Skip to content

Game Theory Models Reference

This page provides a comprehensive reference for all stage games in the NHRA Game Theory toolkit. Each game captures a specific strategic tension in healthcare funding negotiations.

Overview

The simulation uses nine stage games that represent key decision points in NHRA negotiations. Each game is a two-player normal-form game solved for Nash equilibria.

Master Index

Game Row Actions Column Actions Strategic Tension
Definition R (Realism) / E (Efficient) R / E What counts as "efficient"
Bargaining A (Agree) / D (Defer) A / D CAP convergence negotiation
Cost Shifting I (Invest) / S (Shift) I / S Upstream vs downstream investment
Discharge Coordination C (Coordinate) / F (Fragment) C / F Hospital discharge planning
Governance Integration I (Integrate) / S (Separate) I / S UCC and governance structures
Aged Care Interface C (Coordinate) / F (Fragment) C / F ACAT handoff coordination
NDIS Interface C (Coordinate) / F (Fragment) C / F NDIS transition planning
Coding/Audit H (Honest) / U (Upcode) L (Light) / T (Tight) Activity-based funding compliance
Compliance T (Tight) / L (Light) T / L Regulatory intensity choice

Game Parameters

All games receive a GameParams dataclass with the following state variables:

@dataclass
class GameParams:
    pressure: float           # System pressure index (≥~0.8 typical)
    efficiency_gap: float     # Divergence from NEP indexation (0..~0.6)
    discharge_delay: float    # Excess length of stay index (1.0 = baseline)
    political_salience: float # Media/political attention (0..1)
    audit_pressure: float     # Audit intensity (0..1)
    cost_shifting_intensity: float  # Fee-for-service incentives (0..1)
    political_capital: float  # Government negotiating capital (0..1)

Funding Negotiation Games

Definition Game

Strategic tension: What counts as "efficient"?

  • Row player: Commonwealth framing / policy narrative
  • Column player: State implementation reality
Actions Description
R (Realism) Acknowledge cost reality; reduce efficiency gap drift
E (Efficient) Strict efficient-price framing; gap drifts upward

Payoff drivers:

  • realism_benefit = 0.5 + 0.8 × efficiency_gap + 0.4 × (pressure - 1.0)
  • realism_cost = 0.25 + 0.35 × political_salience
  • strict_benefit = 0.35 + 0.45 × political_salience
  • strict_cost = 0.30 + 0.50 × pressure

Equilibrium characteristics: Under high pressure and efficiency gap, (R,R) becomes the welfare-dominant equilibrium. Under low pressure with high political salience, (E,E) may be preferred.

Bargaining Game

Strategic tension: Converge on CAP target or defer?

Actions Description
A (Agree) Converge toward target nominal share
D (Defer) Defer/escalate; slow movement with conflict costs

Payoff drivers:

  • converge_gain = 0.45 + 0.25 × (pressure - 1.0) + 0.20 × political_capital
  • conflict_cost = 0.55 + 0.90 × pressure
  • narrative_gain = 0.25 + 0.50 × political_salience

Equilibrium characteristics: High pressure increases conflict costs, making (A,A) more attractive. Political capital enhances agreement payoffs.

Cost Shifting Game

Strategic tension: Invest upstream or shift costs downstream?

Actions Description
I (Invest) Invest in upstream prevention/primary care
S (Shift) Shift costs to downstream acute care

Payoff drivers:

  • coop_gain = 0.55 + 0.45 × (1.0 - efficiency_gap)
  • shift_gain = 0.35 + 0.75 × efficiency_gap + 1.0 × cost_shifting_intensity
  • pr_cost = 0.65 × pressure

Equilibrium characteristics: High cost_shifting_intensity (fee-for-service incentives) shifts equilibria toward (S,S). Pooled funding (low CSI) promotes coordination.

Discharge and Interface Games

Discharge Coordination Game

Strategic tension: Coordinate discharge planning or fragment care?

Actions Description
C (Coordinate) Joint discharge planning; shared responsibility
F (Fragment) Fragmented care; siloed decision-making

Payoff drivers:

  • benefit = 0.70 + 0.80 × excess_delay (where excess_delay = max(0, discharge_delay - 1.0))
  • cost = 0.30 + 0.10 × (1.0 - min(1.0, excess_delay))
  • pr_penalty = 0.45 × pressure

Equilibrium characteristics: High discharge delay increases coordination benefits. Under pressure, fragmentation becomes increasingly costly.

Governance Integration Game

Strategic tension: Integrate governance (e.g., UCCs) or maintain separation?

Actions Description
I (Integrate) Unified governance structures
S (Separate) Maintain separate governance

Payoff drivers:

  • safety_gain = 0.55 + 0.35 × (pressure - 1.0)
  • integration_cost = 0.20 + 0.35 × political_salience
  • fragmentation_risk = 0.40 + 0.60 × pressure

Equilibrium characteristics: High pressure increases both safety gains from integration and fragmentation risks, generally favouring integration.

Aged Care Interface Game

Strategic tension: Coordinate ACAT handoffs or fragment?

Actions Description
C (Coordinate) Coordinated aged care transitions
F (Fragment) Fragmented handoffs

Payoff drivers:

  • coord_benefit = 0.6 + 0.4 × (discharge_delay - 1.0)
  • frag_cost = 0.5 × pressure

Equilibrium characteristics: A symmetric coordination game where both players prefer (C,C) when discharge delays are high.

NDIS Interface Game

Strategic tension: Coordinate NDIS transitions or fragment?

Actions Description
C (Coordinate) Coordinated NDIS transitions
F (Fragment) Fragmented handoffs

Payoff drivers:

  • coord_benefit = 0.5 + 0.5 × (discharge_delay - 1.0)
  • frag_cost = 0.6 × pressure

Equilibrium characteristics: Similar structure to Aged Care, but with slightly higher fragmentation costs under pressure.

Compliance and Audit Games

Coding/Audit Game

Strategic tension: Provider coding honesty vs auditor vigilance

  • Row player: Healthcare provider
  • Column player: Auditor/regulator
Row Actions Description
H (Honest) Accurate activity coding
U (Upcode) Inflate activity codes for higher reimbursement
Column Actions Description
L (Light) Minimal audit scrutiny
T (Tight) Intensive audit regime

Payoff drivers:

  • upcode_gain = 0.3 + 0.7 × efficiency_gap
  • penalty = 0.8 × audit_pressure
  • audit_cost = 0.2
  • recovery = 0.4 × efficiency_gap

Equilibrium characteristics: Mixed equilibrium common. High audit pressure deters upcoding; high efficiency gap increases temptation.

Compliance Game

Strategic tension: Tight vs light regulatory compliance

Actions Description
T (Tight) Stringent compliance; higher admin cost, less leakage
L (Light) Relaxed compliance; lower cost, more leakage

Payoff drivers:

  • leakage = 0.40 + 0.70 × efficiency_gap
  • admin = 0.18 + 0.45 × audit_pressure

Equilibrium characteristics: Trade-off between administrative burden and revenue protection. High audit pressure makes tight compliance more attractive.

Nash Equilibrium Solver

All games return a TwoPlayerGame object that supports:

from nhra_gt.subgames.nash import solve_all_equilibria

game = definition_game(GameParams(pressure=1.0, efficiency_gap=0.3, ...))
equilibria = solve_all_equilibria(game)

for eq in equilibria:
    print(f"Row: {eq.row_strategy}, Col: {eq.col_strategy}")
    print(f"Expected payoffs: {eq.row_payoff:.3f}, {eq.col_payoff:.3f}")

The solver finds all pure and mixed Nash equilibria using support enumeration.

References