In this article, The Eye Care Advocate looks at Game Theory in Optometry and how the complex field of game theory can be utilised to garner the best outcomes within the optometric setting.

What is Game Theory?

Game theory is a fascinating and influential field of study. At its core, game theory is a branch of mathematics. It deals with the analysis of strategies for competitive situations. In these situations, a participant’s choice of action depends critically on the actions of other participants¹. It’s essentially the science of strategy. It is a way to make rational decisions in scenarios where various players interact with each other.

The principles of game theory apply to a wide range of behavioural relations. They have been used extensively in economics, logic, systems science, and computer science¹. It began with the study of zero-sum games, where one person’s gain is another’s loss. However, it has since expanded to include a variety of complex and cooperative interactions^1,2.

Modern game theory explores how individuals and groups make interdependent decisions. The success of one party’s strategy depends on the strategy adopted by others¹. This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating their own strategy. It’s a powerful tool for analysing the strategic aspects of decision-making, beyond what is left to chance.

In the context of optometry, game theory can help practitioners understand how to navigate competitive markets^2,3. It aids in making pricing decisions³ and managing patient relationships. By considering the potential actions and reactions of patients⁴, competitors, and suppliers³, practitioners can make informed decisions. It’s about anticipating and strategically responding to the moves of all players involved in the ‘game’ of optometry.

Game Theory in Optometry: The Games

There are several concepts or “games” within game theory. We shall now look at some of them to see the relevance of game theory in optometry.

The Stag Hunt

The Stag Hunt is a concept from game theory that provides insight into the problem of social cooperation. It was first introduced by the philosopher Jean-Jacques Rousseau in his work “Discourse on Inequality“^5,6.

In the Stag Hunt, two hunters must decide independently whether to hunt a stag together or to hunt a hare alone^1,6. The dilemma is that hunting a stag requires mutual cooperation, as it’s too challenging for one hunter to accomplish alone. If both hunters go for the stag, they succeed and share a large reward. However, if one decides to go for the hare, a smaller but guaranteed reward, the other hunter is left with nothing if they were still hunting the stag^1,6. This can be seen below in Table 1.

	B Hunts the Stag	B Hunts the Hare
A Hunts the Stag	(10,10)	(0,5)
A Hunts the Hare	(5,0)	(5,5)

The game has two pure-strategy Nash equilibria (that is, where nothing is gained if you switch your strategy, but others keep their strategy): one where both players cooperate (hunt the stag) and one where both players defect (hunt the hare). The cooperative strategy leads to a higher payoff but requires trust that the other hunter will not defect¹. The defecting strategy is safer as it guarantees a smaller reward regardless of the other hunter’s decision.

This scenario is often used as an analogy for various kinds of social cooperation, like business partnerships, international agreements, or even within healthcare environments⁶. It highlights the conflict between safety and social cooperation, and the need for assurance and trust in mutual endeavors⁶.

Applying the Stag Hunt Game Theory in Optometry

With game theory in optometry at the front of our minds, the Stag Hunt can be applied to patient interactions. An optometrist and a patient can be seen as the hunters, where the ‘stag’ is the best possible health and visual outcome for the patient, achievable through cooperation and trust. The ‘hare’ is a suboptimal but immediate solution, like a quick fix for an eye problem that doesn’t address underlying issues. The optometrist must work to build trust with the patient, ensuring they both commit to the long-term health strategy (hunting the stag) rather than settling for short-term solutions (hunting the hare).

A pure example of this could be a patient that has dry eye secondary to meibomian gland dysfunction.  The “stag” would be treating the underlying meibomian gland dysfunction. The “hare” would be just to suggest an ocular lubricant to treat the symptoms of dryness. Successfully hunting this stag would be for the optometrist to suggest the treatment and the patient to follow the treatment plan and remain compliant with it. This cooperative approach can lead to better patient outcomes and higher satisfaction, as it tackles the problem at its source rather than just mitigating the symptoms. It’s a strategic choice that aligns with the principles of game theory, emphasising the benefits of mutual cooperation for a greater reward.

*NB no stags were harmed in the writing of this blog post!

The Prisoner’s Dilemma

In the classic Prisoner’s Dilemma, two individuals are faced with a decision. They must decide whether to cooperate with each other or act selfishly. They cannot communicate¹. The dilemma arises because the best collective outcome is achieved when both parties cooperate.

However, individual incentives may lead each to betray the other. In the original dilemma, two criminals are arrested and interrogated separately. If both stay silent, they get a moderate sentence. If one betrays the other, the betrayer goes free while the betrayed gets a heavy sentence. In the case where both betray each other, they both get a heavy sentence¹. This can be seen in the matrix in Table 2 below.

	B Remains Silent	B Testifies
A Remains Silent	(-1,-1)	(-5,0)
A Testifies	(0,-5)	(-3,-3)

Applying the Prisoner’s Dilemma Game Theory in Optometry

When we apply this to the optometrist-patient relationship, the ‘prisoners’ are the optometrist and the patient, and the ‘dilemma’ is the choice between short-term gains and long-term health benefits:

Cooperation (Silence): If both the optometrist and the patient cooperate, the patient follows the treatment plan, and the optometrist provides ongoing, quality care, leading to the best health outcomes for the patient and a strong, loyal relationship.

Defection (Betrayal): If the optometrist prioritises short-term profits over patient care (e.g., by recommending unnecessary products or services), or if the patient does not adhere to the treatment plan, it leads to a breakdown in trust and potentially poorer health outcomes.

The optimal outcome in the Prisoner’s Dilemma is mutual cooperation. For optometrists, this means fostering a practice environment where patients feel their long-term health is the primary concern. This involves:

Transparency: Being transparent about treatment options, costs, and expected outcomes encourages trust and cooperation from patients.

Communication: Regular, open communication helps patients understand the importance of their treatment plans and the consequences of not following them.

Incentives: Creating incentives for patients to follow through with their treatment plans, such as offering discounts on future services or recognising consistent adherence, can encourage cooperation.

Education: Educating patients about the importance of eye health and the potential risks of neglecting treatment can motivate them to cooperate in the long-term management of their condition.

In repeated games of the Prisoner’s Dilemma, such as ongoing patient visits, trust becomes a crucial factor. Over time, consistent cooperation can lead to a stable relationship where both parties understand that working together yields the best results. This is akin to a tit-for-tat strategy in game theory where each party mirrors the other’s earlier action¹. This promotes an ongoing cycle of cooperation.

By considering the principles of the Prisoner’s Dilemma, optometrists can better understand the importance of building and maintaining trust with their patients, leading to mutually beneficial outcomes and a successful practice. It’s a strategic approach that not only enhances patient care but also strengthens the practice’s reputation and sustainability in the long term.

The Assurance Game

In the Assurance Game, also known as the Coordination Game⁷, two players will choose to cooperate if they have assurance that the other will do the same. The key to success is the mutual belief that both parties will fulfil their part of the agreement^7,8. This game differs from the Prisoner’s Dilemma in that the best outcome does not depend on a dominant strategy but rather on the assurance of mutual cooperation^1,7.

An example of this could be that two friends, Alex and Sam, are deciding whether to go to a concert together. Tickets are expensive, but they would save 10% if ordering 2 tickets or more as a single purchase, so they agree to do this and split the cost.

– If both pay, they enjoy the concert together.
– If one pays and the other doesn’t, the one who pays is out of pocket, and the other misses the concert.
– If neither pays, they both miss the concert.

This idea can be seen in the matrix within Table 3 below:

	Player B Cooperates	Player B Defects
Player A Cooperates	(R,R)	(S,T)
Player A Defects	(T,S)	(P,P)

Applying the Assurance Game Theory in Optometry

An example of the Assurance Game applying in an optometric scenario could be as follows:

Dr. Jones, an optometrist, plans an open day at her clinic to offer a special discount on premium spectacle frames. She hopes that if enough patients commit to buying a complete pair of glasses, she can order the frames in bulk at a reduced cost.

• If patients commit and Dr. Jones orders the frames, both enjoy the benefits of high-quality frames at a lower price.
• If patients commit but later change their mind, Dr. Jones has excess stock that she bought at a higher price.
• If patients don’t commit, Dr. Jones doesn’t order the frames, and both miss out on the potential deal.

In a more healthcare mindset, the Assurance Game suggests that patients will commit to long-term care plans if they are confident that their optometrist is also committed to their well-being. This mutual commitment can be fostered through:

Patient-Centred Care: By focusing on patient-centred care, optometrists make sure that services are tailored to the individual needs and preferences of each patient. This approach respects and responds to the patient’s unique health concerns, thereby building trust and assurance.

Effective Communication: Open-ended questions, active listening, and empathy are crucial communication skills that help in understanding the patient’s perspective. This level of understanding is essential for providing assurance to patients. Note, our eStudy Guide covers this in more detail!

Shared Decision Making: Inviting patients to weigh the pros and cons of different treatment options empowers them and reinforces their assurance in the care process. It’s about reaching shared treatment goals based on the patient’s values.

Education and Engagement: Educating patients about their conditions and the benefits of consistent eye care engages them more deeply in the care process. When patients understand the ‘why’ behind their care plan, they are more likely to commit to it.

Reliability and Consistency: Consistent quality care and follow-ups give patients the assurance that their optometrist is invested in their long-term health. This reliability encourages patients to adhere to recommended care plans.

By incorporating the principles of the Assurance Game into patient interactions, optometrists can create a cooperative environment. In this setting, both the practitioner and the patient work together towards the common goal of optimal eye health. This strategic approach not only improves patient outcomes but also enhances the overall effectiveness of the optometry practice.

The Centipede Game

The Centipede Game is a strategic interaction that can be used to understand and predict behaviors in sequential decision-making scenarios⁹. In the context of optometry, it can provide insights into patient behavior and compliance with treatment plans.

In the Centipede Game, two players take turns deciding whether to continue the game or to take an immediate reward, with the potential reward increasing as the game progresses⁹. The game is typically presented as a series of decisions, where each player must decide whether to “take” a certain payoff now or to “pass” and allow the other player (or themselves in the next round) to potentially take a larger payoff⁹. If they opt to end the game instead of pass, the one who ends it takes the larger reward.

An example of how this plays out can be seen in the diagram below:

Applying the Centipede Game Theory in Optometry

An example of the centipede game theory applied to an optometry setting could include an optometrist advising a patient on the importance of proper contact lens care and compliance to prevent eye infections.

• If the patient follows the advice, they maintain good eye health and comfort, leading to trust in the optometrist’s expertise (patient passes)
• If the patient neglects the advice, they risk eye health issues, which could lead to more visits and costs (patient takes)
• The optometrist’s reputation grows with each patient who follows the care regimen and experiences positive outcomes (optometrist chooses to pass each time)

When applying this game theory in optometry settings, it can model the interaction between an optometrist and a patient over a series of appointments:

Initial Engagement: The game begins with the optometrist offering a treatment plan. The patient can either ‘take’ the plan and commit to it (continue the game) or seek immediate relief elsewhere (take the payoff).

Sequential Decisions: Each subsequent appointment represents a decision point. The patient must decide whether to continue with the long-term treatment (pass) or opt for immediate but less effective solutions (take). This may apply to a short term appointment plan (such as follow ups based on a recovering corneal abrasion or infection), or a long-term plan of lifelong routine eye care at the practice.

Predicting Defection Points: By understanding the points at which a patient might defect, such as skipping appointments, not adhering to treatment, or even defecting to a competitor, the optometrist can anticipate these behaviours and implement strategies to encourage continued engagement.

Intervention Strategies: Optometrists can use reminders, education, and incentives to motivate patients to continue with their treatment plan. This is akin to increasing the potential payoff for continuing the game, encouraging the patient to pass rather than take.

Long-Term Compliance: The ultimate goal is to reach the end of the treatment ‘game’ with the patient fully compliant, resulting in the best possible health outcome. This is the equivalent of reaching the end of the Centipede Game with the highest payoff for both players.

By understanding the dynamics of the Centipede Game Theory in optometry, optometrists can better predict patient behaviours and implement strategies to maintain engagement throughout the treatment process. This approach helps ensure that patients remain on the path to optimal eye health, reflecting the game’s emphasis on strategic foresight and the value of long-term rewards over immediate gratification.

Conclusion: Game Theory in Optometry for Better Outcomes

In the fine balance of patient care and business, game theory provides a strategic framework for optometrists to enhance interactions and outcomes that will benefit both parties. By understanding and applying different game theory models, optometrists can tailor their approach to meet the diverse preferences and needs of their patients.

Customising communication strategies is not just about conveying information; it’s about creating a dialogue that resonates with each individual. Recognising the unique ways in which patients process information and make decisions allows optometrists to craft messages that are both informative and engaging.

The application of game theory in optometry goes beyond mere theory. It’s a practical tool for predicting behaviours, building trust, and fostering long-term relationships. Whether it’s through the mutual cooperation of the Stag Hunt, the reciprocal strategies of the Prisoner’s Dilemma, the shared assurances of the Coordination Game, or the foresight of the Centipede Game, these principles can guide optometrists towards more effective patient interactions.

Ultimately, the goal is to align the practice’s strategies with the best interests of the patients. By doing so, optometrists not only improve the patient experience but also secure the long-term success of their practice. Game theory in optometry, in this light, is not just a set of concepts but a pathway to achieving a harmonious and prosperous optometric practice.

If you have enjoyed this article about how you can utilise game theory in optometry, feel free to comment your thoughts below. I’d be also particularly interested to see if any other optometrists use game theory in optometry! To be notified of further articles, please consider subscribing to The Eye Care Advocate below.

Game Theory in Optometry – Further Reading

Below are some of the articles that were used in forming this post on game theory in optometry. The content predominantly focuses on the aspects of game theory discussed, offering further insight into the games.

1. Marden JR, and Shamma JS (2018). Game theory and control. Annual Review of Control, Robotics, and Autonomous Systems 1: 105-134.
2. Zhang S, and Zhang Y (2003). Introduction to game theory. Chinese Science Bulletin 48: 841-846.
3. Chen JK, and Liu X (2022). A review of logistics pricing research based on game theory. Sustainability 14(17): 10520.
4. Tarrant C, Dixon-Woods M, Colman AM and Stokes T (2010). Continuity and trust in primary care: a qualitive study informed by game theory. Annals of Family Medicine 8(5): 440-446.
5. Rousseau, J. (1754) Discourse on Inequality. Holland
6. McFadden DW, Tsai M, Kadry B, and Souba WW (2012). Game theory: applications for surgeons and the operating room environment. Surgery 152(5): 915-922.
7. Hirsch M (2015). Game theory and international environmental cooperation. Journal of Energy & Natural Resources Law 27(3): 503-510.
8. Kets W, Kager W, and Sandroni A (2022). The value of a coordination game. Journal of Economic Theory 201: 105419.
9. Blavatskyy P (2015). Behavior in the centipede game: A decision-theoretical perspective. Economic Letters 133 [Online].