Theoretical Foundations

Utilizing game theory to simulate issues within medical contexts constitutes a widely employed and well-established methodology [1]. Furthermore, prior investigations have attempted to deduce decisions predicated on game-theoretical principles during the COVID-19 pandemic [2] [3][4]. Although some authors have endeavored to simulate the COVID-19 pandemic as a prisoner's dilemma [5], it appears unsuitable for the scenarios under investigation due to our lack of knowledge regarding the players' payoffs. In discussions concerning the advantages of preventive measures, a topic of significant public concern throughout the pandemic years, we must grapple with numerous variables that remain unknown. Modeling the situation as a game of public goods [6] also proves unhelpful, as it unduly complicates matters by abstractly referencing public goods primarily in the context of non-measurable variables such as freedom and subjective utility, which cannot be readily incorporated into a mathematical formula.

Moreover, one's personal health is not a public good, and individual freedoms do not constitute a common good. The pandemic has demonstrated the public sector's capacity to exclude individuals from the utilization of public goods through restrictive measures (e.g., quarantine, penalties, access restrictions, work prohibitions, etc.). The incongruities and inadequacies of controlling authorities and the associated higher payoffs for players should not be the focus here. When simulating decision nodes among affected citizens regarding preventive measures, a viable simulation approach involves employing a game model with Bayesian equilibria in mixed strategies. In this context, mixed strategies imply that the citizenry makes decisions for or against a measure with a probability denoted as "p" (where "p" ranges between 0 and 1).

Despite the non-numeric nature of the underlying variables, we can work with "p" without the necessity of predefining other variables, as we can derive "p" from real-world data without explicitly specifying the underlying conditioning factors. Indeed, decisions concerning the adoption or rejection of preventive measures, whether pertaining to the prevention of infectious diseases, chronic ailments, or other preventive contexts, invariably constitute personal choices by citizens. Consequently, these choices are contingent upon an array of factors that defy facile measurement or calculation, thereby rendering them outside the purview of this study. The sole exception pertains to preventive constraints imposed without individuals' decision-making agency, with mandatory vaccination serving as an example [7].

The framework within which citizens make these determinations is influenced, among other factors, by stakeholders within the public healthcare sector. Thus, if Player 1 (in our example, representing public healthcare) extends an offer of a preventive measure to Player 2 (in our example, representing the entire population), Player 2 can elect either to accept or decline it. These decision scenarios can be visualized using a tree diagram (see Fig. 1).

The decision to recommend or not recommend a measure, stems from an evaluation process, typically preceded by scientific investigations confirming the advantages of a specific measure. In this context, Player 1 will opt for Strategy A if it offers a favorable outcome (positive payoff) and Strategy B if this measure does not assure a positive outcome. From an objective standpoint, it would consistently be the dominant strategy for Player 2 to cooperate, thereby employing a measure only when offered by Player 1. Nonetheless, objective decision criteria do not invariably govern decision-making, and it remains unpredictable for individual citizens whether they will choose cooperation or defection and which factors hold particular significance for them. It is postulated, however, that these factors remain constant when neither Player 1 nor Player 2 alters the conditions.

Similarly, the subjective payoff of Player 2 cannot be precisely predicted. Regrettably, Player 2 bases their decision on this payoff. For our purposes, this payoff can only be indirectly ascertained through backward induction. As the probability of Player 2 cooperating rises, so does their anticipated (subjective) payoff for cooperation. Of particular interest to us is the question of Player 2's cooperation when Player 1 extends a preventive measure (the left branch of the tree diagram, as depicted in Fig. 2).

The likelihood of Player 2 cooperating or defecting hinges on their subjective payoff, which, in turn, is contingent on individual attributes of Player 2 and the conditions imposed by Player 1, variables that remain undisclosed. Simplistically, it can be inferred that Player 2's individual attributes are rooted in upbringing and past experiences, and thus, they remain constant. Consequently, when Player 1 also maintains the conditions unchanged, the probability of Player 2 cooperating or defecting likewise remains constant. The constancy of the cooperation probability at the decision node assumes paramount importance in repeated games. In such cases, the probability of Player 2 consistently cooperating in each iteration is derived from the initial probability in the first game, denoted as p0, multiplied by the number of repetitions (as illustrated in Fig. 3).

Hence, if the conditions remain unaltered, the probability (px) of Player 2 cooperating in each successive repetition (x) of the game can be calculated based on the initial probability in the first game (p0) using the following formula:

p_x=〖p_0〗^(x + 1)

In the given example, the probability in the 2nd repetition of the initial game for cooperation is calculated as follows:

p_x=〖p_0〗^(x + 1)

p_2=〖0.7〗^3

p_2 = 0.343