transition.matrix.ordered: Build column-stochastic transition matrix for ordered verdict...

In sate: Scientific Analysis of Trial Errors (SATE)

Build column-stochastic transition matrix for ordered verdict options

Description

Constructs the full **column-stochastic** Markov transition matrix \(P\) for a jury deliberation model with an **ordered** set of verdict options (least to most punitive). Transient states are all compositions of 'jury_n' jurors across the 'verdict_options'; absorbing states are the 'K' unanimity vertices (one per verdict), appended at the end in the same order as 'verdict_options'.

The transition from a transient state is built by applying your **2-option step rule** independently at each adjacent *cut* between options and combining those suggestions with **equal weight across cuts**. For cut 'r' (between options 'r' and 'r+1'), let 'g = sum(counts[(r+1):K])' be the number on the **more punitive** side; compute

p_\text{up} = \left(0.5\frac{g-1}{n} + 0.25\right)^2,\quad p_\text{down} = \left(1 - 0.5\frac{g-1}{n} - 0.25\right)^2,\quad p_\text{stay} = 1 - p_\text{up} - p_\text{down}.

Map "up" to moving one juror across the cut toward the more punitive option, "down" toward the less punitive option; pool all "stay" mass (and any illegal move mass at boundaries) into the **self-loop** so each column still sums to 1.

Usage

transition.matrix.ordered(jury_n, verdict_options, digits = NULL)

Arguments

jury_n	Integer. Size of the jury (number of jurors), 'jury_n >= 1'.
verdict_options	Character vector of **ordered** verdict labels from least to most punitive, e.g. 'c("NG","M2","M1")' or 'c("NG","M3","M2","M1")'. Labels must be non-missing, non-empty, and unique. The order defines which options are adjacent.
digits	Optional integer. If supplied, round the returned matrix to this many decimals and then re-normalize each column to remain column-stochastic. Defaults to 'NULL' (no rounding).

Details

* Absorbing columns (the last 'K') are identity columns (unanimity stays put). * Self-loops collect "stay" mass from all cuts and any mass from moves that are illegal at boundaries (e.g., trying to move from an empty option). * Providing 'digits' is meant for tidy printing; for numerical work you may prefer to leave 'digits = NULL' to keep full precision.

Value

A column-stochastic matrix 'P' of size S \times S, where S = \binom{n+K-1}{K-1} is the number of compositions of 'jury_n' into 'K' parts (all states), ordered with **transients first** and then the 'K' unanimity absorbing states in the order of 'verdict_options'. The matrix carries metadata on 'attr(P, "meta")' as a list with elements:

• 'states' - list of length-'K' integer count vectors for each state, in column order;

• 'idx' - environment mapping comma-joined count vectors to column/row indices;

• 'T', 'S', 'K', 'n' - counts (transients, total states, #options, #jurors);

• 'verdict_options' - the label vector you supplied.

See Also

prob.ordered.verdicts for solved absorption probabilities (including appended unanimity starts) built on top of this transition matrix.

Examples

# 3 jurors, 3 options (NG < M2 < M1), equal cut weights
P <- transition.matrix.ordered(3, c("NG","M2","M1"))
dim(P); colSums(P)                        # columns sum to 1
attr(P, "meta")$verdict_options           # labels carried in metadata

# Tidy print:
transition.matrix.ordered(3, c("NG","M2","M1"), digits = 3)

# 4 options (NG < M3 < M2 < M1)
P4 <- transition.matrix.ordered(3, c("NG","M3","M2","M1"))

sate documentation built on March 6, 2026, 1:07 a.m.