Experimental Math: Final Project

My project was in part to investigate the variance of playing the Shepp urn under the strategy which maximizes expected value. To this end after much experimentation I was able to conjecture and subsequently prove that if we play the Shepp urn with p postive balls and β(p) negative balls then the variance of our results grows at a rate of roughly √ p. I didn’t see as obvious of a pattern for other starting values, but the same general ideas may work there as well. We recall the basic definitions of a Shepp Urn. A Shepp Urn is an optimal stopping time game in which a given urn contains m ”minus” balls and p ”plus” balls. The player at each stage may choose to either stop playing, or draw another ball at random from the urn, without replacement. When the player stops, s/he receives p∗ −m∗ dollars where p∗ and m∗ are the number of plus and minus balls drawn respectively. Let V (m, p) denote the expected value of a Shepp Urn game of an urn containing m balls and p + balls Larry Shepp observed that the play which optimizes expected value of the game is to draw until the expected value of the urn with the balls remaining in it is negative.