Abstract
We examine in this article a frequently overlooked, if not ignored, premise underlying the canonical assurance game model: Hunters could potentially bag more than a single hare (or two) in place of the prized stag. Whether a risk-dominant equilibrium is necessarily inefficient or inferior to one that is assumed to be payoff-dominant is the question we seek to address. In doing so, we suggest plausible variations of the model with different game-theoretic realizations. Single-play illustrations drawn from robotic surgery underscore their practical implications for health care economics and management. The robotic technology revolution amplifies the rational and interactive choices available to players under conditions of risk and uncertainty. Like the canonical model, our illustrations involve insulated, self-interested actions arising from the presence or absence of trust and coordination among players. They differ from the canonical model by allowing for multiple, potentially cooperative equilibrium payoffs. Any cooperative action can be considered optimal if players coordinated on it, taking fully into account the quantifiable and multiplicable value of their second best strategies. Nonetheless, we suggest that any dominant solution/s should accommodate best evidence in health care to provide patients with the most suitable treatments and services. There lies the challenge in reconciling theory and practice in health economics.
Introduction
Over the last two decades, game theory has found enormous applications in health care. By modeling complex health-affecting behaviors and decision-making to produce the best outcomes in either competitive or collaborative situations, health economists have relied on it to compare the safety, efficacy, and value of one drug or therapeutic intervention with another, schedule operating rooms, set fees, train technicians, and decide where to build hospitals. Much of the work involves simulation. Some firms, such as General Electric, which is working to address India’s massive health care problems, depend on complex computer modeling for this purpose (Shurkin, 2013).
Pioneered by mathematicians Emile Borel and John Von Neumann in the 1920s, game theory evolved to discover whether a “best” or optimal strategy for any game exists, and to find that optimum. It was initially intended for economists but its applications to politics, psychology, sociology, warfare, recreational games, and other fields, such as evolutionary biology, soon became apparent. Various games and strategy matrices, based on player cooperation with one another, or defection (i.e., competition and conflict), have been identified and developed over time (Smith, 2003). Any game-theoretic model contains five basic elements: (a) competitive players (or decision-makers), (b) rules governing players’ behavior, (c) strategies available to each player within a given set of rules, (d) game outcomes (each of which results from choices made by players at a given point in the game), and (e) payoffs accrued by players as a result of each plausible outcome. Along with a solution concept, these basic elements help game theorists deduce a set of equilibrium strategies for each player. That is, any game with a finite number of players, each of whom has a finite number of options to choose from, would have at least one Nash equilibrium, where no player may profit by unilaterally deviating from a chosen strategy after considering the other player’s decision.
One popular game-theoretic model in health care economics and management is the assurance game, which is invariably referred to as a “stag hunt,” “trust dilemma,” or simply “coordination game.” We discuss a frequently overlooked, if not ignored, premise underlying the model. Whether a risk-dominant equilibrium is necessarily inefficient or inferior to one that is considered payoff dominant is the question we seek to investigate, as we suggest variations in the assurance game with different game-theoretic realizations. Further inspiration came from several assurance game theorists before whom we raised the same question and who encouraged us to write about it. We do so now with single-play illustrations from robotic surgery that offer theoretical and practical insights on health care.
The Model
Any assurance game models conflict between security and cooperation by examining the trade-offs that arise where a larger—but demonstrably costlier or riskier— payoff can be generated from one strategy in contrast to another (Harsanyi & Selten, 1988). Health economists have found it useful, especially in controlling health care costs. With two players (or sets of players) and two choices of actions and outcome-based preferences, assurance games identify a dominant solution/s for a predefined setting based on certain essential prerequisites: (a) players have a common interest to score as high as possible; (b) however, they also have competing interests to increase and protect their respective proportion of scores; (c) each player’s utility-maximizing strategy requires consideration of the other player’s choices without necessarily communicating with each other; and (d) strategy choice is made under conditions of risk and uncertainty (Ashrafian, Darzi, & Athanasiou, 2011; Harsanyi & Selten, 1988).
In the original, brief metaphorical story of philosopher Jean-Jacques Rousseau (1755; Rousseau, in Cranston, 1984), two hunters (the players) are faced with the dilemma of bagging either a stag or a hare. If one hunter chooses a stag, she or he would require the total cooperation of the other to succeed. Having discovered its path in the forest, the hunters can mutually cooperate in capturing the stag, which offers a bigger, tastier meal. However, a long day passes without a trace of the stag, although the hunters are reasonably certain that it will eventually come.
Plausible variations in the story appear at this point: In a few of them, the two hunters see a single hare running back and forth on the same path where they have laid out the stag trap. If they mutually defect (i.e., abandon a joint stag hunt), they divide the prize of that one hare, which may be tasty but substantially less filling as a meal (Golman & Page, 2010). A more common version of the story tells of a pair of hares tempting the starving hunters, who each has an opportunity to take one hare for a meal (Poundstone, 1992). Some versions go beyond two hares, allowing for “a couple of hares” roaming around the same forest, thus further increasing the hunters’ temptation to settle for them after a long, tiring day of waiting for the stag (Harford, 2007). Asserting that “if a hare happened to pass within the reach of one of them, we cannot doubt that he would have gone off in pursuit of it without scruple and, having caught his own prey, he would have cared very little about having caused his companions to lose theirs” (Rousseau, in Cranston, 1984, p. 112), Rousseau did not exactly specify how many hares there were for the hunters to trap. Neither is there any indication among the story’s variations as to how many hares could possibly make up for the prize of bagging a stag (i.e., the meat offered by the stag). This has led one game theorist to ask, “What are the values of a hare and of an individual’s share of the deer given a successful hunt?” (Skyrms, 2001).
At any rate, the risk-utility trade-off in the assurance game is as follows: Risk the stag never coming, or risk another hunter taking the kill (Skyrms, 2004). Two pure strategy Nash equilibria (the best response and second best) exist in the stag hunt, as illustrated in Figure 1. Figure 2 presents the stag hunt in its canonical form.
Payoff matrix in assurance games.
General form of assurance games.
Of the four possible outcomes in Figures 1 and 2, a hunter’s best response is expected to be mutual cooperation (stag–stag). Despite the long, uncertain wait, the hunters will maximize their respective payoffs because the stag offers a larger, tastier meal. But either hunter might opt not to act in tandem with the other. Hence, one outcome is unilateral defection (hare–stag), as the first hunter goes for the hare/s at the expense of the other who waits for the stag. The other is unilateral cooperation (stag–hare), as the first hunter chooses not to go for the hare/s, hoping the other hunter will cooperate in going for the stag (which eventually does not happen). The fourth outcome is mutual defection (hare–hare), which is when both hunters give up on the stag and content themselves with the hare/s out of desperation.
The hunters will perforce figure out which strategy the other will choose, and which equilibrium action to select, based on each one’s anticipated payoff and in light of the attendant risks (Harsanyi & Selten, 1988). It is in this context that an assurance game is said to lead to a dual Nash equilibria. The “efficient” Nash equilibrium is the stag–stag strategy pair, because it is payoff dominant. Game theorists consider it both Pareto-optimal and Hicks-optimal. The fourth strategic outcome (hare–hare), albeit a Nash equilibrium, is considered “inefficient” and often risk dominant (i.e., less risky). Under this scenario, what should not be a dilemma at all (if hunters only mutually cooperate) turns into one because each hunter anticipates the likelihood that, in the absence of mutual trust and communication, the other hunter will rationally defect (Skyrms, 2004). Thus, if one hunter expects that the other will chicken out, she or he will want to chicken out, too, in favor of the second best. Under what circumstances might both hunters rationally defect without necessarily obtaining a less efficient or inferior outcome is the question that the hare–hare strategy pair raises.
Illustrations
Game-theoretic illustrations of the hare question might be drawn from the growing interest and initiatives of many hospitals in developing a robotic surgery program. The robotic technology revolution in the field of surgery amplifies, for our purposes, the rational and interactive choices available to players under conditions of risk and uncertainty. We posit the view that where the so-called inefficient equilibrium strategy is quantitatively valued and multiplicable, the payoff-dominant equilibrium cannot necessarily be held superior and singularly Nash efficient.
At least half a million surgical procedures worldwide are performed robotically each year. American and European hospitals have rapidly adopted robotic technology for use in the treatment of various health conditions, including heart-related surgeries, hysterectomies, prostatectomies, and gynecologic surgical procedures (Rabin, 2013). Although it is often employed for minimally invasive surgery (i.e., procedures performed through tiny or miniature incisions), it has also been used for certain traditional open surgical procedures (e.g., cardiac valve repair). In 2000, robotic surgery with the da Vinci Surgical System—a four-arm robot technology—was approved by the U.S. Food and Drug Administration (FDA). The most widely used robotic surgical system has a camera arm and mechanical arms with surgical instruments attached to them. The surgeon controls the arms while seated at a computer console near the operating table. The console gives the surgeon a high-definition, magnified, 3D view of the surgical site. The surgeon can manipulate miniaturized instruments with greater ease. Computer and robotic technology then translates and scales the surgeon’s hand movements (Mayo Clinic, 2015; Rabin, 2013). Fewer complications (e.g., surgical site infection), less pain and blood loss, quicker patient recovery, and generally smaller or less noticeable scars are, supposedly, the key advantages of using robots in surgery (Mayo Clinic, 2015).
Although the da Vinci system has been investigated by the FDA after a spike in reports of surgical mishaps (Drummond, 2013; Rabin, 2013), its manufacturer (Intuitive Surgical, Inc.) and many hospitals worldwide continue to promote it aggressively for several reasons: (a) more and more physicians complete their residency with robotics training, and many of them require it as a condition for practicing at a hospital; (b) hospitals feel that offering robotic surgery allows them to keep at par with the competition (for patients and funding); and (c) physicians can perform complex and open procedures robotically with greater precision, flexibility, and control than is possible with conventional techniques (DeSmidt, 2013b).
Illustration 1
If substantive and procedural issues concerning robotic surgery were qualitative in nature (i.e., no microeconomic evaluation is necessary to compare unit costs and the consequences of alternative decisions), “hare value” would likely not matter or matter very little. Take, for instance, a situation where a proposed robotics program for a hospital is coming up for decision. There are two players shown in Figure 3: The hospital’s board of trustees, which has to make the final decision and secure funding, and its general surgery department, which vetted and recommended the robotics program following a study that also compared it with the alternative of upgrading conventional laparoscopic equipment. The board directs the chief surgeon to identify (a large) facility space, equipment and accessories, supplies, and physician training/credentialing needs as part of its decision-making process. If both players cooperate by acquiring the da Vinci robot/s, they maximize their respective gains under the best response (2,2). For the board, this means growth in patient referrals and revenues, increased physician applications to practice in their hospital, good publicity, cutting-edge technology, better market share capture, and so on. For the surgery department, this means shorter and more precise surgeries, miniaturization, greater patient comfort and satisfaction (decreased blood loss, less pain, quicker healing time), and increased research support.
Board of trustees and general surgery department. Proposed robotics program.
The trustees subsequently determine that the hospital culture is not at all (or not yet) conducive to robot-assisted surgery. There are, in fact, actual studies showing that a few hospitals have a culture that is strongly supportive of conventional open surgery, while more prefer minimally invasive surgery with improved, laparoscopic technology (Mayo Clinic, 2015). One study reveals two sides of the same coin: The vast majority of patients acknowledge the many benefits of robotic surgery but they still prefer conventional laparoscopy (Boys et al., 2016). Because the trustees in this illustration have chosen to assign primacy to organizational culture, it would be risky for the chief of surgery and his department not to “defect,” and to continue with their preparations without assurance from the board that a suspended robotics project might still be revived. Unilateral cooperation by the surgeons will only produce the nonequilibrium outcome (1,0) in Figure 3. Unilateral cooperation on the part of the board will equally result in a nonequilibrium outcome (0,1) should it go against conventional surgery, and thus create a hostile organizational environment, court adverse publicity, and lose donors and patients.
The trade-off between cooperation and security can lead to the mutual decision of trustees and surgeons to eventually abandon what is otherwise the payoff-dominant Nash equilibrium. Upgrading nonrobotic surgical equipment will certainly be less financially and politically risky but it will yield only moderate gains to each set of players (1,1). Unilateral cooperation or defection, if pursued, will likely be a temporary course of action that is bound to default to the same risk-dominant equilibrium. The extremely limited use of robotic surgery in Asian countries such as the Philippines, where personalism is a distinguishing cultural trait, is attributed in part to similarly unfavorable attitudes of many patients toward computerized devices and unfounded concerns that their doctor is not in control of their treatment. To date, only two private hospitals offer robotic surgery 7 years after it was first introduced in the Philippines. Many hospitals there have chosen to forego their initial plans to establish a robotics program due to lack of public interest as well as sufficient funding.
The strategic choices in Figure 3 simulate classic player dilemmas in utility maximization and interactive decision making in assurance games. Our succeeding illustrations present further variations of the assurance game, with different game-theoretic realizations, particularly when the inefficient Nash equilibrium is unitized and monetized.
Illustration 2
For this illustration, let us assume that the hospital board of trustees is not addressing a purely nominal question (e.g., whether to move forward with a robotics program) but setting priorities and allocating resources within the constraint of limited funding. Assume also that hospital culture is not a controlling factor this time. The trustees are prepared to grant the orthopedics department a one-time budgetary allocation of US$2.0 million to upgrade minimally invasive surgical technology with the purchase of new (nonrobotic) laparoscopic apparatus and instruments. However, the board is open to funding the costlier robotic technology, in lieu of laparoscopic equipment, provided the hospital’s top management and the orthopedics department—the key players in this illustration—can mutually agree and recommend it as a joint resource priority. After all, robotics is a surgical game changer that would also entail an additional capital outlay (i.e., in excess of US $2.0 million).
The da Vinci robot (favored by top management) costs between US$2.2 and US$2.4 million, inclusive of all services, instruments, disposable supplies, and related costs; a da Vinci robot alone costs about US$2.0 million (Scott, 2016). Robotic surgery is projected to cost an orthopedic patient at least 25% more than traditional open surgery, considering that any robotic surgery costs anywhere from US$3,000 to US$6,000 more than regular laparoscopy (Scott, 2016). The newest and most advanced laparoscopic apparatus, priced at less than US$500,000, thus offers a more viable alternative from the perspective of the orthopedic surgeons. Opposing the da Vinci purchase, they cite studies attributing the restricted use of orthopedic robots to their high price tag vis-à-vis limited return on investment (ROI), the need for specialized training, and limited applicability, even as they concede the advantages, including better accuracy levels and precision in the preparation of bone surfaces, more consistent and reproducible results, and better spatial accuracy. While robotics can make a significant difference in closed musculoskeletal surgeries (i.e., without opening the fracture) and improving patient comfort, they assert that orthopedic surgery, unlike many other surgeries, requires the complete involvement of a surgeon for in-depth evaluation, precision, and implantation techniques. Their strong opposition finds empirical support from studies indicating that only 30% of physicians, mostly in the United States, perceive advantages to robotic surgery or would elect it if they needed surgery (Boys et al., 2016). For these compelling reasons, the orthopedic surgeons in this illustration believe that regular laparoscopic surgery as well as conventional open surgery should benefit more from the one-time budget allotment. At least four laparoscopic units can be purchased and will serve way more patients at less expense, including generally lower patient out-of-pocket costs. Any unused amount from the purchases, they assert, can be spent to upgrade open surgery equipment.
Thus, Figure 4 depicts the resulting single-move game between hospital executives and orthopedic surgeons. Maximal gains (≤ 2, ≤ 2) from what is traditionally considered the “efficient” Nash equilibrium would be achieved if both sets of players cooperated and jointly proposed a robot purchase at the budget meeting of the board of trustees. Otherwise, as both of them know, it will take a longtime before the orthopedics department will have another chance to acquire their first robotic apparatus in light of the hospital’s resource constraints. Resembling the stag–hare option, the board will perforce disapprove even a single robot purchase if top management bypassed the surgeons and acted unilaterally in petitioning the board (0,1). The surgeons will instead obtain approval for the purchase of four laparoscopic surgical apparatus. A nonequilibrium outcome will also arise if hospital executives instead opted for the most advanced and innovative laparoscopic equipment, while the surgeons reversed their position in favor of the robot purchase after reconsidering its benefits (1, 0). Any unilateral gains in either case will be moderate in the sense that one set of players will “win” without the support of the other, which is just as critical following board approval and actual equipment purchase (especially during the procurement and installation phases and because of ongoing ecosystem maintenance requirements). Absent a robotic surgical platform, a hospital might be publicly perceived as inferior or not at par with the competition, particularly for patients and grants (Boys et al., 2016; Palladino, 2013).
Executive management and orthopedics surgery department. Robotic versus advanced laparoscopic apparatus.
Without the required consensus on robotics, the inevitable choice for hospital executives and surgeons would be to buy (four units of) the newest and most advanced apparatus for regular laparoscopy (2,2). As both players mutually defect, rather than risk losing the one-time capital outlay, it is evident that purchasing laparoscopic surgical apparatus cannot automatically be dismissed as the Nash-inefficient, second equilibrium in this illustration. Rather, in instances where there are strong, compelling arguments in favor of either equilibrium choice, the second best (2,2) might simply approximate, if not exceed, the payoffs associated in the canonical model with mutually cooperative strategic action. Unitization and multipliability of the second best can spell the difference.
Illustration 3
In this illustration, the question before the hospital board is not whether offering robotic surgery is the optimal funding choice but how to optimize resource allocation in light of the competition that is shaping up in the US$6+ billion market for robotic platforms. Intuitive Surgical remains the dominant and most prominent market player, having pioneered the da Vinci system and turned traditional endoscopy into a futuristic endeavor. However, with an average price tag of almost US$2 million, in addition to hundreds of thousands of dollars in annual supply costs and maintenance fees, the da Vinci robot has been repeatedly criticized for its price. In contrast, the competitive pricing offered by newer brands “would make a much smaller impact on a hospital’s bottom line and could enable a profitable program” (DeSmidt, 2013a). Besides the lower price tags, many competitors are being outfitted with new and specialized capabilities designed to address the long and costly learning curve for physicians using the da Vinci platform. These include increased flexibility and accuracy of surgical instruments, better haptic (or touch) feedback for the surgeon, and improved visualization through superimposed ultrasound images of the patient’s anatomy into surgeons’ real-time view through the endoscope (DeSmidt, 2013a). The Raven, a state-of-the-art, open architecture (software and hardware) surgical robot, exemplifies the competition. Its comparative advantages have been extensively studied (see, for instance, Li, Milutinovic & Rosen, 2016). Like da Vinci, the Raven has six DOFs (degrees of freedom) in each of its four arms, so there is no joint redundancy. Although the “Raven and da Vinci are designed to perform essentially the same tasks, . . . the Raven costs [only] about $300,000” (Greenemeier, 2014). The primary difficulty in using the Raven stems from the lack of encoders on its joints. Instead, “measurements of the joint angles can only be taken from encoders mounted on the motors (i.e., the cable capstans) but not on the joints” (UC Berkeley Robot Learning Lab, 2016).
Against the backdrop of the evolving market competition, let us assume a scenario where the key players are the senior and junior surgeons of the hospital’s general surgery department (Figure 5) and the competing brand names are fully functional for human surgery. The hospital board has initially approved a US$1.8 million budget to enhance the general surgery department’s robotic platform, as its lone (da Vinci) robot is already performing beyond capacity for mostly oncologic procedures. The department is willing to shoulder all operating expenses and fees. Assume further that the hospital’s risk consultants have determined a robust market, particularly in robotic cardiothoracic surgeries, prostatectomies, and other urological procedures due to substantial patient length-of-stay savings (compared with open and laparoscopic procedures), profitability of existing cases, aggregate demand growth, and research and development (R&D) opportunities. For these reasons, the surgery department is confident that it can persuade the board to approve a capital outlay beyond US $1.8 million for its second da Vinci robot if only its entire surgical team would put up a united front and lobby for it (≤ 2, ≤ 2). The rational consequence of interactive decision-making in this case stems from a pair of presumably best response equilibrium choices. Neither senior nor junior surgeons can gain a larger payoff by switching strategies from the “tried and tested” da Vinci robot, provided the other set of players stick with the same strategy.
Senior and junior surgeons, general surgery department. da Vinci versus Raven robots.
However, in view of the uncertain wait time in securing additional board funding for a second da Vinci robot vis-à-vis the immediate option of using the current allotment of US $1.8 million to buy six Ravens, the junior surgeons have grown impatient. They start to consider the feasibility of purchasing and using the Ravens, especially for time-sensitive procedures. The position adopted by the junior surgeons is nonetheless representative of a “wish list” for improvements proposed by many American surgeons, as reported by Fortune, including alternative systems that are “priced low enough to entice hospitals and outpatient surgical centers that have not yet invested in a da Vinci, as well as convince those with established robotic programs to consider a second vendor or switching suppliers altogether” (Tobe, 2016).
Consensus and cooperation among members of the surgery department would have been the best response strategy because it paves the way for the purchase of another da Vinci robot and does away with the high transaction costs associated with acquiring a new robotic system (e.g., staff retraining, refitting facility space, regulatory compliance, etc.). It would be risky for the senior surgeons to press for a second da Vinci robot while their junior colleagues express support for and satisfaction with the current outlay for half a dozen Ravens that can serve more patients (0,1). A nonequilibrium outcome would likewise arise if junior surgeons changed course and opted for the da Vinci robot, while their senior colleagues dissent and find it is no longer efficient to lobby for additional funding due to time, logistical, and clinical constraints in acquiring a second da Vinci robot (1,0). The optimal, albeit risk-dominant, strategy would be for either faction to go for the Ravens after failing to unify and in light of the attractive quantity of these robots that can be purchased with the current US $1.8 million budgetary outlay (2,2).
Optimizing robotics use, particularly in more profitable hospital procedures, to offset their high capital costs can make newer, but less expensive, brands more cost efficient compared with their better known and traditionally dominant market counterparts. With the often difficult-to-justify costs of the da Vinci robot, surgeons might “be better served by having multiple robotic surgery systems to choose from” (Greenemeier, 2014). In China, Japan, and Korea, for example, prominent medical device companies are starting to compete well with Intuitive Surgical after obtaining generous financial support from their governments that enables them to offer hospitals lower prices for prototype surgical robots and conduct robotic R&D at significantly lower cost (Tobe, 2016).
Illustration 4
Our last illustration suggests how a calculated best response might be Pareto-optimal and Hicks-optimal only from the standpoint of asymmetric information (e.g., popular perception arising from a nationwide or worldwide robotics trend). Take the case of colectomies (surgical procedures to remove part or all of the colon). These can be performed using either a minimally invasive, regular laparoscopic technique or a robot-assisted procedure. After a hospital approves and installs a robotic program for surgeries, patient interest and referrals as well as the hospital’s competitive advantage are likely to rise, perhaps exponentially. These will undoubtedly be fueled by trust in and assurance from their doctors concerning the expected benefits of robotic technology (Rabin, 2013).
Following a medical consultation for a colectomy, the two players in this illustration—patient and doctor—can choose to cooperate based on the patient’s (and, of course, the insurer’s) ability and willingness to pay the higher price of robotic surgery and the surgeon’s ability and willingness to perform the colectomy robotically. This is depicted in Figure 6 as the Nash-efficient equilibrium (≤ 2, ≤ 2). Under this scenario, neither patient nor doctor can do any better no matter what. So they have no incentive to go for regular laparoscopy, which is expected to take longer to complete, cause more pain and discomfort, and require a longer hospital stay. But could these players do just as well (or better) with the second best equilibrium, and when?
Surgeons and their patients. Robotic versus conventional laparoscopic colectomy.
Recent empirical research suggests that the da Vinci system may have questionable clinical impact. A 2013 study of 244,129 colectomies by Johns Hopkins University, for example, found that conventional laparoscopy and robotics
had similar complication and mortality rates, and that patients spent similar amounts of time in the hospital following the procedures. The only major difference was cost, with the robotic surgery running an average of nearly $3,000 more due to expensive disposable parts needed for each procedure. (Palladino, 2013).
That figure translates to almost 33% higher than the cost of laparoscopic colectomy (DeSmidt, 2013b). In 2013, the FDA also began formal investigations of Intuitive Surgical as it faced dozens of product liability lawsuits with more than a thousand patients claiming various injuries from the da Vinci robot (Drummond, 2013; Rabin, 2013). The Johns Hopkins study and similar studies have cautioned doctors and patients about believing that any robotic procedure is automatically superior (Juo et al., 2014).
Once information asymmetries are reduced or eliminated (following public dissemination of the results of such studies and investigations), it is fair to assume for the purpose of this illustration that patient incentive (and trust) in electing robot-assisted colectomy will wane. Conventional laparoscopy will likely be valued by patients just as efficiently as (if not more than) a robotic colectomy. Unilateral cooperation will then offer the worst payoff (0) to a surgeon who commits a lot of his or her time and resources in preparing for robotic colectomy only to discover soon that these adverse reports have led the patient to withdraw (0,1). Conversely, unilateral defection by the doctor (1,0) would leave the patient with the worst payoff (0). In this case, the patient will spend (money, time, effort, etc.) for medical consultations that will not lead to a robotic colectomy, as the doctor has chosen to opt out following these reports and to avoid a potential malpractice suit. With trust in and regard for robotics eroded by more and better cost and efficacy information, each player’s optimal strategy can change and depend on his or her expectation of what the other player will do. It is in this context that surgeon and patient are likely to opt for nonrobotic colectomy (2,2). Some reports offer anecdotal evidence of this rational outcome. Many American hospital executives, for instance, at Boston’s Beth Israel Deaconess Medical Center and the University of Illinois Health Sciences System, believe the controversies surrounding da Vinci robots will lead “to further assessments of how the providers talk to patients about surgical options, as well as what kind of training and operational standards are in place” (Carlson & Lee, 2013).
This is one more instance in which mutual defection does not necessarily yield an inefficient, albeit second best, equilibrium. Rather, it lends credence to the view that the assurance or coordination game model “typically either exploits only the information contained in the best response correspondence, or augments this information with risk-dominance and payoff-dominance considerations in order to choose between strict Nash equilibria” (Battalio, Samuelson, & Van Huyck, 2001, p. 749). The “brace of hares” in this illustration may (also) well represent the multitude of issues that hospitals and patients must carefully and thoughtfully consider before going for robotics in colectomies and other surgical procedures, whether minimally invasive or not. These issues include patient safety, technological efficacy, feasibility, and cost savings rather than mere “Star Wars” popularity or trending from which much of the presumed superiority of robotic surgery seems to derive (Palladino, 2013). The quantity and quality of information available to both patient and doctor can substantially change in the course of a medical consultation. The allegory of the hares offers valuable insights on how nuanced comparative technological efficiency is in real life.
Discussion
We have gone full circle. Is there only a hare (or two) for the two hunters to bag and tempt them to abandon a joint stag hunt? As one game theorist reminds us, “Rousseau’s story of the hunt leaves many questions open” (Skyrms, 2001). That (oft-ignored) question is one of them. In raising it here, we sought to examine plausible variations of the stag hunt story based on the number and value of hares, and of an individual’s share of the stag given a successful hunt, for which no consensus yet exists among game theorists. Some of them, in fact, encouraged us to address the question illustratively using abstract modeling.
While an assurance game suggests that the benefits of cooperation can conflict with the security of acting alone, to what extent does the stag offer both hunters a lot more meat than a brace of hares remains speculative. Addressing the issue needs to capture the incentives available to players in the choice of whether to adopt or modify the social contract. By using a few illustrations from what is still largely regarded as the revolutionary field of robotic surgery, we examined how loss avoidance resolves the tension inherent in assurance games, for which payoff-dominant and risk-dominant choices make conflicting predictions. Contrary to received textbook wisdom, we point that the observed preference for the best response or Nash-efficient equilibrium is not absolute and totally stable. This appears to be the case where the quantity of hares is sufficiently large, such that their collective value (or food yield) to each hunter might equal, or perhaps exceed, that of a stag. At that point, our single-play illustrations indicate that players will gain enough incentives to finally cooperate in “bagging hares,” after a long, tiring day of waiting for the stag. Time preferences will affect players’ valuation of the best response as well as their inefficiency attributions to the second best equilibrium.
The first of our four illustrations upholds traditional textbook wisdom. Where a “superior” option exists based on nominal, qualitative assessments, mutual cooperation is the best response, provided there is sufficient trust and assurance among players. Otherwise, it would be risky, for instance, for a hospital board to finance and its surgeons to train for robotic surgery if a hostile culture assumes primacy in utilitarian decision making. Both players will thus prefer one equilibrium to the other. Unilateral choices, on the other hand, are least efficient as one player stands to exploit and gain (often at the expense of the other), with an eventual default to the risk-dominant equilibrium as the likely end result in this case.
The rest of our illustrations—with different sets of players within the same hospital setting—contrast with the outcome of the first, after monetized values were assigned to each available strategic response, and more than a single unit (or two) of the risk-dominant choice was assumed. Illustration 2 suggests that purchases of several units of the newest or most advanced conventional laparoscopic apparatus can vastly improve the payoffs associated with the inefficient Nash equilibrium. It is conceivable that a hospital’s top management and orthopedic surgeons will eventually be swayed by the cost efficiency of acquiring the advanced apparatus in greater quantities compared to a single purchase of the da Vinci robot: More patients can be served in less time and at less cost. It is not inconceivable that patients will opt out of robotic surgery after realizing significant out-of-pocket savings. Some patients might find an (added) incentive to trade off the additional expense of at least US$3,000 for a robotic surgery, if they anticipate that they or their family members will require surgeries in the near future, for which savings in that amount (representing more than 25% of the price of open or regular laparoscopic surgeries) will be of immense value. Further taking into account these patient incentives, we posit the view that mutual trust and assurance might be insufficient in ensuring players’ cooperation toward purchasing surgical robots.
Illustration 3 varies the second best equilibrium while confining player choices to the same (robotic technology) market. We demonstrated that the equilibrium result will nonetheless be similar to the second illustration. Da Vinci robots may have the advantages of market entrenchment and dominance owing to “seniority,” name recall, strong logistics competencies, and aggressive marketing. Yet, newer brands with practically the same features and capabilities might be differently valued by the players, in this case the junior surgeons, provided they are equally cleared by the regulatory agencies. Because more of these newcomers can be purchased for the price of a single da Vinci robot, and in light of clinical concerns raised about the da Vinci robot itself, the market power of what is originally deemed the superior option is not inherently stable or incontestable. Conversely, what might initially be considered the second best equilibrium could be less risky than the traditionally considered efficient equilibrium. In the end, their payoff variance may not be as significant as the canonical model suggests.
Illustrations 2 and 3 take their inspiration from a persistent critique of the assurance game model: “In the ‘real world,’ not all games have decisions of equal value and when studying individuals or populations decisions are not always binary” (Ashrafian et al., 2011, p. 38). Evidence synthesis in health care then needs to identify the benefit ratio of robotic technology (or any advanced technology for that matter) to help a hospital objectively and “successfully choose the ‘better option’ . . . according to the best evidence possible” (p. 38). This is especially critical where interactive, utilitarian choices involve issues of accessibility (i.e., the number of patients served and the affordability of quality treatments and services). In situations like those in illustrations 2 and 3, the quantity and value of hares do matter and can make a difference in the rational outcome of an assurance game.
The fourth illustration indicates how asymmetric information drives incentives that influence the way doctors and patients spend on surgery (and health care services in general). We expanded the canonical model to consider the possibility that some choices might indeed be efficient, while others can be reversed, partly or entirely, by additional or better information about the value attributions of the Nash-efficient equilibrium. Because objective investigations can yield results that cast doubt on the clinical impact of robot-assisted colectomies, what is otherwise a mutual defection from robotics inevitably turns into some sort of mutual cooperation between patient and doctor in favor of conventional laparoscopy, if not open surgery.
Like Illustrations 2 and 3, our last illustration underscores the nuances of utilitarianism and interactive decision-making in producing a dominant equilibrium out of what is pretty much dismissed in the economic literature as Nash inefficient. It is precisely for these reasons that new models, tools, and techniques have been designed and introduced in many health care settings and substantially improved decision making based on evidence synthesis. The multiattribute utility technique (MAUT) is one example. Selected surgical equipment, technology, or procedures receive an overall attractiveness (or utility) score based on the weighted score of each predefined benefit criterion, including surgical applicability, ease of use, patient safety, and treatment outcomes. The overall MAUT value is subsequently compared with the existing instrument using the best evidence available (Ashrafian et al., 2011). We can perhaps hypothesize that tools like the MAUT help or support the rather unconventional outcomes in Illustrations 2, 3, and 4.
Textbook assurance games can thus simplify a more complex reality in which players have to choose between (not one or two but) multiple hares and a stag. By that we refer to a community or society that would often have more than a single mutually cooperative action to pursue with higher-than-expected payoffs. How efficient (or inefficient) strategic rules, choices, and risks are perceived might change once they are quantified and the quantity of the alternative option is increased. There is no gainsaying that any cooperative action will be optimal if players coordinated on it, taking fully into account the quantifiable and multiplicable value of their alternatives. The ensuing payoff from what is otherwise considered the less efficient strategy need not always be lower than the best response. However, if any dominant solution/s cannot or does not accommodate best evidence in health care, “then the set of actions or participants need modification to ultimately provide patients with best-evidence treatments” (Ashrafian et al., 2011, p. 37). There lies the challenge in reconciling theory and practice in health economics. Indeed, as in many real-life situations, one stag may not the finest of dinners make!
Footnotes
Acknowledgements
The author is grateful for the comments and suggestions of two anonymous referees and an associate editor of the journal. He is indebted to Brian Skyrms of Stanford University, whose books on game theory inspired this work and for reviewing the revised manuscript, and Colin Rowat of the University of Birmingham for helpful references. Ellen S. Fansler of Thomas Jefferson University provided research assistance and Nadia Etemad of Drexel University provided copyediting and bibliographic assistance for which the author is equally grateful. As with any work of this nature, the usual caveat applies.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.