Game Theory

Game Theory and Strategic Interaction in Health Care

Game theory gives health economics a way to analyse choices whose consequences depend on what other decision makers do, such as a payer and a manufacturer settling a launch price, hospitals sharing a fixed budget, or prescribers drawing on the shared effectiveness of antibiotics. Because each party's best move depends on the others' expected moves, a model that holds their behaviour fixed can mispredict what a policy achieves.

Players, strategies, payoffs and equilibrium

McAdams and colleagues describe a game as any situation with several decision makers, the players, whose choices affect one another. A model specifies the players, their strategies, what each knows and the payoff from every combination of strategies; Ntais and Talias, in a 2026 review of free riding in health care, add how often the players meet and whether agreements can be enforced.

Colman and colleagues note that payoffs are utilities measured on an interval scale, so dividing them all by a positive constant leaves a game's strategic properties unchanged. In health applications a payoff can be a manufacturer's profit, a payer's net monetary benefit, or the risk of serious illness among a doctor's own patients.

The central solution concept is the Nash equilibrium, a combination of strategies from which no player gains by deviating alone. In the notation of Colman and colleagues, the condition is as follows.

$$\pi_i(p_i^{*}, p_{-i}^{*}) \ge \pi_i(p_i, p_{-i}^{*})$$

where $\pi_i$ is the payoff to player $i$, $p_i^{*}$ is that player's equilibrium strategy, $p_{-i}^{*}$ denotes the equilibrium strategies of all other players, and the inequality holds for every alternative strategy $p_i$ and every player $i$. A strategy is dominant when it gives a player a better payoff whatever the others do. Ross, in the Stanford Encyclopedia of Philosophy, calls the Nash equilibrium a relatively weak solution concept, because applied alone it often cannot choose between several equilibria.

Cooperative, non-cooperative, sequential and repeated games

Non-cooperative game theory, in Ross's summary, assumes each player maximises its own utility while treating the expected responses of the others as constraints, so a predicted outcome has to be a self-enforcing equilibrium. Cooperative game theory starts from an agreement the players have already reached and asks under what conditions it stays stable. The Nash bargaining solution, used to model how a payer and a manufacturer divide the gain from a deal, belongs to the cooperative side and is covered under bargaining theory.

Where order of play affects the outcome, Ross explains, the game tree must be specified, not just the matrix. Sequential games are solved by backward induction from the last decision to the first, giving the subgame perfect equilibrium, a Nash equilibrium in every part of the game. Ross calls a threat incredible when carrying it out would not pay once the time came.

In a one-shot prisoner's dilemma the only Nash equilibrium is mutual defection, but players who expect to meet again can punish a defector in later rounds, and strategies such as tit-for-tat can sustain cooperation. Ross adds that cooperation unravels by backward induction if the players know which round is the last.

Where strategic interaction arises in health care

Ntais and Talias argue that free riding covers different games in different health settings. Analysis should therefore start by identifying the actors, their information, choices, payoffs, frequency of interaction and enforcement.

For voluntary vaccination, Bauch and Earn showed that once enough of the population is immune, even a slight vaccine risk outweighs the individual's risk from infection, so self-interest may prevent eradication; Ntais and Talias note that the game can resemble a threshold public good, a coordination game or a volunteer's dilemma, and the economics is developed under vaccination externalities. For antimicrobial resistance, Colman and colleagues proved that doctors motivated only by their own patients' outcomes will prescribe whatever the level of resistance, although every doctor would do better if all prescribed with more restraint. Porco and colleagues found such a conflict between individual and society in some settings but not all.

Chen and Fan studied Taiwan's global budget, in which a fixed sectoral budget is divided by the total points billed by all hospitals, so the value of each point falls as total volume rises. Expanding volume is each hospital's dominant strategy, and hospitals increased service intensity and shifted towards services with higher price to marginal cost ratios. Modelling China's centralised volume-based drug procurement, Zhang and colleagues concluded that low quality standards induce price competition and less investment in quality, while high standards may leave a single supplier with monopoly power. McAdams and colleagues framed the financing of COVAX as a potential tragedy of the commons in which wealthier countries fail to support shared vaccine access.

Worked example of a launch price game between a payer and a manufacturer

All numbers are illustrative. A manufacturer offers a new medicine to a single payer at a high price of £15,000 or a moderate price of £8,000 per course, and the payer then decides whether to fund it. The medicine treats 1,000 patients, each gaining 0.5 QALYs with no other change in costs, and costs £2,000 per course to supply. The payer values health at a cost-effectiveness threshold of £20,000 per QALY, and the two payoffs are as follows.

$$NMB = N \times \Delta Q \times \lambda - N \times p, \qquad \Pi_M = N \times (p - c)$$

where $NMB$ is the payer's net monetary benefit, $\Pi_M$ is the manufacturer's profit, $N = 1000$ is the number of patients, $\Delta Q = 0.5$ is the QALY gain per patient, $\lambda$ is the threshold, $p$ is the price per course and $c$ is the cost of supply. If the medicine is not funded both payoffs are zero.

1. Fill in the payoffs. The health gained is worth 1,000 × 0.5 × 20,000 = 10,000,000. At the high price, funding gives the manufacturer 1,000 × (15,000 − 2,000) = 13,000,000 and the payer 10,000,000 − 15,000,000 = −5,000,000. At the moderate price the manufacturer earns 1,000 × (8,000 − 2,000) = 6,000,000 and the payer 10,000,000 − 8,000,000 = 2,000,000.

Payoffs are in £ million, manufacturer first.

2. Solve by backward induction. Faced with the high price the payer rejects (0 beats −5); faced with the moderate price it funds (2 beats 0). Foreseeing this, the manufacturer compares 6 with 0 and offers the moderate price. The subgame perfect equilibrium is a moderate price that is funded, paying £6 million and £2 million.

3. Spot the incredible threat. A payer rule of never funding, paired with the high price, is also a weak Nash equilibrium of the strategic form. Against that rule the manufacturer earns zero at either price, and against the high price refusal is the payer's best reply. The threat is not credible, because offered the moderate price the payer would fund and gain £2 million.

4. Remove the payer's credibility. Suppose refusing any medicine costs the payer the equivalent of £6 million in public pressure. The payer now funds at the high price (−5 beats −6) as well as the moderate one, so the manufacturer charges the high price and earns 13 instead of 6. The payer's −£5 million is a net health loss of 5,000,000 / 20,000 = 250 QALYs, because the £15 million displaces 750 QALYs elsewhere, against 500 gained.

5. Let the price vary freely. The payer funds whenever net monetary benefit is not negative, which holds up to $p = \Delta Q \times \lambda = 0.5 \times 20000 = 10000$ pounds. A manufacturer making a take-it-or-leave-it offer would charge £10,000, earning 1,000 × (10,000 − 2,000) = 8,000,000 and leaving the payer nothing. How the split changes when the payer also holds bargaining power is covered under bargaining theory.

Interpretation. The order of moves and the credibility of the payer's refusal decide who captures the value of a new medicine. A payer that cannot say no invites the higher price.

Institutions as the rules of the game, from Baines to the theory of moves

Writing in the Pharmaceutical Journal in 2005, Baines proposed a framework for UK community pharmacy drawn from new institutionalism, citing Douglass North's definition of institutions as "the rules of the game in a society". In his scheme, institutions such as the Medicines Act 1968 set the rules pharmacists played by to earn revenue from the NHS, pharmacies were players with a commercial strategy bounded by professional constraints, and primary care trusts and the Royal Pharmaceutical Society policed their behaviour.

Baines suggested teaching pharmacists strategy through Brams's theory of moves. Brams contrasts it with standard game theory, in which players choose strategies simultaneously. In the theory of moves the players start from an existing state in a payoff matrix, either may switch strategy, the other may respond, and play stops when the player whose turn it is chooses not to move. Players look ahead to countermoves and counter-countermoves, and Brams calls the outcomes reached under this foresight nonmyopic equilibria.

The framing suits health policy, where reforms start from a contract, payment rule or price already in force; choosing such rules is mechanism design, and writing them into agreements is contract theory.

Using game theory in economic evaluation and HTA

A cost-effectiveness model usually takes prices, uptake and provider behaviour as inputs, whereas game theory treats some of them as outcomes of strategic interaction, which matters most when a policy changes incentives. Taiwan's global budget was intended to contain costs, yet service intensity rose; Chen and Fan report that after 2004 providers began to act cooperatively to overcome the dilemma.

Ntais and Talias group policy responses into those that alter payoffs, such as subsidies and mandates, those that alter information and beliefs, and those that change the structure or repetition of the game. In an evaluation or HTA, a game can supply scenarios for how players respond to a reform, which then drive the costs and outcomes compared in scenario analysis. The same review stresses empirical calibration of payoffs and notes that policymakers, insurers, manufacturers and hospitals are themselves strategic actors.

Game theory beside related concepts

The Nash equilibrium page covers the equilibrium concept. Bargaining theory covers how a surplus is divided once a deal is possible.

A game equilibrium is a set of strategies no player wants to leave and need not involve a market equilibrium, where supply equals demand. A game also differs from a decision tree, whose branches beyond the decision maker's own are chance events, not choices by players with their own objectives. Behavioural economics relaxes the assumption that players maximise expected utility, and Ntais and Talias treat it as complementary to game theory.

Where game-theoretic analysis misleads

The commonest error is to read an equilibrium as a good or fair outcome; in the antibiotic and global budget games it leaves players worse off than joint restraint. Another is to label every failure to cooperate as free riding. Ntais and Talias warn that a model treating vaccine refusal solely as strategic free riding may recommend sanctions while missing problems of access and trust.

Results also rest on assumed motives and information. Colman and colleagues assume doctors care only about their own patients, and Porco and colleagues assume perfect information. Illustrative payoff matrices, like the one above, show a mechanism without estimating its size, which is why Ntais and Talias call for empirically calibrated models.

Sources

• Baines D. Is there an appropriate framework for examining UK pharmacy practice? The Pharmaceutical Journal. 2005;275:687-688 (3 December 2005). https://archive.org/details/baines-d-2005-is_there_an_appropriate_framework
• Bauch CT, Earn DJD. Vaccination and the theory of games. Proc Natl Acad Sci USA. 2004;101(36):13391-13394. https://doi.org/10.1073/pnas.0403823101
• Brams SJ. Game theory and the Cuban missile crisis. Plus Magazine, University of Cambridge. 1 January 2001. https://plus.maths.org/content/game-theory-and-cuban-missile-crisis
• Chen B, Fan VY. Strategic provider behavior under global budget payment with price adjustment in Taiwan. Health Economics. 2015;24(11):1422-1436. https://doi.org/10.1002/hec.3095
• Colman AM, Krockow EM, Chattoe-Brown E, Tarrant C. Medical prescribing and antibiotic resistance: a game-theoretic analysis of a potentially catastrophic social dilemma. PLoS ONE. 2019;14(4):e0215480. https://doi.org/10.1371/journal.pone.0215480
• McAdams D, McDade KK, Ogbuoji O, Johnson M, Dixit S, Yamey G. Incentivising wealthy nations to participate in the COVID-19 Vaccine Global Access Facility (COVAX): a game theory perspective. BMJ Global Health. 2020;5(11):e003627. https://doi.org/10.1136/bmjgh-2020-003627
• Ntais C, Talias MA. Free riding in healthcare through a game-theoretic lens: a cross-domain narrative review and conceptual synthesis. Healthcare (Basel). 2026;14(12):1651. https://doi.org/10.3390/healthcare14121651
• Porco TC, Gao D, Scott JC, Shim E, Enanoria WT, Galvani AP, Lietman TM. When does overuse of antibiotics become a tragedy of the commons? PLoS ONE. 2012;7(12):e46505. https://doi.org/10.1371/journal.pone.0046505
• Ross D. Game theory. Stanford Encyclopedia of Philosophy. First published 1997; substantive revision 3 September 2023. https://plato.stanford.edu/entries/game-theory/
• Zhang S, Li J, Li S, Li J, Zhang L, Wang R, Chen C, Zhu L. Unintended consequences of national centralized volume-based drug procurement policy in China: a game theory-based analytical approach for pharmaceutical quality and pricing. Cost Effectiveness and Resource Allocation. 2026;24:101. https://doi.org/10.1186/s12962-026-00794-1