Seed reproducibility is not model validity
Reproducing a run proves that the algorithm and inputs can regenerate the same sequence. It does not show that probabilities, dependencies, payoffs, or omitted pathways represent reality.
Probability
Decision Tree Simulation Calculator
Run a reproducible seeded Monte Carlo simulation of a four-leaf decision tree and compare simulated payoff, success, and loss rates with analytical values.
SEEDED TREE SIMULATION
Use reproducible random trials to test a declared tree-not to replace its exact benchmark
This calculator maps a seeded pseudo-random sequence into four cumulative decision-tree intervals. It reports simulated mean payoff, success rate, and loss rate against the tree's analytical expectation, helping modelers test sampling behavior and reproducibility before scaling to structures that cannot be enumerated easily.
Simulated mean payoff-
Analytical mean payoff-
Simulation minus analytical-
Simulated success rate-
Simulated loss rate-
Simulated successes-
SEEDED TREE SIMULATION
Simulation reconciliation ledger
Use the analytical-versus-simulated gap as a convergence diagnostic; do not choose an option because one finite random run happened to be favorable.
| Simulated metric | Observed numerator / total | Trial denominator / reference | Calculated value | Unit |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
Ui = LCG(seed)i; choose leaf j where Fj-1 <= Ui < Fj; x-barsim = (1/N)sumXi; error = x-barsim - sumpjxj
A deterministic linear congruential generator produces uniform numbers from the seed. Each draw is assigned to the first terminal leaf whose cumulative path probability exceeds it. Trial payoffs and classifications are accumulated, divided by N, and compared with exact enumeration of the same four-leaf model.
HOW TO USE THIS MODEL
Run a simulation that another analyst can reproduce
- Validate the branch structure, probabilities, and common payoff basis before introducing random sampling.
- Choose a trial count large enough for the precision needed, recognizing that rare leaves require more trials to appear consistently.
- Record an integer seed with the scenario so reviewers can reproduce exactly the same pseudo-random sequence.
- Compare the simulated mean and rates with analytical values; investigate large discrepancies before interpreting individual trial outcomes.
- Repeat with additional documented seeds only as a sensitivity check, not until a preferred answer appears, and retain all planned runs.
SEEDED TREE SIMULATION FUNDAMENTALS
What seeded Monte Carlo adds to a finite tree
- Pseudo-random sequence
- A deterministic algorithm produces numbers that behave like random draws for simulation; the seed selects the reproducible sequence.
- Cumulative interval mapping
- Each leaf occupies a segment of the unit interval equal to its path probability.
- Monte Carlo error
- A finite simulation differs from the analytical expectation because the sampled leaf frequencies are not exactly their probabilities.
- Convergence
- As independent trials increase, simulated averages tend to move toward their model expectations, although no finite run guarantees a chosen tolerance.
- Analytical benchmark
- Exact leaf enumeration supplies a zero-sampling-error reference for this small tree and is essential for validating the simulation implementation.
MODEL AND FORMULA
How a seeded uniform draw becomes a terminal payoff
Ui = LCG(seed)i; choose leaf j where Fj-1 <= Ui < Fj; x-barsim = (1/N)sumXi; error = x-barsim - sumpjxj
A deterministic linear congruential generator produces uniform numbers from the seed. Each draw is assigned to the first terminal leaf whose cumulative path probability exceeds it. Trial payoffs and classifications are accumulated, divided by N, and compared with exact enumeration of the same four-leaf model.
DEEPER ANALYSIS
Simulation practices that protect the decision record
Rare events need targeted precision planning
A large overall N can still contain very few observations of a low-probability severe leaf. Precision for tail metrics should be assessed directly; crude Monte Carlo may be inefficient for very rare events.
Stopping rules can bias interpretation
Watching the running mean and stopping when it looks favorable converts random variation into selection bias. Set trial count, seeds, and convergence criteria before viewing the result.
WORKED DECISION CASES
Two reasons to simulate the same tree
Implementation verification
With the default four leaves, 20,000 trials and seed 12345 produce an exactly reproducible sequence. The simulated mean is compared with the analytical 59.6 payoff benchmark; a modest difference is sampling noise, while a persistent large error suggests a mapping or accumulation defect.
Prototype for a larger dependent model
A team begins with the enumerable tree to verify classification, seeding, and reporting. Only after the simulation matches exact leaf probabilities does it add correlated drivers and continuous costs that remove the simple analytical solution.
TECHNICAL LANGUAGE
Monte Carlo tree simulation terminology
- Seed
- An integer initial state that makes the pseudo-random sequence reproducible.
- Trial
- One simulated realization that ends at a single terminal leaf.
- Linear congruential generator
- A recurrence that updates an integer state and scales it to a uniform value in [0,1).
- Cumulative threshold
- The upper unit-interval boundary used to map a uniform draw to a leaf.
- Simulation error
- The difference between a finite-run estimate and the model's analytical quantity.
- Convergence diagnostic
- A comparison used to judge whether sampling variability is acceptably small for the stated purpose.
EVIDENCE AND DATA LINEAGE
Keep seeds, run plans, code version, and analytical controls
Archive the complete input scenario, leaf ordering, random-number algorithm, integer seed, trial count, software version, execution date, and both simulated and analytical outputs. For a multi-run study, predefine the seeds or seed-generation procedure and retain every run. Validate the uniform generator and mapping independently before relying on simulation results in a consequential model.
LIMITS AND EXCLUSIONS
Simulation limitations specific to this implementation
- The built-in generator is intended for reproducible educational and planning simulation, not cryptography or high-assurance random-number certification.
- The page caps trials at 2,000,000 and models only four fixed terminal leaves with independent draws.
- Finite-run frequency can omit rare leaves or produce unstable tail estimates even when the mean appears converged.
- Simulation propagates entered assumptions; it does not discover missing branches, correct biased probabilities, or prove causal mechanisms.
RELIABLE SOURCES
References for this model and its decision limits
FREQUENTLY ASKED QUESTIONS
Questions about seeded decision-tree simulation
Why does the same seed return the same result?
The generator is deterministic. Identical inputs, seed, trial count, leaf order, and implementation produce the same sequence and aggregates.
Should I keep increasing trials until simulation equals the analytical mean?
No finite run must equal it exactly. Choose a precision target in advance and judge the discrepancy against sampling variability and the decision's sensitivity.
Why simulate a tree that can be calculated exactly?
The exact tree is a valuable validation case for simulation mechanics. It also provides a transparent prototype before adding structure that requires simulation.
Can I try many seeds and report the best payoff?
That is selective reporting. Predefine and retain seeds or summarize all planned runs; otherwise random favorable variation is mistaken for evidence.
Does a low mean error prove the tail is accurate?
No. Positive and negative frequency errors can cancel in the mean. Inspect success, loss, and rare-leaf frequencies separately when they support decisions.
Can this model represent correlation between trials?
No. Each trial is drawn independently from the same fixed leaf distribution. Time dependence, common shocks, and parameter uncertainty require an expanded model.
IMPORTANT NOTE
A random-looking output is not independent evidence
This simulation samples the declared tree and is benchmarked against its exact expectation. It is not a validated forecasting system, certified random-number implementation, or substitute for evidence supporting the tree's probabilities, dependencies, consequences, and decision boundaries.