Type of Document Master's Thesis Author Peterson, Joshua Michael URN etd-03312005-200753 Title A Complete Characterization of Nash Solutions in Ordinal Games Degree Master of Science in Electrical Engineering Program Electrical Engineering School School of Engineering Advisory Committee
Advisor Name Title Marwan Simaan Committee Chair Ching-Chung Li Committee Member Luis Chaparro Committee Member Keywords
- ordinal game theory
- cardinal game theory
- non-cooperative games
- Nash solution
Date of Defense 2005-04-11 Availability unrestricted Abstract The traditional field of cardinal game theory requires that the objective functions, which map the control variables of each player into a decision space on the real numbers, be well defined. Often in economics, business, and political science, these objective functions are difficult, if not impossible to formulate mathematically. The theory of ordinal games has been described, in part, to overcome this problem.
Ordinal games define the decision space in terms of player preferences, rather than objective function values. This concept allows the techniques of cardinal game theory to be applied to ordinal games. Not surprisingly, an infinite number of cardinal games of a given size exist. However, only a finite number of corresponding ordinal games exist.
This thesis seeks to explore and characterize this finite number of ordinal games. We first present a general formula for the number of two-player ordinal games of an arbitrary size. We then completely characterize each 2x2 and 3x3 ordinal game based on its relationship to the Nash solution. This categorization partitions the finite space of ordinal games into three sectors, those games with a single unique Nash solution, those games with multiple non-unique Nash solutions, and those games with no Nash solution. This characterization approach, however, is not scalable to games larger than 3x3 due to the exponentially increasing dimensionality of the search space. The results for both 2x2 and 3x3 ordinal games are then codified in an algorithm capable of characterizing ordinal games of arbitrary size. The output of this algorithm, implemented on a PC, is presented for games as large as 6x6. For larger games, a more powerful computer is needed. Finally, two applications of this characterization are presented to illustrate the usefulness of our approach.
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