Strategic Subgames (Nash Equilibria)¶
Strategic behavior in the NHRA is modeled as a set of non-cooperative games between the Commonwealth (Principal), States (Agents), and LHNs (Operators).
1. The Negotiation Game (Hold-Up)¶
Every 5 years, at agreement expiry, a high-stakes bargaining game occurs.
Payoff Matrix (Illustrative)¶
| State: Agree (A) | State: Hold-Up (H) | |
|---|---|---|
| Cth: Concede (C) | (0.9, 1.2) | (0.7, 1.5) |
| Cth: Enforce (E) | (1.0, 1.0) | (0.2, 0.4) |
- Nash Equilibrium: Under high system pressure, the state's threat of failure becomes credible, forcing the Commonwealth to concede higher funding shares (\(\alpha\)).
2. Cost-Shifting Game¶
Models the incentive to shift patients between Commonwealth-funded primary care and State-funded hospital care.
\[
Payoff_{State} = B_{efficiency} - C_{pressure} + G_{shifting}
\]
If \(G_{shifting} > C_{penalty}\), the State chooses to 'Shift' rather than 'Invest'.
3. Solver Implementation¶
We use two primary solvers:
- Discrete Nash: Brute-force enumeration for small 2x2 games.
- Quantal Response Equilibrium (QRE): A JAX-native Logit solver for boundedly rational agents.
4. Evidence Grounding¶
- Strategies: Derived from the Blinded Qualitative Mapping of the NHRA statutory text (see
publications/P1). - Payoffs: Stylised based on vertical fiscal imbalance metrics.