## MEMBERS

### Program Members

##### Optimization Theory and Game Theory

Optimization theory and game theory are adjacent fields in social science, which began in the middle of the twentieth century. Generally speaking, optimization refers to a problem to maximize the profit or minimize the cost under a certain set of constraints, while optimization theory is a mathematical theory to obtain optimal solutions.

On the other hand, game theory aims to present a solution to satisfy players with a rational answer (equilibrium) when each player tries to maximize his or her own profit. Because the problem itself contains optimization problems, game theory uses theories and methods of optimization. However, unlike an optimization problem, game theory needs to answer the fundamental question of what is considered equilibrium. For example, when a cable network is built in an area, the problem of how residents should bear the cost is not clearly answered. The reason for this is evident from the fact that there are parents who refuse to pay for school lunch, but own a luxury car. Therefore, besides presenting a reasonable equilibrium, it is important to develop a system to implement it.

As demonstrated above, game theory needs to address an issue by considering factors that do not fall within the scope of mathematics. Likewise in an optimization problem, the optimum solution can vary, depending on which one of various factors, such as cost, required time, and comfort, are given priority. Hence, to solve a problem, what people want has to be considered.

Roughly speaking, there are two types of optimization methods: one that deals with a continuous quantity, such as linear programming, and another that deals with discrete numbers, such as a graph network. Although these two have been previously studied separately, today they are closely tied together because it is necessary to perform some sort of discretization in order to solve a continuous optimization problem using a computer. The continuous optimization method is being used to solve discrete optimization problems. My current research interests are to associate the continuous structure and discrete structure in optimization, and pure strategy equilibrium using the discrete fixed point theorem in game theory.

Pure strategy, as discussed here, refers to a basic tool to describe a strategic game, such as a rock, paper, and scissors in rock-paper-scissors, while adopting multiple pure strategies by a certain probability is called a mixed strategy. For example, suppose players P1 and P2 play a rock-paper-scissors and their scores are given in the following table.

P1 | P2 | Score of P1 | Score of P2 |
---|---|---|---|

Rock | Scissors | 3 | 0 |

Scissors | Paper | 6 | 0 |

Paper | Rock | 6 | 0 |

Tie | Tie | 0 | 0 |

In this case, the min-max theorem of von Neumann dictates that the optimum strategy is to play rock, scissors, and paper in a ratio of 1:2:1. In other words, throw a dice and play rock if you roll 1, play scissors if you roll 3 or 4,play paper if you roll 4, and throw the dice again if you roll 5 or 6. As in these examples, the strategy to make a decision by throwing a dice is called a mixed strategy, and the strategy to play a certain move is called a pure strategy. Nash has confirmed that even when there are three or more players, an equilibrium always exists in a mixed strategy. However, an equilibrium does not always exist if it is limited to a pure strategy. Thus, I am conducting joint research on equilibrium in pure strategy with one of my graduate student, Jun-ichi Sato.

In addition, I am working to construct a duality theorem, which is important in optimization both from theoretical and practical points of view, for three or more objects. Especifically, I am working on a three-phase partition problem, which is a formulation of grain growth in annealing pre metal and segregation between biological species. In this work, I have proved that the duality problem is a maximization problem of an equilateral triangle divided into three areas. This result is expected to be applicable to support vector machines.

##### Long-Term Internship for Ph.D. Students

The Graduate School of Mathematics of Kyushu University launched its Functional Mathematics Course as part of its Ph.D. Program in April 2006 with the goal of strengthening the education program by initiatives such as a mandatory long-term internship (longer than three months). I am the internship coordinator, and have looked after eighteen Ph.D. students during the last two years. So far, students satisfy the program, and the companies that received our students highly appreciate the program more than expected. The outcomes of this program also include collaborative research projects, which started in FY2008 with two of the companies that participated in the internship program.