How to improve health care using game theory: the Prisoner’s Dilemma

“So take these pills, get a blood report, and see me in a few weeks,” said my doctor. I was somewhat taken aback. It was my first physical as an adult. I was healthy overall, but I was being advised to take a prescription drug as a precaution…for the rest of my life.

I was not sure what to do. Could I trust my doctor?

In a perfect world, patients should not have to ask this question. In reality, patients need to be skeptical, given the economics of health care in America. This is a system where doctors are typically paid for activity rather than outcomes. So even though doctors are generally good people, the system often results in unnecessary drug prescriptions, tests, and surgeries. And along the way, the patient-doctor relationship is strained.

Game theory can help

How can we improve trust between doctors and patients? One answer lies in using game theory to understand the interaction better.

There is a great introductory article on the subject in a 2004 issue of Quality and Safety in Health Care called “Models of the medical consultation: opportunities and limitations from a game theory perspective.” The authors C Tarrant, T Stokes, and A M Colman explain how game theory is a useful tool for understanding strategic interaction and creating better outcomes in medical care. (The full paper is available here pdf)

The authors describe how to analyze a medical consultation using three models in game theory: the Prisoner’s Dilemma, the Assurance game (aka the Stag-hunt game), and the Centipede game.

To give a flavor, I will summarize and comment on the Prisoner’s Dilemma section of their paper. The game will illustrate how doctors and patients are unable to reach the best outcome because of conflicting personal interests. And it will also suggest ways to improve the system.

The Prisoner’s Dilemma

The most famous strategic game, the Prisoner’s Dilemma, is an apt model for many interactions. I’ll first provide an overview on the game. Those of you familiar with the Prisoner’s Dilemma can skip down to the next section where the game is related to medical care.

The Prisoner’s Dilemma describes the incentives of two prisoners being questioned by police. The prisoners have indeed committed a crime, but the police can’t prove this and need at least one confession to make the charges hold.

Each prisoner is interrogated separately and has to choose between concealing information (C) or disclosing it (D). The strategies are usually interpreted as choices between cooperating with one’s partner (C) or defecting from one’s partner (D).

The police want at least one person to disclose information. They tell each prisoner that his fate depends both on his choice and his partner’s choice, in the following structure:

• If both conceal information, then both will be acquitted on lack of evidence
• If both disclose information, then both will be convicted
• If only one discloses, then that prisoner will be set free and given a reward (the best outcome), and the other will receive an especially heavy sentence (the worst outcome)

What will be the outcome of this situation? At first glance, it appears the prisoners have hope that they can be acquitted. If both of them keep mum, then the police have nothing. This is the best outcome for them, and it is completely in their control. But will it happen?

The problem is the outcome does not make sense strategically. The way to approach your decision is by considering what the other person might do, and then calculating your best choice in response. In this particular game, disclosing is always the better choice. Why is that?

There are two cases to consider. First, if the other prisoner discloses information, it is better in response to disclose (normal sentence) than to conceal (heavy sentence). Second, if the other person conceals, then it is also better in response to disclose (acquittal with reward) than conceal (acquittal).

This logic shows why each prisoner is likely to disclose information, and they will both be convicted. The dilemma is that picking the best individual choice precludes the group from achieving the best group outcome, an outcome that seems fair to both parties.

The Prisoner’s Dilemma applied to medical consultation

The Prisoner’s Dilemma can be modified to apply to medical care. I have slightly modified the narrative from the paper.

Consider an obese adult who requires but does not want medical attention. At the request of friends, he visits a doctor to treat his condition. The doctor concludes from reports the adult has elevated blood pressure and elevated cholesterol, risk factors for many diseases.

The doctor has two choices when he meets with the patient. He can choose whether to spend 5 minutes to prescribe mildly effective medicines (D) or he can spend 15 minutes and describe very effective lifestyle changes (C) such as diet and exercise.

The adult also has two choices when he meets the doctor. He can either choose to follow the doctor’s recommendation (C) or he can instead ignore the advice and find a second opinion (D).

There are four possible outcomes from these choices:

• Both cooperate: doctor gives lifestyle advice and patient complies
• Only doctor cooperates: doctor gives lifestyle advice and patient ignores it
• Only patient cooperates: doctor gives medicine and patient takes it
• No one cooperates: doctor gives medicine and patient does not take it

The best outcome is that both cooperate. This is the situation where the doctor gives the best advice and the patient follows it. But will it happen?

Unfortunately, it is not likely in this setup because the payoffs resemble the Prisoner’s Dilemma. Both the doctor and the patient are more likely to defect given the payoffs. Here is why.

The patient will want to find a second opinion regardless. If the doctor prescribes ineffective medicine, he is better off ignoring the advice. If the doctors gives the good advice, then again, the patient can be better by confirming it from another doctor as a double check.

Since the patient always disregards the doctor’s advice, the doctor in turn is better off treating the patient quickly. The doctor will choose to write the prescription and save time to be able to treat other patients.

In the end, doctors write prescriptions and patients don’t take them or they get a second opinion. So in spite of the possible gains, the best outcome where both cooperate cannot be achieved.

The market for lemons

This outcome is not too much of a stretch to what is actually happening. Doctors often cannot trust patients to comply, and patients cannot trust doctors to give the best advice. In aggregate, the entire health system turns into a sort of market of bad health or “lemons.”

(I described this process in my article about the market for lemons.)

Getting to cooperation

There is a large amount of literature on how to achieve cooperation in the Prisoner’s Dilemma. Here are two of the important points.

Cooperation is possible in repeated play (long horizons)

Mathematically it can be illustrated that cooperation can be achieved in infinitely repeated play.

What this means practically is that cooperation is possible in repeated play because both sides desire to share the surplus and they can punish defections.

In medicine, the paper describes the implications:

In the context of the consultation, mutual cooperation becomes a more attractive prospect if future interactions are anticipated. There are incentives for the doctor to spend time finding an appropriate management approach: consultations with the same patient in the future are likely to take up less time and the doctor will have the satisfaction of carrying a management plan through to completion. The patient is likely to follow through with the treatment if there is an expectation that the doctor will monitor his progress in the future. Both the doctor and the patient can anticipate future payoffs from this mutual cooperation, and this model implies that higher quality of care can be achieved when the patient sees the same GP [General Practitioner] repeatedly.

This would suggest it is desirable that patients can choose their doctors rather than having to switch after a job chance or insurance change, as is often the case today.

Cooperation can be achieved by changing the game

If you are not winning the game you are playing, then consider changing the game.

I often think about a situation in my work office where missed calls rolled over to everyone’s phones. Quickly it was observed that no one was picking up the phones. Why? It was sort of like a multi-person game of chicken-it was a better choice not to pick up the phone.

The solution came not from yelling at people but rather changing the game.

The same principle should be used in health care. The Prisoner’s Dilemma arises because patients and doctors are less motivated and poorly compensated for their efforts. We should not blame the behavior but rather look at some of the incentives.

Doctors, for instance, too often have limited time due to reimbursement schedules. They simply have to rush and they do not have time to tell comprehensive changes. Plus, they are very worried that patients won’t follow their advice.

There are several ways to change this situation. One can reward doctors for better health outcomes so that lifestyle changes become a more profitable decision. Additionally, the public needs to be motivated and educated to follow such advice, as improvement is a two way street.

Conclusion

The doctor-patient interaction is a complicated game but one that can be improved using the tools of game theory. The Prisoner’s Dilemma is a good starting point to see how interests of patients and doctors cannot reach an optimal outcome because of competing individual interests. Whatever health care reforms are made should at minimum address these concerns.

Other games: the Assurance game (aka the Stag-hunt game), the Centipede game

The authors C Tarrant, T Stokes, and A M Colman additionally describe how the Assurance game and the Centipede game can be used to model the medical consultation. These are slightly more involved games, but the article makes them accessible.

Check them out in the full paper (pdf)