Nash Equilibrium

Nash Equilibrium in Health Care Decisions

A Nash equilibrium is a stable combination of choices, which holds when every decision maker's choice is a best reply to what the others are doing, so nobody regrets their own choice once the others' choices are known. Health economists use it to predict where strategic situations settle, such as how many parents vaccinate, how doctors prescribe antibiotics or how hospitals set quality, and a recurring finding is that the settled outcome can be worse for everyone than an alternative nobody can reach alone. The concept rests on Nash's definition and existence proof, and it separates strict from weak equilibria and equilibrium from efficiency, which an illustrative hospital investment game makes concrete. Sequential and repeated games are covered under game theory.

Nash's definition and the best-reply test

In his 1950 note in the Proceedings of the National Academy of Sciences, Nash defined a game in which each of n players has a finite set of pure strategies and each combination of strategies gives a definite payment to every player. One combination counters another if each player's strategy in it gives that player the highest obtainable expectation against the others' strategies in the second, and a combination that counters itself is an equilibrium point. In current notation the condition reads as follows.

$$u_i(s_i^{*}, s_{-i}^{*}) \ge u_i(s_i, s_{-i}^{*}) \quad \text{for every strategy } s_i \in S_i \text{ and every player } i$$

where $u_i$ is the payoff to player $i$, $s_i^{*}$ is that player's equilibrium strategy, $s_{-i}^{*}$ denotes the equilibrium strategies of all the other players, and $S_i$ is the set of strategies open to player $i$. Holt and Roth call this an announcement test, because if all players announced their strategies at once, nobody would want to reconsider. In health applications the players may be hospitals, prescribers or parents, and a payoff may be a hospital's surplus or a family's health risk.

Pure strategies, mixed strategies and existence

A pure strategy is a definite choice. Nash defined mixed strategies as probability distributions over the pure strategies, with payoffs then measured as expected values. Holt and Roth give a second reading that suits health care, in which a mixed strategy can describe a population in which some proportion makes each available choice, so an equilibrium mixed strategy of 0.7 can be read as 70% uptake of a vaccine.

A mixed equilibrium has a defining property. Slantchev's lecture notes on strategic-form games show that a player who randomises in equilibrium must be indifferent between all the pure strategies it mixes over, because otherwise shifting probability towards the better one would raise its expected payoff. Each player's mixing probability is therefore set by the need to keep the other indifferent.

Nash proved that an equilibrium exists, using Kakutani's fixed point theorem at David Gale's suggestion. Myerson summarises the result by saying that equilibria in randomised strategies exist for any finite normal-form game. Existence is not a promise of a pure equilibrium, and in the pursuit game the Stanford Encyclopedia of Philosophy entry describes, only randomised strategies are in equilibrium.

Strict and weak equilibria

An equilibrium is strict when each player's equilibrium strategy is its unique best reply, so the inequality above holds strictly for every alternative. It is weak when some player has an alternative that does exactly as well. Slantchev notes that a mixed equilibrium can never be strict, since a player willing to randomise has more than one best reply, and that pure equilibria, especially strict ones, are the more compelling predictions in a one-off encounter.

The antibiotic prescribing game analysed by Colman and colleagues has a strict equilibrium. The state in which every symptomatic patient receives antibiotics is a strict Nash equilibrium and attracts play from every other state.

Multiple equilibria and coordination

Nash observed that in two-person zero-sum games any two equilibria give the players the same expectations, but that this need not occur in general. The Stanford Encyclopedia entry illustrates with a "Hi-lo" game in which both players matching on one option pays 10 each, matching on the other pays 1 each and mismatching pays nothing. Both matched outcomes are Nash equilibria, and the definition alone cannot say which will occur. Tadelis's textbook makes the same point with the stag hunt, where a safe equilibrium persists alongside one that is better for both, because each player's belief that the other will play safe is self-fulfilling.

Myerson describes Schelling's answer. With multiple equilibria, any publicly known factor that draws the players' attention to one of them, such as a shared tradition or a social authority, can lead them to implement it.

Refinements narrow the set. Holt and Roth explain that Selten treated the Nash conditions as necessary but not sufficient for rational play, so superfluous equilibria can be removed by further requirements. The Stanford Encyclopedia gives elimination of weakly dominated strategies (a comparison of one player's strategies, unrelated to dominance between interventions in cost-effectiveness analysis) as one refinement, and in sequential games the standard refinement is subgame perfection, covered on the game theory page.

Equilibrium is not efficiency

Holt and Roth describe social dilemmas as games in which a socially desirable action is not a Nash equilibrium; in the prisoner's dilemma each player's best reply is to defect, although both would earn more if both cooperated. Tadelis states that Nash equilibrium does not guarantee Pareto optimality, and an outcome can be stable yet fail the test set out under Pareto efficiency.

Health care supplies cases. Colman and colleagues proved that their prescribing game has a unique Nash equilibrium and is a social dilemma of the tragedy-of-the-commons type, which matters for antimicrobial resistance. Shim and colleagues show that universal vaccination cannot be a Nash equilibrium, because one person who declines enjoys herd immunity without the cost or side effects of the vaccine; Bauch and Earn conclude that individual self-interest might preclude complete eradication, and the vaccination case is developed under vaccination externalities. Between countries, Klepac and colleagues model each country choosing its own vaccination strategy, finding that the Nash equilibrium can leave coverage below the global optimum, and voluntary coalitions can do better if they can deter free riding.

Efficiency also depends on whose payoffs are counted. In the hospital quality game used by Han and colleagues, three hospitals at regulated prices would maximise joint profit at the minimum quality level of 1, while the symmetric Nash equilibrium is at quality 10 and the patient optimum at 13. The equilibrium is inefficient for the hospitals, yet their failure to collude benefits patients, and in the experiment average quality ended above the Nash level.

Worked example with two hospitals and one angioplasty service

All numbers are illustrative. Two hospitals, A and B, each decide whether to open a 24-hour primary angioplasty service for heart attacks. The local population supports one service. A hospital that invests alone earns a surplus of GBP 4 million a year, the other earns nothing, and if both invest they split the volume and each loses GBP 2 million.

Payoffs are in GBP million a year, hospital A first.

1. Find the pure equilibria. If B stays out, A's best reply is to invest (4 beats 0); if B invests, A's best reply is to stay out (0 beats −2), and the game is symmetric. The pure Nash equilibria are therefore (Invest, Stay out) and (Stay out, Invest). Both are strict, because each equilibrium choice beats its alternative by at least GBP 2 million, while in (Invest, Invest) and (Stay out, Stay out) a hospital gains by switching.

2. Find the mixed equilibrium. Let $q$ be the probability that B invests. A will mix only if investing and staying out pay the same, as shown below.

$$U_A(\text{Invest}) = -2q + 4(1 - q) = 0 = U_A(\text{Stay out}) \quad \Rightarrow \quad q^{*} = \tfrac{4}{6} = \tfrac{2}{3}$$

where $U_A$ is A's expected payoff from each choice and $q^{*}$ is B's equilibrium probability of investing. By symmetry A also invests with probability $\tfrac{2}{3}$, and each hospital's expected payoff is zero.

3. Count the outcomes of mixed play. Both invest with probability $\tfrac{2}{3} \times \tfrac{2}{3} = \tfrac{4}{9}$, exactly one invests with probability $\tfrac{2}{3} \times \tfrac{1}{3} \times 2 = \tfrac{4}{9}$, and neither invests with probability $\tfrac{1}{3} \times \tfrac{1}{3} = \tfrac{1}{9}$, or about 11%.

4. Value the outcomes for the health system. Suppose patients' health gain, net of what the commissioner pays the hospitals, is worth GBP 10 million a year with one service and GBP 11 million with two. Adding the hospitals' surpluses, one service yields 14, two services yield $-4 + 11 = 7$ and none yields 0. The pure equilibria deliver 14, while the mixed equilibrium delivers the following on average.

$$E[W] = \tfrac{4}{9} \times 14 + \tfrac{4}{9} \times 7 + \tfrac{1}{9} \times 0 = \tfrac{84}{9} \approx 9.33$$

where $E[W]$ is the expected total surplus in GBP million a year, a loss of about 4.67 against either pure equilibrium.

5. Let one hospital move first. If A can commit before B decides, B stays out after an investment (0 beats −2), so A invests and the subgame perfect outcome is (Invest, Stay out).

6. Make duplication break even. If two services left each hospital with zero instead of a loss, investing would never do worse than staying out. All three outcomes with at least one investor would then be weak equilibria, and eliminating the weakly dominated choice of staying out would select (Invest, Invest), worth 11 to the health system against 14.

Interpretation. The game has three equilibria, the two pure ones are equally stable, and coordination or commitment decides between them. A commissioner who designates one centre supplies a focal point in Schelling's sense; without one, if the hospitals play the mixed equilibrium, the region has a duplicated service four times in nine and no service one time in nine.

Using equilibrium analysis in evaluation and policy

Holt and Roth read a Nash equilibrium as advice from which no player would want to depart, a resting point of a process in which players adjust towards better replies, and a self-enforcing agreement. Cost-effectiveness models usually take uptake, prescribing or provider behaviour as fixed inputs, whereas equilibrium analysis predicts how they respond when incentives change; Bauch and Earn's analysis shows that a rise in perceived vaccine risk tends to cause larger falls in uptake for infections with more secondary cases, and that coverage is hard to restore after a vaccine scare.

Where a model has several equilibria, each is a candidate outcome, and an evaluation can carry them as separate scenarios in a scenario analysis. Holt and Roth also describe institutions invented to turn dilemmas into games in which cooperation is an equilibrium, and choosing such rules is mechanism design.

Limits and common misreadings

A common error is to treat an equilibrium as an optimum or a fair result. A second is to treat it as an accurate forecast. Tadelis notes that the concept requires each player's beliefs about the others to be correct, a far stronger demand than rational choice given those beliefs. Holt and Roth report that players in prisoner's dilemma experiments often cooperate to some degree, and that in some games equilibrium is a persistently poor predictor; Han and colleagues predicted above-equilibrium quality from providers who care about patients, a motive a profit-only payoff leaves out, and their results were in line with that prediction.

Mixed equilibria also need care, because a player indifferent between its options has no reason of its own to choose the equilibrium probabilities, which is why the population reading is often more natural in health care. Finally, the Nash equilibrium differs from the Nash bargaining solution covered under bargaining theory, which divides a surplus between parties who can make a binding agreement, and from a market equilibrium, where supply equals demand at a price.

Sources

• 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
• 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
• Han J, Kairies-Schwarz N, Vomhof M. Strategic behaviour and decision making in competitive hospital markets: an experimental investigation. Int J Health Econ Manag. 2024;24(3):333-355. https://doi.org/10.1007/s10754-024-09366-3
• Holt CA, Roth AE. The Nash equilibrium: a perspective. Proc Natl Acad Sci USA. 2004;101(12):3999-4002. https://doi.org/10.1073/pnas.0308738101
• Klepac P, Megiddo I, Grenfell BT, Laxminarayan R. Self-enforcing regional vaccination agreements. J R Soc Interface. 2016;13(114):20150907. https://doi.org/10.1098/rsif.2015.0907
• Myerson RB. Nash equilibrium and the history of economic theory. Journal of Economic Literature. 1999;37(3):1067-1082. https://doi.org/10.1257/jel.37.3.1067
• Nash JF. Equilibrium points in n-person games. Proc Natl Acad Sci USA. 1950;36(1):48-49. https://doi.org/10.1073/pnas.36.1.48
• Ross D. Game theory. Stanford Encyclopedia of Philosophy. First published 1997; substantive revision 3 September 2023. https://plato.stanford.edu/entries/game-theory/
• Shim E, Meyers LA, Galvani AP. Optimal H1N1 vaccination strategies based on self-interest versus group interest. BMC Public Health. 2011;11(Suppl 1):S4. https://doi.org/10.1186/1471-2458-11-S1-S4
• Slantchev BL. Game theory: dominance, Nash equilibrium, symmetry. Lecture notes, University of California, San Diego; 23 May 2008. http://slantchev.ucsd.edu/courses/gt/04-strategic-form.pdf
• Tadelis S. Pinning down beliefs: Nash equilibrium. Chapter 5 in Game Theory: An Introduction. Princeton: Princeton University Press; 2013. https://assets.press.princeton.edu/chapters/s5-10001.pdf