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.

The card game of “(In-)Betweenies” or “In-Between” [

KC is famous for suggesting an optimal betting strategy which, at the same time, eliminates the possibility of the gambler getting ruined [

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 (which we designate as event

There are several possible variations to the rules above.

Multiple decks of cards may be used.

Aces may always carry their face value 1.

Assuming that the three dealt cards are all equal, a payout of 11 : 1 may be paid to the player (i.e, the player gets the original bet back plus 11 times the bet).

Assuming that the two dealt cards have consecutive values, the bet is returned to the player at no loss.

The bet payout may vary depending on how many cards apart the two dealt cards are: 5 : 1 to for one card apart, 4 : 1 for 2 cards apart, 2 : 1 for three cards apart, and 1 : 1 for 4 or more cards apart.

The unit contribution to the pot may be nonapplicable.

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 [

Kelly's criterion (KC) [

Consider a random variable

One possible approach is to maximize the expected wealth: assuming that the player's initial total wealth is

This strategy is, however, highly risky, as, with probability

Note that, without loss of generality, we may consider that

KC suggests maximizing the exponential growth factor of the wealth, or, equivalently, the log-return of the game:

What happens when

Setting

As a historical note, let us mention that the KKT theory, formulated in 1951, predates KC, published in 1956. Unfortunately, (

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

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 12. Our analysis above concerned

When

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

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

The most important feature of KC to keep in mind is that the betting strategy it proposes maximizes the player's wealth

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

Assuming then that

Assuming now that

When aces are present, things get complicated by the low-high option. Let

Let

There is clearly no point in betting when

We see that

Note that we do not imply that the player should invariably use

We may similarly consider

We will now determine the optimal betting strategy for a given hand using KC, as described in Section

More specifically, at the end of the round, the player's wealth is

Note that

If the restriction imposed by the available pot amount

If

In particular, the formula indicates that

Assuming that the player places a bet

Alternatively, we may consider the mean wealth

In order to consider how the mean wealth varies over different rounds, we let

We mentioned above that if the player's original wealth is

Suppose that the player's intention is to play successive rounds of the game in order to eventually accumulate wealth (unboundedly) greater than the original wealth: the relevant quantity to consider is the probability of ruin

In order to both simplify analysis and make it more realistic at the same time, we will henceforth assume that wealth and bets can only be multiples of 1 (namely, integers), and that bets

To determine

For all other values of

In order to compute an approximation of

(a)

Figure

Our results indicate that

From now on, we consider all options mentioned in the list of Section

Assuming

We define the spread of the hand

The probability

Unlike the party version, there is no possibility of certain loss here: when

Once more, following the discussion of Section

In general, then, assuming that

To sum up, we observe that, despite the many superficial differences between the casino and the party versions, the betting strategy for both, under KC, is exactly the same. In particular, the increased odds considered for the rarer cases

Since the betting strategy is identical in the party and casino versions, the analysis of Section

As

Figure

A sequence of rounds of Betweenies (party version) leading to ruin (a) and a large wealth (b), with a starting wealth of

Furthermore, long periods where the wealth slope is equal to

These considerations bring us back to the criticism of KC in Section

Violating KC and hyperbetting 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. Of course, if the player gets greedy and overbets without withdrawing when wealth exceeds 80, the probability of ruin increases to 77.83%.

We described two variations (called the party and the casino version) of the card game of Betweenies (also known by several other names), in which the player bets on whether the value of the card he is about to receive will lie between the two cards he has already been dealt. After a brief introduction to Kelly's criterion (KC), a method to determine the optimal betting scheme in a game of chance based on Information Theory, in a sense that the logarithm of the ratio of the player's wealth before and after the game is maximized, we applied it to both described versions of the game, essentially coming up with the same betting strategy in both cases. In the party version, where every player is required to contribute a fixed amount of money to the pot in the beginning of each round, there is an initial wealth-dependent probability of ruin, which we studied in two ways: by simulation and by solving the equations numerically. The latter method suggested that the probability of ruin is asymptotically inversely proportional to the initial wealth, and we verified this by direct substitution in the equations.

We finally provided some simulations of gameplay of Betweenies (one leading to ruin and another to a large wealth), assuming that players follow the strategy laid down by KC, and we demonstrated how alternative strategies can perform better in short term, by reducing the probability of ruin and increasing the probability of gain, albeit reducing the expected gain. We also showed that, in the long term, such strategies increase the probability of ruin.