Universal Properties of Poker Tournaments Persistence, the leader problem and extreme value statistics. Clément Sire

Transcription

1 Universal Properties of Poker Tournaments Persistence, the leader problem and extreme value statistics Clément Sire Laboratoire de Physique Théorique CNRS & Université Paul Sabatier Toulouse, France

2 Introduction Statistical physicists are more and more interested in systems out of the realm of physics (networks, competitive systems, epidemics, finance, biology ) Poker tournaments are perfectly isolated systems, a priori obeying purely human laws (bluff, prudence, aggressiveness ) Link with other physics/probability problems: persistence, the leader problem, extreme value statistics

3 The Science of Poker Head-to-head game La Relance introduced by Émile Borel (1938): Player A and B put 1$ in the pot and are each distributed a hand of value c2[0,1]; A starts: A starts A raises R$ if c A >r A B calls R$ if c B >r B Best hand wins (R+1)$ B folds if c B <r B A wins 1$ A folds if c A <r A B wins 1$ Borel obtained the best strategy for A and B

4 The Science of Poker John von Neumann (1944) modifies the rules of La Relance to include bluff Player A and B put 1$ in the pot and are each distributed a hand of value c2[0,1]; A starts: A raises R$ B folds A wins 1$ A starts if c A >r A or if c A <s A B calls R$ if c B >r B Best hand wins (R+1)$ A checks if s A <c A <r A Best hand wins 1$ The best A-strategy now involves random bluffing

5 The Science of Poker Strategies for more complicated/realistic rules for head-to-head poker haven been obtained Poker bots now learn to adapt and mix their strategies (two-player game) Poker bots in multiplayer games are still bad In this work, we are interested in the overall statistical properties of poker tournaments (strategies are irrelevant)

6 WSOP 2009: 57 tournaments ( players) Main event: 6494 players; 10000$ buy-in; winner prize: 8.5 M$

7 Texas Hold em Tournaments players (8 10 per table) Buy-in: $ for x 0 chips (x 0 ~ ) A player is eliminated when x(t)=0 (NB: N(t)x(t)=N 0 x 0 ) Play: Each player receives 2 cards; the two players after the dealer post their blind bets; first round of betting (fold, call, raise, re-raise, re-re-raise ) 3 common cards are dealt (the flop ); 2 nd betting round A 4 th common card is dealt (the turn ); 3 rd betting round A 5 th common card is dealt (the river ); last betting round The best hand made of 5 cards (among 2+5 common cards) wins the pot

8 Example of Texas Hold em Play

9 Main Ingredients of a Poker Tournament The blind b(t) increases exponentially (for x 0 =3000, b=40, 60, 100, 200, 300, 400, ) Most of the times, the winner of the hand wins a pot of a few blinds; but sometimes, two or more players raise each other aggressively and bet a large fraction if not all of their stack: all-in! A simple but faithful model of poker tournaments should incorporate these features

10 A Model of Poker Tournaments N 0 players distributed on tables of T=10 seats receive x 0 chips At the start of any deal, the blinder posts the blind b(t)=b 0 exp(t/t 0 ) (x 0 Àb 0 ) Each player receives a hand (a random number c [0;1]) Players can call (i.e. bet b) with probability p(c), if c<1-q, go all-in, if c>1-q (and can be called by other players having a hand c>1-q), and fold otherwise NB: p(c) is irrelevant; for instance, p(c)=c n The best hand wins the pot; time is updated to t+1 Tables run in parallel; players are redistributed on empty seats in order to minimize the number of active tables

11 Poker Model without All-in (q=0) The stack x(t) of an individual player is a generalized random walker: x(t+1)=x(t)+b(t) (t) where t is the number of hands played, and h (t)i=0 (no winning strategy/chips are conserved) h (t) (t )i= t,t 2 (successive deals are uncorrelated) The considered player is eliminated when x(t)=0 The number P(x,t) of players still in game with a stack x obeys the Fokker-Planck equation: with the absorbing boundary condition P(x=0,t)=0 Link to the persistence problem

12 Physical Applications of Persistence Persistence exponent of the Ising model: local and global persistence; a new critical exponent for lattice spin systems Persistence exponent for the diffusion equation Persistence for any level M: distribution of the minima and maxima of a process, Many other experimental measurements (breath figures, Si surfaces, soap bubbles, rare gas )

13 Solution of the Master Equation (q=0), P( x 0, t) 0 Stack (fortune) distribution:

14 Scaling Regime (q=0) Stack distribution: Exponential decay of the number of players controlled by the growth rate of the blind: Universal scaling function: Estimate of the duration of a tournament:

15 Stack Distribution (q=0) The cumulative distribution gives the rank as a function of

16 All-in Decay Rate Probability of an all-in event at a given table (q 1): Population decay due to all-in events:

17 The Optimal All-in Probability If, the tournament is dominated by all-in events (players are taking stupid risks just to win the blind) If, the players (especially those with small stacks) do not take enough the opportunity to double their stack with a very good hand The optimal q is such that

18 Master Equation Including All-in Events All-in Kernel (time argument dropped): à Wins against a bigger stack All-in decay rate: à Wins and eliminates a poorer player à Looses but survives against a poorer player

19 Scaling Equation Scaling stack distribution: Scaling equation (integro-differential and non-linear; no parameter):

20 Internet Tournaments (22-55$ buy-in)

21 WPT 2006 data (10000$ buy-in)

22 Properties of the Chip Leader The number of successive chip leaders growths logarithmically with the number of initial players

23 The Leader Problem Competing agents get a time-dependent score where s is any continuous increasing function (arises in evolutionary biology) The fitnesses v i are independently drawn from the same distribution f(v) Then, the average total number of leaders and its variance verify: where the constants and are universal and only depend on the large v properties of f(v) (3 universality classes) Ref.: C. Sire, S.N. Majumdar, and D.S. Dean, J. Stat. Mech., L07001 (2006) ; reprint on arxiv.org/abs/cond-mat/

24 Distribution of the Chip Leader Stack Gumble s distribution of extreme values:

25 Conclusion Successful modeling of poker tournaments after identifying their two main properties (exponential growth of the blind, all-in events) and the irrelevance of individual strategies Link to persistence, the leader problem, and extreme value statistics Refs: C. Sire, J. Stat. Mech. P08013 (2007); reprint on arxiv/physics/ Articles in the Scientific American, the New Scientist, PhysOrg.com, Poker Live Magazine, Libération, BBC Focus, and hundreds of poker web sites Other recent game theory applications: contest on a random polymer, baseball and football championships

26 Persistence of a non-markovian stationary Gaussian random walker A stationary Gaussian random walker x( ) is fully characterized by f( )=hx( )x(0)i The persistence is the probability that x(s) remains above (or below) a certain level M, for all s2[0, ]; P( )»exp(-µ ) f( )=exp(- ) means that x( ) is Markovian If hy(t)y(t )i=g(t/t ), then x( )=y(exp ) is stationary in the new variable =ln t; hence, P(t)»t -µ Many physical quantities of interest are Gaussian or well approximated by a Gaussian process

27 Approximate calculations (M=0) Perturbation theory around a Markovian process (S.N Majumdar, CS) Independent Interval Approximation (IIA) (A. Bray, S. Cornell, S.N Majumdar, CS & B. Derrida et al.)

28 Physical applications Persistence exponent of the Ising model: local and global persistence (PT); a new critical exponent for lattice spin systems Persistence exponent for the diffusion equation Persistence for any level M (CS): distribution of the minima and maxima of a Gaussian process, Many other experimental measurements (breath figures, Si surfaces, soap bubbles, rare gas )

29 References Perturbation theory: - S.N. Majumdar, C. Sire, Phys. Rev. Lett. 77,1420 (1996); reprint on arxiv:cond-mat/ S.N. Majumdar, A.J. Bray, S.J. Cornell, C. Sire, Phys. Rev. Lett. 77, 3704 (1996); reprint on arxiv:cond-mat/ C. Sire, S.N. Majumdar, A. Rüdinger, Phys. Rev. E 61, 1258 (2000); reprint on arxiv:cond-mat/ Independent Interval Approximation: - S.N. Majumdar, A.J. Bray, S.J. Cornell, C. Sire, Phys. Rev. Lett. 77, 2867 (1996); reprint on arxiv:cond-mat/ B. Derrida, V. Hakim, R. Zeitac, Phys. Rev. Lett. 77, 2871 (1996); reprint on arxiv:cond-mat/ Distribution of the extrema of a smooth Gaussian walker: - C. Sire, Phys. Rev. Lett. 98, (2007); reprint on arxiv:condmat/

30 Other competitive systems: baseball and soccer (CS & S. Redner) Fraction of wins vs rank during the two major baseball eras ( ; ) & the corresponding streaks pdf (one parameter model)

31 Understanding performance vs rank in soccer Number of points/fraction of home wins/draws/away wins in major European leagues vs rank (here, UK and Germany; 2 parameters model)