Related papers: Integer Compositions Applied to the Probability An…
We describe the probability theory behind a casino game, blackjack, and the procedure to compute the optimal strategy for a deck of arbitrary cards and player's expected win given that he follows the optimal strategy. The exact blackjack…
Blackjack or "21" is a popular card-based game of chance and skill. The objective of the game is to win by obtaining a hand total higher than the dealer's without exceeding 21. The ideal blackjack strategy will maximize financial return in…
Chances of a gambler are always lower than chances of a casino in the case of an ideal, mathematically perfect roulette, if the capital of the gambler is limited and the minimum and maximum allowed bets are limited by the casino. However, a…
Compositional Game Theory is a new, recently introduced model of economic games based upon the computer science idea of compositionality. In it, complex and irregular games can be built up from smaller and simpler games, and the equilibria…
In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function…
In this paper, we calculate a basic strategy for several variations of a simplified version of Blackjack. In short, for these variants the player has only the two options of hit or stand, and they may only make either decision once. Other…
Automatic chess problem or puzzle composition typically involves generating and testing various different positions, sometimes using particular piece sets. Once a position has been generated, it is then usually tested for positional…
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number…
We study the compositions of an integer n whose part sizes do not exceed a fixed integer k. We use the methods of analytic combinatorics to obtain precise asymptotic formulas for the number of such compositions, the total number of parts…
Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of…
Computer calculations for most exact expected values in blackjack have been available since the 1960's, but exact results for pair splitting and resplitting have previously been too computer intensive. This paper describes a new algorithm…
For a balanced cardcounting system we study the random variable of the true count after a number of cards are removed from the remaining deck and we prove a close formula for its standard deviation. As expected, the formula shows that the…
We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we…
Consider a gambling game in which we are allowed to repeatedly bet a portion of our bankroll at favorable odds. We investigate the question of how to minimize the expected number of rounds needed to increase our bankroll to a given target…
We introduce a variant of the classic poker game blackjack -- the continuous blackjack. We study the Nash Equilibrium as well as the case where players deviate from it. We then pivot to the study of a large class of adaptive strategies and…
Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference…
We give an extension of de Finetti's concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to…
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(\l_1, \l_2, \dots, \l_k) \in \Z_{> 0}^k$ such that $n = \l_1 + \l_2 + \dots + \l_k$. Our goal is to enumerate those compositions whose parts $\l_1, \l_2, \dots, \l_k$ avoid a…
We consider how to make probability forecasts of binary labels. Our main mathematical result is that for any continuous gambling strategy used for detecting disagreement between the forecasts and the actual labels, there exists a…
The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a…