Existence of Nash equilibria for optimum location problems and its applications

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In the present research, I have proved the following mathematical results in location problems.1. Under suitable conditions, the two party system is naturally realized under the single member constituency system.2. There is a Nash equilibrium solution in the optimum location problem in a two dimensional plane by three competitors if all members pursue the top share.These two results give answers to open problems in the rational voting theory. I have also obtained the following results in related areas.3. Bid-ask spread is reliably measured by a Kalman filter approach (joint work with Ms. Zheng).4. Downward bias is reported in the parameter estimation of heteroskedastic time series models. I have found that this bias is substantially amplified if the model is misspecified (joint work with Dr. Chu).5. There is a sufficient condition for assuring that an eigenvalue is the second largest one.