Alan Zame

On fair coin-tossing games

Let [Omega] be a finite set with k elements and for each integer n [greater, double equals] 1 let [Omega]n = [Omega] - [Omega] - ... - [Omega] (n-tuple) and 11 and aj [not equal to] aj+1 for some 1 [less, double equals] j [less, double equals] n - 1}. Let {Ym} be a sequence of independent and identically distributed random variables such that P(Y1 = a) = k-1 for all a in [Omega]. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in [Omega]n and in [Omega]n with respect to the stochastic process {Ym}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.

Limiting distributions of two random sequences

For fixed p (0 = 2, with probability p let {Ln, Rn} = {Ln - 1, Xn} or = {Xn, Rn - 1} according as , with probability 1 - p let {Ln, Rn} = {Xn, Rn - 1} or = {Ln - 1, Xn} according as , and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n >= 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n --> [infinity]. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 - y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability).

An Optimal Acceptance Policy for an Urn Scheme

An urn contains m balls of value -1 and p balls of value +1. At each turn a ball is drawn randomly, without replacement, and the player decides before the draw whether or not to accept the ball, i.e., the bet where the payoff is the value of the ball. The process continues until all m+p balls are drawn. Let

On discounted subfair primitive casino

\overline{V}(m,p)

A Random Version of Shepp's Urn Scheme

denote the value of this acceptance

Orthogonal sets of vectors over Zm

(m,p)

Transversal theory and rank functions

urn problem under an optimal acceptance policy. In this paper, we first derive an exact closed form for \overline{V}(m,p) and then study its properties and asymptotic behavior. We also compare this acceptance (m,p) urn problem with the original (m,p)

Bandit Problems With Infinitely Many Arms

urn problem which was introduced by Shepp [Ann. Math. Statist., 40 (1969), pp. 993--1010]. Finally, we briefly discuss some applications of this acceptance (m,p) urn problem and introduce a Bayesian approach to this optimal stopping problem. Some numerical illustrations are also provided.

On weak mixing of cascades

SummaryAs is well known, in a subfair primitive casino a gambler with an initial fortune f, 0<f<1, desiring to reach 1 (his goal) should play boldly since there is no other strategy that can provide him with a higher utility (the probability of reaching his goal). Now suppose the game is modified by adding a discount factor which is used to motivate the gambler to recognize the time value of his goal and complete the game as quickly as is reasonably consistent with reaching his goal. Then one would intuitively suspect that again the bold play would be optimal. We will show in this paper that for certain subfair or fair primitive casinos the bold play is always optimal regardless of the discount factor; however, for some subfair or fair primitive casinos, there exist some discount factors for which the bold play is no longer optimal.

A Note on the First Occurrence of Strings

In this paper, we consider the following random version of Shepps urn scheme: A player is given an urn with n balls. p of these balls have value +1 and n-p have value -1. The player is allowed to draw balls randomly, without replacement, until he or she wants to stop. The player knows n, the total number of balls, but knows only that p, the number of balls of value +1, is a number selected randomly from the set {0, 1,2,...,n}. The player wishes to maximize the expected value of the sum of the balls drawn. We first derive the players optimal drawing policy and an algorithm to compute the players expected value at the stopping time when he or she uses the optimal drawing policy. Since the optimal drawing policy is rather intricate and the computation of the players optimal expected value is quite cumbersome, we present a very simple drawing policy, which is asymptotically optimal. We also show that this random urn scheme is equivalent to a random coin tossing problem.