Bellman-Isaaks equations Bellman-Isaaks equations for differential games with random durationfor differential games with random duration

Ekaterina ShevkoplyasKatya_shev@mail.ru

Supervisor Leon A. Petrosjan

REVIEW

E.J. Dockner, S. Jorgensen, N. van Long, G. Sorger. 2000 [1]• Hamilton-Jacobi-Bellman equation for differential games with prescribed duration or infinite time horizon • A game-theoretic model of nonrenewable resource extraction with infinite time horizon

Petrosjan L.A., Murzov N.V., 1966 (in Russian) [3]• 2- person game of pursuits with random duration (terminal payoffs)

Petrosjan L.A., Zaccour G., 2003 [5]• Non-standard algorithm of characteristic function values calculating was proposed («Nash equilibrium» approach: if k players form coalition K, then the remaining players stick to their feedback Nash strategies) • Time-Consistency of the Shapley Value was proved.

Petrosjan L.A. , 1977 [2]•The notion of the time-consistency for differential games solution (prescribed duration of the game)

Definition of the gameDifferential n-person game (x0). The final time instant T is a random variable with distribution function F(t). Let hi(x(τ)) be an instantaneous payoff of the player i. Then the expected integral payoff of the player i, i=1,...,n is as follows:

.)())((),..,(0 0

10

t

t

t

ini tdFdxhuuxK

Cooperative game )( 0x

).,()())(*(*)*,..,(),..,(max 011

101

10

0 0

xNVtdFdxhuuxKuuxKu

n

i t

t

t

i

n

ini

n

ini

Non-standard problem of dynamic programming (see form of functional)!

An Example (A Game-Theoretic Model of Nonrenewable Resource Extraction)Let x(t) and ci(t) denote respectively the stock of the nonrenewable resource and player i's rate of extraction at time instant t [1].The final time instant of the game is random variable T with the exponential frequency distribution. The utility function (or instantaneous payoff) for player i at time instant τ

.)ln())(( BcAch iii

Results1. We introduced a new class of differential games such that differential games with

random duration.

2. We derived the Isaak-Bellman equation (or the Hamilton-Jacobi-Bellman equation) for the problem with random duration.

,,),(

),(max),(

),()(1

)(

)g( uxx

txWuxH

t

txWtxW

tF

tfu

3. Using non-standard approach of Petrosjan and Zaccour [5] we proposed an algorithm of the characteristic function construction with the help of the new Hamilton-Jacobi-Bellman equation .

4. We applied our algorithm to game-theoretical model of nonrenewable resource extraction with random duration.

5. We introduced a notion of time-consistency for differential games with random duration. IDP (imputation distribution procedure) was derived in analytic form. Moreover we proposed a method of optimality principle regularization if instantaneous payoffs are positive.

6. We proved the time-inconsistency of the Shapley Value in the game of nonrenewable resource extraction with random duration.

The questions and the way forward

1. What about regularization under condition of arbitrary sign of instantaneous payoffs of the players (not only positive)?

2. Can we use incentive strategies in our model of nonrenewable resource extraction?

We are going

to consider another form of utility function ( instantaneous payoff) in the model of resource extraction such as follows:

.1,1

1

Bc

Ah ii

to calculate PMS-value for all examples. to consider asymmetric payoffs of the players in the resource extraction game. to investigate the agreeability of the optimality principle in the game with random duration.

References1. E.J. Dockner, S. Jorgensen, N. van Long, G. Sorger. Differential Games in

Economics and Management Science. Cambridge University Press, 2000.

2. Petrosjan L.A.. Differential Games of Pursuit. World Sci. Pbl.2003. p. 320.

3. Petrosjan L.A., Murzov N.V. A Game Theoretic Model in Mechanics // Litovskyi matematicheskyi sbornik, vol.VI, Vilnjus, Litva, 1966, pp. 423- 432. (in Russian)

4. L.A. Petrosjan, E.V. Shevkoplyas. Cooperative Solutions for Games with Random Duration. Game Theory and Applications, Volume IX. Nova Science Publishers, 2003, pp.125-139.

5. Petrosjan L.A., Zaccour G. Time-consistent Shapley Value Allocation of Pollution Cost Reduction. // Journal of Economic Dynamics and Control, Vol. 27, 2003, pp. 381-398.

6. E.V. Shevkoplyas. On the Construction of the Characteristic Function in Cooperative Differential Games with Random Duration. International Seminar "Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations" (CGS'2005), Ekaterinburg, Russia. Ext.abstracts, Vol.1,pp. 262-270. (in Russian)