game of cards

BETWEENIES CARD GAME

Introduction
The exact rules vary, the main concept is that the player is dealt two cards and bets on whether the value of a third card dealt about to be dealt will be between the values of the two previously dealt cards. In this work we calculate the probabilities associated with the game, namely, the probability that a given hand is dealt and the probability of winning given the hand dealt, and i suggest a betting strategy based on Kelly’s criterion (KC).
HOW TO PLAY
Each player is first dealt 2 cards face down. Taking it in turns, players turns their cards over. Each player must bet a certain number of fingers of drink on whether they think the next card turned over from the pack will have a value in-between the two cards they have in front of them (aces are low in this game).
If the card turned over is equal to either of the two cards then the player must drink double the stake bet. This is called “Hitting the posts”.
Cards are left turned up on the table until everyone has had one turn each. The whole deck is then re-shuffled.
Strategy
To maximize your bets, bet when there are at least 8 cards between you two. For example, 2 & J…3 & Q….4 & K…5 & A.
If your cards are closer together, pass or bet zero.
The Rules
The game is played in rounds and with a standard deck of 52 cards. The cards from 2 to 10 are associated with their face values, while Jack, Queen, and King with the values 11, 12, and 13, respectively. Aces can be associated with either 1 or 14, subject to further rules stated below. In the beginning of the round, every player contributes a fixed amount of money (henceforth assumed to be equal to 1), known as the “ante,” to the pot in order to play. Subsequently, each player is dealt two cards, one at a time, face up.
Assuming that the first card is an ace, the player must declare it “low” (of value 1) or “high” (of value 14).
Assuming that the second card is an ace, it is automatically declared “high.” After the two cards are dealt to a player, the player bets an amount of money, ranging from 0 to the pot amount, that the third card dealt will be strictly between the two already dealt cards.
If the player wins, the player receives the amount of money bet from the pot; otherwise, the player contributes the amount of money bet to the pot. If a player’s wealth becomes zero, the player quits the game

Kelly’s Criterion
Put, assuming independent trials in a game of chance, it suggests a betting strategy, based on which a player can expect an exponential increase of his wealth. The rate of this increase is, more precisely, equal to the information gain between the two underlying probability distributions of the game: true outcome probabilities and projected outcome probabilities, as suggested by the advertised odds. We demonstrate it here by repeating the example in Kelly’s original paper , which is worth working out in full detail, as the original exposition is rather terse, omitting, however, the extra information provided by the “private wire” considered therein, which is an additional complication not directly relevant to our discussion.

Description of a General Game and the Classical Approach
Consider a random variable X with n possible mutually exclusive outcomes xi with probabilities pi, i = 1,…..,n. Assume further that the odds placed on x(i) are t(i) ÷ 1, whereby it is meant that, if a player places a bet bi on xi, and xi is indeed the outcome, he will receive a wealth of t(i)b(i) back including the original bet hence the use of the new symbol “÷” instead of the previous “:”); otherwise the bet is lost. How should the various bi be determined? One possible approach is to maximize the expected wealth: assuming that the player’s initial total wealth is w, the total wealth after betting and assuming outcome xi)

Assuming p(i)t(i) ≤ 1 for some i, for any bet with bi = b > 0, the new bet with all bj , j/= i left
unchanged and bi = b −epsilon > 0, e> 0 is at least as profitable; hence the optimal bet can be taken to have bi = 0. Focusing on an i such that p t > 1, for any bet with bi = b < w, the bet with all b j , j/_ i left unchanged and bi _ b _ _ 0 is at least as profitable

This strategy is, however, highly risky, as, with probability 1−pi∗ the bet is lost and the player is ruined. Furthermore, the probability that the player is not ruined after m rounds of the game is _pi∗_m; assuming that pi∗ 0 may actually be advantageous.

Simulations of card Gameplay and Short-Term Considerations to explain reasonable.
The mean wealth after one round given a hand of spread s ≥ 7 and initial wealth w is

Fraction eventually decreases towards 29151/28561 ≈ 1.02066 as n increases.

Figure A sequence of rounds of Betweenies (party version) leading to ruin (a) and a large wealth (b), with a starting wealth of w = 50. Note the large lost bet in the final rounds of (b).
Figure: shows two actual games of the party version, one of which resulted in ruin and one in a very large wealth over 1000 rounds (which could have been even larger had it not been for a large lost bet in the final rounds.)

We assume that the player played solo with an infinite pot. The player’s initial wealth was taken to be 50. The simulation on the left is typical of games resulting in ruin and gives some insight into the mechanisms that cause ruin. More specifically, we see that the player started by doing well, and three successful bets in rounds 61, 65, and 71 boosted his wealth to 386, an almosteight-fold increase. In round 77, however, an unsuccessful large bet reduced his wealth to 152, followed by two more unsuccessful sizeable bets in rounds 78 and 79, finally reducing his wealth to a meager 74. In short, ruin was partly caused by large unsuccessful bets, namely, localized large losses. Furthermore, long periods where the wealth slope is equal to −1 are clearly visible in the figure, and they correspond to the periods where the player places no bets due to unfavorable hands, but pays the ante in the beginning of each round
These considerations bring us back to the criticism of KC. Is it possible to reduce uncertainty of wealth growth conceding a decrease of the expected final wealth? Let us consider the following two sets of simulations, each consisting of 10,000 games of (at most) 100 rounds each, and where the initial wealth is always w = 50. In the first set bets are placed according to KC: the probability of ruin is 52.13%, the mean wealth is 525.9, and the probability of no gain (end wealth is less than or equal to the initial wealth) is 60.55%, but the median wealth assuming no loss is 392 and the 5% and 95% quantiles lie at 67 and 4,699, respectively. In the second set the bets placed are double the bets suggested by KC, but the player has moderate expectations and withdraws at any point his wealth exceeds 80: the probability of ruin is 35.62%, the mean wealth is 64.34, the probability of no gain is 35.72%, but the median wealth assuming no loss is 94 and the 5% and 95% quantiles lie at 82 and 138, respectively.
Violating KC and hyper betting at low wealth allows the player to leave faster the low wealth zone where ruin is likely to occur due to gradual loss of wealth. Alas, it makes ruin due to sudden loss of wealth much more likely when wealth is high and bets are high… except that now the player pulls out of the game before such high values of wealth are reached! This strategy then outperforms KC in short term, leading to smaller probability of ruin and loss: gain is now small but almost certain, and its effective value is constrained in a much narrower range than before, so that the variation in the expected wealth is smaller.If the player gets greedy and over bets without withdrawing when wealth exceeds 80, the probability of ruin increases to 77.83%.

References
Article on In Between, http://www.onlinepoker.net/Card-Games/In-Between.php

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