A game theory application : the interaction between physicians and patients

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Politecnico di Milano

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea in Ingegneria Matematica

A Game Theory application:

the interaction between physicians and

patients

Relatore: Prof. Roberto LUCCHETTI

Tesi di Laurea Magistrale di: Margherita VIGORELLI Matr. 804337

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Abstract

The purpose of this work is to describe and analyse a typical clinical interaction between a doctor and a patient, via Game Theory. The conceptual tools provided by Game Theory can be used, in medical consultation, to analyse optimal deci-sions and to highlight the dynamics of the doctor-patient interaction.

After a brief review of the basic concepts of Game Theory, the Prisoners’ Dilemma game is applied to different situations in the medical consultation field. Later, since every clinical encounter is fundamentally based on trust, the Prisoners’ Dilemma game is modified into a trust version of it (incorporating regret, guilt and frustration) to make it relevant to describe a typical clinical interaction. There-fore, it is employed to analyse the participation of a patient to a randomized controlled trial.

Moreover, two simultaneous games by B. Djulbegovic are used to describe the simplest clinical interaction: a doctor has to decide whether or not to prescribe a treatment and a patient has to decide whether or not to accept it, in conditions of diagnostic uncertainty. After a deep analysis of the models, two sequential games are suggested, in order to describe a more realistic situation. The analysis shows that the solutions of the games strictly depend on the probability of disease of the patient, which is assessed and known by both the patient and the doctor. It appears clear that the most reasonable model to describe an everyday interaction between a patient and a doctor is a sequential game: the doctor chooses whether or not to treat a patient and, later, in both circumstances, the patient chooses whether or not to trust him.

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Sommario

L’obiettivo di questa tesi `e quello di descrivere e analizzare, attraverso la Teoria dei Giochi, l’interazione tra un medico ed il suo paziente. I modelli offerti dalla Teoria dei Giochi permettono di analizzare le strategie ottimali degli agenti e os-servare le dinamiche alla base della relazione medico-paziente.

Dopo una breve introduzione alla Teoria dei Giochi, sono presentate alcune ap-plicazioni del Dilemma del Prigioniero in campo medico. Dal momento che ogni interazione tra medico e paziente si basa sulla fiducia, il Dilemma del Prigioniero `e successivamente modificato e riproposto in una versione che integra i concetti di pentimento, frustrazione e colpa, strettamente legati alla fiducia. Questo nuovo modello descrive meglio l’interazione clinica ed `e mostrata una sua applicazione alla scelta di un paziente di partecipare a un esperimento clinico.

Successivamente, è presa in considerazione una semplice ma interessante inte-razione clinica, in cui un medico, in caso di diagnosi incerta, deve valutare se prescrivere o non prescrivere un trattamento a un paziente, che a sua volta può decidere se fidarsi del medico oppure no. Dopo una dettagliata analisi di due giochi simultanei, proposti da B. Djulbegovic, sono suggeriti due giochi sequen-ziali, per descrivere in modo più realistico un incontro clinico. L’analisi mostra che le soluzioni dei giochi dipendono strettamente dalla probabilità che il paziente sia malato, nota ad entrambi gli agenti. Risulta che il modello più ragionevole per descrivere una tipica interazione tra medico e paziente è un modello sequenziale: il medico decide se prescrivere un trattamento oppure no e, dopo aver osservato la scelta del medico, il paziente, in entrambi i casi, decide se fidarsi.

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Introduction

A clinical encounter, between physicians and patients, is a two-way social inter-action. Typically, the physician gets information from the patient, provides a diagnosis and, if needed, prescribes a treatment [15]. However, the physician is not the only agent in the decision analysis, since the patient can choose what infor-mation to tell the doctor (influencing the diagnosis) and even more importantly, can choose to reject the physician’s advice or treatment. Hence, the outcome of the consultation depends on the choice of both the patient and the doctor. Game Theory provides conceptual tools to describe and analyse optimal decisions, in situations in which the outcome depends on more interactive individual with common and divergent interests, as in medical consultations [1].

Game Theory first applications have been to economics, to work out how events occur when agents (business leaders, organisations, companies) behave in what they think is their best interests. Later, it has found applications to various dis-ciplines as psychology, political science, computer science and biology and, today, it applies to a wide range of human behaviours.

In the medical field, Game Theory has already been employed in several settings. For example, it has been used to study the evolution of tumours (carcinogenesis), to predict under which conditions cancer cells have a better chance of emerging [11]. Even in medical imaging, image segmentation, which is the process of de-composing an image into homogeneous regions to delineate anatomical structures for medical analysis and intervention, can be formulated as a game [7]. More-over, Game Theory is used in the field of kidney transplantation to determine a

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Introduction 2

systematic way of selecting, for each incompatible patient-donor pairs, a set of compatible transplants [14].

One interesting application to Game Theory is the interaction between physicians and patients: Game Theory models can be used to understand health care deci-sions, highlighting the doctor-patient dynamics.

In recent years, several studies have been presented in this field. In particular, in this work, we analyse three Game Theory models, presented by Benjamin Djul-begovic et al. in their articles [3, 4].

The purpose of the following work is to present some models describing the in-teraction between physicians and patients, to suggest modified models and to compare the solutions.

In Chapter 1, we provide the basic concepts of Game Theory, useful in the fol-lowing analysis. Specifically, non cooperative games are described. In Chapter 2, the most known strategic game, the Prisoners’ Dilemma, is applied to different situations in the medical consultation field. Since trust is fundamental in every clinical encounter, the Prisoners’ Dilemma game is modified to make it relevant to describe a typical clinical interaction: specifically, trust, regret, guilt and frus-tration are integrated in the pay-off function of the players. Therefore, in Chapter 3, we describe an application of the trust version of the Prisoners’ Dilemma game to a specific clinical interaction: the participation of a patient to a randomized controlled trial (RCT) [3]. Then, in Chapter 4, firstly, we present two simulta-neous games describing a clinical encounter [4], where the doctor has to decide whether or not to prescribe a treatment and a patient has to decide whether or not to accept it, in conditions of diagnostic uncertainty. Since the aim of the models is to analyse a typical clinical interaction, the simultaneous approach does not seem the most suitable (even if it is applicable to specific situations). For this reason, we introduce and analyse, in this thesis, two sequential games, describing the same situation, but with a radically different assumption, that the patient

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Introduction 3

is aware of the decision of the doctor. The analysis shows that the solutions of the games strictly depend on the probability of disease of the patient, which is assessed and known by both the patient and the doctor. In the simultaneous games, the solution is a pure or mixed strategies for both the players, while in the sequential games, the doctor’s optimal choice is a pure strategy, while for the patient, it is a couple of pure strategies. The solution concept is what makes the models essentially different.

From the analysis, it appears clear that the most reasonable model to describe an everyday interaction between a patient and a doctor is a sequential game: the doctor chooses whether or not to treat a patient and, later, the patient chooses whether or not to trust him, in both circumstances.

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Chapter 1 Introduction to Game Theory

Game Theory is a branch of mathematics that deals with the study of models of conflict and cooperation, between several decision-makers. The concepts of Game Theory provide a language to formulate, analyse and understand strategic scenarios. The object of the study is the game, which is a formal model of an interactive situation with two or more players: thus, a game is a simplified, yet efficient, representation of real life situations.

The formal definition of a game lays out the players, their preferences, their knowledge about each other choices, the strategic actions available to them, and how these influence the outcome.

In a game, each player has some alternatives to choose from and the combination of choices of each player leads to a possible outcome of the game. Each player has preferences on the outcomes and this is expressed by means of a pay-off (or utility): the pay-off associated to a particular outcome is the payout a player receives, by getting there.

A central assumption, in Game Theory, is that players are rational: a rational player always chooses the action that gives him the outcome he most prefers, given what he expects his opponents to do.

Thus, Game Theory is the study of taking optimal decisions in presence of multiple players.

There are, roughly speaking, two categories of games: the cooperative games and 4

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Chapter 1, Introduction to Game Theory 5

the non cooperative games.

In cooperative games, the players have the possibility to form coalitions: for each potential coalition, we specify the pay-off that the coalition can obtain if its members cooperate. Thus, the game is played between coalitions of players, rather than between individual players, even if what analysis provides is always the utility for the single players.

In contrast, in non cooperative games, players cannot form binding commitments and they act strategically to get the best for themself.

1.1 Non Cooperative Games

A non cooperative game can be represented in two different forms: the strategic form and the extensive form.

A game in strategic form (or normal form) lists each player’s strategies and the outcomes that result from each possible combination of choices.

In a two player game, let X and Y be the sets of strategies for the players and let f, g : X × Y → R be the pay-off functions of the players. Thus, a two player game in strategic form is a quadruplet (X, Y, f : X × Y → R, g : X × Y → R). In case the strategy sets are finite, the game can be efficiently represented by a bimatrix which shows players, strategies, and pay-offs.

  (a11, b11) ... (a1m, b1m) ... ... ... (an1, bn1) ... (anm, bnm)  

The first player and the second player have n (rows) and m (columns) possible strategies, respectively. When the first player chooses the i -th row and the second one chooses the j -th column, they produce the outcome assigning utility aij to

the first player and bij to the second one.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually

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presented in extensive form.

The extensive form of a game is a complete description of all relevant information about the game and of how the game is played over time (in sequential games). Therefore, we specify the initial setting, all possible evolutions, all final outcomes of the game (and relative pay-offs), the order in which players take actions and the information that players have at the time they are called to take those actions. Usually, this is made by constructing the tree of the game.

r s p t u q I II II (aps, bps) (apr, bpr) (aqt, bqt) (aqu, bqu)

Figure 1.1: Example: an extensive game with two players

The game in Fig.1.1 consists of two players (I, II). Each node represents a point of choice for one of the players and each branch out of the node represents an available action for that player. The combination of players’ choices leads to all the available outcomes of the game, whose pay-offs are specified at the bottom of the tree.

A game in extensive form may be analysed directly, or can be converted into an equivalent strategic form.

1.1.1 Games in strategic form

Strictly dominated strategies The strategic form of a game is useful to anal-yse strictly dominated strategies. Since all players are assumed to be rational, they make choices which result in the outcome they prefer most, given what their opponents do. Thus, the following assumption is quite natural:

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him to get strictly more, no matter which choice will make the other players.

Then strategy Z is said to strictly dominate strategy X and a rational player will never choose to play a dominated strategy. In some games, elimination of strictly dominated strategies results in a unique rational outcome of the game.

Nash equilibria In many games, however, the rational outcome cannot be singled out by means of the procedure of eliminating dominated strategies. Nash equilibrium is a more general concept of solution, which includes dominant strategy equilibria as special cases of it.

Definition 1.1. Let (X, Y, f : X × Y → R, g : X × Y → R) be a two player non cooperative game in strategic form. A Nash equilibrium for the game is a pair (x, y) ∈ X × Y such that:

f (x, y) ≥ f (x, y) for all x ∈ X g(x, y) ≥ g(x, y) for all y ∈ Y.

A Nash equilibrium recommends a strategy to each player, which any player can-not improve upon unilaterally: each player, taking for granted that the other one will play what he is recommended to play, has no incentive to deviate from the proposed strategy.

Since the other players are also rational, it is reasonable for each player to expect his opponents to follow the recommendation as well.

Let BR1 (best response) be the following multifunction:

BR1 : Y → X : BR1(y) = max {f (·, y)} .

In order to maximize his utility, the first player, once he knows that the second player plays a given strategy y, will choose a strategy x, belonging to

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The same argument follows for the second player best response BR2. Let BR be

such that

BR : X × Y → X × Y : BR(x, y) = (BR1(y), BR2(x)).

Thus, (x, y) is a Nash equilibrium for the game if and only if (x, y) ∈ BR(x, y).

Pure and mixed strategies A game in strategic form does not always have a Nash equilibrium that allows each player to deterministically choose one of his strategies (the players play pure strategies). Players may instead randomly select from among these pure strategies with certain probabilities and, thus, play a mixed strategy.

Definition 1.2. A mixed strategy is a probability distribution over the set of the pure strategies.

A mixed strategy assigns a probability to each pure strategy and randomizes play-ers’ choices. Thus, we need to consider expected pay-offs.

A Nash equilibrium in mixed strategy recommends a mixed strategy for each player, where any players cannot gain on average, by unilateral deviation. More-over, it always exists.

Theorem 1.3. If mixed strategies are allowed, then every game with a finite num-ber of players, in which each player can choose from finitely many pure strategies, has at least one Nash equilibrium.

1.1.2 Games in extensive form

While, in a game in strategic form, the players act simultaneously and do not know others’ choices (there is not a temporal component), the extensive form of a game allows to formalize also interactions where the players are informed about the actions of others (sequential games).

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for every possible situation throughout the game, even those not reached by the actual play of the game because of a former choice specified by the strategies itself. A game is called of complete information if the features of the players (strategies and utility functions) are common knowledge.

Perfect information games and backward induction A game may be of perfect information: every player knows at any time what is the current situation, the past history and all possible future evolutions. Every player is at any point aware of the previous choices of all other players and only one player moves at a time.

Extensive games with perfect information can be analysed by backward induction, in order to understand how rational players will play. This technique solves the game by first considering the last possible choices in the game: it analyses what happens at every terminal node, i.e. a node such that all branches, going out from it, lead to a final situation. Every player knows what to do at every node he is called to make a decision, and the other players know what he will do (they are all rational players and they act in their own self-interest). Once the last moves have been decided, backward induction proceeds to the players making the next-to-last moves (and then continues in this manner).

Usually, backward induction provides a unique outcome of the game. The only exception is if a player is indifferent among several alternatives.

Backward induction solution specifies the way the game will be played, starting from the root of the tree and proceeding along a path to an outcome. Because backward induction looks at every node in the tree, it specifies for every player a strategy, a complete plan of what to do at every node in the game where the player can make a move, even though that node may never be reached in the course of play. Thus, an extensive game can be described by his strategic form. Backward induction may not always find all Nash equilibria of a game of perfect information, described in strategic form.

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Imperfect information games The extensive form can also describe games with simultaneous moves. These games are called of imperfect information, since a player does not know the decision of the others, once is called to decide. However they know who the other players are, what their possible actions are and the preferences of these other players (the information is complete). Thus, a player knows to be in one of possible nodes, but not in which one. To represent it, a dotted line connects these nodes and the set of nodes is called information set.

r s p t u q I II II (aps, bps) (apr, bpr) (aqt, bqt) (aqu, bqu)

Figure 1.2: Example: an extensive game with an information set for the second player

A strategy specifies the choice of the player at each information set, labelled by the name of the player (in games of perfect information, the information sets are singletons).

Uncertainty Extensive games may include states where neither player makes a choice: a random decision is made by a player (the chance) who has no strategic interests in the outcome. For rationality, the players will evaluate their utility functions in terms of expected values. This allows players having no uncertainty about past moves and possible evolutions: the presence of the chance, once its moves are observed by all the players, does not imply that the information is imperfect. Rather, pay-offs will be evaluated using expected values.

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1.2 A strategic game: the Prisoners’ Dilemma

As described in the previous sections, Game Theory provides a means of abstract-ing the fundamental structure of an interaction and representabstract-ing it in terms of a strategic game.

The most famous of all strategic games with two players, in an interaction in-volving cooperation and competition, is the Prisoners’ Dilemma game. It is a standard example of game of complete and imperfect information: the structure of the game and the pay-off functions of the players are common knowledge, but each player has to choose strategically without knowing the other player’s choice. The Prisoners’ Dilemma shows why two rational players might not cooperate, even if cooperation would be in their best interests.

The concept of this game was originally developed by Merrill Flood and Melvin Dresher working at RAND Corporation in 1950 and later formalized, with pay-offs as prison sentence rewards, by Albert W. Tucker (that is the reason why it is called Prisoners’ Dilemma).

In the standard formulation of the dilemma, two people are suspected of being responsible of a serious crime. They are arrested and kept in solitary cell, there-fore they cannot communicate (what is important is that they cannot make a binding commitment to play in a certain way). The judge lacks enough evidence to convict the pair on the principal charge, but he can prove that they are guilty of a lesser crime. Each prisoner has two possible strategies: to confess the crime (specifically, he confesses the responsibility of both) or to remain silent. Seven years in jail is the punishment for the crime they are suspected of. The judge offers each prisoner a bargain:

• if one confesses and the other does not, the collaborator will be set free and the other one will be sentenced to 7 years in jail

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sentence: collaboration is appreciated)

• if no one confesses, they both will be sentenced to 1 year in jail (they are guilty of a lesser crime)

The Prisoners’ Dilemma is a non cooperative game. It is useful to present the game in strategic form, by means of a bimatrix, showing players’ strategies and pay-offs. Each element of the matrix stands for the pay-off, obtained by the players for a specific outcome of the game: in this particular situation they define the number of years prisoners are sentenced to.

Given the game, the goal is to define, for each player, the optimal strategy, that is the one minimizing the pay-offs (pay-offs are years in jail).

The available strategies for each players are ”to confess” (C) and ”not to confess” (NC).

C NC

C (5,5) (0,7) NC (7,0) (1,1)

Table 1.1: Prisoners’ Dilemma Bimatrix

C NC C C NC NC I II II (0,7) (5,5) (7,0) (1,1)

Figure 1.3: Prisoners’ Dilemma Tree

Eliminating all dominated strategies can solve this game: each prisoner analyses his best strategy given the other prisoner’s possible strategies.

Specifically, if a player confesses, it is in the best interest of the other player to confess too, in order to avoid the worst punishment (7 years). Otherwise if a

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player decides not to confess, the other one, confessing, can be free.

It results that to confess is a dominant strategy for both players (it is better than another one for a player, no matter how the opponent may play): this way, both prisoners are sentenced to 5 years (C-C). This is the unique Nash equilibrium of the game.

Clearly (NC-NC) would lead to a better outcome for both the players than (C-C) (only 1 year in jail instead of 5) but it is not in equilibrium: this is because the best replay to NC is to confess (a prisoner who discloses information, while the other conceals it, is rewarded) and, by choosing NC, a prisoners exposes himself to the worst possible outcome for himself. Because confessing offers a greater reward than keeping silent, all rational self-interested prisoners would confess, and so the only possible outcome for two purely rational prisoners is for both of them to confess.

The interesting part of this result is that pursuing individual reward leads both of the prisoners to confess, when they would get a better reward if they both kept silent: the rational solution of the game does not lead to the best possible outcome. The dilemma is that picking the best individual choice precludes the group from achieving the best common outcome.

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Chapter 2 A doctor-patient interaction

Modern clinical practice is founded on doctor-patient relationship: through this interaction, doctors gather information and data, make diagnoses and therapeutic decisions, treat the patient and help preventing illness, using clinical judgement. Usually, a doctor checks the health condition of a patient, through a medical examination and through a discussion about symptoms: therefore, the doctor-patient relationship directly determines the quality of information gathered and the care provided, since the more the patient trusts his doctor, the more he will disclose complete information. Once the doctor has all the data, he has to decide whether or not it is necessary to do something and, specifically what to do, in the patient’s best interest: therefore, he presents findings and options to the patient. The patient, deciding to consult a doctor, relies on him but he may expect his needs to be met in the way himself defines. Usually, he has his own treatment preferences or expectations (for example, among different treatment options, he would prefer a specific one) [6].

Often, interests of doctors and patients partly diverge and it may happen that a patient does not get the treatment he wants. As we have seen in Chapter 1, in this specific situation, Game Theory allows to analyse the strategic behaviour of a patient and a doctor, interacting in a clinical encounter. This analysis underline the doctor-patient dynamics and it can guide decision making in health care. Firstly, we present two applications of the most known strategic game, the

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Chapter 2, A doctor-patient interaction 15

oner’s dilemma game, described in Section 1.2. In his famous formulation, it may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature that have the same pay-off matrix. Actually, it can be used as a model for many real situations involving cooperative behaviour and is therefore of interest to the social sciences such as economics, politics, and sociol-ogy, as well as to the biological sciences.

With some simplifications, medical consultations in primary care may have an underlying structure corresponding to the Prisoners’ Dilemma game.

2.1 The Prisoners’ Dilemma in a doctor-patient

encounter

Prisoners’ Dilemma and consultation A first possible hypothetical scenario can be constructed with a busy doctor and his patient [15, 3].

Suppose that the doctor is consulted by a patient with a sore problem. After a brief examination, the doctor has two options: on one hand, he can prescribes antibiotics, hence dealing with the patient in less than 5 minutes, even if he is not sure it is the best option for the patient, or, on the other, he can undertake a full assessment of life style and other contributing factors and give the patient a prescription and self-management advise after a detailed discussion of benefit and risk of treatment, which would prolong the consultation to over 10 minutes. The patient can choose to follow the advise and the prescription or to ignore the prescribed treatment and seek a second opinion (consulting another doctor). There are four possible outcomes:

• (C,C) the doctor spends times giving advice, the patient follows the advice; • (C,D) the doctor spends times giving advice, the patient does not follow the

advice;

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ad-Chapter 2, A doctor-patient interaction 16

vice;

• (D,D) the doctor prescribes briefly a treatment, the patient does not follow the advice.

In this example, the best collective outcome is (C,C): the doctor acts in the patient’s best interest and the patient follows his advice without taking up valuable time of other doctors. But this is not a Nash equilibrium.

Individually, the patient prefers seeking a second opinion (D) to avoid possible medical errors, therefore the doctor’s best option is to spend little time on the patient (D). On the other hand, if the doctor chooses to deal with the patient quickly (D), than the patient prefers not to follow the treatment and to get a second opinion.

Therefore, the patient obtains antibiotics for his sore problem and the doctor can move on to the next patient. Both the patient and the doctor, by choosing D, avoid the worst possible outcome for themself, but their choices lead to (D,D), which is a negative outcome in good quality care.

Prisoners’ Dilemma for prescribing opioids to potentially drug-seeking patients Another paradigmatic situation, that can be described through the Prisoners’ Dilemma game is the description and analysis of a clinical encounter, where a doctor prescribes opioids to potentially drug-seeking patients [8, 4].

I am addicted to [opioids], and it is doctors’ fault because they prescribed them.

But, I will sue them if they leave me in pain.

Opioids are synthetic narcotics that affect on the brain to decrease the sensation of pain. They are typically used in medicine as analgesics (painkillers) with severe restrictions on their use, as most of them are extremely addictive.

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abusers and it results that sixty percent of the abused opioids have been obtained directly through a physician’s prescription. In many instances, doctors are fully aware that their patients are abusing these medications or diverting them to oth-ers for non-medical use, but they prescribe them anyway. Prisonoth-ers’ Dilemma models this paradigmatic situation.

Until the 19th _{century, pain was considered a sign of physical vitality, important}

to the healing process. Nowadays, as the availability of painkillers increases, re-lieving pain and suffering seems to be what the doctors are trained and almost obliged to do: society expects them to treat pain and it seems that the patients’ subjective experience of pain prevails other considerations.

Recently, the importance of the subjective experience of pain has increased by the practice of assessing patient satisfaction after the medical attention: patients have to fill out surveys about the care they received, which include specific questions about how their physicians have behaved in regard of their pain. In some institu-tions, physicians’ compensation may depend on patients’ satisfaction scores and this score can affect even the physician’s reputation. Obviously, doctors achieve great satisfaction and professional gratification in relieving patients’ pain.

Thus, doctors are pushed on treating pain. However, when a patient asks for opioids to alleviate suffering, the doctor has to evaluate whether the patient is in real pain or he is pretending to be, in order to avoid prescription to patients who ask for opioids for non-medical uses (for example, illegal uses or addiction habits). If the patient has real pain, the rational choice for the doctor is to treat him. If he fakes pain and the doctor suspects it, prescribing opioids is inappropriate as it would continue to feed addiction or illegal uses: the professionally right choice would be to diagnose and treat addiction. Besides having disastrous consequences for patient with addiction, prescribing opioids when there is no need represents also squandering resources for the health system.

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are likely to get a poor satisfaction score from those patients (which eventually means poor reimbursement and bad reputation for the doctor). Moreover, the professionally right choice not to prescribe pain medication to those who are not in pain will not improve doctors’ satisfaction score. Therefore, it seems that it is in doctor’s best interest to treat the pain, may it be real or fake.

This particular situation can be described by the following game tree.

Real Fake Prescribe Real Fake Do Not Prescribe Doctor N N S/NR S/R D D/R

Figure 2.1: S satisfied, D dissatisfied, R professionally rewarding, NR profes-sionally less rewarding

Specifically, in Figure 2.1 the first player is the doctor, who decides whether or not to prescribe opioids, while at the nodes, labelled by N, Nature chooses: the patient may be in real or fake pain.

The patient, asking for opioids, is assumed to be more satisfied if he gets opioids rather than if does not get them, no matter what his state of health is.

Moreover, the pay-offs in figure refer to the doctor:

Doctor : S/R indicates that the doctor gets high satisfaction score and he is pro-fessionally rewarded (he prescribed opioids to a patient in real pain), D indicates that the doctor gets poor satisfaction score from the patient (he did not prescribe opioids to a patient in real pain), S/NR indicates that the doctor receives a high satisfaction score but he is not professionally re-warded (he prescribed opioids to a patient faking pain), D/R indicates that the doctor gets a poor satisfaction score but he is professionally rewarded (he did not prescribe opioids to a patient faking pain).

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S/R > S/N R > D/R > D

It results that, for the doctor, Prescribe is a dominant strategy.

This is a classical Prisoners’ Dilemma: the best strategic move for individual (to prescribe opioids, no matter the state of health of the patient) is not good for the society, which actually is damaged.

2.2 The trust version of the Prisoners’ Dilemma

game

The Prisoners’ Dilemma game has been applied to two different situations but it cannot be assumed as an appropriate model of doctor-patient interactions. For example, the pain example, just considered, is a simplification of a typical interac-tion between physicians and patients, because it neglects doctors’ professionalism and ethical obligations [4]. Moreover, Prisoners’ Dilemma does not take into ac-count trust, which is essential in every clinical enac-counter: without it, patients would not commit to doctors in the first place.

In both the considered situation, lack of trust intensifies the Prisoners’ Dilemma, while considering trust may help avoiding it.

Thus, Prisoners’ Dilemma can be modified, adding trust (this way, the pay-offs change), in order to overcome paradoxical outcomes and to be a more realistic description of a clinical interaction.

Trust is important because it allows agents to form relationship with others and to depend on others in order to get something, yet it is risky because trusting someone does not guarantee to actually obtain what needed [12].

Clearly, every clinical encounter is asymmetrical: patients have to expose themself in a position of vulnerability and their trust can be abused. Sometimes it hap-pens that patients’ interests may diverge, partially or completely, from doctors’ interests. Moreover, facing with decisions involving uncertain outcomes, only in retrospect it is possible to see whether a decision proves right or wrong: doctors

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cannot guarantee a priori that the therapy chosen for the patients will be effective (in diagnostic uncertainty, he can make mistakes).

Therefore both patient and doctor may regret their choices: after making a de-cisions under uncertainty, they can discover that another alternative would have been better, for example, it happens when the doctor does not prescribe a treat-ment to a patient needing it.

Moreover, the doctor feels guilty when he abuses patient’s trust, for example when he gives unnecessary treatment to a patient who trusted him.

Eventually, both patients and doctors, can experience frustration when they are not able to do something, because of other’s resistance. For example it occurs when the patient refuses a treatment (frustration for the doctor) or when the doctor refuses to prescribe a treatment to a patient that demands it (frustration for the patient).

These concepts can be formalized in Game Theory models: regret, guilt and frus-tration lead to smaller utilities and satisfaction for patients and doctors.

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Chapter 3 The participation of a patient to

a Randomized Controlled Trial

The article When is it rational to participate in a clinical trial? A game theory ap-proach incorporating trust, regret and guilt by Djulbegovic and Hozo [3], provides a game theoretic analysis of a dilemma researchers and patients face, whether to test new medical treatments on humans, through Randomized Controlled Trials. Randomized Controlled Trials (RCTs) are the standard form of clinical trials: they are scientific experiments, done in clinical research, to discover new treatments and test standard ones. Specifically, they are used to test the efficacy of various types of medical interventions, i.e. experimental treatments, standard treatments or placebo: people, participating in a trial, are randomly allocated one or other of the different treatments under study, and through human experimentations, they generate data on safety and efficacy.

A patient can participate to a Randomized Controlled Trial only if the principle of clinical equipoise is verified, that is to say that there is genuine uncertainty about the preferred treatment. This principle provides the ethical basis for med-ical research that involves assigning patients to different treatments.

RCTs have always raised ethical concerns whether a researcher risks putting clin-ical research ahead of his patients’ best interests.

In clinical research that uses Randomized Controlled Trial, there is an interaction

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Chapter 3, The participation of a patient to a RCT 22

between two agents, a patient and a clinical researcher, with common and con-flicting interests. If a patient provides informed consent for their participation in an RCT, he is not certain to receive treatment that is best for him personally, but he prides himself in his contribution to clinical research. Obviously, he also hopes to improve his own health conditions. Similarly, a clinical researcher is motivated to help his own patient, through RCT, even if the main purpose of the RCT is to potentially improve healthcare for the good of future patient.

Thus, the matter of patients’ participation to RCTs can be formulated as a Game Theory game with two players that act strategically to advance their interests, in conflict and cooperation situations.

In a model describing a RCT situation, trust has to be taken into account, since every clinical encounter is based on it and specifically, trust is essential for the participation of patients in experimental clinical trials.

A patient, by trusting the researcher and by participating to a RCT, accepts some level of risk and vulnerability. Once enrolled in the trial, the patient may discover that his trust has been abused (for the sake of research) and therefore, regret his choice to participate. In the same way, the researcher may feel guilty because he did not honour his patient’s trust.

Concepts of regret and guilt are formalized in Game Theory models, to under-stand when it is rational to participate in a Randomized Controlled Trial, from the point of view of patients and researchers. We will see that the analysis leads to the same conclusions of the Prisoners’ Dilemma game.

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3.1 A sequential game with perfect information

The model used to describe the participation of a patient to a Randomized Con-trolled Trial is a sequential game with perfect information: the patient first has to decide whether or not to trust the researcher and once the researcher is trusted, he has to decide whether to honour the patient’s trust or to abuse it.

It is considered a Randomized Controlled Trial, used to test a new experimen-tal treatment, by comparing it to a standard one. In the RCT, the patient is randomly assigned to the experimental treatment or to the standard one, with probability of randomization r, associated to the experimental treatment.

Since no treatment is always successful, it is assumed that there is a certain prob-ability of success of experimental and standard treatment, respectively e and s.

Patient STD 1-s s No T_rust Researcher EXP 1-e e Abuse STD 1-s s 1-r EXP 1-e e r Honour Trust (U4− R(U1− U4), N A) (U3, N A) (U2− R(U3 − U2), V2 − G(V2− U2)) (U1, V1) (U4, V4) (U3, V3) (U2, V2) (U1, V1)

RCT

Figure 3.1: A sequential game with perfect information to model a RCT clinical research - Tree structure

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Firstly, the patient decides whether or not to trust the researcher: if he trusts him, he accepts to get the new, unproven, experimental treatment (EXP) within the context of the trial, otherwise, if he does not trust the researcher he chooses to directly get the standard treatment (STD). Once the patient trusts the researcher and decides to participate in the RCT, the researcher has to decide whether to offer the experimental treatment only within the context of the RCT (honour-ing patient’s trust) or to offer it outside the trial (abus(honour-ing trust). The last case happens if the researcher believes that the new treatment is superior or if they invested considerably effort in developing it.

Available strategies for the patient are Trust and No Trust, while, for the re-searcher, they are Abuse and Honour.

At every possible outcome of the game, each player associates a pay-off, which represents his preference on the outcome of the game. Patient and researcher’s utilities are denoted by U and V, respectively.

Specifically:

V1, U1 are the pay-offs associated with the success of experimental treatment

V2, U2 are the pay-offs associated with the failure of experimental treatment

V3, U3 are the pay-offs associated with the success of standard treatment

V4, U4 are the pay-offs associated with the failure of standard treatment

If the patient chooses No Trust, he gets the standard treatment, directly: the researcher is not called to decide, therefore his utility is not defined (NA).

It is assumed that the patient favours success over failure of a treatment and he gets more satisfaction in the experimental success rather than in the standard one, since, this way, he feels to contribute to clinical research on new treatments.

U1 ≥ U3 ≥ U2

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The researcher favours success over failure of a treatment, but does not prefer a treatment success over the other: data on experimental treatment are significant, even in case of unsuccessful testing. Thus, researcher’s pay-off, associated with the failure of experimental treatment is greater than analogous patient’s.

V1 ≥ V2 V3 ≥ V4 V2 ≥ U2

Since in clinical experiments patients cannot be guaranteed successful outcomes with any treatment, clinical trials, as any other clinical encounter, are based on trust. Thus, concepts of regret and guilt have been integrated in the model: they all lead to smaller utilities and satisfaction for the players.

Specifically, after a patient has volunteered participation in the trial, he may discover that his trust has been abused and, therefore, he will regret his choice: regret (R) is defined as a fraction of the difference between the utility of the taken action and the utility of the best action he should have taken, a posteriori. For the sake of simplicity, R is the same in all possible scenarios (No Trust and Abuse). Similarly, a researcher may feel guilty when he abuses the patient’s trust. Guilt (G) diminishes the researcher’s utility by a fraction of the difference between his and the patient’s utility corresponding with the same outcome, but obtained in the RCT scenario.

For example, the path Trust - Abuse - Failure is considered: the patient trusts the researcher, who abuses his trust and offers the experimental treatment outside the RCT, without success.

P atient : U2 − R(U3− U2)

Researcher : V2− G(V2− U2)

The patient:

1. gets the experimental treatment without success → U2

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of the difference between the utility he gets and the utility he should have achieved, a posteriori → R(U3 − U2)

The researcher:

1. gives the patient the experimental treatment without success → V2

2. feels guilty: his utility diminishes by a fraction of the difference between his and the patient’s utility corresponding with the same outcome (unsuccessful experimental treatment), but obtained in the RCT scenario. → G(V2− U2)

3.2 Analysis

Success of experimental treatment, success of standard treatment and tion in RCT introduce uncertainty in the model. Once probability of randomiza-tion is fixed, e and s will be the key parameters to determine optimal strategies. In order to solve the game, the model is reduced by evaluating the utility functions of the players in terms of expected utilities.

Specifically, the expected values of all possible scenarios are calculated, denoted by P and R, for patient and researcher respectively: P1 and R1 are the expected

values of scenario Honour (they are randomization-weighted averages), P2 and R2

are the expected values of scenario Abuse, P3 and R3 are the expected values of

scenario No Trust. P1 = r · [e · U1+ (1 − e)U2] + (1 − r) · [s · U3+ (1 − s) · U4] R1 = r · [e · V1 + (1 − e) · V2] + (1 − r) · [s · V3+ (1 − s) · V4] P2 = e · U1+ (1 − e) · U2− (1 − e) · R · (U3− U2) R2 = e · V1+ (1 − e) · V2− (1 − e) · G · (V2− U2) P3 = s · U3+ (1 − s) · U4− (1 − s) · R · (U1− U4) R3 = N A

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Chapter 3, The participation of a patient to a RCT 27 Patient STD No Trust Researcher EXP Abuse RCT Honour Trust (P3, R3) (P2, R2) (P1, R1)

Figure 3.2: A sequential game with perfect information to model a RCT clinical research - Reduced tree structure

Solving the game means to determine patient’s and researcher’s optimal strategies. Let p be a variable such as

p = (

1 if the researcher honours trust 0 if the researcher abuses trust and let τ be a variable such as

τ = (

1 if the patient trusts the researcher

0 if the patient does not trust the researcher

Since it is a perfect information game, the method used to solve the game is backward induction. This technique starts from the terminal nodes of the tree structure and proceeds up to the root.

Researcher The unique terminal node is the one associated to the researcher: his choice depends on which expected utility is greater, R1 or R2.

Specifically, the researcher honours patient’s trust (p = 1) if R1 > R2.

Let

EV[Exp] = e · V1 + (1 − e) · V2 EV[Std] = s · V3+ (1 − s) · V4

be the researcher’s expected utilities of experimental and standard treatment, respectively.

Using the expressions of utilities, the researcher honours patient’s trust (p = 1) if r · (EV[Exp] − EV[Std]) ≥ EV[Exp] − EV[Std] − (1 − e) · G · (V2− U2)

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If EV[Exp] > EV[Std], i.e. the researcher prefers the experimental treatment over

the standard one, he will choose to honour patient’s trust if r > 1 −(1 − e) · G · (V2 − U2)

EV[Exp] − EV[Std]

Otherwise, if EV[Exp] < EV[Std], the researcher will choose to honour patient’s

trust if r < 1 −(1 − e) · G · (V2 − U2) EV[Exp] − EV[Std] Let r∗ = 1 − (1 − e) · G · (V2− U2) EV[Exp] − EV[Std]

be the randomization probability, that makes the researcher indifferent between choosing to honour or abuse trust.

Thus, the researcher’s best reply is

BRr =          p = 1 if r > r∗ ∧ EV[Exp] > EV[Std] p = 0 if r < r∗ ∧ EV[Exp] > EV[Std] p = 1 if r < r∗ ∧ EV[Exp] < EV[Std] p = 0 if r > r∗ ∧ EV[Exp] < EV[Std]

The researcher’s best strategy is defined, once all parameters are fixed.

Patient To conclude the analysis, root node is analysed: the patient is called to move. Since it is a game of perfect information, the patient knows what is the researcher’s rational choice.

Let

EU[Exp] = e · U1+ (1 − e) · U2 EU[Std] = s · U3 + (1 − s) · U4

be the patient’s expected utilities of experimental and standard treatment, re-spectively.

In order to choose, the patient has to compare the expected utility of No Trust (P3) with the expected utility of Trust, which depends on the researcher’s choice.

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1. if r > r∗ ∧ EV[Exp] > EV[Std] the researcher honours trust (p = 1).

Therefore, the patient’s decision depends on the sign of P1− P3.

Specifically, he trusts the researcher (τ = 1) if P1 > P3

r · EU[Exp] + (1 − r) · EU[Std] > EU[Std] − (1 − s) · R · (U1− U4)

2. if r < r∗ ∧ EV[Exp] > EV[Std] the researcher abuses trust (p = 0).

EU[Exp] − (1 − e) · R · (U3− U2) > EU[Std] − (1 − s) · R · (U1− U4)

3. if r < r∗ ∧ EV[Exp] < EV[Std] the researcher honours trust (p = 1).

4. if r > r∗ ∧ EV[Exp] < EV[Std] the researcher abuses trust (p = 0).

5. if r = r∗, the researcher is indifferent between choosing Honour or Abuse. In this situation, backward induction does not provide a unique outcome of the game.

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3.2.1 Data

To compute the solution, it is necessary to estimate variables in the game: players’ utilities, trust variables such as regret and guilt and the randomization probabil-ity. Once they are fixed, best strategies for the players depend on values of the variables e and s, the probability of success of the experimental and standard treatment, respectively.

There are not empirical data that can precisely inform the values of each of the utilities of the model: usually they are obtained by surveying a sample of ex-perienced clinical investigators, asking them to provide the values of each of the utilities in the model, first from a patient and then from a researcher point of view. Regret and guilt are quantified using psychometric measurement approach. Djulbegovic and Hozo, in their article [3], provide values for utilities and trust variables, which satisfy the conditions

U1 ≥ U3 ≥ U2 U1 ≥ U3 ≥ U4 V1 ≥ V2 V3 ≥ V4 V2 ≥ U2 Specifically, Researcher Patient V1=95 U1=90 V2=54 U2=16.3 V3=70 U3=84 V4=44 U4=16.9 G =0.2 R=0.2

Moreover, let r = 0.5: it is assumed equal probability of being assigned to the standard or to the experimental procedure.

Using data, it results that

EV[Exp] − EV[Std] > 0 ⇔ e > 26 41s − 10 41 r > r∗ =: 1 − (1 − e) · G · (V2 − U2) EV[Exp] − EV[Std] ⇔ e < 26 56.08s + 5.08 56.08

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Thus, all the results are rewritten, highlighting the dependence on e and s. Figures are used to display the results of the patient’s and the researcher’s best strategy over all possible values of the success of experimental and standard treatment.

Researcher Firstly, the researcher’s best reply is analysed. It results that e > 26 56.08s + 5.08 56.08 ∧ e < 26 41s − 10 41

are incompatible. It corresponds with conditions r < r∗ ∧ EV[Exp] < EV[Std]: it

is not possible that, under the condition that the researcher prefers the standard treatment over the experimental one (EV[Exp] < EV[Std]), his best strategy is to

honour trust (p = 1).

Thus, the researcher’s best reply is

BRr =          p = 1 if e < _56.0826 s + _56.085.08 ∧ e > 26₄₁s − 10₄₁ (3.1a) p = 0 if e > _56.0826 s + _56.085.08 ∧ e > 26₄₁s − 10₄₁ (3.1b) p = 0 if e < _56.0826 s + _56.085.08 ∧ e < 26₄₁s − 10₄₁ (3.1c) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 s e p=1 Honour HaL p=0 Abuse HbL p=0 Abuse HcL

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A researcher should enrol a patient in a RCT when there is genuine uncertainty about the preferred treatment, i.e. the probability of success of the experimental treatment is equal to the probability of success of the standard one. It means that, in figure, for values of e and s close to the diagonal e = s, the researcher’s best strategy should be to honour trust.

Instead, as shown, close to the diagonal, the researcher’s best strategy is to abuse the patient’s trust and to give him the experimental treatment outside the RCT. The researcher’s best strategy is to honour the patient’s trust (p = 1) if:

- both probabilities e and s are close to zero; - ratio e

s is close to 0.5: the probability of success of the standard treatment

is, at least, twice the probability of success of the experimental treatment. For all other combination of e and s, the researcher abuses the patient’s trust (p=0 ) and gives him the new experimental treatment outside the trial.

Specifically, it happens if:

- the probability of success associated to the experimental treatment is large, let s be small or large;

- both probabilities are large (the higher the probability that the experimental treatment will be successful, the more incentive the researcher has to abuse the patient’s trust);

- the probability of success associated to the standard treatment is large and the probability of success associated to the experimental one is close to zero.

Patient To derive the patient’s best reply, it is necessary to distinguish four cases:

1. if the probabilities of success associated with experimental and standard treatment, e and s, are such as

e < 26 56.08s + 5.08 56.08 ∧ e > 26 41s − 10 41, (a)

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the researcher honours trust (p = 1) and enrols the patient in the RCT. Thus, the patient trusts the researcher (τ = 1) if P1 > P3.

Using data, it results that the patient trusts the researcher (τ = 1) if

e > 96.34 73.3 s − 28.64 73.7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Τ=1, Trust Τ=0, No Trust

Figure 3.4: Patient’s best reply if p = 1

Under the conditions that make Honour the researcher’s best choice, the patient chooses to trust the researcher if both the probabilities of success of the treatments, e and s, are close to zero. Otherwise, for high values of the probability of success associated with the standard treatment, the patient’s best choice is No Trust: he gets directly the standard treatment.

e > 26 56.08s + 5.08 56.08 ∧ e > 26 41s − 10 41 (b)

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the researcher abuses trust (p = 0) and directly offers the experimental treatment to the patient.

Thus, the patient trusts the researcher (τ = 1) if P2 > P3.

Using data, it results that the patient trusts the researcher (τ = 1) if e > 81.72 87.24s − 0.48 87.24 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Τ=1, Trust Τ=0, No Trust

Figure 3.5: Patient’s best reply if p = 0 (b)

Under the conditions r < r∗ ∧ EV[Exp] > EV[Std], that make Abuse the

researcher’s best choice, the patient chooses to trust the researcher and to volunteer participation in the trial if the probability of success associated with the experimental treatment is larger than the one associated with the standard treatment.

e < 26 56.08s + 5.08 56.08 ∧ e < 26 41s − 10 41 (c)

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the researcher abuses trust (p = 0) and directly offers the experimental treatment to the patient.

Thus, the patient trusts the researcher (τ = 1) if P2 > P3.

Using data, it results that the patient trusts the researcher (τ = 1) if

e > 81.72 87.24s −

0.48 87.24

However, e > 81.72_87.24s − _87.240.48 and e < 26₄₁s − 10₄₁ are incompatible, therefore this is an empty region: it is not possible that, under the conditions that the researcher prefers the standard treatment over the experimental one (EV[Exp] < EV[Std]), the patient’s optimal choice is Trust.

Under these conditions, the patient chooses No Trust always.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Τ=0, No Trust

Figure 3.6: Patient’s best reply if p = 0 (c)

4. if e = 26 56.08s +

5.08

56.08, the researcher is indifferent between choosing Honour

or Abuse. In this situation, backward induction does not provide a unique outcome of the game.

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Once the players’ utilities, trust variables and the randomization probability are fixed, players’ best strategies depend on the probabilities of success e and s. The final situation is depicted in the following figure, which displays the results of the patient’s and the researcher’s best strategy over all possible values of the success of experimental and standard treatment.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Τ=1, p=0 Τ=0, p=0 Τ=1, p=1 Τ=0, p=1

Figure 3.7: Players’ best strategies, function of e and s. The dot shows the most likely values of e and s

The best possible outcome would be for the patient to trust the researcher and for the researcher to honour trust (τ = 1 and p = 1, yellow field in figure): it is favourable both for the researcher to have people enrolling to clinical trial and for the patient to be able to trust his own doctor.

It happens for values of e and s close to zero: the researcher honours the patient’s trust if both treatments have little probability to be successful.

From RCTs performed over 50 years in the field of cancer, the most likely values of the probability of success e and s are assessed to be (e, s) = (0.41, 0.59). As shown, the most likely situation is such that neither the patient trusts the researcher, nor the researcher honours trust (τ = 0 and p = 0, the dot in the

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purple field in figure). Thus, under randomization of 50%, the most rational strategy for the player is not to cooperate, which is a negative and very socially inefficient outcome.

This situation describes a Prisoners’ Dilemma: the players’ optimal strategies are Abuse and No Trust.

The results of this model may explain the low rate of patients enrolling to clinical trials, because they believe the researcher will not honour their trust. Therefore, the model highlights the necessity to find a way to induce an optimal behaviour of the researcher, in order to lower abuses and increase participation to clinical trials.

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Chapter 4 Game Theory models for

healthcare decisions

It is now interesting to analyse Game Theory application in healthcare decisions, specifically in a generic and common interaction between a patient and a doctor, to analyse the strategic behaviour of the agents, interacting in a clinical encounter. Therefore, we consider the easiest clinical encounter between a patient and a doc-tor. The doctor faces the difficult choice of whether or not to administer a certain treatment to a patient who may have a disease or not. It is supposed (and this happens in the majority of clinical interactions) that there is uncertainty about the presence of a given disease and no further diagnostic testing is available. Ad-ministering a treatment, known to be effective for the disease under consideration, is advantageous if the disease is actually present, otherwise it could be damaging if the disease is absent. On the contrary, failing to prescribe the treatment is good if there is not disease, but it may worsen the patient’s condition if the disease is present [5].

The patient has to decide whether or not to trust the doctor. Moreover, he may demand treatment when the doctor does not recommend it (and he may get it or not).

In this chapter, several different models are presented, analysed and compared, specifically simultaneous games and sequential games. Simultaneous games have

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Chapter 4, Game Theory models for healthcare decisions 39

been provided by already existing studies [4], while sequential ones are new mod-els. The choice of modifying models aims to create a more realistic game for everyday clinical encounters.

Since every clinical interaction is fundamentally based on trust, concepts of regret, guilt and frustration have to be integrated in Game Theory models.

4.1 Simultaneous games for healthcare decisions

The first models we consider are two simultaneous games with two players, the doctor and the patient, as Djulbegovic, Hozo and Ioannidis present them in the article Modern health care as a game theory problem [4].

They are games of complete and imperfect information: the structure of the game and the pay-off functions of the players are commonly known, but the players do not see all the moves made by others (they play simultaneously).

This is a strong hypothesis: it is assumed that the patient does not know if the doctor prescribes the treatment or not. Even if it is realistic in some situation, as in a surgical context, this hypothesis is not verified in everyday clinical inter-actions, since usually, the doctor shares the diagnosis and the benefit and risk of a treatment with the patient.

The situation is the following: a doctor sees a patient, there is uncertainty about the presence of a certain disease and a specific treatment is known to be efficient for that disease. It is not possible to obtain further information, as all diagnostic testing is exhausted.

The doctor’s strategies are:

R: he recommends the treatment to the patient

No R: he does not recommend the treatment to the patient Instead, the patient’s strategies are:

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No Trust: he does not agree with the doctor’s choice

In the specific situation when the doctor does not recommend the treatment and the patient does not trust him (he disagrees with the doctor’s decision not to recommend the treatment), the patient himself can ask for the treatment.

It leads to two slightly different models:

Model A : the patient asks for the treatment and he does not get it Model B : the patient asks for the treatment and he gets it

4.1.1 Simultaneous game, model A: the patient demands

treatment and he does not get it

Firstly, we consider model A: the patient asks for the treatment that the doctor does not recommend and he does not get it.

Since it s a simultaneous game, its extensive form has an information set for the patient (dashed line in the tree structure): the first player is the doctor and the second player, the patient, when called to decide knows to be in one of the two nodes of the information set, but not in which one.

Doctor Patient D− D+ No Trust (Demand R) D− D+ Trust No R Patient D− D+ No Trust D− D+ Trust R (V4, U4− Fp(U4− U2)) (V3− G(U1− V3) − Rd(V1− V3), U3− (Rp+ Fp)(U1− U3)) (V4, U4) (V3− G(U1− V3) − Rd(V1− V3), U3− Rp(U1− U3)) (V4, U4) (V3− Fd(V1− V3), U3− Rp(U1− U3)) (V2− G(U4− V2) − Rd(V4− V2), U2− Rp(U4− U2)) (V1, U1)

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D+ states that the disease is present (the treatment is known to be effective), while D− states that the disease is absent (the treatment may be harmful). Each path of the tree (or course of action) is associated with the pay-offs, which refer to how the players quantify various clinical outcomes such as life expectancy, mortality rates, absence of pain, satisfaction with care and cost. Doctor and patient’s utilities are denoted by V and U, respectively.

Specifically:

V1, U1 are the pay-offs associated with the prescription of the treatment in presence

of the disease

V2, U2 are the pay-offs associated with the prescription of the treatment in absence

of the disease

V3, U3 are the pay-offs associated with the non-prescription of the treatment in

presence of the disease

V4, U4 are the pay-offs associated with the non-prescription of the treatment in

absence of the disease

It is assumed that the doctor gets more satisfaction in treating a patient with disease (V1) than in non-treating a patient without disease (V4): action is valued

better than no action and the patient expects the doctor to do something. The administration of no treatment to a patient without disease (V4) is valued more

than the unnecessary administration of the treatment to someone without disease (V2). The worst outcome, however, is associated with failing to administer

treat-ment to a patient with disease (V3). Similarly, patient’s outcomes are ordered in

the same way.

0 ≤ V3 < V2 < V4 < V1

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The patient’s utility of receiving the treatment in presence of disease (U1) is greater

than the doctor’s utility when he does not recommend the treatment to a patient with disease (V3).

V3 ≤ U1

The patient’s utility of non-receiving the treatment when there is not disease (U4)

is greater than the doctor’s utility when he recommends a treatment to a patient without disease (V2).

V2 ≤ U4

This game models a clinical encounter and, since every doctor-patient interaction is fundamentally based on trust, concepts of regret, guilt and frustration have been integrated in the model: they all lead to smaller utilities and satisfaction for the players. Doctor Patient D− D+ No Trust (Demand R) D− D+ Trust No R Patient D− D+ No Trust D− D+ Trust R (V4, U4−Fp(U4− U2)) (V3−G(U1− V3) −Rd(V1− V3), U3− (Rp+Fp)(U1− U3)) (V4, U4) (V3−G(U1− V3) −Rd(V1− V3), U3−Rp(U1− U3)) (V4, U4) (V3−Fd(V1− V3), U3−Rp(U1− U3)) (V2−G(U4− V2) −Rd(V4− V2), U2−Rp(U4− U2)) (V1, U1)

Figure 4.2: Simultaneous Model A tree structure, highlighting regret, guilt and frustration

When a player regrets his decision because he realizes that another course of action would have been preferable, he has a loss of potential utility: R is defined as a fraction of the difference between the utility of the taken action and the utility of the best action he should have taken, a posteriori. For the sake of simplicity, R

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is the same in all possible scenarios, but it may differ from doctor (Rd) to patient

(Rp).

Doctor may feel guilty when he abuses the patient’s trust: he fails to evaluate the health condition and makes a mistake in the therapeutic decision. G diminishes the doctor utility by a fraction of the difference between his and the patient’s utility.

Eventually, frustration occurs when a player cannot do something because of the resistance of the other player. Similarly to regret,Fis defined as a fraction of the difference between the utility of the taken action and the utility of the best action he should have taken, a posteriori.

For example, we consider the path No R - No Trust - D+ in Fig. 4.2: in presence of the disease, the doctor does not recommend the treatment to a patient, who asks for it, without getting it.

Doctor : V3−G(U1− V3) −Rd(V1− V3)

P atient : U3− (Rp+Fp)(U1− U3)

The doctor:

1. does not prescribe the treatment in presence of the disease → V3

2. feels guilty: his utility diminishes by a fraction of the utility that the patient could have achieved and his utility → G(U1− V3)

3. regrets his decision (he should have chosen R): his utility diminishes by a fraction of the difference between the utility he should have achieved and the utility he gets → Rd(V1− V3)

The patient: