On some blackjack type optimal stopping problem
Andrzej Z. Grzybowski · 2008
In the paper a class of optimal stopping problems which have some blackjack game features is considered. Both a value of the problem and an optimal stopping rule are found in some special case. Some examples and practical questions are considered as well. Introduction Blackjack (also known as “twenty-one) is the most popular casino table card game in the world. Blackjack is played on a points system that gives numeric values to every card in a single deck of playing cards. The cards are given to a player sequentially until he decides to stop (stand). His score is the sum of the values in hand. 21 is the best score one can achieve in the game, and players should be focusing on getting as close to that number as possible without busting. However, if a player’s cards exceed 21, then he has gone bust the player loses and its bet is immediately taken by the dealer once this happens. The feature of the game we are interested in is the following: the player with the highest total wins as long as it doesn't exceed a given limit number. In the paper we consider similar problem. Let X1, X2,..., XN be a finite sequence of independent nonnegative random variables. A player observes sequentially the values and decides whether to stop or to continue. If he decides to stop at the moment k he gains a value ) ( 1 ∑ = k i i X W , where + + → R R W : is a given nonnegative function. We assume that the function W is positive and increasing on the interval (0, T] and is equal to zero for arguments greater than T. It means that the player obtains positive payoff which is the greater, the greater the sum ∑ = k i i X 1 is, unless the sum exceeds a positive number T a limit given in the problem (in blackjack game T = 21). If so, then the player gains 0. Our aim is to find a stopping rule which maximizes the expected payoff for a player. Such a problem can be a model for various real world situations which can be observed in economics, finance, politics and social life. One specific problem of the type will be considered in detail in the sequel.