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Role of Information in Non-Equilibrium Situations and the Game of Gambling

Explore the critical role of information theory in both non-equilibrium situations and gambling. Discover how information influences capital growth in gambling, the application of Kelly criterion, physics principles like Jarzynski equality, and the intriguing dynamics of biased coin tossing. Delve into strategies for maximizing capital growth rate and the impact of side information on betting decisions. Uncover insights on card counting in blackjack, revealing how players can gain an advantage by leveraging information on dealt cards. Gain a deeper understanding of the dynamics of betting, capital evolution, and the significance of information in complex decision-making scenarios.

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Role of Information in Non-Equilibrium Situations and the Game of Gambling

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  1. Role of information in non-equilibrium situations Yuji Hirono [Stony Brook Univ.] In collaboration with Yoshimasa Hidaka [RIKEN]

  2. Role of information in gambling Yuji Hirono [Stony Brook Univ.] In collaboration with Yoshimasa Hidaka [RIKEN]

  3. Why is gambling interesting? 3. 1. Information theory Capital growth in gambling Jarzynski equality 2. [Kelly (1956)] 4. Physics

  4. Biased coin tossing 100 百 head tail Lose Win

  5. Biased coin tossing Capital evolution Win Lose : fraction of money to bet

  6. Biased coin tossing Capital evolution Win Lose : fraction of money to bet How should a gambler choose f ?

  7. Kelly criterion Capital growth rate Choose f to maximize : “Kelly criterion” [Kelly (1956)] Surpasses other strategies in the long run [Breiman (1961)]

  8. Kelly criterion When all the tosses are independent & identical The player wants to maximize

  9. Kelly criterion

  10. Kelly criterion [Kelly (1956)] Channel capacity of a binary symmetric channel

  11. Biased coin tossing with side information 100 百 Informant head tail “side information” Lose Win

  12. Side information boosts the capital growth [Kelly (1956)] Mutual information

  13. Outline Information theory Capital growth in gambling Jarzynski equality 2. [Kelly (1956)] Physics

  14. Jarzynski equality [Jarzynski(1997)] : entropy production Jensen’s inequality - Holds for the processes far from equilibrium - 2nd law of thermodynamics can be derived

  15. Jarzynski equality for systems with feedback controls (demons) [Sagawa & Ueda(2010,2012)] :change in the mutual information : entropy production Jensen’s inequality

  16. Outline 3. Information theory Capital growth in gambling Jarzynski equality [Kelly (1956)] Physics

  17. Jarzynski-type equality for biased coin toss [Hirono & Hidaka (2015)] : Shannon entropy Jensen’s inequality

  18. Binary gambling with side information [Hirono & Hidaka (2015)]

  19. Binary gambling with memory effects and side information [Hirono & Hidaka (2015)] : Shannon entropy : Directed information [Massey (1990)]

  20. Response of capital growth rate to the deviation from Kelly bet “Fluctuation-response relation”

  21. Outline Information theory Capital growth in gambling Jarzynski equality [Kelly (1956)] 4. Physics

  22. Blackjack Players can beat the casino by using information of dealt cards card counting [E. Thorp (1960)] [ “21” (film) (2008)] [ “Rain man” (film) (1988)] How much money can we make by counting?

  23. Rule of blackjack Dealer vs player. Other players are irrelevant Purpose: reach a final score closer to 21 without exceeding 21 “go bust” 1 or 11 10 Score number on the card

  24. Rule of blackjack Dealer “shoe” 6 decks “hit” draw a card Player end one’s turn “stand”

  25. Rule of blackjack Dealer “shoe” Player

  26. Rule of blackjack Dealer “shoe” Dealer has tohit until (score) > 16 Player Dealer’s action is completely specified by rules!

  27. Recipe of counting – “high-low” +1 “low cards” Large count 0 More high cards in the shoe -1 High return for players “high cards”

  28. Time evolution of count = random walk Bet more money when the count is high!

  29. Various counting systems

  30. True count knows expected return True count Approximately, indicates the performance of a counting system [Werthamer (2009)]

  31. Capital growth under Kelly betting : the fraction at which cards are reshuffled

  32. Capital growth under Kelly betting : counting vector-dependent coefficient

  33. Capital growth under Kelly betting

  34. Average number of rounds to double the capital

  35. To double the capital, • We also have to consider: • Effects of minimum bet 50k rounds 800hours 60 rounds / h 100days 8 hours / day

  36. Summary & outlook • Derived Jarzynski-type equalities for gambling • Constrain the fluctuations of capital growth • Reproduce the Kelly bound • Financial value of side information is quantified by • Mutual information (memoryless) • Directed information (with memory effects) • Analysis of blackjack • Capital growth rates under Kelly betting for various counting vectors • [Outlook] • Application to stock investing • More practical analysis of blackjack

  37. Backup slides

  38. Capital growth rate under counting & Kelly bet

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