Differential Sensitivity Analysis of Monte Carlo Poker Equity: Jacobian and Hessian Approaches

Dierential Sensitivity Analysis of Monte Carlo Poker Equity: Jacobian and Hessian Approaches Ashlin Darius Govindasamy ADGSTUDIOS Department of Computer Science and Mathematics Contents 1 Introduction 3 2 Mathematical Model 3 3 Monte Carlo Estimator 4 4 Teaming and Collusion 4 4.1 Derivation........................................ 4 4.2 MonteCarlovalidation................................. 5 4.2.1 Pythonsimulation ............................... 5 4.2.2 Csimulation .................................. 5 4.3 Results.......................................... 5 4.4 Discussion........................................ 6 5 Jacobian of the Equity Function 6 6 Hessian 7 7 Convergence 7 8 Normal Distribution Behaviour 7 8.1 Hypothesis testing and normality checks . . . . . . . . . . . . . . . . . . . . . . . 9 8.2 Actionprobabilitymodel................................ 9 9 Tensor Field Representation 9 10 Counterfactual Regret Minimisation (CFR) 10 11 Conclusion 11 12 Appendix 11 12.1Notation......................................... 11 12.2 A. Lemma: Symmetric Random-Hand Equity . . . . . . . . . . . . . . . . . . . . 11 12.3 B. Monte Carlo Variance Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 12 12.4 C. Explicit Jacobian and Hessian Matrices . . . . . . . . . . . . . . . . . . . . . . 12 12.5 D. Exponential Survival Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 13 12.6 E. Additional Probability Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 13 12.7 F. Matrix Representation of Expected Value Updates . . . . . . . . . . . . . . . . 13 12.8G.Figures........................................ 14 1 13 Proof 15 2 Abstract We establish a rigorous mathematical framework proving that heads-up play in poker tournaments yields a strictly higher advancement and win probability compared to any n -player conguration with n > 2 . Our analysis combines (1) combinatorial probability, (2) Monte Carlo convergence theory, (3) partial dierential sensitivity of the equity estimator, and (4) regretminimising dynamics derived from Counterfactual Regret Minimisation. Using blackboard notation and formal theoremlemmacorollary structure, we derive explicit percentage advantages, asymptotic convergence rates, and structural dominance theorems. This constitutes an original contribution to incomplete-information game theory. 1 Introduction Poker tournaments are sequential elimination games with incomplete information. Player survival depends on accumulation of chip equity over repeated stochastic interactions. It is empirically observed that reducing table sizeespecially to heads-up conguration ( 2 players) dramatically improves tournament survival probability. We formalise this phenomenon using rigorous probability theory and derive:  strict dominance of heads-up equity: E2>En for all n>2 ,  Monte Carlo convergence properties under N(µ, σ2) asymptotics,  a dierential characterisation of equity sensitivity,  explicit percentage advantage of order Θ(n) ,  and a regret-based justication showing faster utility accumulation. This follows the foundational work of Monte Carlo simulation (Metropolis & Ulam 1949) and strategic analysis in poker Wilson (2019). 2 Mathematical Model Let (Ω,F,P) be a probability space modelling the poker deal. Let H denote the hero's private hand and Hi the hand of opponent i . Denition 1 (Equity) . The equity of the hero at an n -player table is: En=P(H > H1, . . . , H > Hn−1). Under the classical symmetric random-hand assumption: En=1 n. This setting is sucient for theorem-level results and forms the base of our analysis. Model simplications and limitations For analytical clarity we adopt an i.i.d.Uniform(0,1) model for the relative hand strength where each player's hand strength is sampled independently. This simplication omits important poker phenomena such as card removal eects, dependencies induced by shared community cards, and the intricacies of betting-driven information. The Monte Carlo methods and proofs below carry over to more realistic deck-sampling models (52card deck, community cards, explicit hand-evaluation) at the cost of increased computational complexity; where applicable we remark on such extensions. 3 3 Monte Carlo Estimator The Monte Carlo equity estimator is dened as: b EN=W+1 2T N, with W, T, L representing wins, ties, losses. By the Central Limit Theorem: b ENd −→ NE,E(1 −E) N. Thus variance strictly decreases with higher true equity. 4 Teaming and Collusion This section studies the eect of players forming a colluding team on the probability of the team winning a hand. We rst derive a closed form using symmetry and order-statistics for the case where hand strengths are independent and identically distributed. Then we validate the formula using Monte Carlo simulation and provide small Python and C implementations to reproduce the experiments. 4.1 Derivation Consider an n -player table, and suppose a subset A of size m form a team; the complement B contains n−m independent opponents. Let X1, . . . , Xn be i.i.d.continuous random variables representing hand strengths with CDF F(x) . The team wins if the maximum of its members exceeds the maximum of the opponents. Let XA= max i∈AXi, XB= max j∈BXj. The CDFs are FA(x)=F(x)m, FB(x) = F(x)n−m. The team win probability is P(XA> XB) = ZR P(XB< x)dFA(x) = ZR FB(x)dFA(x). Assuming F is the standard uniform distribution on [0,1] (a common and convenient choice for comparing relative strengths) gives F(x) = x and hence P(XA> XB) = Z1 0 xn−md(xm)=mZ1 0 xn−1dx =m n. Therefore, under the i.i.d.unbiased draw assumption, the team's probability of having the best hand equals m/n . Intuitively, each of the n raw hands is equally likely to be the overall maximum and the team contributes m such hands. Note on ties: ties among maxima can be handled similarly by integrating the probability of an exact match; for continuous distributions tie probability is zero. If ties are non-negligible they can be included by adding half-credit or other tie-breaking rules matching the Monte Carlo scoring used in the main text. 4 4.2 Monte Carlo validation The following simulation uses simple uniform random draws (interpreting the hand strength by a uniform score in [0,1] ). We report the empirical fraction of trials where at least one of the m colluding players holds the largest score among all n players; this empirically validates the formula. 4.2.1 Python simulation Listing 1: Python Monte Carlo simulation for teaming win probability. 1 import numpy as np 2 3 def team_win_prob(n, m, trials =200000): 4 # n total players , m in colluding team 5 draws = np.random.rand(trials , n) 6 team_max = draws[:, :m].max(axis=1) 7 overall_max = draws.max(axis=1) 8 return (team_max >= overall_max).mean() 9 10 cases = [ (8,2), (8,3), (6,2), (10,2) ] 11 for n,m in cases: 12 p = team_win_prob(n, m, trials=200000) 13 print(f"n={n}, m={m}, empirical={p:.6f}, theory={m/n:.6f}") 4.2.2 C simulation Listing 2: C Monte Carlo simulation for teaming win probability (gives the same empirical result). 1 #include <stdio.h> 2 #include <stdlib.h> 3 #include <time.h> 4 5 double rand01(){ return (double)rand() / (double)RAND_MAX; } 6 7 int main(){ 8 srand((unsigned)time(NULL)); 9 int trials = 200000; 10 int n = 8, m = 2; // example 11 int team_wins = 0; 12 for(int t=0;t<trials;t++){ 13 double max_team = -1.0, max_all = -1.0; 14 for(int i=0;i<n;i++){ 15 double v = rand01(); 16 if(i < m){ if(v > max_team) max_team = v; } 17 if(v > max_all) max_all = v; 18 } 19 if(max_team >= max_all) team_wins++; 20 } 21 printf("n=%d, m=%d, empirical=%.6f, theory=%.6f\n", n, m, 22 (double)team_wins / trials , (double)m/n); 23 return 0; 24 } 4.3 Results Running the Python experiment yields values that match the theoretical m/n to within Monte Carlo error. A compact summary is shown in Table 1. 5 Table 1: Monte Carlo validation of m/n formula (trials=300000). Total players n Team size m Empirical / Theory 8 2 0.24873 / 0.25 8 3 0.37539 / 0.375 6 2 0.33257 / 0.33333 10 2 0.19948 / 0.2 Concrete examples Here are a few explicit numeric examples derived from the formula and the Monte Carlo runs:  Two collaborators versus six opponents (2 vs 6, total n= 8 ): P= 2/8 = 0.25 .  Two collaborators versus four opponents (2 vs 4, total n= 6 ): P= 2/6≈0.3333 .  Three collaborators versus ve opponents (3 vs 5, total n= 8 ): P= 3/8=0.375 . 4.4 Discussion The simple analytical result m/n shows collusion in the form of pooling players or coordinated play does not change the probability that the team contains the strongest raw hand  only how prize money or chips are distributed changes. Teaming aects expected revenue per player (team members can capture more of the pot by coordinating betting), but does not increase the probability that the team holds the best hand under independent random draws. The simulation validates the theory and can be extended to more realistic poker models (shared community cards, card removal eects, and betting strategy) by replacing the simple uniform draw with a deck-simulation engine (e.g., modeling 52-card deck draws and evaluating hand strengths). Reproducing the experiments The included scripts reproduce the Monte Carlo results locally:  Python: 1 python scripts/teaming_sim.py  C (compile and run): 1 gcc c/teaming_sim.c -O2 -o teaming_sim.exe 2 ./teaming_sim.exe 8 2 200000 5 Jacobian of the Equity Function The estimator is dierentiable with respect to (W, T, L) . Let N=W+T+L . Then: ∂E ∂W =1−E N,∂E ∂T =1 2N,∂E ∂L =−E N. Thus the Jacobian is: J(E) = 1−E N,1 2N,−E N. 6 0.0 0.2 0.4 0.6 0.8 1.0 Equity −4 −2 0 2 4 Sensitivity 1e−6 Jacobian of Poker Equity Estimator ∂E/∂W ∂E/∂T ∂E/∂L Figure 1: Jacobian sensitivity of the equity estimator. 6 Hessian The second derivatives yield a constant negative curvature in W and L : ∂2E ∂W 2=−1 N2,∂2E ∂L2=−1 N2,∂2E ∂T 2=−1 4N2. Thus the Hessian is negative semidenite, implying concavity of the estimator. 7 Convergence We have: b EN−E=OP(N−1/2). Since E2>En , variance satises: σ2 2=E2(1 −E2) N<En(1 −En) N=σ2 n. Heads-up converges strictly faster. 8 Normal Distribution Behaviour Monte Carlo estimator density: f(x) = 1 √2πσ2exp−(x−E)2 2σ2. Plots in Figures 2a2b illustrate narrowing variance for large N . See Appendix 12.3 for the Monte Carlo variance expansion and small-sample corrections. Table 2 summarizes sample statistics computed from the C++ dist sampling CSV (see also Figure 3). 7 0.0 0.2 0.4 0.6 0.8 1.0 Equity 0 1 2 3 4 5 6 7 8 Density Small N (higher variance) (a) Small sample size 0.0 0.2 0.4 0.6 0.8 1.0 Equity 0 5 10 15 20 25 30 35 40 Density Large N (lower variance) (b) Large sample size Figure 2: Normal approximation to the equity distribution for dierent sample sizes. 0.0 0.2 0.4 0.6 0.8 1.0 Equity 0 2 4 6 8 10 12 14 Density C++ fast-sim hand equity distribution (n=6) normal fit mu=0.116 sigma=0.104 Figure 3: Empirical distribution of hand equity for randomly sampled two-card holes against 5 opponents (n=6) using the C++ dist-mode sampled CSV. Statistic Value Sample size 200 Mean 0.115780 Std (sample) 0.103796 Median 0.087500 Q1 0.026000 Q3 0.193000 95 Skewness 0.728044 Excess Kurtosis -0.596361 Jarque-Bera statistic 20.632 Table 2: Sample statistics for hand equity distribution (nplayers=6) 8 8.1 Hypothesis testing and normality checks The central limit theorem justies using a Normal approximation for the sample mean of Monte Carlo estimators, but the underlying empirical distribution of per-hole equities may exhibit skew or heavy tails. We compute summary statistics  mean, sample standard deviation, median, interquartile range, skewness, and excess kurtosis  and apply the JarqueBera test for normality (a two moment-based test combining skewness and kurtosis) to the sampled equities. The JarqueBera statistic is dened as JB =n 6S2+(K−3)2 4, where S and K are the sample skewness and kurtosis respectively; under the null of normality it is approximately chi-square distributed with 2 degrees of freedom. A small p-value (e.g. p < 0.05 ) rejects the normality assumption. Results for the C++ sample indicate the JarqueBera statistic and p-value reported in Table 2. For hypothesis testing about means across dierent table sizes (e.g. comparing n= 2 versus n= 6 ), the two-sample t-test can be used (assuming independent samples and approximately equal variances). A more robust approach uses Welch's t-test for unequal variances. If users desire exact permutation tests or nonparametric checks, those are easy to implement against the generated CSVs. Note on Monte Carlo standard errors: for any quantity estimated as a mean across independent Monte Carlo trials, the standard error of the estimator is bσ/√N and a 95% condence interval for the mean is given by ¯x±1.96 bσ/√N . These intervals assume independent draws of the underlying samples and are valid asymptotically. 8.2 Action probability model Given an approximately normal distribution of equity for a given hole and table size with mean E[b E] = µ and standard deviation σ , we model action probabilities using decision thresholds. Let Fold if X < l, Check if l≤X < r, Raise if X≥r. Under the normal approximation, P( Raise ) = 1 −Φr−µ σ, P( Check )=Φr−µ σ−Φl−µ σ, P( Fold )=Φl−µ σ, where Φ(·) denotes the standard normal CDF. These formulas provide closed-form estimates of action probabilities using the distributional t. The script scripts/hand_eval.py computes the empirical mean and standard deviation and generates histogram plots (e.g., Figure ?? ). 9 Tensor Field Representation Let E(W, T, L) = W+1 2T W+T+L. The tensor eld (W, T )7→ E(W, T, 1−W−T) is visualised in Figure 4. 9 0.0 0.2 0.4 0.6 0.8 1.0 Equity −4 −2 0 2 4 Sensitivity 1e−6 Jacobian of Poker Equity Estimator ∂E/∂W ∂E/∂T ∂E/∂L Figure 9: Jacobian matrix (graphical depiction). 0.0 0.2 0.4 0.6 0.8 1.0 Equity −2.50 −2.25 −2.00 −1.75 −1.50 −1.25 −1.00 −0.75 Curvature 1e−11 Hessian Curvature of Poker Equity Estimator ∂²E/∂W² ∂²E/∂T² ∂²E/∂L² Figure 10: Hessian matrix (graphical depiction). Proof of Theorem Substitute E2=1 2 and En=1 n : Adv(n) = 1 2−1 n 1 n 100 = n 2−1100. Proof of Corollary Survival across k hands satises: Sk= k Y i=1 Eni. Replacing Eni with E2 increases every term, hence S(HU) k> S(n>2) k. 16 W 0.0 0.2 0.4 0.6 0.8 1.0 T 0.0 0.2 0.4 0.6 0.8 1.0 E 0.0 0.2 0.4 0.6 0.8 3D Tensor Field E(W,T,L) Figure 11: Tensor eld visualisation used in the theoretical derivation. References Metropolis, N. & Ulam, S. (1949), `The monte carlo method', Journal of the American Statistical Association . Wilson, A. (2019), Game Theory and Poker , Academic Press. 17