Summer 2005 REU Program: The Mathematics of Games of Chance

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Spencer Bagley: Chemin de Fer and Sampling Without Replacement
The optimal strategy at chemin de fer has been known for some time, but all analyses that have been presented up to this point have used the simplifying assumption of sampling with replacement. Without this simplification, chemin de fer is a $2^5 \times 2^{484}$ matrix game and strategies are composition-dependent. In my paper, I present a solution to the game in the case of sampling without replacement.

Joseph Houpt: Unbalanced Card Counting
This paper analyzes one type of unbalanced card counting systems that are used in blackjack. These unbalanced counting systems start with a negative count and then have a set of cards counted as one more than in a normal High-Low count, so as to offset that original negative count. To get an idea for the relationship between the true count and unbalance count, the more simple game of Evens and Odds is used. The formulas are derived for the expected value of the player's advantage given the unbalanced count. A computer-generated table of these values is given at the end.

Polina Milyavskaya: Composition Dependent Strategy at Blackjack: Hit or Stand on Hard Sixteen
In my paper, I analyze the game Blackjack with the purpose to identify optimal strategy for players. I assume following rules of the game: the game is played with a single deck of cards; Player and Dealer are dealt two cards. If one of them has a ``natural'' (2-card 21) --- the game is over; otherwise, player is dealt cards until the total is greater than or equals to 17. Player stands on 17 or higher, Dealer stands on soft 17. Player is allowed to double down on the first two cards, and is not allowed to double down after splitting. Surrender is not allowed. Player is allowed to split aces once and to split non-ace pairs up to 3 times. In addition to these rules, following assumptions are made: Dealer does not have ``natural,'' and player makes one-unit bets. Using conditional probabilities and expectations, I find an optimal composition-dependent strategy for the player in case when he/she has hard 16 on hand.

Matthew Reimherr: Realtionship between the Gambler's Ruin and Run Formulas
This paper will attempt to accomplish two goals. The first is to establish the similarities between the Run formula and the GamblerŐs Ruin formula and to give at least a simplistic ex-planation for the relationship. The second goal is to examine the gamblerŐs ruin formula and formulate some basic methods of con-structing the formula symbolically to see if we can further understand the connection between the two equations.

Sithparran Vanniasegaram: Le Her with s Suits and d Denominations
In my paper, I analyze the two player card game Le Her played with a deck consisting of arbitrary numbers of suits and denominations instead of the standard 4 suits and 13 denominations. Le Her (with a standard deck) is played in the following way: Player 1 and Player 2 are each given a card from the deck with neither of them knowing what the other has. If Player 2 has a king, the game immediately ends with victory for Player 2. Otherwise, Player 1 has the option of exchanging his card with Player 2. Next, Player 2 has the option of exchanging his card with the top card of the deck. If he does decide to exchange his card and the top card is a king, the exchange is void and Player 2 must keep his card. If Player 2 has the higher ranking card or has a card of the same rank as Player 1, he wins. Otherwise, Player 1 wins. Using the convexity of the payoff matrix, I find optimal strategies for each player as well as the value of the game. (This paper is not available online since it will be submitted for publication.)
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