Bellman-Isaaks equations for differential games with random duration Ekaterina Shevkoplyas

1. Bellman-Isaaks equations for differential games with random duration Ekaterina Shevkoplyas Katya_shev@mail.ru Supervisor Leon A. Petrosjan

2. 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)

3. Definition of the game Differential n-person game (x0). The final time instant T is a random variable with distribution function F(t). Let hi(x(τ)) be an instantaneouspayoff of theplayer i. Then the expected integral payoff of the player i, i=1,...,n is asfollows: Cooperative game is defined as 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

4. Results • We introduced a new class of differential games such that differential games with random duration. • We derived the Isaak-Bellman equation (or the Hamilton-Jacobi-Bellman equation) for the problem with random duration. • 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 . • We applied our algorithm to game-theoretical model of nonrenewable resource extraction with random duration. • 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. • We proved the time-inconsistency of the Shapley Value in the game of nonrenewable resource extraction with random duration.

5. The questions and the way forward • What about regularization under condition of arbitrary sign of instantaneous payoffs of the players (not only positive)? • 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: • 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.

6. References • E.J. Dockner, S. Jorgensen, N. van Long, G. Sorger. Differential Games in Economics and Management Science. Cambridge University Press, 2000. • Petrosjan L.A.. Differential Games of Pursuit. World Sci. Pbl.2003. p. 320. • 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) • 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. • 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. • 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)