Nash Arbitration Methods: Game Theory for students without calculus
William P. Fox
Naval Postgraduate School
We illustrate a method to obtain the solution to Nash arbitration using only methods involving college algebra. A necessary component of the Nash arbitration is the status quo point which is usually the security levels found from the Prudential Strategies. We use linear programming to find these security levels using the SimplexLP of the Solver. We show how golden section search, used as an iterative numerical method with Excel, is used to find the solution value for the x coordinate of the Nash arbitration scheme. Then we show how we find the y coordinate using the equation developed from the Pareto optimal line. Having…show more content… His interests include applied mathematics, optimization (linear and nonlinear), mathematical modeling, statistical models for medical research, game theory, and simulation models. He is currently the Past-President of the Military Application Society of INFORMS.
We teach a course in decision making titled modeling in conflict. Our main topics in this course are applications and algorithms in decision theory and game theory. In our game theory course, we cover such main topics as utility theory, zero sum games & constant sum with their methods of solutions, non-zero sum games with Nash equilibriums and strategic moves, three person zero sum and non-zero sum games, and Nash arbitration scheme for bargaining. To somewhat complicate our teaching is that the course requirement is college algebra, not calculus. Our other main issue is that we are limited to the technology that we use. Currently that technology is Excel since Excel is the software that our students will have after graduation in the careers.
Our colleague, Frank Giordano, developed a geometric approach to Nash arbitration involving triangles, mid-points, and similar triangles to obtain both the arbitration point and how strategically to play the arbitration game. We found that often geometry is harder to extract from our students memories than the basic algebra concepts. We also wanted to give the students some technology using Excel to assist them in their efforts also
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