#### Date of Submission

6-28-1989

#### Date of Award

6-28-1989

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Parthasarthy, T. (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

A mathematical theory of Games of Strategy was born in several stages between 1928 and 1941. John von Neumann is known as its father. The culmination of the pioneering work of von Neumann and Morgenstern was the publication of the Theory of Games and Economic Behavior (Ref.28] in 1944. Ik is snid of the book . posterity may regard this book as one of the major scientifie achievements of the first half of the twentieth century. Emphasizing a new approach to competetive behaviour through a mathematical reduction to suitable games of strategy, this giant work laid bare a host of problems in the mathematical theory of games.A game is simply a set of descriptive rules. A play of the game includes every particular instance in which the game is played from beginning toend. The participanta are the players. A game consists of a sequence of moves of the players, while a play comprises a sequence of choices made by them. The decisive otep in the mathematieal trentment of gamen in the normalization achieved by introduction of pure strategies (sctions). A pure strategy is a plan formulated by a player prior to the start of a play, which covers all of the possible decisions which he may face during any play permitted by the rulen of the game. The expected course of a play is thereby completely determined by the selection of a pure strategy by each player in ignorance of that chosen by any other player. von Neumann firet considered games with a finite number of pure strategies i.e. finite gamen. The theory wha Inter extended to games with an infinite number of pure ntrntegies. Both the notions will be useful in the nequel.1.2 (Finite) Zero Sum Two Person Games: Sach games are played by two players and what one player wins, the other loses. A zero sum two person game with a finite number of pure strategies Can be described by an m xn matrix A= ((a,)), where a;( or a(i, j)) in the payment to plnyer I by player II if I chooses his is pure strategy and II chooses his j* pure strategy. However, simplest games like matching pennies show that a player is worse off if he alwnys uses the same pure strategy. Instend, during repented plays of the game, he can be better off by mixing his various pure strategies randomly, but with chosen relative frequencies. Subesequently, a pasenge from such a snmple of playa of the gaine to the underlying populationrevenls the notion of a mixed strategy(mixed actions)as a probability distribution over the net of pure strategien. Mized strategies for I and II will be denoted by a and y respectively, where 2 = (21, 7m), y- (. ); 2, 20 for all i and p; 20 for al j with E. 2, = 1 =E, v,. When I plays a mixed strategy a and II plays a mixed strategy y.

#### Control Number

ISILib-TH173

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Sinha, Sagnik Dr., "Contribution to the Theory of Stochastic Games." (1989). *Doctoral Theses*. 146.

https://digitalcommons.isical.ac.in/doctoral-theses/146

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842922