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  • 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


    http://www.lpt-tlse.fr/ http://www.lpt-tlse.fr/ http://www.lpt-tlse.fr/

  • 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

  • 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:

    Borel obtained the best strategy for A and B

    A starts

    A raises R$ if cA>rA

    B calls R$

    if cB>rB

    Best hand wins (R+1)$

    B folds

    if cB

  • 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:

    The best A-strategy now involves random bluffing

    A starts

    A raises R$

    if cA>rA or if cArB

    Best hand wins (R+1)$

    A checks

    if sA

  • 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)

  • WSOP 2009: 57 tournaments (200 – 6500 players)

    Main event: 6494 players; 10000$ buy-in; winner prize: 8.5 M$

  • Texas Hold’em Tournaments

    10–35000 players (8–10 per table)

    Buy-in: 0–25000$ for x0 chips (x0~1000–20000)

    A player is eliminated when x(t)=0 (NB: N(t)x(t)=N0x0)


    • 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”); 2nd betting round

    • A 4th common card is dealt (the “turn”); 3rd betting round

    • A 5th common card is dealt (the “river”); last betting round

    • The best hand made of 5 cards (among 2+5 common cards) wins the pot

  • Example of Texas Hold’em Play

  • Main Ingredients of a

    Poker Tournament

    The blind b(t) increases exponentially (for x0=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

  • A Model of Poker Tournaments

    N0 players distributed on tables of T=10 seats receive x0 chips

    At the start of any deal, the “blinder” posts the blind b(t)=b0exp(t/t0) (x0Àb0)

    Each player receives a hand (a random number c[0;1])

    Players can call (i.e. bet b) with probability p(c), if c1-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)=cn

    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

  • Poker Model without All-in (q=0)

    The stack x(t) of an individual player is a generalized random walker:


    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

  • 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…)

  • Solution of the Master Equation (q=0)

    Stack (fortune) distribution:

    ( 0, ) 0P x t ,

  • 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:

  • Stack Distribution (q=0)

    The cumulative distribution gives the rank as a function of

  • All-in Decay Rate

    Probability of an all-in event at a given table (q¿1):

    Population decay due to all-in events:

  • 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

  • All-in Kernel (time argument dropped):

    All-in decay rate:

    Master Equation Including All-in Events

    Ã Wins against a bigger stack

    Ã Wins and eliminates a poorer


    Ã Looses but survives against a

    poorer player

  • Scaling stack distribution:

    Scaling equation (integro-differential and non-linear; no parameter):

    Scaling Equation

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

  • WPT 2006 data (10000$ buy-in)

  • Properties of the “Chip Leader”

    The number of successive “chip leaders” growths logarithmically with the number of initial players

  • The “Leader Problem”

    Competing agents get a time-dependent “score”

    where s is any continuous increasing function (arises in evolutionary biology)

    The fitnesses vi 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/0606101

    http://dx.doi.org/10.1088/1742-5468/2006/07/L07001 http://dx.doi.org/10.1088/1742-5468/2006/07/L07001 http://dx.doi.org/10.1088/1742-5468/2006/07/L07001 http://arxiv.org/abs/cond-mat/0606101 http://arxiv.org/abs/cond-mat/0606101 http://arxiv.org/abs/cond-mat/0606101

  • Distribution of the Chip Leader Stack

    Gumble’s distribution of extreme values:

  • 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/0703122

    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…

    http://www.iop.org/EJ/abstract/1742-5468/2007/08/P08013 http://arxiv.org/abs/physics/0703122

  • 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

  • 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.)

  • 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