transition.matrix.ordered: 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 → 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)

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.

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")`. 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.

Examples

Run this code

# 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"))