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The Ohio State University

1. Heyman, Joseph Lee. On the Computation of Strategically Equivalent Games.

Degree: PhD, Electrical and Computer Engineering, 2019, The Ohio State University

This dissertation is concerned with the efficient computation of Nash equilibria (solutions) in nonzero-sum two player normal-form (bimatrix) games. It has long been believed that solutions to games in this class are hard, and recent results have indicated that this is indeed true. Thus, in this dissertation we focus on identifying subclasses of bimatrix games which can be solved efficiently. Our first result is an algorithm that identifies nonzero-sum bimatrix games which are strategically equivalent to zero-sum games via a positive affine transformation. Games in this class can then be solved efficiently by well-known techniques for solving zero-sum games. The algorithm that we propose runs in linear time to identify games in this class, representing a significant improvement compared to existing techniques.The second result uses the theory of matrix pencils and the Wedderburn rank reduction formula to develop a generalized theory of rank reduction in bimatrix games. The rank of a bimatrix game is defined as the rank of the sum of the payoff matrices of the two players. Under certain conditions on the payoff matrices of the game, we devise a method that reduces the rank of the game without changing the equilibrium of the game.The final result applies the general theory to the subclass of strategically equivalent rank-1 games. We show that for this subclass, which may include games of full rank, it is possible to identify games in the subclass and compute a strategically equivalent rank-1 game in linear time. These games can then be solved in polynomial time by relatively recent results for solving rank-1 games. Overall, our results significantly expand the class of bimatrix games that can be solved efficiently (in polynomial time). Advisors/Committee Members: Gupta, Abhishek (Advisor).

Subjects/Keywords: Applied Mathematics; Computer Science; Electrical Engineering; Economics; game theory; algorithmic game theory; nonzero-sum games; wedderburn rank reduction; strategic equivalence in games

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Heyman, J. L. (2019). On the Computation of Strategically Equivalent Games. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1561984858706805

Chicago Manual of Style (16th Edition):

Heyman, Joseph Lee. “On the Computation of Strategically Equivalent Games.” 2019. Doctoral Dissertation, The Ohio State University. Accessed December 11, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1561984858706805.

MLA Handbook (7th Edition):

Heyman, Joseph Lee. “On the Computation of Strategically Equivalent Games.” 2019. Web. 11 Dec 2019.

Vancouver:

Heyman JL. On the Computation of Strategically Equivalent Games. [Internet] [Doctoral dissertation]. The Ohio State University; 2019. [cited 2019 Dec 11]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1561984858706805.

Council of Science Editors:

Heyman JL. On the Computation of Strategically Equivalent Games. [Doctoral Dissertation]. The Ohio State University; 2019. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1561984858706805

2. Mu, Rui. Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations.

Degree: Docteur es, Mathématiques, 2014, Le Mans

Cette thèse traite les jeux différentiels stochastiques de somme non nulle (JDSNN) dans le cadre de Markovien et de leurs liens avec les équations différentielles stochastiques rétrogrades (EDSR) multidimensionnelles. Nous étudions trois problèmes différents. Tout d'abord, nous considérons un JDSNN où le coefficient de dérive n'est pas borné, mais supposé uniquement à croissance linéaire. Ensuite certains cas particuliers de coefficients de diffusion non bornés sont aussi considérés. Nous montrons que le jeu admet un point d'équilibre de Nash via la preuve de l'existence de la solution de l'EDSR associée et lorsque la condition d'Isaacs généralisée est satisfaite. La nouveauté est que le générateur de l'EDSR, qui est multidimensionnelle, est de croissance linéaire stochastique par rapport au processus de volatilité. Le deuxième problème est aussi relatif au JDSNN mais les payoffs ont des fonctions d'utilité exponentielles. Les EDSRs associées à ce jeu sont de type multidimensionnelles et quadratiques en la volatilité. Nous montrons de nouveau l'existence d’un équilibre de Nash. Le dernier problème que nous traitons, est un jeu bang-bang qui conduit à des hamiltoniens discontinus. Dans ce cas, nous reformulons le théorème de vérification et nous montrons l’existence d’un équilibre de Nash qui est du type bang-bang, i.e., prenant ses valeurs sur le bord du domaine en fonction du signe de la dérivée de la fonction valeur ou du processus de volatilité. L'EDSR dans ce cas est un système multidimensionnel couplé, dont le générateur est discontinu par rapport au processus de volatilité.

This dissertation studies the multiple players nonzero-sum stochastic differential games (NZSDG) in the Markovian framework and their connections with multiple dimensional backward stochastic differential equations (BSDEs). There are three problems that we are focused on. Firstly, we consider a NZSDG where the drift coefficient is not bound but is of linear growth. Some particular cases of unbounded diffusion coefficient of the diffusion process are also considered. The existence of Nash equilibrium point is proved under the generalized Isaacs condition via the existence of the solution of the associated BSDE. The novelty is that the generator of the BSDE is multiple dimensional, continuous and of stochastic linear growth with respect to the volatility process. The second problem is of risk-sensitive type, i.e. the payoffs integrate utility exponential functions, and the drift of the diffusion is unbounded. The associated BSDE is of multi-dimension whose generator is quadratic on the volatility. Once again we show the existence of Nash equilibria via the solution of the BSDE. The last problem that we treat is a bang-bang game which leads to discontinuous Hamiltonians. We reformulate the verification theorem and we show the existence of a Nash point for the game which is of bang-bang type, i.e., it takes its values in the border of the domain according to the sign of the derivatives of the value function. The BSDE in this case is a coupled…

Advisors/Committee Members: Hamadène, Saïd (thesis director).

Subjects/Keywords: Jeux différentiels stochastiques de somme non nulle; Equations différentielles stochastiques rétrogrades; Point d'équilibre de Nash; Nonzero-sum Stochastic Differential Games; Backward Stochastic Differential Equation; Nash Equilibrium Point; 515.35

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Mu, R. (2014). Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations. (Doctoral Dissertation). Le Mans. Retrieved from http://www.theses.fr/2014LEMA1004

Chicago Manual of Style (16th Edition):

Mu, Rui. “Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations.” 2014. Doctoral Dissertation, Le Mans. Accessed December 11, 2019. http://www.theses.fr/2014LEMA1004.

MLA Handbook (7th Edition):

Mu, Rui. “Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations.” 2014. Web. 11 Dec 2019.

Vancouver:

Mu R. Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations. [Internet] [Doctoral dissertation]. Le Mans; 2014. [cited 2019 Dec 11]. Available from: http://www.theses.fr/2014LEMA1004.

Council of Science Editors:

Mu R. Jeux différentiels stochastiques de somme non nulle et équations différentielles stochastiques rétrogrades multidimensionnelles : Nonzero-sum stochastic differential games and backward stochastic differential equations. [Doctoral Dissertation]. Le Mans; 2014. Available from: http://www.theses.fr/2014LEMA1004

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