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Role of information in non-equilibrium situations

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

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Role of information in non-equilibrium situations

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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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