Game theoretic analysis of persons, the pharmaceutical industry, and donors in disease contraction and recovery

The article studies how individuals’ choices of risky versus safe behavior, the pharmaceutical company’s decision to develop a drug, and donor subsidies jointly shape disease contraction and recovery outcomes. The central research questions are: (1) which strategies N persons adopt (safe vs. risky behavior and, conditional on infection and availability, buying vs. not buying the drug), and (2) whether and when the pharmaceutical company develops a drug, as a function of parameters such as probabilities of infection and recovery, costs and subsidies, and utilities of health outcomes. A five-period game is constructed in which Nature determines infection and recovery probabilities, the donor sets subsidy rates for R&D and purchases parametrically, and the game is solved by backward induction to identify parameter regions producing five distinct person-level outcomes. The study aims to fill a gap in largely non-game-theoretic health policy analyses by offering an integrated strategic framework applicable to behaviorally driven infectious diseases such as HIV/AIDS and COVID-19.

The literature on disease control is mostly non-game-theoretic, emphasizing either prevention, treatment, or both. Limited game-theoretic work includes analyses of resource allocation between prevention and treatment and international vaccine coordination, but typically assumes vaccines/drugs are available or focuses on policy makers rather than firms’ development incentives and individuals’ risky behavior. Non-game-theoretic treatment-focused studies highlight that incentives for developing therapeutics often exceed those for vaccines, influencing prevalence and resource allocation (Kremer and Snyder; Kremer and Glennerster; Thomas). Empirical and economic evaluations document ART’s global health and economic benefits (Forsythe et al.), R&D cost magnitudes (DiMasi et al.), and expected revenues (West and Schneider), as well as financing challenges in low- and middle-income countries (Hecht et al.; Izazola-Licea et al.), and cost-effectiveness of treatment strategies (Goldie et al.). Prevention and combined approaches stress flexibility in modeling, uncertainty, and program design (Alistar and Brandeau; Bärnighausen et al.; Boily et al.; Granich et al.; Hogan et al.; Bertozzi et al.; Gonsalves; Kumaranayake et al.). Biological/epidemiological modeling relevant to recovery probabilities and drug effectiveness appears in influenza and COVID-19 analyses (Moxnes and Hausken; Sun et al.). This article integrates these threads by endogenizing behavior and firm development decisions within a unified game while treating donor support and Nature’s probabilities parametrically.

A complete-information, five-period game is formulated among N persons and a pharmaceutical company, with a donor and Nature acting parametrically/probabilistically. Sequence: Period 1: each person i chooses risky or safe behavior. Period 2: given risky behavior, Nature determines disease contraction with probability q. Period 3: the firm decides whether to develop a drug, incurring development cost d of which fraction X is subsidized by the donor; hence firm cost is (1−X)d. Period 4: if no drug is available, Nature assigns recovery probability x (death with 1−x). If the drug is available, person i chooses to buy it at price C with donor subsidy S (out-of-pocket (1−S)C) or not; if not purchased, recovery probability remains x. Period 5: if the drug is purchased and applied, Nature assigns recovery probability w (with 0 ≤ x ≤ w ≤ 1). Persons’ utilities: H for safe behavior, E for risky behavior without infection, R_i for recovery, D_i (< R_i) for death/severe morbidity; expected utilities follow from probabilities along realized paths. The person’s buy decision in period 4 follows the threshold (w−x)(R_i − D_i) ≥ (1−S)C. The firm develops the drug in period 3 if mC − (m c)^k ≥ (1−X)d ≥ 0, where m is the number of buyers among infected persons and (m c)^k is the production cost function with k capturing scale (k<1 concave, k=1 linear, k>1 convex). Risky vs. safe behavior choice in period 1 depends on whether the drug would be bought or not: if not buying (or no drug), risky is chosen when (1−q)E_i + q[(1−x)D_i + xR_i] ≥ H_i; if buying, risky is chosen when (1−q)E_i + q[(1−w)D_i + wR_i − (1−S)C] ≥ H_i. Backward induction yields five person-level outcomes: (1) safe behavior; (2) risky behavior without infection; (3) infection without drug availability; (4) infection with drug availability but not buying; (5) infection with drug availability and buying. The donor’s utility aggregates persons’ expected utilities and subtracts subsidy costs m S C + X d; the donor’s choices X and S are treated as parameters. A parameter estimation procedure is proposed using HIV/AIDS data: treatment costs (e.g., generic first-line ART ≈ $75–$160 per person-year), R&D costs (≈ $2.6B), revenues, prevalence (adult 15–49 HIV prevalence varies from <0.1% to >20% in some countries), and resource availability by funding source to inform C, c, k, d, q, x, w, X, S, and to illustrate feasible values for m, G, L, M, and N.

- Backward induction characterizes conditions for each outcome via inequalities: buy decision threshold (w−x)(R_i − D_i) ≥ (1−S)C; firm development threshold mC − (m c)^k ≥ (1−X)d; risky-vs-safe thresholds depending on whether the drug would be bought (Eqs. (6) and (7)). - The model yields five person outcomes and two firm outcomes (develop vs. not develop). The firm develops the drug only when expected profits are nonnegative and at least one buyer exists; persons’ decisions depend on utilities, probabilities, and subsidies. - Illustrative parameterization (example): C = $100/year, c = $80/year, k = 0.5, d = $2.6B, X = 0.5, S = 0.5; utilities: D_i = −$7,000,000, E_i = $1,000,000, H_i = $500,000, R_i = $200,000; probabilities: q = 0.1, x = 0.1, w = 0.9; m = 14,000,000. With these values: • Person utilities for the five branches (Eq. (9)) are: $500,000 (safe), $1,000,000 (risky, no infection), −$6.28×10^6 (infected, no drug), −$6.28×10^6 (infected, drug available, not buying), and −$520,050 (infected, buying drug). Drug availability and purchase induce higher expected payoff relative to no drug if infected, and can make risky behavior attractive ex ante. • Firm profit (Eq. (10)) if developing is approximately $9.99665×10^7; otherwise zero. Thus development can be profitable given sufficient buyers and subsidies. • Donor illustration with N = 100,000,000, G = 40,000,000, L = 30,000,000, m = 14,000,000, M = 10,000,000 gives donor utility V ≈ −$1.384×10^14 without development vs. −$5.7762×10^13 with development, indicating higher (less negative) utility when development occurs, despite subsidy costs, due to improved outcomes for buyers. - The analysis highlights potential risk compensation: when an effective, subsidized drug is expected and affordable, some persons may rationally choose more risky behavior ex ante. - The framework maps parameter regions to outcomes (Table 3), offering clear testable thresholds for policy levers (S, X), firm costs (d, c, k), prices (C), and epidemiological factors (q, x, w).

The findings show how individual behavior, firm development decisions, and donor subsidies interact to shape disease and treatment outcomes. By explicitly modeling incentives and probabilities, the framework explains when drugs will be developed and purchased, when individuals prefer safe versus risky behavior, and how subsidies shift these thresholds. The example demonstrates that sufficiently effective and subsidized drugs can both enable profitable development and increase donors’ net welfare, while potentially encouraging risky behavior (risk compensation). The results are relevant for policy: adjusting purchase (S) and R&D (X) subsidies can alter firms’ investment incentives and individuals’ buy thresholds, while pricing and cost reductions (C, c, k) and improvements in effectiveness (w) expand regions where development and purchase are optimal. Parameter changes (e.g., epidemiology, technology, costs) can move populations across outcome regions. The model thus provides a structured tool for anticipating strategic responses to policy and market changes and for designing interventions that balance prevention incentives with treatment access.

The study develops and solves a five-period game integrating persons’ risky vs. safe behavior, a pharmaceutical company’s drug development decision, donor subsidies, and Nature’s infection and recovery probabilities. Backward induction yields clear inequality conditions for five person-level outcomes and two firm outcomes. A parameter estimation procedure grounded in HIV/AIDS data illustrates how realistic values map to outcomes and shows that under plausible conditions drugs are developed and purchased, potentially inducing risk compensation while improving expected outcomes for infected individuals. The framework provides a tool for policymakers and stakeholders to assess how changes in costs, subsidies, effectiveness, and epidemiology alter strategic behavior and market outcomes. Future research should extend the model to multiple firms and donors, heterogeneous populations, dynamic expectations about drug availability, endogenized epidemiological dynamics, and disease-specific calibrations beyond HIV/AIDS.

Key limitations include: (1) parametric treatment of donor behavior (X, S) and Nature’s probabilities (q, x, w) rather than fully endogenized strategic/epidemiological dynamics; (2) representative timing abstraction (e.g., long drug development lags compressed into a period-3 decision from the individual’s perspective); (3) simplified utility representation of outcomes (death vs. recovery with fixed utilities) and homogeneity assumptions in illustrative examples; (4) exclusion of additional healthcare system actors (doctors, hospitals, regulators) and broader market/political constraints; (5) uncertainty in empirical parameter estimates (e.g., costs, prevalence, effectiveness) and external validity across diseases and contexts. These constraints may affect generalizability and precision of quantitative predictions, motivating extensions with richer heterogeneity, multiple firms/donors, and endogenous epidemiology.