A game theoretic approach identifies conditions that foster vaccine-rich to vaccine-poor country donation of surplus vaccines

The study addresses when strictly self-interested vaccine-rich countries would donate surplus COVID-19 vaccine doses to vaccine-poor countries. In the COVID-19 pandemic, vaccines were developed rapidly but supply scarcity and highly unequal global allocation persisted. Initiatives like COVAX sought equitable distribution but fell short amid vaccine nationalism, leaving large unvaccinated populations in low- and middle-income countries and increasing the risk of variants of concern (VOCs) that can undermine vaccines. Vaccine-rich countries have self-interested incentives to donate: maintaining open economies dependent on international travel and reducing the probability of VOC emergence abroad that could trigger costly domestic outbreaks requiring boosters. However, donations create a public good and face free-riding incentives because any country’s donation lowers risk for all, while the donor may lose domestic surplus needed for rapid response. The paper develops a game-theoretic model to identify donation strategies that are socially optimal for vaccine-rich countries collectively and to assess whether such strategies are stable (Nash equilibrium or self-enforcing agreements), focusing on key parameters: the maximum fraction of the global unvaccinated population that could be covered by full donation (Vmax), the baseline expected annual rate of VOCs (λ), and the fraction of outbreak cost unavoidable despite surplus (α).

Prior epidemic game-theory work largely examined individual-level vaccination decisions assuming vaccine availability (the vaccination game). Historical efforts at international vaccine redistribution include smallpox eradication subsidies, which differed due to lack of dose scarcity and the question centering on which country would pay. Related work on managing mobile public bads (e.g., invasive species) showed that in Nash equilibrium fewer contributing countries can sometimes perform better, but those settings assumed a single or few countries could eliminate the problem alone. Pre-vaccine COVID-19 policy interactions were likened to weakest-link games, where inadequate measures by a few can sustain global spread. The present vaccine donation game differs: it focuses on cross-country surplus vaccine allocation under scarcity, VOC risk externalities, and conditions for voluntary cooperation (Nash or self-enforcing agreements) among multiple vaccine-rich countries.

The authors model N vaccine-rich countries, each having surplus sufficient to vaccinate its population twice beyond initial full vaccination. Each country i chooses s_i ∈ {0,1,2}, representing the number of surplus full-population rounds donated (0 means retain two domestic surplus rounds; 1 donate one round; 2 donate both). Let v denote the fraction of the global unvaccinated population that becomes vaccinated due to total donations, ranging from 0 (no donation) up to Vmax (if all donate both rounds). If Vmax ≥ 1, full donation could vaccinate the entire unvaccinated global population; if Vmax < 1, residual risk remains even under full donation. VOC emergence in the unvaccinated population follows a Poisson process: with no donation, the expected annual number of VOCs is λ. With donation, the expected rate becomes λ(1 − v) for v ≤ 1 and 0 if v > 1, assuming independence of variant emergence events. Thus, the probability of exactly k VOCs is Poisson with mean λ(1 − v). Donations reduce global VOC risk but impose a cost on donors if they forgo domestic surplus. If a country faces an outbreak and has sufficient surplus to cover that many variants, it still bears a fraction α (0 < α ≤ 1) of the outbreak cost due to unavoidable delays and administration time; without surplus, it bears full cost. The expected cost to country i, C_i, depends on its retained surplus (as determined by s_i) and on the probabilities P1, P2, P≥3 derived from the Poisson distribution with mean λ(1 − v). The social planner’s problem minimizes the sum of expected costs of all vaccine-rich countries by choosing s_i for all i. Stability is assessed via: (1) Nash equilibrium—no single country can reduce its own expected cost by unilateral deviation from the optimal profile; and (2) self-enforcing international agreement (SEA)—given n signatories following the welfare-maximizing strategy for signatories and N−n non-signatories contributing less, no signatory benefits from opting out and no non-signatory from opting in. Numerical analysis enumerates all combinations of s_i ∈ {0,1,2}, computes v and the corresponding Poisson probabilities (P1, P2, P≥3), evaluates country and total costs, identifies the optimal profile, and then tests for Nash and SEA conditions by comparing utilities for potential deviations and signatory changes. Parameter exploration considers ranges motivated by COVID-19: 2 ≤ N ≤ 10, 0 ≤ Vmax ≤ 1, a range of λ values, and 0 < α ≤ 2 (with principal illustrations for α in [0.1, 0.4]). No statistical methods are used; results are fully reproducible via the described algorithm and shared code.

- Optimal donation levels are symmetric: all countries optimally choose the same s_i (0, 1, or 2), depending on parameters λ (expected VOC emergence rate), α (unavoidable outbreak cost fraction despite surplus), Vmax (maximal coverage attainable by full donation), and N (number of vaccine-rich countries). - Full donation (s_i = 2) is optimal when Vmax is sufficiently high and/or α is large. Above a Vmax threshold (illustrated around Vmax > ~0.75 in one case), the optimality of full donation becomes largely insensitive to λ. - Partial donation (s_i = 1) is optimal when λ is small and Vmax and α take intermediate values. - No donation (s_i = 0) is optimal when λ is sufficiently large and both Vmax and α are small, as donations cannot meaningfully reduce VOC risk and retaining surplus yields better domestic protection. - Stability: Some optimal solutions are Nash equilibria (e.g., regions where s_i = 2 or s_i = 1 is both optimal and each country’s best response). In adjacent parameter regions, the optimal solution is not Nash but is a self-enforcing agreement (SEA), because a unilateral opt-out triggers a change in optimal contributions by remaining signatories that leaves the deviator worse off. - As N increases, the parameter space where optimal donations are stable shrinks: with more countries, an individual’s marginal benefit from donating falls while its private cost remains, reducing incentive compatibility. For N = 2, optimal solutions are stable across explored parameters; for N = 10, some optimal donation regions are unstable. - Where optimal full or partial donations are unstable (regions II and V), the SEA yields fewer donating countries than socially optimal, creating a measurable social efficiency deficit (cost of anarchy). - Policy implication: When the aggregate surplus can cover a large share of the unvaccinated world (high Vmax) and when rapid domestic response reduces little of outbreak costs (higher α), full or partial self-interested donations are both optimal and, in many cases, stable. If the aggregate surplus is small (low Vmax), self-interested donations are unlikely.

The findings delineate the conditions under which self-interested vaccine-rich countries will voluntarily donate surplus vaccines. Donations reduce the global probability of VOCs, which in turn lowers expected domestic outbreak costs for donors. When the potential impact of donations is high (Vmax large), full donation is optimal and often stable; notably, above an α-dependent Vmax threshold, this result is largely insensitive to λ. Below that threshold, λ plays a bigger role in determining whether partial donation or no donation is optimal. Stability analysis clarifies that in many practically relevant scenarios, optimal strategies are Nash equilibria or at least SEAs, meaning coordinated agreements can persist without external enforcement. However, as the number of vaccine-rich countries grows, free-riding incentives intensify, narrowing the range where optimal donations are self-enforcing. These insights inform international coordination: prioritize efforts when Vmax is high (ample aggregate surplus relative to global need) and where domestic stocking yields limited cost mitigation (higher α), as cooperation is both efficient and implementable. Where optimal strategies are unstable, mechanisms that align private and collective incentives (e.g., binding agreements, side payments, or coordination to reduce N of key donors) may be needed to approach the social optimum.

The study contributes a general game-theoretic framework to evaluate when vaccine-rich countries will, out of self-interest, donate surplus vaccines to vaccine-poor nations. It shows that donations—full or partial—are optimal across sizable parameter regions and can be stable as Nash equilibria or self-enforcing agreements, particularly when aggregate surplus can cover a large share of global need (high Vmax) and when domestic surplus only modestly reduces outbreak costs (higher α). Conversely, when aggregate surplus is small, self-interested donations are unlikely. These results guide policymakers on when coordination initiatives are most likely to succeed and suggest that designing agreements to reduce free-riding is crucial as the number of potential donors increases. Future research should incorporate heterogeneous country sizes and capacities, richer representations of stockpiling motives and mechanisms, and empirical calibration to specific pandemic contexts to refine policy recommendations.

- The model assumes countries stock surplus primarily to respond rapidly to VOC-driven outbreaks; other motives (uncertain duration of immunity, portfolio diversification across vaccine platforms) are not explicitly modeled and could reduce donation incentives. - Stockpiling mechanisms (physical inventories, priority purchase contracts, dedicated budgets) differ in response times; these are abstracted into a single parameter α. Real-world heterogeneity in logistics could alter effective α across countries. - Countries are modeled as equal in size and capacity. Heterogeneity in population size, economic exposure, or health system capacity could change optimal and stability results and may yield asymmetric optimal strategies. - Parameter values (e.g., λ, Vmax, α) are uncertain and context-dependent; while the analysis explores broad ranges, precise thresholds are illustrative and not empirical estimates. - The model focuses on self-interest and does not incorporate altruism, geopolitical considerations, or dynamic learning about variants and vaccine effectiveness beyond the Poisson assumption.