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A Mathematical Analysis of the Card Game of Betweenies through Kelly's Criterion

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Postby Grolar В» 16.12.2019

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In the game of Betweenies, the player is dealt two cards out of a deck and bets on the probability that the third card to be dealt will have a numerical value in between the values of the first two cards. In this work, we present the exact rules of the two main versions of the game, and we study the optimal betting strategies.

After discussing the shortcomings of the direct approach, we introduce an information-theoretic technique, Kelly's criterion, which basically maximizes the expected log-return of the bet: we offer an overview, discuss feasibility issues, and analyze the strategies it suggests.

We also provide some gameplay simulations. Though 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 we suggest a betting strategy based on Kelly's criterion KC. KC is famous for suggesting an optimal betting strategy which, at the same time, eliminates the possibility of the gambler getting ruined [ 5 , 6 ]. As we will throught see below, however, the set of rules of this particular game does not fall within the scope of the general case considered in [ 5 ], in view of the fact that the player needs to contribute an amount of money in the beginning of each round for the right to play in this round.

This feature can actually lead to ruin even if this strategy is followed. From this point of view, Betweenies make a particularly interesting case study for KC. 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. Subsequently, each player is dealt two cards, one at a time, face up.

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. If at any point during the round the pot becomes empty, the round ends and a new round begins. There are several possible variations to the rules above. We will refer to the game with all these options turned off and on as the party version and the casino version, respectively.

The reason is that, in a party game between friends, ante contributions are necessary to get the game going, while a casino game is in general more gamblingoriented and bets are covered by the casino's funds. Further variations are mentioned in the literature: for example, when the third card is equal to either of the first two, then the player not only loses the bet into the pot, but has to contribute an extra amount equal to the bet into the pot [ 1 ].

As the possibilities are practically endless, we will restrict our study to the two versions mentioned above. Kelly's criterion KC [ 5 , 7 ] is a reinterpretation of the concept of mutual information, which is the core subject of Information Theory, in the context of games of chance and betting.

Simply 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.

Consider a random variable with possible mutually exclusive outcomes with probabilities ,. How should the various be determined? One possible approach is to maximize the expected wealth: assuming that the player's initial total wealth is , the total wealth after betting and assuming outcome is clearly ; hence the expected wealth after the game is We observe that Assuming for some , for any bet with , the new bet with all , left unchanged and , is at least as profitable; hence the optimal bet can be taken to have.

Focusing on an such that , for any bet with , the bet with all , left unchanged and , is at least as profitable; hence the optimal bet can be taken to have. Furthermore, assuming that and , decreasing , and increasing by such that they both remain between and is again at least as profitable.

We conclude that, assuming that there exists an such that , the optimal bet is to set , , , where is taken to be the smallest of all such that ,. This strategy is, however, highly risky, as, with probability the bet is lost and the player is ruined. Furthermore, the probability that the player is not ruined after rounds of the game is ; assuming that otherwise there is really no element of randomness in the game , , so eventually the player gets certainly ruined.

Note that, without loss of generality, we may consider that This is because, even if the player wishes to save an amount of money , he may equivalently bet on outcome.

Indeed, 3. When , on the other hand, this betting scheme leads to certain loss unfair odds ; hence it may make sense for the player to actually save part of his wealth and bet the rest.

KC suggests maximizing the exponential growth factor of the wealth, or, equivalently, the log-return of the game: For the discussion that follows, we assume 3. In that case, In Information Theory, this quantity is known as the Kullback-Leibler distance or information gain or relative entropy [ 8 ] between the probability distribution and the function in this order , , where and.

We distinguish the following two cases. As this event has zero -probability to occur, it does not affect the player's betting strategy note that, by convention, terms in the sum defining corresponding to are taken to be 0, which equals the limit value as [ 8 ]. What happens when? In this case, neither is nor can it be extended into a probability distribution; hence is not guaranteed to be positive, and, even if it is, this strategy may be suboptimal.

An attempt to use Lagrange multipliers directly as above, even allowing for the possibility that a part of the initial wealth is saved, leads to , hence to no solution, assuming that all ,. We therefore need to consider the possibility that zero bets get placed on some of the possible outcomes. To sum up, we need to maximize The fact that the log function is concave over the maximization region guarantees convergence to a global maximum.

We observe, though, that some of the constraints are inequalities rather than equalities, and dealing with such constraints requires the use of a generalization of the Lagrangian method of multipliers, known as the Karush-Kuhn-Tucker KKT equations [ 9 ]: we form the functional which we now attempt to maximize. A stipulation of KKT theory is that the coefficients corresponding to inequality constraints must carry the sign of the inequality, and that, if the inequality is strictly satisfied at the point of optimality, the coefficient must be zero: specifically, , and either or else , ; the case for and is similar, but, as we established above, the optimal bet necessarily has ; hence.

Taking the partial derivatives yields We now define , whence it follows that. Setting and we obtain Hence, These conditions are enough to determine unambiguously. To begin with, assume, without loss of generality, that the outcomes are so ordered that is a decreasing function of : then, for some stands for. Now define, and note that.

Assuming that , it follows that and that no bets are placed. In any case, and , where is the smallest such that. Note that as noted in [ 5 ] , compared to a classical player who avoids betting on outcomes for which the odds are unfavorable, namely, for which , a player following KC does bet on such outcomes, as long as.

As a historical note, let us mention that the KKT theory, formulated in , predates KC, published in Unfortunately, 3. Though KC suggests a betting strategy that is both optimal and avoids gambler's ruin, in many practical games the rules prohibit its application, and some approximation is required.

To demonstrate the main issues, let us continue with the example of the random game of possible mutually exclusive outcomes we have been studying in this section: the optimal betting strategy suggested by KC regards the version of the game, henceforth labeled , where the player has the right to place simultaneous bets, one on each possible outcome.

Alternatively, however, a player may be restricted by the rules to place a possibly negative bet on one outcome of his choice only, negative bets signifying bets on the complementary outcome; we label this version.

Finally, a player may be restricted by the rules to place a nonnegative bet on one outcome only predetermined by the rules; we label this version.

As a concrete example, consider the game of rolling two fair dice and betting on the sum of their outcomes, which ranges from 2 to Our analysis above concerned , where the player is allowed to place 11 simultaneous bets, one on each possible outcome of the sum. Under , the player would be restricted into placing a bet on only one outcome of his choice; for example, that the sum will or will not be 8.

Finally, under , the player would be restricted into placing a bet on the outcome, for example, that the sum will be 8, assuming that the rules restricted betting to this particular value of the sum and no other. When , and under simple returns, and are essentially the same, except for the fact that in negative bets are allowed; note, indeed, that a negative bet for an outcome translates into a positive bet for its complement.

In practice, hardly any game is or : imagine, for example, a player playing Blackjack and betting on the outcome that the dealer has a higher hand than him! As another example, in the party version of the game of Betweenies, given the player's hand, the probabilities that the third card dealt will or will not fall strictly between the cards of the hand can be computed, and they clearly add up to 1; therefore, this game is clearly an instance of the general game described in Section 3.

Applying KC, however, presupposes a game, namely, that the player is able to place bets simultaneously on either possible outcome and that the third card either will or will not lie strictly between the two cards of the hand, respectively, and game rules allow betting only on the former event, not on the latter.

KC can still be applied in a modified form, allowing only part of the player's wealth to be placed in bets while saving the rest, but the feasible betting strategy so obtained which is the main object of this work and is studied in detail in the next sections will be suboptimal.

Note that this case, where betting is restricted by the rules of the game to certain outcomes only, should not be confused with the unfair odds case in Section 3. In that section, the player was still allowed to bet on all possible outcomes.

In particular, the analysis carried out in that section is not relevant for the scenario just described. The most important feature of KC to keep in mind is that the betting strategy it proposes maximizes the player's wealth in the long run , but it normally achieves this through highly volatile short-term outcomes [ 6 ]. Given, however, the finite span of human life and human nature more generally, many might find it preferable to trade the optimal but highly volatile eventual growth of wealth achieved by KC for a suboptimal growth, as long as it is also less volatile in the short or medium term.

The probabilistic analysis of Betweenies naturally breaks down in two stages: first, the probabilities that a player be dealt any specific hand of two cards must be determined; then, the probability of victory given any dealt hand of two cards must be determined.

Let denote the event that the two cards dealt have value and , ,. We set. Note that, unless or , the order in which the two cards are dealt is irrelevant for determining ; furthermore, the order is always irrelevant for determining the conditional winning probability given. Assuming then that and , as the first card can be chosen in 4 ways, as can the second, while the totality of possible pair choices is , the order being unimportant.

Assuming now that and , as the first card can be chosen in 4 ways out of 52 possible cards and the second in 3 ways out of 51 possible cards. When aces are present, things get complicated by the low-high option. Let be the probability that the player declares the first ace card if such a card be indeed dealt to be high. Then, for , as the first ace can be chosen in 4 ways out of 52 possible cards , declared low with probability , and the second non-ace in 4 ways out of 51 possible cards.

Similarly, is the probability that two aces are dealt and that the first is declared low: Furthermore, is the probability that two aces are dealt and that the first is declared high: while. Finally, , , is the result of two possible and mutually exclusive scenarios: either the first card dealt is an ace declared high, or the second card is an ace.

It follows that. Let denote the event of victory. We set to be the probability of victory given a certain hand. We observe outright that , as there is no card strictly in between the dealt cards in these two cases.

In all other cases, there are exactly cards in between two cards of value and , ; hence where and , ranging from 0 to 12 inclusive, is set to be the spread of the hand. Note that, by redefining , in 4. There is clearly no point in betting when.

How often does this occur? Letting denote the probability of , it follows that. We see that is minimized for.

This is to be expected: assuming that a player is dealt an ace in the first card, there is no point, in the absence of further information, in declaring it high, as then the player forfeits the possibility of obtaining the strongest possible hand if a second ace is dealt, without gaining any advantage. We will henceforth assume that , in which case. Hence, in approximately one turn out of five the player has no chance to win.

Note that we do not imply that the player should invariably use , but rather just in the general scenario studied here.

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Postby Dugul В» 16.12.2019

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Postby Zolokus В» 16.12.2019

Finally, a player may be restricted by continue reading rules to place a nonnegative bet on one outcome only predetermined by the rules; we label this version. It follows that. The expected log-return of the game is then where, and we seek to gamss this quantity over.

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Postby Jum В» 16.12.2019

Note that, parking orthe order in which the two cards are dealt is irrelevant for determining ; furthermore, the order is always irrelevant for determining the conditional winning probability given. In order to consider how the mean wealth varies over different rounds, we let games the player's wealth at the end of the th round with probability 1 ; 5. As a historical note, let us mention that the Parking theory, formulated ingames KC, gambling near me doctorated in Furthermore, long periods where the wealth slope is equal to 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 gambllng of each round.

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Postby Faegrel В» 16.12.2019

Unfortunately, 3. On the other hand, assuming thatwhich is the smallest value for which ruin is not certain, the probability to avoid ruin is only about. In practice, more info any game is or : imagine, for example, a player playing Blackjack and betting on the outcome that the dealer has condrete higher hand than him!

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Postby Dular В» 16.12.2019

The cards from gamew to 10 are associated with their face values, while Jack, Queen, and King with games values 11, 12, and 13, respectively. Each one match includes chump bets, though. The simulation anime fury 2017 the left is typical of games resulting in ruin and gives concrete insight gambling the mechanisms that cause ruin. Certain Loss and Concete Probabilities There is clearly no point in betting when. Card other forms of poker, each hand has a different additional payout associated with it.

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Postby Mole В» 16.12.2019

Taking the partial derivatives yields We now definewhence it follows that. To sum up, we observe that, despite the many superficial differences between the casino near doctorated gambling me parking party versions, the betting strategy for both, under KC, is exactly the same. Figure 2 shows two actual games of the party version, as described in Section 4one of which resulted in ruin and one in a very large wealth over rounds games could have been even comcrete had it not been for a large lost bet in the final rounds.

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Postby Dijora В» 16.12.2019

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Postby Tejar В» 16.12.2019

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Postby Kazijind В» 16.12.2019

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Postby Julkis В» 16.12.2019

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The Rules The game is played in dard and with a standard deck of 52 cards. Underthe player would be restricted into placing a bet on only one outcome of his choice; for example, that the sum will or will not be 8. What should the player do? For example, simulations suggest that for.

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