Championship Probabilities

16 teams, 4 rounds, 240 transitions. Three independent methods — ODE simulation, Monte Carlo sampling, and closed-form analytical propagation — all compute the same probabilities from the incidence matrix C.

Team Seed Analytical ODE MC (10k)
Duke128.76%28.32%29.4%
Arizona118.13%17.86%17.9%
Michigan111.21%11.04%11.1%
Houston210.25%10.09%9.8%
Florida18.86%8.73%8.9%
Purdue25.39%5.31%5.3%
Iowa State23.56%3.50%3.5%
UConn23.33%3.28%3.3%
Illinois32.57%2.53%2.5%
Gonzaga32.16%2.13%2.2%
Michigan St31.52%1.50%1.5%
St John's51.36%1.34%1.3%
Virginia31.13%1.11%1.1%
Alabama40.95%0.93%0.9%
Kansas40.49%0.48%0.5%
Nebraska40.33%0.33%0.3%

Three Nets, Two Methods

We built three Petri net models with different topologies. The surprise: coupled transitions don't cause ODE/MC disagreement. Binary tokens do.

Three test case net structures

When Do ODE and Monte Carlo Agree?

Two axes determine agreement: coupling (how many inputs per transition) and token count (binary vs multi-token). Only the bottom-right cell disagrees.

Agreement characterization matrix

The Incidence Matrix as Bridge

The incidence matrix C defines all three methods. ODE flows probability mass through C continuously. MC routes tokens through C discretely. The analytical formula reads the answer directly from C.

Incidence matrix as universal bridge

Three Methods Converge

Championship probabilities from analytical (exact), ODE (asymptotic), and Monte Carlo (sampled) methods. All agree because they compute the same function of C.

Three-way comparison bar chart

Data Sources

Team strength metrics normalized to 0–100 scale from three sources:

  • Barttorvik — Adjusted offensive/defensive efficiency, Barthag, SOS, WAB
  • NCAA API — Win-loss records, conference records
  • Sports-Reference — Minutes distribution, bench scoring, roster depth

Five facets: Offense (20%), Defense (25%), Record (20%), Momentum (20%), Depth (15%). Win probability via logistic function: P(A beats B) = 1 / (1 + exp(-0.15 * (str_A - str_B))).

Further Reading