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

Background: Scarcity in supply of COVID-19 vaccines and severe international inequality in their allocation present formidable challenges. These circumstances stress the importance of identifying the conditions under which self-interested vaccine-rich countries will voluntarily donate their surplus vaccines to vaccine-poor countries.
Methods: We develop a game-theoretical approach to identify the vaccine donation strategy that is optimal for the vaccine-rich countries as a whole; and to determine whether the optimal strategy is stable (Nash equilibrium or self-enforcing agreement). We examine how the results depend on the following parameters: the fraction of the global unvaccinated population potentially covered if all vaccine-rich countries donate their entire surpluses; the expected emergence rate of variants of concern (VOC); and the relative cost of a new VOC outbreak that is unavoidable despite having surplus doses.
Results: We show that full or partial donations of the surplus stock are optimal in certain parameter ranges. Notably, full surplus donation is optimal if the global amount of surplus vaccines is sufficiently large. Within a more restrictive parameter region, these optimal strategies are also stable. Conclusions: Our results imply that, under certain conditions, coordination between vaccine-rich countries can lead to significant surplus donations even by strictly self-interested countries. However, if the global amount that countries can donate is small, we expect no contribution from self-interested countries. The results provide guidance to policy makers in identifying the circumstances in which coordination efforts for vaccine donation are likely to be most effective.

Introduction

The COVID-19 pandemic has shone a harsh light on the urgent need to better understand the drivers of policy solutions to global infectious disease crises. By November 2021, the pandemic had claimed the lives of more than 5.23 million people worldwide[1], and had led to severe economic losses. During most of 2020 and before effective vaccines and treatments were available, containing the spread of the virus compelled countries to adopt lockdowns and social distancing policies. In a remarkable scientific achievement, safe and effective vaccines were developed in record time[2,3]. The availability of COVID-19 vaccines offers countries and the international community important new means to combat the pandemic. At this particular stage in pandemic cycles, the key questions are how best to utilize the available vaccines on a global scale, and what strategies are likely to receive the required international cooperation for their implementation.

Scarcity in supply, coupled with unequal allocation of the available vaccines across countries, present the most formidable challenges for maximizing the potential benefits of these vaccines[4]. Allocating doses internationally in proportion to countries’ population sizes is estimated to be a close-to-optimal strategy in terms of averting deaths worldwide[5] and benefiting the economy, both globally and in donor countries[6,7]. Nevertheless, extremely unequal vaccine distribution has typified the availability of vaccines across countries[2,8]. This situation means that most people in the world’s poorest countries might not have access to COVID-19 vaccines until at least mid-2023[8]. Since about 85% of the global population resides in low- and middle-income countries, most of humanity remains exposed to continued outbreaks[8]. This situation increases the risk that further virus variants will emerge, possibly undermining the efficacy of existing vaccines[2,3].

A notable attempt to distribute vaccines from high-income to low-income nations is COVAX - the COVID-19 Vaccine Global Access Facility. Led by the World Health Organization and coordinated jointly with the Coalition for Epidemic Preparedness Innovations and the Global Alliance for Vaccines and Immunization, COVAX is a pooled procurement initiative that aims to provide all countries with COVID-19 vaccines at differential prices[2,3,9]. The guiding principle of COVAX was to prioritize vaccination globally by sub-populations: from older adults, healthcare workers, and other high-risk individuals to the wider sections of the population. It stipulates that no country should vaccinate more than 20% of its population until all countries have vaccinated 20% of their populations[3].

However, the COVAX initiative has so far failed to come close to its objectives[10], due to the aggregate consequences of vaccine nationalism[11]. Many of the world’s wealthiest countries have adopted procurement strategies that prioritize widespread inoculation of national populations ahead of the vaccination of health care workers and high-risk populations in low-income countries[2,3].

The challenges facing a globally effective supply of vaccines through supranational initiatives such as COVAX have led to initiatives that focus on country-level and ad hoc intergovernmental approaches, such as donations by high-income countries of some of their pre-purchased vaccines to middle and low-income countries[12]. The G7 summit of June 2021 jointly pledged to provide 1 billion COVID-19 vaccine doses until June 2022[9]. Such initiatives focus on ways to redistribute the excess stocks of doses accumulated by vaccine-rich countries after they have vaccinated large shares of their own populations. They echo similar attempts in previous global health crises, such as smallpox in the 1970s[13], HIV in the 1980s[2], and H1N1 in 2009[3].

This study adopts a game-theoretical approach to identify the conditions under which self-interested vaccine-rich countries would donate their surplus vaccine doses (beyond those needed for the initial vaccination of their own populations) to vaccine-poor nations, rather than stocking these surplus doses domestically for their own future use. (Note that, to avoid expiration of stored vaccine vials, stocking in practice might be based on pre-purchased doses that can be supplied on short notice from the manufacturer[14]. Vaccine-rich countries have at least two self-interested reasons for sharing their vaccine surpluses with vaccine-poor countries. First, they have large open economies that are dependent on international trade and thus require significant levels of international travel and open borders. Second, their efforts to stop the pandemic within national borders could be undermined by the emergence of new variants of concern (VOCs). Such a VOC (e.g., Delta and Omicron variants of COVID-19) may require increasing the level of immunization in the population by administering a booster[15]. In such circumstances, having a surplus stock can significantly shorten time-to-delivery, thereby better containing the outbreak. The probability of a VOC is a crucial factor in understanding vaccine-rich governments’ willingness to re-allocate vaccines to vaccine-poor countries. The fear of VOC can either increase short-term national self-interest and a tendency to reserve health resources for domestic purposes or create awareness that long-term pandemic control can only succeed by effective global vaccination. Another complicating element is that donation by one country benefits all others by reducing the probability of future variants of the virus and their consequent costs; however, such a donation incurs a cost to the donating country by leaving it without available surplus in the case of a VOC outbreak.

Previous game-theoretical models applied to the challenge of epidemics, notably the “vaccination game”[16,17], have addressed individual-level decisions to vaccinate or not, assuming the availability of vaccines for the studied population. The current study points to the significance of international vaccine inequality in the context of a pandemic (defined as “an epidemic occurring worldwide or over a very wide area, crossing international boundaries”[18]), notably the fact that in many countries, vaccines are in short supply or practically unavailable, while other countries possess surplus stocks. These surplus stocks are controlled by the governments of vaccine-rich countries, and these country-level decisions are the focus of our model.

This study models (1) the vaccine donation strategy that is optimal for all vaccine-rich countries combined (a social planner’s perspective); and (2) whether the optimal solutions could be adopted by the relevant countries, assuming strictly self-interested motivation on their part. Given that vaccine donations are dependent on the choices of vaccine-rich countries, the analysis strictly takes these countries’ perspectives – both as a collective and as independent actors. In particular, we examine how the answers to these questions depend on key pandemic parameters: (1) the fraction of the global unvaccinated population potentially covered if all vaccine-rich countries fully donate their surplus (); (2) the baseline expected annual rate of VOCs (λ); and (3) the fraction of the total cost of a new VOC outbreak that is unavoidable despite having surplus doses (α). The game-theoretical model we develop is general in the sense that it identifies the conditions under which a minority of vaccine-rich countries is likely to donate a costly remedy in the context of a pandemic. By considering the realistic ranges of the model parameters in the case of COVID-19, we can cautiously infer more specific implications for potential international cooperation in coping with this particular pandemic.

Methods

The vaccine donation game

We consider the strategies of N vaccine-rich countries (hereafter: “the countries”). Beyond the vaccine doses needed for vaccinating their own entire population, each vaccine-rich country is assumed to have purchased additional stock sufficient to vaccinate its population two additional times. We assume that the benefit of having this surplus full-population stock is that it could be administered in case a VOC emerges. Each country can decide how much of the two extra doses per capita to donate to vaccine-poor countries: , or 2, where denotes the country. Each country can donate zero extra doses and thus keep two extra doses per capita for itself; donate one dose and keep 1 for itself; or donate both extra doses and keep none (1 dose may protect against 1 variant; 2 doses may protect against 2 variants).

We denote v as the fraction of the global unvaccinated population that could be vaccinated due to vaccine-rich countries’ donations. Specifically, assuming that countries will donate only after fully vaccinating their domestic population (see Fig. 1 as a motivational illustration justifying our assumptions), “global unvaccinated population” refers to the unvaccinated in vaccine-poor countries. v could vary from zero if no country donates ( for all i) to if all vaccine-rich countries donate both doses ( for all i). Specifically, if , the countries could vaccinate the entire unvaccinated world population and be fully protected against any variants. However, if , there is still a chance that variants will emerge even if all vaccine-rich countries contribute all their extra doses.

Fig. 1

The share of population fully vaccinated.

The share of the population fully vaccinated against COVID-19 across high-, lower middle-, and low-income countries (source: Our World in Data, https://github.com/owid/covid-19-data/tree/master/public/data).

In turn, we denote λ as the expected number of VOCs that emerge in unvaccinated populations within one year if no doses are donated. We assume that variants can emerge independently with some small probability in each unvaccinated person[19]. Therefore, if vaccines are donated, the expected annual rate of VOCs is given by if and 0 if . Also, we assume that variants occur independently of one another: The probability of a variant to emerge does not depend on the number of other variants that emerge. Equivalently, the emergence of a variant does not change the probability that other variants emerge. Therefore, the probability distribution of the number of variants is given by the Poisson distribution[20]. Specifically, if , the probability distribution of the number of VOCs that emerge is given by a Poisson distribution with a mean . It follows that the probability for the occurrence of exactly k variants, , is given by the Poisson coefficient

Donation by country i benefits all countries because it reduces the probability of future virus variants and their consequent costs. However, a donation also comes with a cost to the donor country because it might be left without an effective vaccine if variants do occur. Specifically, if a country has stocked enough doses to cover the occurrence of a variant, it will have to bear only a fraction α () of the total cost of the corresponding outbreak. Note that α is expected to be greater than zero – i.e., a certain cost of a future outbreak is practically unavoidable, since some time is required for administering surplus vaccines to the population. For example, if a booster shot is required, the time and direct costs of administering the booster shot remain. Note that α may, more generally, capture the time and costs required to apply any stocked resource. For simplicity, we assume that if there is more than one outbreak, the cost α is the same in the first and the second outbreaks. During this interim period, the country is expected to bear the cost of the variant outbreak. Therefore, the expected cost of future outbreaks to country i is given bywhere β is the cost due to an outbreak if the country does not have extra vaccine doses, and is the probability that three or more variants emerge. Note that P1, P2 and depend on v (Eq. 1), therefore they depend on s and on the strategies of all other countries.

Analysis of the model: Optimal and stable solutions

Using this model, we address the following two tasks. First, we identify the optimal solutions from the point of view of a ‘vaccine-rich-world’ social planner who aims to maximize the social welfare of all vaccine-rich countries combined. Specifically, we assume risk-neutral countries, namely, the optimal solution is given by the choice of s for each i that minimizes the total expected cost of outbreaks to all vaccine-rich countries:

After identifying the optimal solution, we ask whether this solution is stable. Assuming that each vaccine-rich country chooses the strategy s that minimizes its own expected cost, will the optimal solution be adopted? To answer this question, we consider two solution concepts. The first concept is the Nash equilibrium. Assume, for example, that for each country is the optimal solution. This is also a Nash equilibrium if and only if it does not benefit any of the countries to unilaterally change its strategy to or . However, the Nash equilibrium solution concept is somewhat restrictive, because it does not take into account the responses of other countries to the deviation of a given country.

Another widely used solution concept in the context of international cooperation is a self-enforcing international agreement (henceforth SEA)[21]. The assumption is that each country chooses whether to be a signatory or not. Non-signatories do not contribute (or contribute less), while signatories adopt the strategy that maximizes their welfare (minimizes their expected cost) as a whole. An agreement is self-enforcing if no signatory has an incentive to opt-out and become a non-signatory, and no non-signatory has an incentive to opt-in and become a signatory[21]. Note that if the optimal solution is a Nash equilibrium, it is also a SEA. However, there could be cases in which the optimal solution is a SEA but not a Nash equilibrium. For example, in the context of our game, consider the case in which by all countries is not a Nash equilibrium: if all countries adopt , it may benefit country j to deviate to, say, . However, following this opting out by country j, the best strategy of the remaining countries (the signatories) may become instead of . If country j anticipates that the remaining countries will reduce their contribution to in response to its own deviation, it might be beneficial for country j to persist with .

Numerical methods

To calculate the optimal solution, we calculated the expected cost to a given country i, , given the set of strategies that all the countries adopt, we first calculate P1 and P2 (Eq. 1), and then we set and substitute the results in Eq. (2). In turn, we examine all possible sets of strategies, where in each set, each country can contribute a different amount (s equals 0, 1, or 2 for each i). For each set, we calculate the total cost (Eq. (3)), and we find the set for which the total cost is minimized.

Next, we did the following to check whether the optimal solution is a Nash equilibrium. Without loss of generality, we examine whether country 1 can benefit from deviating if all the countries adopt the optimal solution, . Specifically, country 1 has 2 possible deviations: to or if , and to or if . We first calculate , the expected cost to country 1 if it does not deviate (and the other countries adopt ). Then, we calculate and , the expected cost to country 1 following each of its possible deviations. We conclude that the optimal solution is a Nash equilibrium if and only if and .

In turn, we did the following to check whether the optimal solution is a self-enforcing international agreement. For every , we consider n signatories and non-signatories, and we calculate the utility (minus the cost – Eq. (2)) of a signatory, , and the utility of a non-signatory, . (Note that, for each n, [21]). In turn, an agreement with signatories is by definition the optimal solution. This solution is a self-enforcing international agreement if a country cannot benefit from opting-out when all N countries are signatories. Specifically, if a country remains a signatory, its utility is , while if it opts-out, its utility becomes . Therefore, the optimal solution is self-enforcing if and only if .

Statistics and reproducibility

This study is quantitative mathematical and computational. No statistical method is used. All results are fully reproducible: The methods are fully described in the Methods section and parameter values are given in the caption of each relevant figure.