2021

These authors contributed equally to this work.

1]Zhongtai Securities Institute for Financial Studies, Shandong University,
Street, Jinan, 250100, Shandong, China 1]National Center for Applied
Mathematics, Shandong, China

# Long bet will lose: demystifying seemingly fair gambling via two-armed
Futurity bandit

Zengjing Chen<EMAIL_ADDRESS>Huaijin Liang<EMAIL_ADDRESS>Wei Wang
<EMAIL_ADDRESS>Xiaodong Yan<EMAIL_ADDRESS>[  [

###### Abstract

No matter how much some gamblers occasionally win, as long as they continue to
gamble, sooner or later they will loss more to the casino, which is the so-
called “long bet will lose”. Our results demonstrate the counter-intuitive
phenomena that gamblers involve “long bet will lose” but casinos always
advertise their unprofitable circumstances. Here we expose the law of
inevitability behind “long bet will lose” by theoretically and experimentally
demystifying the profitable mystery behind casinos under two-armed antique
Mills Futurity slot machineEthier2010 . The main results straightforwardly
elucidate that all casino projects are seemingly a fair gamble but essentially
unfair, i.e., the casino’s win rate is greater than 50%. We anticipate our
assay to be a starting point for studying the fairness of more sophisticated
multi-armed Futurity bandits based on mathematical tool. In application,
fairness study of the Futurity bandits not only exposes the fraud of casinos
for gamblers but also discloses the discount marketing, bundled sales or other
induced consumption tactics.

###### keywords:

Fairness, Multi-armed bandit, Futurity slot machine

## Introduction

The history of human gambling is estimated to be about the same as that of
civilization. Tracing back to the history of human gambling, people had the
habit of “taking chances” as early as the late Paleolithic Age, such as
divination was widely used to judge the bad and the good, including major
wars, through divination in prehistoric civilization of China. The appearance
of casinos has boosted the prosperity of the gambling industry, which has been
enduring. Various forms of gambling have been derived, such as horse racing
and lottery in Hong Kong, baccarat and dice in Macao, baccarat, slot machine,
roulette, blackjack, etc., and the Philippine government supports and
encourages the development of the gambling industry because they found that
the gambling industry stimulates domestic economic development under the
situation of global economic downturn, which essentially elucidates the
profitability of gambling industry but a secret that the casino never said, no
matter how they are changed, causing a situation of the so-called “long bet
will lose” for the players.

Gambling activity attracts amounts of players by the fairness illusion that
the casino is unprofitable, then their enthusiasm is enlightened by the so-
called honest advertisement, even they indulge in fantasy of earning more
money. The rigged casino appears to be especially absorbing to players with
gambling-related pathologyDixonmike  who becomes completely immersed in the
playing, then depression symptomatology, gambling expectancy and dark flow
ratings are also emergingDixon . Such honest advertisement about the fairness
can be attributed to the casino loyalty programmes Hollingshead  offering the
same rewards to gamblers for their equivalent input. The aim of a loyalty
programme is to increase attitudinal and behavioural loyalty. Attitudinal
loyalty refers to the degree that people trust and are satisfied with the
casino. It also involves a sense of identification with the casino brand.
Behavioural loyalty refers to people’s actual behaviours that show their
loyalty. For example, visiting the casino again and again to gamble. However,
all seemingly fair casino projects are essentially unfair, because it has
never been heard that the casinos are defeated by players, until the
occurrence of Kelly formulaKelly  with long reputation in “Las Vegas” and
“Wall Street” delivers the optimal proportion that should be wagered in each
periodThorp  in a repeated gamble of Blackjack (21 points) or repeated
investment with more than 50% winning rate.

Enlightened by the boom of gambling industry nevertheless the attitudinal
loyalty of the players on the casino, this article explores the potentially
profound mystery with a multi-armed Futurity bandit mathematically. Because
multi-armed bandit (MAB)feller ; Sut2018 , as a popular entertainment tool,
has been meticulously designed by the casinos for looking seemingly fair and
attracting the gamblersEthier2010  . MAB has also been widely studied
theoretically to analyze amounts of complex decision problemschenepstein ; ACF
; Chen-Feng-Zhang  in science, society, economy, management and other areas
and plays a central role in reinforcement learningMorris ; Narisawa ; Aga1995
; Collins  . In detail, this work introduces the two-armed Futurity bandit to
demystify that such all-encompassing absorption of the gamblers on the casino
can be attributed to the seemingly fair one-armed slot machine, because it can
be unprofitable for casino by the futurity reward design, such as the futurity
slot machine, when the current number of consecutive failures reaches $J$, all
$J$ coins will be returned at this time. However, two-armed slot machines
break the fairness and shows Parrondo’s paradox phenomenonParrondo  that the
casino becomes profitable when players alternate arms in certain random or
nonrandom ways as well as being advertised that two arms are fair.

People just learned from experience that “long bet will lose”, but the law of
inevitability behind “long bet will lose” is determined by the “law of large
numbers” in probability theory and the mystery behind casinos will be unknown
without the probability. This article adopts probability tool the law of large
numbers and two-armed antique Mills Futurity slot machine, designed by the
Chicago Mills Novelty CompanyEthier2010 . The Futurity slot machine applies
fair compensation after consecutive failure to reveal the seemingly fair fraud
of gambling, which alludes to “fairness illusion”. The compensation rule
satisfies that two coins will be returned to the player when the consecutive
failures reach two correspondingly, and the fraud can be stated by the
followings. If we play one-armed Futurity bandit, the smart casinos always
honestly advertise that the deviced slot machines are unprofitable through
advocating the fairness, which implies that the casino isn’t capable of making
money in the long run through using one-armed Futurity bandit for gambling.
This trick designed on one-armed Futurity bandit increases the reputation of
casino in terms of the fairness, whilst artifacts occur for two-armed Futurity
bandit by alternatively playing the left and right arms, which demonstrates
that the casino can be always profitable under the same rule that if
consecutive failures occur two times, the player will obtain the common return
with two coins. This results are corresponds to the conjecture proposed by S.
N. Ethier and Jiyeon Lee’sEthier2010  showing that this two-armed Futurity
bandit obeys Parrondo’s paradox when it repeatedly executes any non-random
mixed strategy $D$ and $J=2$. Specifically, a non-random strategy decided by
the player before playing including two arms, such as $D=ABB$, and then
repeats the game according to the strategy $D$, returning two coins for every
two consecutive failures during the game. This article mathematically proves
this conjecture and uses experiment to verify the theoretical results.

The main contributions of this article include the followings:

  * •

This the first development of nonrandom two-armed Futurity bandit to reveal
the profitable law of seemingly fair design of gambling slot machine,
enlightening inspiration source of the gambling boom and the miserable lot
“long bet will loss” for the players.

  * •

This work proposes a novel probability framework for a nonrandom sequences of
arm-playing rule and provides the explicit expressions of profitable return
for casinos, which elucidates the changing law of the return along with the
considered parameters.

  * •

This assay would be a starting point for more sophisticated practical studies
by using two-armed Futurity bandit, such as disclosing the discount marketing,
bundled sales or other induced consumption tactics.

## Results

Model. As a casino project, the Futurity slot machine was designed by the
Chicago Mills Novelty Company. A player consumes 1 coin per coup and there are
two screens on the slot machine. The pointer in one screen records the current
number of consecutive failures. When the number reaches 10 (it can also be
other values set by the casino), all 10 coins will be returned at this time.
This payoff is called the Futurity award. Another screen will display the
current mode. Its internal structure is a periodic cam with several fixed
modes. Different modes will have different winning conditions and rewards for
the current game. Each time the player plays a game, the cam rotates to the
next pattern. The player chooses to pull the arm once before playing the game,
and each arm has a separate mode cam that is different from each other, which
means that each arm has an independent payoff distribution. For the Futurity
award, regardless of the order in which the player plays the slot machine, the
current number of consecutive failures will be recorded, and when the pointer
reaches the value $J(J\geq 2)$ set by the casino, the $J$ coins will be
returned to the player. The casino owner honestly advertises that all arms on
his multiple-armed machine are “fair” in the sense that either of the arms has
a 50 $\%$ chance of making the player profitable. The gambler is allowed to
play either arm on every play in some deterministic order or at random. An
important question would be whether such a casino still obeys the rule that
“long bet will lose”, how much profit can casino owner could win?

For simplify, we consider a simple two-armed Futurity bandit, where the two
arms denoted as $A$ and $B$ and each arm has a different i.i.d payoff
sequence. For the convenience of analysis, we can regard it as a Bernoulli
distribution, that is, the probability of winning a game is $p_{A}$ (resp.
$p_{B}$), and $0<p_{A},p_{B}<1$. The player needs to pay 1 coin to the casino
at each game and shakes the two arms alternatively under a pre-determined
playing rule. The casino also brings futurity reward to the player who suffers
consecutive failure in gambling, that is, if the player fails in $J$
consecutive games, the casino will return all the coins spent in the $J$
games. The casino usually advertise to design $J=2$, which is the considered
case in this work and the most attractive for the gamblers. Before the game
starts, the player needs to choose a pre-formulated non-random mixed strategy
$D$, where $D$ contains at least 1 $A$ and 1 $B$, such as $D=ABB$ meaning that
the players will proceed to pull the arms $A$, $B$ and $B$ repeatedly and
indefinitely. This work consider a “fairness” design for one-armed Futurity
bandit by adjusting the payoff distribution of each arm, where the reward is
assumed to be $(3-2p)/(2-p)$ under the wining probability $p=p_{A}$ and
$p_{B}$ for Arms $A$ and $B$ respectively. Once the gambler plays the game
according to the above rules, it seems that the gambler is fairly playing the
game with no loss, but the casino will definitely make a profit in the long
run, which is demystified in the following theorem.

Subtly mathematical deduction can conclude that any non-random mixed strategy
$D$ can be always arranged by the following same asymptotic form
$D(\boldsymbol{a}(h,r,s))$, that is

$D(\boldsymbol{a}(h,r,s))=\underbrace{A...A}_{r_{1}}\underbrace{B...B}_{s_{1}}...\underbrace{A...A}_{r_{k}}\underbrace{B...B}_{s_{k}}...\underbrace{A...A}_{r_{h}}\underbrace{B...B}_{s_{h}}$
(1)

where $r_{k}>0$, $s_{k}>0$, $\sum\limits_{k=1}^{h}r_{k}=r$,
$\sum\limits_{k=1}^{h}s_{k}=s$ and the vector
$\boldsymbol{a}(h,r,s)=(a_{1},a_{2},...,a_{2h})=(r_{1},s_{1},...,r_{k},s_{k},...,r_{h},s_{h})$,
which is used to describe the structure of any strategy $D$. In order to make
our results more concise, we define the function $b_{i}$ of the vector
$\boldsymbol{a}$ for $1\leq i\leq 4h$ as follows:

  * •

${b_{2j-1}}=(-1)^{a_{2j-1}}(1-p_{A})^{a_{2j-1}}$,
${b_{2j}}=(-1)^{a_{2j}}(1-p_{B})^{a_{2j}}$ for $1\leq j\leq h$.

  * •

${b_{i}}={b_{i-2h}}$ for $2h+1\leq i\leq 4h$.

######  THEOREM 1.

The casino’s asymptotic profit expectation $R$ is $2QS$, where

$Q:=Q(D(\boldsymbol{a}(h,r,s)))=h+\sum\limits_{m=1}^{2h}\sum\limits_{j=1}^{2h-1}(-1)^{j}\prod\limits_{i=m}^{m+j-1}{b_{i}}+h\prod\limits_{i=1}^{2h}{b_{i}}$
$S:=S(r,s,p_{A},p_{B})=\frac{(p_{A}-p_{B})^{2}(1+(-1)^{r+s}(1-p_{A})^{r}(1-p_{B})^{s})}{(r+s)(2-p_{A})^{2}(2-p_{B})^{2}(1-(1-p_{A})^{2r}(1-p_{B})^{2s})}$

Theoretical results elucidate that the casino would be always profitable if
$p_{A}\neq p_{B}$, and the asymptotic profit expectation $R=0$ if and only if
$p_{A}=p_{B}$, which also implies that the discrepancy of the two arms in
terms of the distinct winning probability favors the casino. In detail, the
expression of casino asymptotic profit expectation $R$ consists of three
parts, where the first part is the number 2, representing the settlement rule
$J=2$ of the Futurity bandit reward. The second part function $Q$ denotes the
playing rule across the two arms in the internal structure of strategy $D$.
The last part function $S$ characterizes the profitable change along with the
considered parameters $p_{A},p_{B}$ and the considered playing number $r,s$.

Fig. 2 shows the three-dimensional surface of the casino’s payoff along with
the winning probability $p_{A}$ and $p_{B}$ of Arms $A$ and $B$ respectively
under four different but representative nonrandom strategy $D$. The four sub-
images shows that different nonrandom strategies generate distinct profit
modes with commonly positive profit of casino. Fig. 2(a) implies that
alternatively playing the two arms can be capable of generating more profits
for the casino. Equally playing the two arms guarantees the symmetric form of
the payoffs (see Fig.s 2(a) and (b)).

If the gambler plays the two-armed Futurity bandit according to a random
strategy $C$ with the probability $p_{\gamma}$ playing Arm $A$ and
correspondingly playing the Arm B with probability $1-p_{\gamma}$. The
asymptotic profit expectation $R_{C}$ of the casino has been concluded
Ethier2010  is

$R_{C}=f(p_{\gamma}(1-p_{A})+(1-p_{\gamma})(1-p_{B}))-p_{\gamma}f(1-p_{A})-(1-p_{\gamma})f(1-p_{B}),$

where $f(z)=\frac{2z^{2}}{1+z}$. Since $f(z)$ is a convex function, the casino
will be profitable in the long run with $R_{C}\geq 0$ and $R_{C}=0$ if and
only if $p_{A}=p_{B}$. Fig. 4 shows the payoff performance under random
strategy with probability $p_{\gamma}=\\{0.1,0.3,0.5,0.7,0.9\\}$ selecting Arm
$A$. All the sub-figures display non-negative payoffs under any combination of
$p_{A}$ and $p_{B}$. $p_{\gamma}=0.5$ meaning equal numbers of playing the two
arms generates a asymmetric images of the payoffs, which implies the same
results corresponding to Fig.s 2(a) and (b) in terms of the symmetry.

## Experiment

### The simulated proof of the theoretical results

This section designs Monte Carlo simulations to verify the theoretical results
under four cases corresponding to the non-random mixed strategies $D=AB$,
$D=AABB$, $D=AAABB$, $D=AAAABBBBAAAAAABBB$ with the number of playing
$M=100000$ and the sample mean of profit for casino equals $(M-W-J*C)/M$,
where $W$ represents the total number of win for the gambler and $C$ denotes
the count of consecutive $J$ failures, where $J=2$. The 10000 replications are
conducted in each simulations.

As stated above, the casino can adjust the probability distribution of the two
arms so that it can not only ensure the fairness of the two arms but also
adjust its own profit. In the first simulation of single arm slot, Fig. 1
shows that, the payoff of player or casino always centres around zero across
different probability in interval [0,1] for the single arm, representing the
fairness for each arm. In particular, the payoff of gambler or casino is zero
without uncertainty, while the probability is 0 or 1.

Due to $Q$ in the theoretical is also related to the probability distribution
of the two arms, the impact of $Q$ on profits should also be considered when
adjusting the distribution of two arms. Fig. 2 shows the three-dimensional
image of casino’s payoff across the various winning probability of the two
arms, which vividly elucidates that the casino could select the the winning
probabilities of $A$ and $B$ to gain the largest profit.

Next, we aim to compare the theoretical payoff and its simulation results by
cutting four slice of the three-dimensional surface in Fig. 2. Without loss of
generality, we fixed the probability of Arm B with $p_{B}=0.5$. Fig. 3 shows
the theoretical and simulation’s curves for those four nonrandom strategy,
which obviously implies that the theoretical conclusions are highly consistent
with the simulated results and then verifies Theorem 1.

Figure 1: The payoff of single arm across probability $[0,1]$.

(a)

(b)

(c)

(d)

Figure 2: The payoffs across varying winning probability of Arms $A$ and $B$
under four different nonrandom strategies.

(a)

(b)

(c)

(d)

Figure 3: The sample mean payoff of casino across probability of Arm $A$,
under the fixed probability of Arm $B$, $p_{B}=0.5.$

(a)

(b)

(c)

(d)

(e)

Figure 4: The sample mean payoff of casino under the random strategy with
probability $p_{\gamma}$ selecting Arm $A$.

### A empirical study with a real two-armed futurity slot machine

This part considers the antique Mills Futurity slot machine, designed in 1936
by the Chicago Mills Novelty CompanyEthier2010  . There are two screens on the
slot machine, one screen recording the current number of consecutive failures
and the other screen displays the current mode. In detail, a player consumes 1
coin per coup and when the number of consecutive failures reaches $J=10$, all
10 coins will be returned at this time, but this experiment still considers
$J=2$ extremely attracting the players. Its internal structure is a periodic
cam with Mode E or Mode O, corresponding to the Arm $A$ and Arm $B$ and
essentially both follows multi-points distribution shown in Table 1. Both
modes have distinct winning probabilities and rewards and the two-armed
machines are “fair” in the sense that either of the arms has a 50% chance of
making the player profitable, advertised by the casino. However, the casino
does not allow you to play one arm and this experiment would check the trick
that alternatively using the two “fair” arms will make money for the casino.

In this application, we transform the multipoint distribution of each mode
into two-point distribution. For each mode, we split it into positive payoff
and loss and then the reward of each model are obtained, such that the single
mode is fair. Particularly, we show the casino’s mean profit for the four
strategies in Fig. 5, showing that the sample mean profit of casino converges
to positive value after long play, which still confirms the conclusion that
the casino can always earn money in the long run under the two-armed Futurity
bandit under a equivalent compensation rule that consecutive failures reaching
2.

$reward\backslash probability$ | 0 | 3 | 5 | 10 | 14 | 18 | 150
---|---|---|---|---|---|---|---
Mode E | 0.968 | 0.003 | 0.007 | 0.018 | 0.004 | 0 | 0
Mode O | 0.357 | 0.576 | 0.064 | 0 | 0 | 0.002 | 0.001
Table 1: The multipoint distribution of reward in Mode E and Mode O of the
futurity slot machine.

(a)

(b)

(c)

(d)

Figure 5: The cumulative payoff of casino across the number of coups and
different strategies $D$ with a two-armed antique Mills futurity slot
machineEthier2010  .

## Discussion

This article implies that the root cause of losing games for gamblers rests
with the sufficiently mathematical logic of any gambling equipment and the
extremely sophisticated program design based on probability modeling and
random calculation. This work rigorously demystifies the so-called casino
loyalty programmes designing fair returns with one-armed Futurity bandit to
attract the gamblers and the hoax making a buck from the gamblers continuously
by using two-armed Futurity bandit, which implies the seemingly fair fraud of
two-armed Futurity bandit. Explicit mathematical expression of casino profits
has been concluded in the results and the corresponding images vividly
elucidate the functional change along with the considered parameters, which
also implies that the the casino can be always profitable in the long run.
Experiments are designed to check the correctness of the theoretical results
by simulation and a real two-armed futurity slot machine with more complex
output has also been checked to verify the conclusion. We anticipate the assay
to be a benefit for the players to thoroughly recognize the gambling industry
who usually design the so-called loyalty programme with fairness to attain the
profitable purpose. However, we have no wish for using the theoretical
findings to design the slot machine by the casinos, even by other businesses
such as discount marketing, bundled sales or other induced consumption
tactics. This article may be a starting point for studying the profit
achievement of casinos with more sophisticated multi-armed Futurity bandits
based on the probabilitic tool.

## Online content

Any methods, additional references, nature research reporting summaries,
supplementary information, acknowledgements, peer review information; details
of author contributions and competing interests; and statements of data and
code availability are available at online.

## References

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## Method

Below we will uncover the secrets behind the long-term profitability of this
slot machine step by step. The player will always be interested in the
question of why casinos can claim that every arm is fair, which is what
attracts the player to this game. Although the strategy $D$ provided by the
player has many results, it seems that the player can formulate a strategy
that is beneficial to him in advance, but in fact the casino will treat many
of the player’s strategies as the same strategy. So what is the relationship
between the different strategies? After the casino gets the strategy $D$, it
needs to find out how different strategies affect the casino’s profit. This
slot machine can also be regarded as a confrontation game between the casino
and the player, so how is the asymptotic profit expectation difference between
the casino and the player calculated? After the casino calculates the
asymptotic profit expectation, we can still ingeniously calculate whether the
value of this profit is strictly positive, which explains why the casino will
be profitable in the long run.

### Why casinos can claim every arm of the slot machine is fair?

The source of the casino’s profit is only 1 coin paid by the player before
each game. The player’s profit from the slot machine is divided into two
parts, one part is the payoff $u$ obtained by winning a single game, and the
other part is the refund of the casino obtained by losing two consecutive
games. We can choose one arm for analysis. If the player only play the $A$
arm, $p^{\circ}_{A}$ represents the asymptotic probability of the player
getting the futurity reward per game, then the player’s expectation of the
asymptotic revenue per game is $\mu^{*}_{A}=p_{A}u_{A}+2p^{\circ}_{A}$ where
$u_{A}$ is the payoff the player gets from winning a single game. The casino
can make $\mu^{*}_{A}=1$ by adjusting the parameters $p_{A}$ and $u_{A}$. This
means that the player’s asymptotic payoff expectation per game is equal to the
1 coin payoff received by the casino before each game. S.N Either and Jiyeon
Lee calculated the value of $p^{\circ}_{A}$, that is,
$p^{\circ}_{A}=\frac{p_{A}q^{2}_{A}}{1-q^{2}_{A}}$ where $q_{A}=1-p_{A}$. Then
in order to ensure fairness, the casino needs to ensure
$u_{A}=\frac{3-2p_{A}}{2-p_{A}}$ while modifying the arm’s payoff
distribution. In the same way, the casino can also make the $B$ arm fair, but
the set parameters need to be $p_{A}\neq p_{B}$, $0<p_{A},p_{B}<1$.

### What is the relationship between different non-random mixing strategies?

Fixing the general $r,s>0$, no matter what strategy $D$ the player offers, the
casino only cares how this strategy affects the casino’s asymptotic profit
expectations. We denote $p^{\circ}_{D}$ as the asymptotic probability of
player getting the futurity reward under strategy $D$ per game, and
$p^{D}_{i}$ denotes the winning probability of the $i$th game under strategy
$D$. S.N Either and Jiyeon Lee preliminarily gave the form of $p^{\circ}_{D}$.
On this basis, we preliminarily calculated the casino’s asymptotic profit
expectation and the value of $p^{\circ}_{D}$.

######  LEMMA 1.

The casino’s asymptotic profit expectation

$R=2(p^{\circ}_{D}-\frac{r}{r+s}p^{\circ}_{A}-\frac{s}{r+s}p^{\circ}_{B})$

where

$p^{\circ}_{D}=\frac{1}{r+s}\sum\limits_{k=1}^{r+s}\big{(}\sum\limits_{j=1}^{r+s}p^{D}_{j}\prod\limits_{i=j+1}^{j+2k}q^{D}_{i}\big{)}\frac{1}{1-(q^{r}_{A}q^{s}_{B})^{2}}$

where $q^{D}_{i}=1-p^{D}_{i}$.

From the above lemma, we can observe that different strategies only affect the
value of $p^{\circ}_{D}$, and the value of $p^{\circ}_{D}$ will affect the
casino’s asymptotic profit expectation $R$. From the expression of
$p^{\circ}_{D}$, we observe that the value of $p^{\circ}_{D}$ is equal between
some strategies.

######  LEMMA 2.

Fix the general r, s and l, where $l=1,2,...,r+s$. Consider nonrandom-pattern
strategies $D_{1}$ and $D_{2}$, where $p^{D_{1}}_{i}=p^{D_{2}}_{i+l}$ for any
$i=1,2,...,r+s$. Then $p^{\circ}_{D_{1}}=p^{\circ}_{D_{2}}$

To understand the above lemma more intuitively, we can think of steps $A$ and
$B$ in the strategy as $r$ $A$ balls and $s$ $B$ balls. These balls are placed
one by one on a round table, then the value of $p^{\circ}_{D}$ for the same
arrangement is equal. For example, if $r=4$, $s=2$, the following two
arrangements are the same value of $p^{\circ}$, that is
$p^{\circ}_{AABABA}=p^{\circ}_{ABABAA}$

$\begin{matrix}\ &\ &A&\longrightarrow&A&\ &\ \\\ \ &\nearrow&\ &\ &\
&\searrow&\ \\\ A&\ &\ &\ &\ &\ &B\\\ \ &\nwarrow&\ &\ &\ &\swarrow&\ \\\ \ &\
&B&\longleftarrow&A&\ &\ \end{matrix}$ $\begin{matrix}\ &\
&A&\longrightarrow&B&\ &\ \\\ \ &\nearrow&\ &\ &\ &\searrow&\ \\\ A&\ &\ &\ &\
&\ &A\\\ \ &\nwarrow&\ &\ &\ &\swarrow&\ \\\ \ &\ &A&\longleftarrow&B&\ &\
\end{matrix}$
In this way, any strategy provided by the player can be regarded by the casino
as a strategy starting from Arm $A$ in the process of calculating the profit.
And use the vector $\boldsymbol{a}(h,r,s)$ to represent the structure of this
strategy, that is (1). Where $h$ is the number of times the arm is switched in
a single strategy. We conjecture that in the strategy for fixed $r$,$s$
provided by the player, the more frequently switching arms, the higher the
profit of the casino, that is, $R$ and $h$ are positively correlated.

### How is the casino’s asymptotic profit expectation calculated?

We notice that only the value of $p^{\circ}_{D}$ is the factor that can affect
the profit of the casino. From the above form of $p^{\circ}_{D}$, we need to
analyze the arrangement of each step in strategy $D$ and the next
$2k$($k=1,2,...,r+s$) steps. This calculation seems overly complicated as the
strategy varies, even though we have simplified the analysis of strategy $D$.
But for the simplest policy $D_{1}=AA...ABB...B$ i.e. $\boldsymbol{a}=(r,s)$,
we are not that complicated to calculate because it has the least number of
different cases between adjacent steps. Let $q_{A}=x$, $q_{B}=y$, then we have

$R_{D_{1}}=2S(1-(-1)^{r}x^{r})(1-(-1)^{s}y^{s})$

######  LEMMA 3.

For any $h\in\mathbb{Z}^{*}$, consider two nonrandom-pattern strategies with
pattern $D$, $D^{\prime}$.

$D=\underbrace{A...A}_{r_{1}}\underbrace{B...B}_{s_{1}}...\underbrace{A...A}_{r_{h-1}}\underbrace{B...B}_{s_{h-1}}\underbrace{A...A}_{r_{h}}\underbrace{B...B}_{s_{h}}$
$D^{\prime}=\underbrace{A...A}_{r_{1}}\underbrace{B...B}_{s_{1}}...\underbrace{A...A}_{r_{h-1}}\underbrace{B...B}_{s_{h-1}}\underbrace{B...B}_{s_{h}}\underbrace{A...A}_{r_{h}}$

where $1\leq k\leq h$, $r_{k}>0$ , $s_{k}>0$, $\sum\limits_{k=1}^{h}r_{k}=r$,
$\sum\limits_{k=1}^{h}s_{k}=s$ Then

$R_{D}-R_{D^{\prime}}=2S(1-{b_{2h-1}})(1-{b_{2h}})\big{(}\sum\limits_{j=0}^{2h-3}(-1)^{j}\prod\limits_{i=1}^{j}{b_{i}}+\sum\limits_{j=1}^{2h-2}(-1)^{j}\prod\limits_{i=1}^{j}{b_{2h-1-i}}\big{)}$

where $\boldsymbol{a}=(r_{1},s_{1},r_{2},s_{2},...,r_{h},s_{h})$

We observe that strategy $D^{\prime}$ is obtained by exchanging the strategy
segment in $D$ with only $A$ with the immediately following strategy segment
with only $B$. The reason for choosing these two special strategies is that
most of the product terms $q^{D}_{i}$ in the expression of $p^{\circ}_{D}$ are
equal to $q^{D^{\prime}}_{i}$ in the expression of $p^{\circ}_{D^{\prime}}$.
Due to the symmetry of $A$ and $B$, we can also exchange the strategy segment
with only $B$ in $D$ with the immediately following strategy segment with only
$A$, and we can get similar results. As can be seen from the previous section,
any strategy provided by the player can always be written in the form (1). We
first exchange the strategy segment $B$ of the $2h-2$-th segment and the
strategy segment $A$ of the $2h-1$-th segment to obtain a new strategy
$D_{h-1}$. Then exchange the strategy segment $B$ of the $2h-4$-th segment and
the strategy segment $A$ of the $2h-3$-th segment in $D_{h-1}$, so that we can
get a new strategy $D_{k}$, that is

$D_{k}=\underbrace{A...A}_{r_{1}}\underbrace{B...B}_{s_{1}}...\underbrace{A...A}_{r_{k-1}}\underbrace{B...B}_{s_{k-1}}\underbrace{A...A}_{r_{k}+r_{k+1}+...+r_{h}}\underbrace{B...B}_{s_{k}+s_{k+1}+...+s_{h}}$

We keep exchanging until we finally get $D_{1}=AA...ABB...B$, which happens to
be the strategy for which we calculated the casino’s asymptotic profit
expectations in advance. The difference between the profits of each exchange
of the previous strategy can be calculated by the lemma 3, so that we sum them
all up and prove by induction, we get the casino asymptotic profit expectation
$R=2QS$.

### Why casinos can be profitable in the long run?

In theorem 1, we note that if $x\neq y$, $0<x,y<1$, then $S>0$. Then we only
need to observe the positive and negative of $Q$. We can think of $b_{1}$,
$b_{2}$,…,$b_{2h}$ as $2h$ points and draw them on the round table in
clockwise order and we number the points in the order of $\\{b_{k}\\}$’s
numbering. For the convenience of the explanation of the following formula, we
find a representative example $\boldsymbol{a}=(2,1,2,2,1,2,1,2,2,1)$. We first
pick out $k$ that satisfy $b_{k}<0$ and renumber them, they represent the odd
number of $A$ or $B$ in the original strategy segment, we call these points
“negative” points, the rest are called “positive” points. Define a function
$c_{i}(j)$, $i,j\in\mathbb{Z}$ that satisfies the following properties:

  * •

$c_{1}:=\min\\{i\mid i>0,b_{i}<0\\}$

  * •

$c_{k}:=\min\\{i\mid i>c_{k-1},b_{i}<0\\}$

  * •

$c_{\delta}:=\max\\{c_{k}\mid c_{k}\leq 2h\\}$

Now we consider the case where $\delta>1$. In our example we try to relabel
elements where $b_{k}<0$ as follows

$\begin{matrix}\ &\ &b_{2}&\longrightarrow&b_{3}&\longrightarrow&b_{4}&\ &\
\\\ \ &\nearrow&\ &\ &\ &\ &\ &\searrow&\ \\\ b_{1}&\ &\ &\ &\ &\ &\ &\
&b_{5}\\\ \uparrow&\ &\ &\ &\ &\ &\ &\ &\downarrow\\\ b_{10}&\ &\ &\ &\ &\ &\
&\ &b_{6}\\\ \ &\nwarrow&\ &\ &\ &\ &\ &\swarrow&\ \\\ \ &\
&b_{9}&\longleftarrow&b_{8}&\longleftarrow&b_{7}&\ &\ \end{matrix}$
$\begin{matrix}\ &\
&\boldsymbol{b_{c_{1}}}&\longrightarrow&b_{3}&\longrightarrow&b_{4}&\ &\ \\\ \
&\nearrow&\ &\ &\ &\ &\ &\searrow&\ \\\ b_{1}&\ &\ &\ &\ &\ &\ &\
&\boldsymbol{b_{c_{2}}}\\\ \uparrow&\ &\ &\ &\ &\ &\ &\ &\downarrow\\\
\boldsymbol{b_{c_{4}}}&\ &\ &\ &\ &\ &\ &\ &b_{6}\\\ \ &\nwarrow&\ &\ &\ &\ &\
&\swarrow&\ \\\ \ &\
&b_{9}&\longleftarrow&b_{8}&\longleftarrow&\boldsymbol{b_{c_{3}}}&\ &\
\end{matrix}$

We observe that $Q$ is formed by the sum of products of different numbers of
$b_{k}$ that are adjacent. We factor the product term in the following two
ways. First we can choose $i$ as the “starting” point and $i+j$ as the “end”
point, and a formula is generated from this, that is,
$\xi(i,i+j):=(-1)^{j+1}\prod\limits_{k=i}^{i+j}{b_{k}}$ for $0\leq j<2h$ and
$\xi(i,i-1):=1$. And if the end point is positive point, that is $b_{i+j}>0$,
then the first factorization method is $\xi(i,i+j)=-\xi(i,i+j-1)b_{i+j}$. If
we find two negative points $c_{i}$ and $c_{i+j}$ as starting and end points,
and similarly a formula is generated from this, that is,
$\eta(i,i+j)=\xi(c_{i},c_{i+j})$ for $0\leq j<\delta$. And we can obtain the
following lemma by proof by contradiction.

######  LEMMA 4.

For $\delta>1$, $0<j<\delta$, if $\eta(i,i+j)<0$, then we have $m(i,i+j)$ or
$n(i,i+j)$ satisfies $\eta(i,i+m)<0$ or $\eta(i+n,i+j)<0$ for some $0<m\leq j$
or $0\leq n<j$ and $m$, $j-n$ are odd. Meanwhile it satisfies the following
properties:

  * •

There is no $0<m_{0}<m$, $0<n_{0}<m$ so that $\eta(i,i+m_{0})<0$ or
$\eta(i+n_{0},i+m)<0$.

  * •

There is no $n<m^{\prime}_{0}<j$, $n<n^{\prime}_{0}<j$ so that
$\eta(i+n,i+m^{\prime}_{0})<0$ or $\eta(i+n^{\prime}_{0},i+j)<0$.

Then the second factorization method is
$\eta(i,i+j)=-\xi(c_{i},c_{i+n}-1)\lvert\eta(i+n,i+j)\rvert$, where $n$ is
found by the above lemma and it is assumed that such $n$ exists. For example
$\eta(1,3)=-\xi(2,4)\lvert\eta(2,3)\rvert<0$, where $\xi(2,4)>0$ and
$\eta(2,3)<0$. This way if we fix the starting point $i$, some product terms
starting from $i$ can be written as a polynomial. According to the above two
factorization methods, we start the factorization from the $\xi(i,i+2h-1)$
with the most product terms, that is, the end point $i+2h-1$(i.e $i-1$)
farthest from the starting point. And record the new end point obtained by
factoring, and then continue to factorize according to the above methods from
the new end point until it cannot be decomposed. The end point at this time
must be a negative point. Then we select the farthest end point that did not
appear just now, and continue to decompose according to the above method, so
that the sum of all product terms starting from $b_{i}$ can be written as the
sum of several polynomials. Going back to our example, the sum of the product
terms starting at $b_{2}$ can be reduced to the sum of the following
polynomials.

$\displaystyle\sum\limits_{j=0}^{9}\xi(2,2+j)$ $\displaystyle=$
$\displaystyle-
b_{2}\big{(}1-b_{3}(1-b_{4}(1-b_{5}b_{6}b_{7}(1-b_{8}(1-b_{9})))\big{)}+\xi(2,5)(1-b_{6})+\xi(2,10)(1-b_{1})$

We notice that the value of the second term of the product in the above
summation formula is between 0 and 1, that is, when we study the positive and
negative of the product term, we only need to study the positive and negative
of the first term of the product. For this part, we noticed that its end point
can no longer be factored according to the above methods, but we can try to
factorize its starting point, and we also have the following two methods.

  * •

If $i$ and $i+j$ as starting and end points, and if $b_{i}>0$, then the third
factorization method is $\xi(i,i+j)=-b_{i}\xi(i+1,i+j)$.

  * •

If $c_{i}$ and $c_{i+j}$ as starting and end points, the fourth factorization
method is $\eta(i,i+j)=-\lvert\eta(i,i+m)\rvert\xi(c_{i+m}+1,c_{i+j})$ where
$m$ is found by the above lemma and it is assumed that such $m$ exists.

For the function $Q$, we have

$Q=\sum\limits_{l=1}^{h}(1-{b_{2l}})\big{(}\sum\limits_{j=0}^{2h-1}\xi(2l+1,2l+j)\big{)}$

We first take all negative points as endpoints, and observe the addition
between two adjacent endpoints. Without losing generality, we observe the
addition between $c_{i}$ and $c_{i+1}$. According to the new form of $Q$, we
consider partial summation
$\sum\limits_{c_{i}<2l<c_{i+1}}(1-{b_{2l}})\big{(}\sum\limits_{j=0}^{2h-1}\xi(2l+1,2l+j)\big{)}$.
In addition, on the basis of the original sum, if $c_{i+1}$ is even, we also
need to consider
$-{b_{c_{i+1}}}\big{(}\sum\limits_{j=0}^{2h-1}\xi(c_{i+1}+1,c_{i+1}+j)\big{)}$.
Also, if $c_{i}$ is even, we also need to consider
$\sum\limits_{j=0}^{2h-1}\xi(c_{i}+1,c_{i}+j)$. Then according to the
different parity of $c_{i}$, $c_{i+1}$, it can be divided into four addition
cases. It just so happens that the four segments summation formula that our
example is divided into corresponds to four different cases.

$\begin{matrix}case&1:&\boldsymbol{b_{c_{1}}}&\longrightarrow&b_{3}&\longrightarrow&\boldsymbol{b_{4}}&\longrightarrow&b_{c_{2}}\\\
\\\
case&2:&b_{c_{2}}&\longrightarrow&\boldsymbol{b_{6}}&\longrightarrow&b_{c_{3}}&\
\\\ \\\
case&3:&b_{c_{3}}&\longrightarrow&\boldsymbol{b_{8}}&\longrightarrow&b_{9}&\longrightarrow&\boldsymbol{b_{c_{4}}}\\\
\\\
case&4:&\boldsymbol{b_{c_{4}}}&\longrightarrow&b_{1}&\longrightarrow&\boldsymbol{b_{c_{1}}}&\
\\\ \end{matrix}$

The end point $c_{4}$ of the line in case 3 is exactly the starting point of
the line in case 4, and the end-to-end connection is unique, then according to
the decomposing method we discussed earlier, $Q$ can be divided into the above
four cases. After the above discussion, all negative polynomials starting from
$i$ and $b_{i}>0$ can always find positive polynomials starting from $i+1$.
According to the third factorization method, the sum of the two is positive.
Let’s take case 1 as an example.

$\displaystyle\sum\limits_{j=0}^{9}\xi(3,2+j)+(1-b_{4})\sum\limits_{j=0}^{9}\xi(5,4+j)$
$\displaystyle=$ $\displaystyle
1-b_{3}\big{(}1-b_{4}(1-b_{5}b_{6}b_{7}(1-b_{8}(1-b_{9})))\big{)}+\xi(3,5)(1-b_{6})+\xi(4,10)(1-b_{1})(1-b_{3})$
$\displaystyle+$
$\displaystyle\big{(}1-b_{5}b_{6}b_{7}(1-b_{8}(1-b_{9}(1-b_{10}b_{1}b_{2})))\big{)}(1-b_{4})-b_{5}(1-b_{6})(1-b_{4})+\xi(5.10)(1-b_{1})$
$\displaystyle>$ $\displaystyle
1-b_{3}(1-b_{4}(1-b_{5}b_{6}b_{7}(1-b_{8}(1-b_{9})))+\xi(3,5)(1-b_{6})+\xi(5,10)(1-b_{1})$

If there are more than one $l$ in the summation segment satisfying
$c_{i}<2l<c_{i+1}$, then the polynomial sum generated by the starting and end
points of the same summation segment is still positive. Now our summation
formula is greater than the sum of the negative polynomial starting from
$c_{i+1}$ and the positive polynomial starting from $c_{i}+1$. We also get
similar results in the other 3 cases, only the difference of whether or not
$\xi(1,2h)$ is included. This way we don’t have to consider negative
polynomials starting from any positive points. As we discussed earlier, the
first product term that makes up a negative polynomial starts and ends at
negative points. Without loss of generality we write this product term as
$\eta(i+1,i+j)$. Then according to our factoring method and Lemma 4, we must
be able to use the fourth factoring method to get
$\eta(i+1,i+j)=-\lvert\eta(i+1,i+m)\rvert\xi(c_{i+m}+1,c_{i+j})$.
Corresponding to our four summation cases, the negative polynomial generated
at the end of the summation segment must have a unique correspondence with the
positive polynomial generated at the beginning of some segment, and the
addition of the two is positive. In this way, either a closed loop is found so
that the sum of all polynomials in the closed loop is positive, or there are
no more negative polynomials in a certain summation segment. Then we get
$Q>0$. In our example, the negative polynomial $\xi(5,10)(1-b_{1})$ of the
summation section of case 1 can be found in the summation section of case 3
with its unique corresponding positive polynomial $\xi(8,10)(1-b_{1})$, such
that the sum of the two is positive, that is
$\xi(8,10)(1-b_{1})(1-b_{5}b_{6}b_{7})>0$. Next, the negative polynomial in
the summation segment of case 3 can also find the corresponding positive
polynomial in the summation segment of case 1, thus forming a closed loop.
Also in our example the summation segments of case 2 and case 4 do not contain
negative polynomials. Hence we get $Q>0$.

For the relatively simple case $\delta=0$, we can directly concatenate all the
expressions into a polynomial, that is

$Q=\sum\limits_{l=1}^{h}(1-{b_{2l}})\big{(}1-b_{2l+1}(1-b_{2l+2}(...(1-b_{2h+2l-2}(1-b_{2h+2l-1}))...))\big{)}>0$

And for $\delta=1$, without losing generality we suppose that ${b_{m^{*}}}<0$
and $m^{*}$ is odd, then

$\displaystyle Q=$
$\displaystyle\sum\limits_{l=1}^{h}(1-{b_{2l}})\big{(}(1-b_{2l+1}(...(1-b_{m^{*}-2}(1-b_{m^{*}-1}))...))$
$\displaystyle+$
$\displaystyle\lvert\xi(2l+1,m^{*})\rvert(1-b_{m^{*}+1}(...(1-b_{2h+2l-2}(1-b_{2h+2l-1}))...))\big{)}>0$

From this, we have revealed the secret of the long-term profitability of the
Futurity slot machine in a mathematical sense.

## Data Availability

There are no data underlying this work.

## Code availability

The DAQ program is freely available for academic use from Github at
https://github.com/yanxiaodong128/datareconstruction.

## Acknowledgements

This work was supported in part by the National Key R$\&$D Program of China
(Grant No. 2018YFA0703900) and the National Natural Science Foundation of
China (Grant Num. 11901352).

## Author contributions

Zengjing Chen conceived and supervised the study. Huaijin Liang supplied the
results and method. Wei Wang conducted the experiment, Xiaodong Yan drafted
the parts of abstract, introduction and discussion. All the authors read and
approved the manuscript.

## Competing interests

The authors declare no competing interests.