本論文集為賭場(casino)經營及優勢玩家(advantage players, APs)的一系列研究，第一篇探討優勢玩家(之後簡稱APs)如何透過賭場提供的優惠合約獲利，研究採用模擬搭配實際賭場優惠合約內容，並提出賭場經營困境及解套方法；第二篇回顧了繼菲律賓賭場在提供的優惠合約導致巨額損失之後，透過國際比較，指出下一個被APs攻擊國家為何?；第三篇探討，賭場在吸引顧客的優惠合約與承擔APs套利風險之間的兩難，提出模擬結果以及解方。以下我們詳述各篇的內容綱要。

第一篇:
本篇文章闡述，優勢玩家(APs)或也可稱為職業玩家，在賭場的獲利機制。首先，賭場做為莊家，所有的賭局一定有利於莊家，透過長期累積的莊家優勢，可以確保賭場正常經營。然而，隨著民眾的知識水準提高，大部分人都知道十賭九輸的道理，因此，從賭場的財務結構來看，資金的收入來源已經逐漸由賭桌(table)項目，轉移到其他購物與休閒娛樂項目。為了吸引源源不絕的賭客，特別是一擲千金的大客戶，賭場提出各式讓利的措施，只要投注的資金達到門檻即可享有該讓利優惠，也因此逆轉了莊家優勢成為專業玩家的優勢，透過數學的推理，只要嚴格遵守各種下注及停利條件，經由大量的人海戰術，即可扳倒莊家成為玩家優勢，懂得使用此戰略的玩家通常為一團隊，我們就稱之為優勢玩家(APs)。不同的賭博(gambling)項目適用的條件都不盡相同，主要在於每下注一塊錢，期望損失的金額大小與標準差的不同，以此來測量賭博項目的波動程度，當波動程度越大的情況之下，APs要準備的資金也會相對比較多，資本優勢顯得更重要。因此，職業玩家能在賭場套利，主要有兩個方法：一是賭場所提供的優惠合約，第二個是透過技術，如算牌(card counting)的方式，找出規律在即將有利時巨額投注，扭轉期望值劣勢。本研究主要著重在於前者，僅利用賭場提供的優惠。雖然過程中，期望值為負；但因為賭場的額外優惠，導致此額外的優惠大於賭博項目本身的期望值，使整個賭博項目的莊家優勢扭轉為玩家優勢。相關的數學基礎，我們使用隨機過程模型的推導計算出最佳的下注及出場點，搭配模擬說明優勢玩家在參與賭博項目的過程中如何利用賭場的優惠合約獲利。並且進一步說明，所需要的啟動資金。

第二篇:
本篇文章，根據國際上賭場提供給貴賓(very important person, VIP)的優惠條件，做國際性的比較，探討美國、韓國以及菲律賓。目前，從菲律賓的賭場已取消提供VIP優惠的教訓。此舉證明強力促銷爭取獲利的潛在風險，向全球賭場管理層提供了一個示警的信號。賭場在設計VIP合約的過程中，須要注意合約的相關金額以及所提供賭博項目的下注上限，這個比例如果沒有設計好，將會造成賭場的損失。本文運用隨機過程模型，進行一些變化，使得職業玩家在套利過程中不容易被賭場識破，更能夠隱藏其身份，持續套利。我們針對VIP合約以及賭博項目的管理，做出了下列建議，美國賭場管理的穩健性是最好的，其次是韓國，而菲律賓則是曝露在最大的風險之下。因此，在韓國或類似菲律賓的其他國家的VIP優惠，所產生的運營風險，可能成為APs攻擊的下一個目標。

第三篇:
本篇旨在解釋並分析近年來合法賭場的運營部門收入急劇下降的主要原因。APs在賭場獲利，主要有兩個方法：一個是賭場所提供的優惠，第二個是透過算牌方式。我們證明並披露了，職業玩家或是APs如何合法地，應用隨機過程、數學和模擬，來利用賭場提供的折扣優惠。本詳細展示了，APs如何利用，賭場提供給VIP折扣優惠並長期從賭場中獲利。此外，基於經驗證據，我們針對賭場管理所面臨的兩難，提出了一些建議方案。這些方案，可以有效減少APs的優勢，確保賭場運營部門的收益。若賭場運營部門業務穩定和恢復活力，無疑也會對酒店客房、飲料、零售和其他設施的收益，產生協同效應。換句話說，賭場型酒店的盈利能力得以提高。

Essay1: This paper uses computational and stochastic measures to demonstrate the negative impacts on the casino business of advantage players that legally rely on their knowledge to continuously make profits at licensed casinos worldwide. We present the stochastic process utilized by the advantage players to make profits from premium offers. We show in detail how players estimate the probability of reaching the win goal, the net-expected-win value, and the total expected return. In addition, a simulation is conducted to determine the amount of the starting capital that is needed before participating in games. The findings provide significant implications that advantage players are capable of exploiting premium offers and profiting from the casinos.
Essay2: This paper provides a useful signal to the global casino management by using the lesson from the cancellation of premium offers at Philippine casinos to emphasize the potential risks to profitability. A smaller ratio of bankroll/table maximum and the newly interpreted version of Brownian motion proves the net-expected win resulted in the negative net-expected win of casinos. Eventually, casinos in the Philippines canceled premium offers in 2015. Furthermore, we combine simulation results, the mean of the maximum win of casinos, the minimum of the maximum loss, the standard deviation of the accumulated-net loss. These indicate that the soundness of the management of casinos in the USA is the best, South Korea comes next and the Philippines was the riskiest. More importantly, operational risks with premium offers at casinos in South Korea, or other countries similar to the Philippines could be the next target of exploitation by advanced players.
Essay3: This paper aims to explain one of the main reasons why and how earnings of the casino-operation department of licensed casinos have been decreasing sharply in recent years. We prove and disclose how skillful players legally apply Brownian motion theorem and simulations to exploit the premium offer provided by casinos. The contributions of this paper show, in detail, how the skillful players have been taking advantage of premium offers and have been making profits from casinos for a long time. Furthermore, based on the empirical evidence, we also propose some solutions to the enigma faced by the casino management. These solutions can effectively reduce the advantage taken by skillful players and improve the earnings of the casino-operation department. If the casino-operation department business stabilizes and rejuvenates, there can certainly be synergy effects on the hotel rooms, beverage, retailing and other facilities’ earnings. In other words, the profitability of the hotel-casino can improve. 

Contents                                                                  І
List of Tables                                                             IV
List of Figures                                                             VII
Chapter 1.   Introduction                                                   1
Chapter 2.   How do Advantage Players Exploit Casinos’ Premium Offers? Simulations and Solutions                                               2
2.1	     Background                                           2
2.2	     Literature Review                                           3
2.3	     The Method                                               4
2.3.1        The loss-rebate theory and application of the stochastic process and optimization                                                4
2.3.2        Application to the Blackjack Game                                7
2.3.3        An Example of the Premium Offer                                10
2.3.4        The Behaviours of APs                                         10
2.3.4.1      The mediocre APs                                             14
2.3.4.2      The advanced APs                                             15
2.4          Discussion                                                  21
2.5          Simulation results for the APs’ starting capital                        22
2.5.1        Method                                                      22
2.5.2        The simulation results of the blackjack game                         25
2.6	     Discussion and Concluding Remarks                                 31
2.7	     Practical implications and future research                            33
Chapter 3.   What is the Next After the Collapse of Philippine Casinos’ Premium Offers? International Comparisons                                      35
3.1	     Background                                                   35
3.2	     Literature Review                                         38
3.3	     Applications of the newly Interpreted Brownian motion        40
3.3.1        The Brownian Motion Model                                  40
3.3.2        Brownian Motion Models for Baccarat in Three Countries          40
3.3.3        Multiple-Round Brownian Motion Models for Game 21            44
3.3.4        Casinos’ Ignorance of Baccarat                                  48
3.4	     Simulation                                  49
3.4.1        Comparing Loss Rebate Offer Risks Confronted by Casinos in Three Different Countries                                                   50
3.4.2        Illustration of Risks and Earnings Confronted by Casinos in the Philippine 52
3.4.3        Further Evaluation of Risks and Earnings in the Three Countries          55
3.4.4        Explanation and Coordination of the Asymmetry between Brownian motion and Simulations                                              61
3.4.5        Risk Measurement and Prevention of APs’ Exploitations on Loss Rebate Offers 62
3.5	     Conclusions                                                   63
Chapter 4.   The Enigma between Premium Offer of Casinos and Skillful Players: Simulations and Solutions                                    66
4.1          Background                                                   66
4.2          Literature Review                                               69
4.3	     The Mathematical Model and Simulation Results                      71
4.3.1        The Mathematical Model                                         72
4.3.2        The results of the newly interpreted Brownian motion model            98
4.4          Discussions on the Surveillance of Casinos                          106
4.5          Concluding Remarks                                          110
Chapter5.    Conclusions                                               114
Reference                                                                115
Appendix                                                                117

List of Tables                         
Table 2.1 Results of Blackjack Game based on Combinations of Bet Size and Loss Rebate      Rates                                                          9
Table 2.2 Contracts of premium offers based on countries and casinos                  12
Table 2.3 Total expected returns for two types of APs                               17
Table 2.4 Illustration of total expected returns with various bankrolls and bet sizes        21
Table 2.5 Confidence intervals, actual and theoretical relative frequencies of the mediocre APs 27
Table 2.6 Confidence intervals, actual and theoretical relative frequencies of the advanced APs                                                                          27
Table 2.7 Starting capital and the probability of bankruptcy for blackjack for mediocre & advanced APs                                                   28
Table 3.1 APs’ Net-expected-win of Baccarat under Various Premium Offers in Three Countries, House Advantage: -1.0579, Standard Deviation: 0.927372     43
Table 3.2 Total Expected Return Without Accumulation                            46
Table 3.3 Calculation Process of Accumulated Total Expected Returns for The Four Rounds                       derived from Table 3.2                                              47
Table 3.4 Accumulated Total Expected Returns of the Ordinary AP and the Advanced AP  48
Table 3.5 The Case of Casinos in the Philippine                                  50
Table 3.6 The Case of Casinos in South Korea with the Bankroll USD$50,000, Table    Maximum USD$20,000                                           51
Table 3.7 The Case of Casinos in South Korea with the Bankroll USD$50,000, Table Maximum USD$50,000                                           51
Table 3.8 The Case of Casinos in the USA                                       52
Table 3.9 The Frequency Distribution of Maximum Loss of Casinos in the Philippine    56
Table 3.10 The Frequency Distribution of Maximum Win of Casinos in the Philippine    57
Table 3.11 The Frequency Distribution of Maximum Loss of Casinos in South Korea, Table Maximum USD$20,000                                           58
Table 3.12 The Frequency Distribution of Maximum Win of Casinos in South Korea, Table Maximum USD$20,000                                           58
Table 3.13 The Frequency Distribution of Maximum Loss of Casinos in South Korea, Table Maximum USD$50,000                                           59
Table 3.14 The Frequency Distribution of Maximum Win of Casinos in South Korea, Table Maximum USD$50,000                                           59
Table 3.15 The Frequency Distribution of Maximum Loss of Casinos in the USA        60
Table 3.16 The Frequency Distribution of Maximum Win of Casinos in the USA          60                    
Table 4.1 The Net-Expected Win of SPs on Game 21 from Results of Brownian Motion (in        USD$)                                                         75
Table 4.2 The Win Goal of SPs on 21 from Results of Brownian Motion               77
Table 4.3 The Probability of Reaching the Win Goal of SPs on 21 from results of Brownian Motion                                                         78
Table 4.4 The Hands Played by SPs on Craps from Results of Brownian Motion         80
Table 4.5 Total Returns of 21 of the Ordinary and Highly SPs                        81
Table 4.6 The Net-Expected Win of SPs on Craps from Results of Brownian Motion      88
Table 4.7 The Win Goal of SPs on Craps from Results of Brownian Motion             88
Table 4.8 The Probability of Reaching the Win Goal of SPs on Craps from Results of Brownian Motion                                                         89
Table 4.9 The Hands Played by SPs on Craps from Results of Brownian Motion         90
Table 4.10 Total Returns of the Craps of the Ordinary and the Highly SPs              91
Table 4.11 Total Returns of the Craps with various Bankrolls and with different Table Maximum                                                      92
Table 4.12 Starting Capital of Blackjack obtained from 2Million simulation results for Ordinary & Highly SPs                                           101
Table 4.13 Starting Capital of Craps obtained from 2million simulation results for Ordinary & Highly SPs                                                     104

List of Figures
Figure 2.1 The win-goal versus probability, player’s and house edge, play-time, and net expected win                                       8
Figure 2.2 The Accumulated Capital Along the 2,000 Contracts                       25
Figure 2.3 Accumulated Capital for the Infimum, Supremum, Mean and Standard Deviation Along the 2,000 Contracts for the Mediocre APs                        29
Figure 2.4 Accumulated Capital for the Infimum, Supremum, Mean and Standard Deviation Along the 2,000 Contracts for the Advanced APs                       30
Figure 3.1 The Probability Distribution of Casinos’ Maximum Loss in the Philippine      53
Figure 3.2 The Probability Distribution of Casinos’ Maximum Win in the Philippine      54
Figure 4.1 Maximum Win and Maximum Loss on 21, 2Million Simulation Results, Ordinary SPs                                                           102
Figure 4.2 Maximum Win and Maximum Loss on 21, 2Million Simulation Results, Highly SPs                                                     102
Figure 4.3 Maximum Win and Maximum Loss on 21, 2Million Simulation Results, SPs  103
Figure 4.4 Maximum Win and Maximum Loss on Craps, 2Million Simulation Results, Ordinary SPs                                                    105
Figure 4.5 Maximum Win and Maximum Loss on Craps, 2Million Simulation Results, Highly SPs                                                             105
Figure 4.6 Screening SPs by dealers of the Table Department at Casinos               109

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