t) = Vδ (N,x(t),T — t) = Vζ (N,x(t),T — t) =

(6)

= —Dn (T — t)x(t) + 2 Bn (T — t) — 1 Dn Bn (T — t)2 + 2 Dn (T — t)3,

3^3

where Bn = Σ bi, Bn = Σ b2.

i=1 i=1

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

max

Ui

ui — di x

dt,

1, 2, 3.

We apply the PMP to this maximization problem. The players’ Hamiltonians

are

H,

1 3

(x, u, ψ) = (bi - 2u,) u, - d,x + ψ^Μ,, i = 1, 3,

i=1

and their first order partial derivatives w.r.t. u,’s are dH,

du,

■(x, u, ψ) = bi - u, + ψ, i = 1,3.

The Hessian matrices are negative definite hence we conclude that Hamiltonians

Hi are concave w.r.t. u,·.

d2 Η, du2

(x, u, ψ) = —1 < 0, i = 1, 3.

The adjoint equations and the related transversality conditions are

^ = d·

ψ(Τ ) = 0.

We get the Nash equilibrium strategies

uNE(t) = (bi - di(T - t), b2 - d2(T -t), Ьз - d3(T - t)). (7)

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

max

Ui,Uj J t0

Uk =uNE

çT 1 1

J ((bi - 2Щ)u, + (bj - 2uj)uj - (di + dj)x)dt.

Following PMP we get

if = bi - (d, + dj)(T - t), uf = bj - (d, + dj)(T - t).

(8)

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

f t 1 1

min у ^ ^bi 2 u^ ui + (bj 2 uj) uj (di + dj )x^dt*

By using PMP we get

uk = bfc.

(9)

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

max

Ui,Uj J t0 uk —bk

/j 1 1

((bi - 2u,)u, + (bj - 2uj)uj - (di + dj)x)dt.

Applying PMP in the same way we get

bi - (d, + dj)(T - t), uf = bj - (d, + 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 a—characteristic function:

1 1 1

V“({i},x(t),T—t) = —di(T—t)x(t) + -b2(T—t) — -BNd,(T—1)2 + -d2(T—t)3, (11)

2 2 6

V “({i, j},x(t),T — t) = —(di + dj)(T — t)x(t) + 2(b2 + b2)(T — t) —

— 1BN (di + dj )(T — t)2 + 2 (di + dj )2(T — t)3·

We get a δ—characteristic function from (3), (7), (8):

Vδ ({i}, x(t), T —1) = —di(T — t)x(t) + 2 b2(T — t) — 2 Bn d (T —1)2 +

+ 1 di (2DN — di )(T — t)3 ;

(12)

(13)

Vδ({i, j},x(t), T — t) = —(di + dj)(T — t)x(t) + 2(b2 + b2)(T — t) —

— ^ BN (di + dj )(T — t)2 + 2 (dfc (di + dj ) + (di + dj )2)(T — t)3 ·

(14)

Finally, for a ζ—characteristic function from (4), (5), (9) we obtain: Vζ ({i}, x(t), T — t) = —di (T — t)x(t) + 2 b2(T — t) — 2 Bn di(T — t)2 — 6 dn (dn — 2di )(T —1)3,

(15)

Vζ({i,j}, x(t), T — t) = —(di + dj)(T — t)x(t) + ^(b2 + b2)(T — t) —

(16)

— 2 Bn (di + dj )(T — t)2 — 3 Dn (Dn — 2(d + dj ))(T — t)3·

3.2. Superadditivity of the c.f.

Check that a—characteristic function (2) is superadditive (1). From (6), (11), (12) we have

V“(N) — V“({*}) — V“({i, j}) = 33dk(di + dj)(T — t)3 > 0,

V “({i,j }) — V “({i}) — V “({j}) = 2 di dj (T — t)3 > 0,

since t є [to,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

Vδ(N) — Vδ({k}) — Vδ({i,j}) = 6(DN + d2)(T — t)3 > 0,

Vй({ij}) - Vй({i}) - Vй({j}) = 1(dj + dj)(T -1)3 > 0,

since t Є [to, 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

Vζ(N) - Vζ({k}) - Vζ({i,j}) = 1 Dn(di + dj + 2dk)(T - t)3 > 0,

Vе({i,j}) - Vζ({i}) - Vζ({j}) = 3Dn(di + dj)(T -1)3 > 0,

since t Є [to, T]. This concludes that ζ-characteristic function is also superadditive (as must be in general).

3.3. Comparison of characteristic functions

Let us investigate how the constructed c.f.s relate to each other. Obviously, we have

Vй (N) = Vа (N),

also from (11), (13) and (12), (14) we get

Vй ({i}) = V a({i}) + 1 di (dj + dk )(T -1)3,

Vй({i,j}) = Va({i,j}) + 3dk (di + dj)(T -1)3.

Thus,

Vй(·) > Vα(·). (17)

Furthermore, from (11), (15) and (12), (16) we have

V “(N) = Vζ (N),

V“({i}) = Vζ({i}) + 6(dj + dk)2(T -1)3,

V“({i,j}) = Vζ({i,j}) + 1 dk(T -1)3.

Thus,

V“(·) > Vζ(·). (18)

Finally, (17) and (18) imply that

Vй(·) > V“(·) > Vе(·).

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 (Petrosyan and Murzov, 1966, Petrosyan and Shevkoplyas, 2000, Petrosyan and Shevkoplyas, 2003, Marin-Solano and Shevkoplyas, 2011, Shevkoplyas, 2014) (T — to), where T is a random variable with exponential distribution function F(t),t є [to,Tf ]. The strategy of each player is to choose the amount of pollution emitted to the atmosphere, щ є [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

We assume that di > 0 Vi = 1, 2,3 and there are additional constraints called the

3

regularity constraints: bi > -^т, where Dn = Σ di.

According to (Shevkoplyas, 2014, Kostyunin and Shevkoplyas, 2011, Gromova and Tur, 2017) the payoff of each player i є N can be represented as

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.

3

X(t) = ^2 ui(t), x(to ) = xo.

The expectation of the payoff of players i = 1,2, 3 are calculated as

bj — 2Ui(r)) Ui(τ) — djx(t)j e λ(τ to)dr

The Hamiltonian is

its first order partial derivatives w.r.t. ui’s are

Ui)e t + ψ, i = 1, 3.

λt

The Hessian matrix is negative definite hence we conclude thet Hamiltonian H is concave w.r.t. u*.

d 2H ~duf

(x, u, ψ) = —e λ < 0, i = 1, 3.

The adjoint equations and the related transversality conditions are

= Y' de-At

dt = d*e ,

i= 1

lim ψ(ί) = 0.

t—— ^O

We have optimal controls in the following form

*/.N (l Dn , Dn Dn \

u (t) = Г — ~,b2 — — ~λ

(19)

Given the initial conditions (t, x), те compute the c.f. for the grand coalition N from (19).

Va(N, x(t), T — t) = Vй(N,x(t),T — t) = Vζ(N,x(t),T — t) = A?( BN λ2 — Bn Dn λ + 12 DN — Dn A2x),

(20)

3 _ 3

where Bn = Σ Ъи Bn = Σ b2.

i=1 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.

max ^ ((bi — 2 Ui(r )j щ(г) — d*x(r )j e-Ar dr.

According to the PMP the Hamiltonian has the form

Hi (x,u, ψ)

3

ui — di'Xj e-At + ψ^2 Щ,

i= 1

i = 1, 3

and its first order partial derivatives w.r.t. ui’s are

dHi

dui

(x, u, ψ)

(bi — ui)e At + ψ, i = 1, 3.

The Hamiltonians Hi are concave w.r.t. u-, because the respective Hessian matrices are negative definite.

β2 H-

„ 2 (x, u, ψ) = —e-At < 0, i = 1, 3.

du2

The adjoint equations and the related transversality conditions are

= de-At dt = die ,

lim ψ^) = 0.

t — TO

We get the Nash equilibrium strategies

di d,2

(21)

uNE<*>_ (bi - f · - f) ■

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

/* t і і

^max I ^ ^bi 2 ui^ ui + (bj 2 U j ^ uj (di + dj )x^dt·

Ui,Uj Jto

Uk =uNE

Following PMP we get the controls in form of

uS _ uS _ b di + dj

ui bi д · uj bj д ■

For the case of α—, ζ—d. (2), (4) we will have to solve the minimization problem

(22)

min e

Uk

„Atn

' to

From the PMP we get

1 N \ Λ 1

2 Ui2 )ui + ^j --- 2

uk _ bk ■

- — At„

(23)

According to the definition of a-c.f. (2), to calculate it we have to solve the maximization problem having in mind (23):

^ ^bi 2 ui^ ui + (bj 2 uj^ uj (di + dj)x^e dt·

Ui,Uj J t0 uk —bk

Applying the PMP to this problem we get

di + dj

u, _ bi -

λ

us _ b. _ di + dj uj _ bJ д

(24)

Given the initial conditions (t, ж), we calculate the value of three described above characteristic functions for one- and two-players coalitions.

With (2), (23), (24) we calculate the a—c.f.

V a({i},x(t),T — t) _ Дз (2 Ь2Д2 — Bn di λ + 1 d2 — d^2 x(t)), (25)

V “({ij }, x(t), T — t)

i

λ3

(2 (b? + b2)V

Bn (di + dj )λ + (di + dj )2

(26)

— (di + dj)X2x(t)j ■

According (3), (21), (22) the δ—characteristic function could be calculated as Vδ({i}, x(t), T — t) _ (2Ь2Д2 — Bnd^ + (dj + dfc)di + 2d2 — diλ^^)) , (27)

VЙ ({*; j}j x(t) T — t) — λ (1 (bi + ^2)λ2 — BN (di + dj)λ + dk (di + dj) + +(di + dj)2 — (di + dj)X2x(t)j.

(28)

For the Z—characteristic function from (4), (19), (23) we obtain Vζ({i}, x(t), T — t) — 1 b2λ2 — Bndλ — IDN + diDN — d;A2x(t)), (29)

Vζ ({i,j },x(t),T — t) — ^ (1 (b2 + &2)λ2 — Bn (dj + dj )λ—

— DN + 2DN (di + dj ) — (di + dj ^2x(t))

(30)

4.2. Superadditivity of c.f.

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

V “(N) — V “({k}) — Vа ({i,j}) — 2^ ((di + dj )2 + 2 % + 6 dk (dj + dj )) > 0,

Vа({i,j}) — V“({i}) — V“({j}) — 2L (d? + d2 + 4djdj) > 0, since t Є [to, T].

Additionally we could prove that δ—characteristic function is superadditive using (20), (27), (28).

Vй(N) — Vй({k}) — Vй({i,j}) — 2^ ((di + dj)2 + 2dk + 2dk(dj + dj)) > 0,

Vй({i, j}) — Vй({i}) — Vй({j}) — 2^3 (di + j > 0, since t Є [t0, T],

Also Z—characteristic function is superadditive too (20), (29), (30).

Vζ(N) — Vζ({k}) — Vζ({i,j}) — λ,Dn (di + dj + 2d^ > 0,

Vζ({i, j}) — Vζ({i}) — Vζ({j}) — λ,Dn(di + dj) > 0, since t Є [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

Vй (N) = V a(N),

also from (25), (27) and (26), (28) we get

Vй ({i}) = V a({i})+(dj +A3dfc )d,

Vй ({ij }) = V “({*,j}) +dk (d;+dj}.

Thus,

Vй(·) > V“(·).

(31)

Apart from that, from (25), (29) and (26), (30) we have

Vа (N) = Vе (N),

Va({i}) = Vζ({i}) + 2A3(Dn - di)

Vα({i,j}) = Vζ({i,j}) + A3 dk

Thus,

Vа(·) > Vе(·).

Inequalities (17) and (18) imply that

Vй(·) > Vα(·) > Vе(·).

(32)

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 = {0},

Σ Ki( u* uNE ·) S c N,

Vη(S, ·) = < ieS uS, uN\S, ), (33)

max Σ Ki(ui,.. . , un? •)7 S = N.

“1,···-“" i=1

where u* = {u* }ieN is the profile of strategies for which the maximal value of payoff function is achieved for all players, u*S = {u*}ieS.

In (33) for players from S we use (obtained earlier) strategies u*S from optimal profile u* (as in Z—c.f.) and for players from N \ S we use (obtained earlier) strategies uN(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.

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