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This thesis is about game theory, its general application to different games, and its different notions. We are going to focus on how different solutions can be found depending on how we view the game in question. We will particularly focus on games where the information is incomplete. In order to do so, we will use a game named ”The battle of sexes” as a guide throughout the entire thesis. We will adapt that game to the different kinds of games we will talk about. The importance with game theory and why it is an interesting field of study is that it gives a way to mathematically understand and analyse various situations in different other fields and real life areas. In this thesis we will not go deeper into the applications of game theory, but will mostly focus on the mathematical part of the field, and understanding how to find different desired outcomes for a given game. We will primarily lay out the fundamentals of game theory, in order for the reader to understand the basic notions of a game, a equilibria, and have the necessary knowledge to explore the field. Guillermo Owen’s book Game theory covers the field in a wider sense than this thesis, and is an interesting lecture to broaden ones knowledge on game theory.

We study a stochastic game of hiring and firing in which the employee has a random capacity which is not directly observable for the employer, and only par- tially observable for the employee. In this setting, we derive a Nash Equilibrium of strategies.

The purpose of this paper is to present a new nonparametric test statistic for testing against ordered alternatives in a Balanced Incomplete Block Design (BIBD). This test will then be compared with the Durbin test which tests for differences between treatments in a BIBD but without regard to order. For the comparison, Monte Carlo simulations were used to generate the BIBD. Random samples were simulated from: Normal Distribution; Exponential Distribution; T distribution with three degrees of freedom. The number of treatments considered was three, four and five with all the possible combinations necessary for a BIBD. Small sample sizes were 20 or less and large sample sizes were 30 or more. The powers and alpha values were then estimated after 10,000 repetitions.The results of the study show that the new test proposed is more powerful than the Durbin test. Regardless of the distribution, sample size or number of treatments, the new test tended to have higher powers than the Durbin test.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22984号 / 理博第4661号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 藤 定義, 教授 佐々 真一, 教授 早川 尚男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Ph. D.

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No description available.

We know that the best linear combination of the intra- and inter-block estimates is Intra-estimatexInter-variance+Inter-estimatexIntra-variance / Intra-variance + Inter-variance ; however, this combined estimate is merely theoretical, since we do not know in practice the exact inter- and intra-variances. A reasonable solution is to use a random weight which can be computed from the data of our experiment, but so far there has been no practical solution without severe restrictions on the size of the experiment, and no solution at all for a clear answer to the question of how much we recovered. In fact, the experimenter applying the methodology available to him now, cannot be sure that he is really improving the accuracy of his estimation. This research has achieved the following: 1. A new method of combining two independent estimates has been developed. This method has its use in incomplete block designs, in similar experiments, and in randomized block designs with heterogeneous variances. The improvement introduced by this method is very satisfactory, compared with the utmost possible theoretical improvement. 2. A procedure for recovering the inter-block information in B.I.B. designs was given, which is applicable in experiments as small as t = 4. 3. It has been proven that the practical utilization of inter-block information is possible in any P.B.I.B. with seven treatments or more. 4. A general procedure for recovering the inter-block information in P.B.I.B.'s with two associate classes was given. 5. An inter-block analysis of singular and semi-regular group divisible designs was discussed, which makes a partial utilization of the inter-information possible. In general, this work has two merits: 1. It makes possible the utilization of the interblock information in small and moderate size experiments. 2. As a ratio of the utmost possible theoretical recovery (by combining linearly), either exactly or a lower bound of the ratio of recovery is always computable. Tables which enable the experimenter to use the procedures described in this dissertation were given. The ratios of recovery listed in these tables show that the new method gives good results where the old method is not applicable, and when the old method starts, hopefully, to be valid, the ratio of recovery achieved by the new method starts to approach the theoretical value that can be achieved, assuming the intra- and inter-variance are known. / Ph. D.

The work of Kramer and Bradley on the use of factorials in incomplete block designs bas been extended to permit both the intra-block and combined intra- and inter-block analyses of factorials in balanced and partially balanced incomplete block designs. In particular, we have obtained a combined intra- and inter-block analysis for factorials in balanced incomplete block designs, group divisible designs, and Latin Square types of partially balanced, incomplete block designs. For the class of Latin Square sub-type L₃ designs both the intra-block and combined intra- and inter-block analyses have been developed. In general, factorial treatment combinations were assigned to the association schemes by permitting the rows of the association schemes to represent the levels of one factor and the columns to represent the levels of a second factor. The extension to multifactor factorials was then carried out by sub-dividing the levels of the basic two-factor factorial, the levels in the sub-divisions representing the levels of the multi-factor factorials. Estimators for the factorial effects have been obtained along with their variances and covariances. Sums of squares in terms of the factorial estimators have been derived and can be used to carry out tests of significance. These sums of squares were shown, for the combined intra- and inter-block analyses, to be independently distributed as χ²-variates with the appropriate numbers of degrees of freedom. Suitable sums of squares for tests of significance are not possible in general for Latin Square sub-type L₃ designs. In situations such as these, we can only consider contrasts among the estimators and use their variances to perform tests of significance. However, for the special cases of factorials in the 4 x 4 Latin Square sub-type L₃ design, a complete analysis yielding the adjusted sums of squares for the factorial effects is possible if the factorial treatments are applied to the association scheme in a different manner. Single-degree-of-freedom contrasts are obtained in much the usual way as for factorials in complete block designs. The method of incorporating a fractional replicate of a factorial into incomplete block designs is also considered. Numerical examples have been worked in detail for a group divisible design and a Latin Square sub-type L₂ design. The procedure for analyzing a Latin Square sub-type L₃ design is also discussed. / Ph. D.