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Saddle squares in random two person zero sum games with finitely many strategies

Description

By the von Neumann min-max theorem, a two person zero sum game with finitely many pure strategies has a unique value for each player (summing to zero) and each player has a non-empty set of optimal mixed strategies. If the payoffs are independent, identically distributed (iid) uniform (0,1) random variables, then with probability one, both players have unique optimal mixed strategies utilizing the same number of pure strategies with positive probability (Jonasson 2004). The pure strategies with positive probability in the unique optimal mixed strategies are called saddle squares. In 1957, Goldman evaluated the probability of a saddle point (a 1 by 1 saddle square), which was rediscovered by many authors including Thorp (1979). Thorp gave two proofs of the probability of a saddle point, one using combinatorics and one using a beta integral. In 1965, Falk and Thrall investigated the integrals required for the probabilities of a 2 by 2 saddle square for 2 × n and m × 2 games with iid uniform (0,1) payoffs, but they were not able to evaluate the integrals. This dissertation generalizes Thorp's beta integral proof of Goldman's probability of a saddle point, establishing an integral formula for the probability that a m × n game with iid uniform (0,1) payoffs has a k by k saddle square (k ≤ m,n). Additionally, the probabilities of a 2 by 2 and a 3 by 3 saddle square for a 3 × 3 game with iid uniform(0,1) payoffs are found. For these, the 14 integrals observed by Falk and Thrall are dissected into 38 disjoint domains, and the integrals are evaluated using the basic properties of the dilogarithm function. The final results for the probabilities of a 2 by 2 and a 3 by 3 saddle square in a 3 × 3 game are linear combinations of 1, π2, and ln(2) with rational coefficients.

ContributorsManley, Michael (Author) / Kadell, Kevin W. J. (Thesis advisor) / Kao, Ming-Hung (Committee member) / Lanchier, Nicolas (Committee member) / Lohr, Sharon (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)

Created2011

Relationships Between School Music Ensemble Participation and Academic Achievement in the USA

Description

The purpose of this study was to examine relationships between school music participation and a composite measure of academic achievement in American schools. The authors examined the first (baseline) year of a longitudinal study commissioned by the National Center for Education Statistics, conducted in 1988 (NELS:88/94 Data Analysis system, with Additional Systems for High School and Beyond and the National Longitudinal Study of 1972, NCES, 1996). The data set contains information on more than 23,000 American middle school students, and is thought to be representative of the national as a whole.

Several previous researchers employing causal modeling techniques have found relationships between extracurricular activities and education attainment as measured by standardized academic achievement tests. In music, students with school band and orchestra experience make significantly higher grades in high school mathematics, English, and social science than do non-performing students, and instrumental music instruction seems to improve scores on tests of spatial-temporal ability, which is thought to correlate with ability in mathematics.

The present study confirmed previous findings on relationships among music participation and academic achievement. Significantly more students in school music choral or instrumental groups ranked above the 50th percentile in academic grades than did other students (p < .05). For instrumental students, these results held true for all four socioeconomic quartiles, for both boys and girls, and for all races/ethnicities. For choral students, only students in the third and fourth quartiles and Caucasian students ranked significantly higher than other students.

ContributorsBush, Jeffrey E. (Author) / Humphreys, Jere Thomas (Author) / Allen, Douglas (Author) / Lohr, Sharon (Author)

Created2000

National Levels of School Music Ensemble Participation and Relationships with Academic Achievement

DescriptionThe purpose of this study was to examine relationships between school music participation in choral and instrumental ensembles--by socioeconomic status, gender, and race--and a composite measure of academic achievement.

ContributorsBush, Jeffrey E. (Author) / Humphreys, Jere Thomas (Author) / Allen, Douglas (Author) / Lohr, Sharon (Author)

Created2001