Game-Theoretic Simulation Checklist (Batch 5): Stage-Game Timeline & State Vector¶

Source: User provided text (December 2025) Context: Formal stage-game specification, compact state vector, transition functions, and payoff templates for an agent-based or Markov game implementation.

1) Time structure¶

Layer	Index	Purpose	What changes here
Operational	month `t`	queues, capacity use, coding/claims, cashflow	demand shocks, admissions/LOS, coding, interim payments, public signals
Financial	year `Y`	reconciliation + cap/redistribution + true-ups	annual true-up, cap kink/redistribution, annual audit outcomes
Constitutional	agreement cycle `G` (e.g., 5y)	renegotiation / schedule extension	contribution shares, cap design, side-payments, rule tweaks

2) Monthly stage game (within-year repeated game)¶

Stage timeline (one month `t`)¶

Stage	Move order	Information at this point	Decision variables (examples)	State updates (examples)
0	World	Everyone sees last month’s published metrics + own internal data	–	advance calendars; apply lags; roll forward cumulative counters
1	Nature	Demand shock realised: `ε_t` (may be partially observed)	–	arrival streams for ED + elective demand; inflation/price drift (if monthly)
2	State/Territory (principal)	Knows own cash position, queues, capacity, election clock; uncertain about others’ internal constraints	`B_jt` internal allocations to LHNs; elective target rules; surge directives; capacity investment orders (with lags); disclosure stance	internal budgets/targets set; investment pipeline updated
3	LHNs / Hospitals (agents)	Observe own queues, staffing, demand signals; imperfectly observe future audits and next-year rules	admissions thresholds; scheduling; LOS management; substitution settings; coding governance effort `g_pt`; “classification shifting” choices (ABF vs block vs other stream)	episodes generated; throughput; queue changes; quality pressure metrics
4	Claims submission	Claims reflect coding intensity `θ_pt` within feasibility bounds	choose coding intensity (bounded); documentation effort; claim timing	NWAU computed from episodes + coding; measurement error introduced
5	NHFP payment engine (referee)	Applies deterministic rules to submitted activity	–	interim payments; cash positions; cumulative cap-used counters
6	Audit / integrity (optional explicit player)	Sees anomaly signals; chooses targeting	audit targeting `A_t`; thresholds	detections/repayments/penalties; update perceived audit risk
7	Dispute & Arbitration (Clauses 127-130)	Unresolved audit/payment disputes	Move to formal arbitration; legal evidence	final true-up; political capital hit
8	Public signal release	Everyone observes published signals (some lagged/aggregated)	–	belief updates; reputational payoffs (rankings) update
9	Transition	–	–	carry-forward state to next month

3) Annual stage (reconciliation + cap/redistribution + reset)¶

At end of each year Y (or at a reconciliation boundary), run this “year-end game within the repeated game”:

Step	What happens	Inputs	Outputs/state updates
A1	Annual reconciliation	submitted vs revised activity; lags; dispute outcomes	true-up payments, revised entitlements, corrections to cumulative counters
A2	Cap + redistribution	cap design parameters; total growth and shares	marginal funding kink realised; redistribution amounts assigned
A3	Audit outcomes finalised	audit pipeline, disputes, repayments	net penalties/repayments; update compliance reputations
A4	Reputational true-up	annual performance league tables	update `reputation_score`; political utility impacts
A5	Budget closure	state fiscal constraints; Commonwealth fiscal stance	end-year deficits/surpluses; political utility impacts
A6	Next-year expectations formed	published year-end results; policy chatter	belief priors for next year (NEP path, enforcement stance, negotiation probabilities)

4) Agreement-cycle stage (renegotiation / extension bargaining)¶

Run every G years (or whenever a “schedule extension / crisis” trigger occurs):

Stage	Game form	Players	What’s chosen	What updates
G1	Alternating-offers bargaining (deadline)	Commonwealth ↔ States/Territories	contribution shares; cap design; side-payments; rule changes; integrity regime	new parameter set for next cycle
G2	Outside options realised	same	walk-away / interim extension	fallback rule-set (extension), political costs
G3	Ratification & signalling	same	public commitments	belief resets + reputational terms

5) Compact state vector (code-ready)¶

Global / rule-set state `R_t`¶

Symbol	Type	Meaning
`NEP_t`	float	national efficient price (or price index)
`W_t`	dict	weight system / grouper parameters for NWAU calculation
`elig_t`	dict	eligibility rules (ABF vs block vs other stream)
`alpha_t[j]`	float	Commonwealth share / contribution parameters by jurisdiction
`cap_t`	dict	cap design params (rate, base, redistribution rule, kink shape)
`recon_t`	dict	reconciliation lag structure, revision probabilities, dispute rules
`audit_policy_t`	dict	baseline audit intensity, targeting logic, penalty schedule
`bargain_state_t`	dict	where you are in the agreement cycle (time-to-deadline, offers on table)

For each jurisdiction `j`: fiscal + political + beliefs `J_t[j]`¶

Symbol	Type	Meaning
`cash_jt`	float	cash position / liquidity
`own_spend_jt`	float	state-funded spend this year-to-date
`cum_nwau_jt`	float	cumulative NWAU (or eligible NWAU) this year-to-date
`cap_used_jt`	float	cap consumption metric needed for kink/redistribution
`queue_metrics_jt`	dict	headline KPIs (elective breaches, ED LOS, ramping proxy, etc.)
`pol_jt`	dict	political state: election clock, salience weights, reputation
`beliefs_jt`	dict	beliefs about demand growth, others’ constraints, audit risk, next-year rules

For each provider/LHN `p` within `j`: operations + coding + capacity `P_t[p]`¶

Symbol	Type	Meaning
`cap_pt`	dict	capacity state (beds/ICU/theatres/workforce)
`inv_pipe_pt`	dict	capacity investments in pipeline with lags
`q_ed_pt`	float	ED queue state (or arrival + backlog)
`q_el_pt[u]`	float	elective queue by urgency class `u`
`mix_pt`	vector	case-mix distribution over DRGs/service lines
`cost_pt`	dict	cost structure (fixed/variable, efficiency parameter)
`theta_pt`	float	coding intensity (bounded)
`g_pt`	float	governance/documentation effort (affects feasible coding + audit risk)
`quality_pt`	dict	endogenous quality state (pressure → adverse events proxy)
`contract_pt`	dict	internal contract terms/targets from jurisdiction
`beliefs_pt`	dict	local beliefs about demand, audits, budgets

Lags / measurement layer `M_t`¶

Symbol	Type	Meaning
`claims_lag_t`	queue	pipeline of submitted claims pending reconciliation
`signal_lag_t`	queue	what gets published when (lags/aggregation)
`revision_model_t`	dict	probability of later revisions by class/provider

6) Transition functions (what updates the state)¶

Function	Inputs	Outputs
`demand_step(S_t, seed)`	`S_t`	arrivals `λ_t`, realised shocks `ε_t`
`policy_step(S_t)`	`S_t`	updates to any monthly policy knobs (if any)
`jurisdiction_move(S_t)`	`S_t`	`B_jt`, targets, directives, investment orders
`provider_move(S_t)`	`S_t`	admissions/scheduling/LOS decisions; `theta_pt`, `g_pt`, stream-shifting
`ops_queue_update(S_t)`	decisions + capacity + arrivals	updated queues, throughput, congestion metrics
`nwau_calc(S_t)`	episodes + coding + rules	`nwau_pt`, eligible splits by stream
`payment_engine(S_t)`	eligible NWAU + cap state + rules	interim payments, cash updates, cap_used updates
`audit_step(S_t, seed)`	activity + anomalies + audit_policy	detections, penalties/repayments, updated perceived risk
`publish_signals(S_t)`	signal lag rules	observed signals for belief updates
`belief_update(S_t)`	observed signals + priors	updated beliefs (Bayesian or heuristic)
`year_end_reconciliation(S_Y)`	claims pipeline + revisions/disputes	true-up payments, corrected NWAU, redistribution, reset counters

7) Invariants (sanity checks you should enforce)¶

Invariant	Check
Non-negative entitlements	payments ≥ 0 unless explicit recovery/repayment
Mass-balance of activity	NWAU derived only from generated episodes; no “free” NWAU
Capacity feasibility	throughput ≤ feasible capacity given staffing and beds
Bounded coding	`theta_pt` within [min,max] and constrained by documentation/governance
Queue consistency	queues update with arrivals − served + deferrals; never negative
Cap accounting consistency	cap_used updates match eligible growth definition exactly
Reconciliation traceability	every year-end adjustment maps to a claim in the lag pipeline