The two Lagrange multipliers in Mucho+MOTA's MaxEnt tournament simulator that internally control the cash-frequency and heads-up sufficient statistics. Lambda is the dual representation of the (CF, HU) archetype parameters.
Lambda (λ₁, λ₂) are the two Lagrange multipliers in SALSA’s maximum-entropy construction. They are the dual representation of the two structural sufficient statistics — cash frequency (CF) and heads-up edge (HU) — that define a player’s archetype on the 2D skill plane.
Mathematically, the MaxEnt finish distribution p(i) over positions i in a field of N satisfies:
p(i) ∝ exp(λ₁ · φ_CF(i) + λ₂ · φ_HU(i))
where φ_CF and φ_HU are the indicator/score functions for the cash-position and heads-up-position constraints. Solving the constrained optimization finds the (λ₁, λ₂) pair such that the expected CF and HU under p match the input archetype values. This is the standard exponential-family / Gibbs-distribution form that drops out of Jaynes’ MaxEnt program.
In practice, lambda values are an internal representation — users see archetypes (e.g., “ROI 25% / variance 2.5σ”) rather than raw lambdas. But lambdas are what get cached, transferred between buy-in buckets, and updated by Bayesian inference as new tournament results arrive. The shrinkage rule operates on lambdas, not on the surface ROI numbers.
This is a paid-tier internal — the free version of Mucho+MOTA exposes neither lambdas, the SALSA solver, nor the per-bucket transfer functions.
A statistical model that infers a player's tournament-poker skill from observed results by combining a population prior with personal history, returning a probability distribution rather than a point estimate.
The bet-sizing formula that maximizes the long-run growth rate of a bankroll. In tournament poker it determines what fraction of bankroll to risk per buy-in given expected ROI, variance, and bankroll size.
In poker staking, the running balance a backed player owes their backer from past unprofitable sessions; future winnings cover this balance before any profit chop occurs.
A model that converts tournament chip stacks into expected dollar equity, used for final-table deal-making, bubble play, and any decision where chip EV diverges from $-EV.
Tournament formats where part of the buy-in is awarded for eliminating other players. Three common variants — Standard knockout, Progressive (PKO), and Mystery — have meaningfully different variance profiles.
A capital reserve a backed player posts so the backer can recover funds if the player abandons the arrangement mid-makeup. In Mucho+MOTA the bond is invisible to honest players — settlement happens automatically via oracle.
A staking package where multiple investors buy shares of a player's action across a defined slate of tournaments, with profit and makeup pooled at the slate level rather than per-tournament.
Mucho+MOTA's MaxEnt-based tournament simulator that produces an entire ROI distribution from two structural parameters (cash frequency and heads-up edge), the field size, the payout vector, and the rake.
The vector of prize amounts assigned to each finishing position in a tournament. The shape of this vector — top-heavy versus flat — dominates ROI variance more than skill at most stake levels.
In tournament poker, average net profit per dollar of buy-in invested. ROI is the headline skill metric but is dominated by variance over realistic sample sizes — confidence intervals, not point estimates, are what matter.
The dispersion of tournament outcomes around expected value. In tournament poker variance is heavy-tailed and dominates short-run results — most observed deviation from expected ROI is variance, not skill change.