Universal Properties of
Poker TournamentsPersistence, the “leader problem”
and extreme value statistics
Clément SireLaboratoire de Physique Théorique
CNRS & Université Paul Sabatier
Toulouse, France
www.lpt.ups-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<rB
A wins 1$
A folds
if cA<rA
B wins 1$
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 cA<sA
B folds A wins 1$
B calls R$
if cB>rB
Best hand wins (R+1)$
A checks
if sA<cA<rA
Best hand wins 1$
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 mixtheir 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)
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”); 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
–
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 x0chips
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 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)=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:
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
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 maximaof a process,
Many other experimental measurements (breath figures, Si surfaces, soap bubbles, rare gas…)
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:
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
à Wins against a bigger stack
à Wins and eliminates a poorer
player
à Looses but survives against a
poorer player
Scaling stack distribution:
Scaling equation (integro-differential and non-linear; no parameter):
Scaling Equation
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
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 onarXiv/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…
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): distribution of the minima and maxima of a Gaussian process,
Many other experimental measurements (breath figures, Sisurfaces, soap bubbles, rare gas…)
References
Perturbation theory:
- S.N. Majumdar, C. Sire, Phys. Rev. Lett. 77,1420 (1996); reprint on arXiv:cond-mat/9604151
- S.N. Majumdar, A.J. Bray, S.J. Cornell, C. Sire, Phys. Rev. Lett. 77, 3704 (1996); reprint on arXiv:cond-mat/9606123
- C. Sire, S.N. Majumdar, A. Rüdinger, Phys. Rev. E 61, 1258 (2000); reprint on arXiv:cond-mat/9810136
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/9605084
- B. Derrida, V. Hakim, R. Zeitac, Phys. Rev. Lett. 77, 2871 (1996); reprint on arXiv:cond-mat/9606005
Distribution of the extrema of a smooth Gaussian walker:
- C. Sire, Phys. Rev. Lett. 98, 020601 (2007); reprint on arXiv:cond-mat/0606145
Other competitive systems:
baseball and soccer (CS & S. Redner)
Fraction of wins vs rank during the two major baseball
eras (1900-1960; 1961-2007) & the corresponding
streaks pdf (one parameter model)