Balancing the benefits of vaccination: An envy-free strategy

Abstract The Covid-19 pandemic revealed the difficulties of vaccinating a population under the circumstances marked by urgency and limited availability of doses while balancing benefits associated with distinct guidelines satisfying specific ethical criteria. We offer a vaccination strategy that may be useful in this regard. It relies on the mathematical concept of envy-freeness. We consider finding balance by allocating the resource among individuals that seem heterogeneous concerning the direct and indirect benefits of vaccination, depending on age. The proposed strategy adapts a constructive approach in the literature based on Sperner’s Lemma to point out an approximate division of doses guaranteeing that both benefits are optimized each time a batch becomes available. Applications using data about population age distributions from diverse countries suggest that, among other features, this strategy maintains the desired balance, throughout the entire vaccination period. We discuss complementary aspects of the method in the context of epidemiological models of age-stratified Susceptible - Infected - Recovered (SIR) type.


Introduction
The unprecedented situation in which a vaccine was successfully developed amid the disease pandemic, the case of Covid-19, brought about urgent questions related to the possible strategies for allocation of the doses made available gradually in very small batches that do not cover the entire population in a community all at once [1], [2].Given the high transmissibility of the virus and widespread infection, the problems associated with vaccine allocation highlighted the urgent need to elaborate and put in practice certain guidelines that best satisfy a given set of ethical requirements [3], [4].In many countries, the prioritization followed protocols suggested by the WHO [1] to allocate the first doses that became available to the oldest and to those with comorbidities since these individuals are the most likely to develop severe forms of the disease [5], [6].Other groups at maximum risk such as health care workers and, in some places, members of disadvantaged groups deprived of minimal protection against direct exposure to the virus, the homeless for instance, in addition to members of indigenous populations and isolated small communities, among others, have also been eligible for doses of the vaccine from the first batches [1].
It is worth noticing that, after the most elderly and other most-at-risk groups receive their doses, virtually the entire population remains unvaccinated while new doses continue to be available in very limited numbers.In the remaining susceptible individuals, comprising the large majority of the population, the ability to transmit the disease and the severity of the symptoms are widely dispersed, and these generally correlate with age.We restrict the contribution offered here to this scenario.
The elderly within this remaining population would still be mostly benefited directly from receiving the doses because they tend to develop the most severe forms of the disease compared to the younger that are likely to present with only mild symptoms, although this is not a rule [5].On the other hand, due to their mobility and intense daily activity, younger people have a major capacity to transmit the virus compared to that of the elderly [3].Therefore, vaccinating younger people would greatly benefit the entire population, as an indirect effect.
Direct and indirect effects of vaccination, regarding mainly the interplay between decreasing disease severity and its transmissibility, raise questions about the possibility of balancing these two factors that seem to compete with each other in making decisions about allocating vaccine doses [3].We believe that any solution to this problem should comprise the following points: 1) a measure to evaluate proximity to a balanced condition that enables comparing results among different strategies of vaccination, and 2) a methodology to implement dose allocation that maintains such balance on time until vaccine coverage of the entire population is achieved.
We argue that this can be approached following an envy-free type of strategy for a fair division of doses among individuals possessing different utilities.The concept of envy-freeness is often illustrated in the literature by the classical cake-cutting problem [7], [8].This refers to a partition among agents of a certain resource (the cake), generally heterogeneous, such that each one of the agents evaluates their parts as being the best among the parts chosen by the others.The heterogeneity of the resource is usually expressed through a utility function that assigns to the different parts of the cake, different values satisfying additivity.In general, each agent has its own utility function.Here, we use the notion of utility to quantify both the direct and indirect benefits of vaccinating the diverse age groups of the population.We then formulate such a strategy to address the case of vaccine dose allocation by the agents to the individuals adapting the constructive approach presented in [9] and reviewed in [10] which is based on Sperner´s Lemma.For this, we assume that the vaccine is a desirable good and also that individuals and vaccine doses can be conceived as divisible quantities being represented by densities, defined appropriately.Accordingly, each age group receives a score named utility in agreement with the various views and plans of certain consultants or counselors expressing their priority criteria in line with the current public policies in the considered community.
We examine the case at issue regarding transmissibility and severity of the disease as a prototype to explain our ideas, although the model is not restricted to it.In keeping with this, it will be sufficient to consider the expertise of only two counselors, each one in charge of scoring all individuals according to their ages.One of the counselors referred to as C A (Ana), is an expert in predicting the ways of spreading the disease.The other counselor, C B (Bob), is an expert in disease symptomatology.Specifically, C A represents the allocation guideline that accounts for the benefit of vaccinating to control transmission -the indirect effect of vaccination.C B represents another allocation guideline that accounts for the benefit at the individual level -the direct effect of decreasing the severity of the symptoms.Both C A and C B are interested in balancing the two aspects; none of them wants to dispute vaccine doses.Therefore, whenever a new batch becomes available to this community, the doses will be allocated in such a way that each counselor agrees on the distinct groups of individuals to be vaccinated to optimize separately the benefit envisaged by each one.We claim that this characterizes an envy-free division of the vaccine doses.
The way that this may be accomplished is our main proposal and will be developed in the following Sections.We emphasize that unlike utilitarian models [11], [12], our strategy is not based on a single score system for which the priorities for doses allocation are evaluated in terms of the total score received by each individual from the different counselors.Rather, we conceive the model in such a way that each counselor optimizes the benefit according to their particular view.Our approach differs also from the reserve system strategies [13] for which the total vaccine supply from each batch is distributed according to pre-assigned proportions to certain reserved categories.In our model, the proportions of doses attributed to the management of each counselor are dynamic quantities, resolved along the process.
Our proposal is presented in Section 3. Illustrative examples are considered in Section 4. We compare the results from the application of the envy-free strategy using data comprised of certain population age distributions, with those predicted by the other two strategies examined, referred to as oldest-first, and maximize-benefit, as detailed below.We have also considered a strategy named minimize-benefit to set a scale to measure the efficiency with which the benefits are acquired by each strategy.These comparative results indicate that the envy-free leads to a considerable improvement in keeping the benefits related to C A and C B close together over time conferring support to this strategy as a way to pursue the desired balance.A discussion of these results and considerations about the extent of the applicability of the model are presented in Section 5.In the Supplementary Material we outline the numerical procedure used to implement the model dynamically.

Model
Our formulation is decomposed into two parts.The first part consists of preliminary definitions to build up the relevant simplex as a basis for the choice of individuals to be vaccinated at each time.It is assembled using accessible data about population age distribution, taken in connection with the utility attributed to all groups of individuals by each counselor.The second part consists in building up the dynamics that drives this choice to achieve the required balance between the two guidelines.

The simplex
The age group distribution in a community with N susceptible individuals will be considered for the account of two counselors C A and C B , each one of them endowed with a utility function ρ η (I), η = A, B hypothesized in such a way to attribute a score to each individual according to the corresponding age-related priority criteria.This can be performed by ordering these N individuals in such a way that their age I(x) is a monotonic increasing surjective In order to build up the 1-simplex of interest over which we perform our considerations about the choices of the counselors, we map the interval [0, N ] into the interval [0, 1] and variable x into a real variable y = x/N such that y ∈ [0, 1].The age at position y within this map shall now be calculated as I(yN ).We may assume that all individuals within the same age group are equally valued by each counselor though most likely, the value varies between counselors.It will be convenient though, to deal with continuous utility functions ρ η (I(yN )) ≡ ρ η (y) represented by a combination of smoothed step functions for each counselor η = A, B, as detailed in Section 5.The utility densities u η (y) defined for all y ∈ [0, 1] as are the functions that allow for considerations about envy-free divisions based on Sperner´s Lemma, as will be explained next.Observe also that any region ω of the considered simplex may be decomposed into a number M of disjoint sub-regions {v j }, j = 1, 2, ...M .Each v j , extended between endpoints y jI (initial) and y jF (final) with

The dynamic
Consider a population that at a certain instant of time t comprises N (t) susceptible individuals to whom a batch of V < N vaccine doses shall be allocated.We suppose that the availability of the batches occurs at a certain frequency of 1/T until the entire population is vaccinated.We also assume that individuals achieve full protection after receiving a single dose.The time t shall then be better measured in terms of the interval T between batches as t = nT, for n ∈ Z + .The question posed here regards the choice of the V individuals to receive the doses at each time t in order to balance the current guidelines.
We the constructive approach [8] based on Sperner´s Lemma as presented in [9] and reviewed in [10].
From the utility density functions u η (y), η ∈ {A, B} we define the benefit U ω(t) η according to the referred counselor perspective, that results from vaccinating the individuals inside a region ω(t) of the simplex at time t: The total benefit that is reached after vaccinating the entire population of susceptible amounts to one, according to both counselors: We proceed by partitioning the 1-simplex, at a time t, into a number d of identical parts, each of which comprised between a pair of neighbor points (p i , p i+1 ) at the positions (y pi , y pi+1 ), respectively, with y pj = j/d, for j = {0, 1, 2, ...d}.We then assign to the endpoint p 0 at y p0 = 0 a label arbitrarily chosen between A and B so called in reference to the counselors, and then proceed by labeling each of the following points p i of the sequence as A or B, alternately.
Observe that each point p i at y pi splits the ordered population into two parts, part I on the left of p i and part II on the right of p i , comprising respectively N i I and N i II individuals at time t, such that At each of these points p i we also consider splitting the batch of vaccines available at a certain time into two parts, V i I and V i II .We choose V i I and V i II proportional to N i I and N i II , respectively: To the extent that the simplex is arranged in this way, both quantities, i.e. individuals and vaccine doses, are evaluated using the single continuous variable y.This allows each counselor to express, at the corresponding labeled points p i , what would be their preferential side to proceed with vaccination.We assume that V i I and V i II are intended necessarily to vaccinate individuals, respectively, on sides I and II defined for each i.Explicitly, to maximize benefit at each A-labeled point p i , counselor C A is asked to express her preference, based on u A , about vaccinating V i I individuals on the side I or V i II individuals on the side II.The same for counselor C B at each B-labeled point p i ,based on u B .One should notice that it is implicit in this procedure, regardless of the counselorsć hoices, that neither of them would be able to vaccinate the entire population at once with the corresponding amount II of doses available on each side.Moreover, at an A-labeled point p i where counselor C A is in charge of choosing the side and decides for say, side II, she is supposed to make use of all of the V i II doses pre-assigned to that side.This implies that counselor C B would necessarily vaccinate individuals on the other side using all of the V i I doses, even though it may not be his preferential´s.Despite this, C B would look for individuals amounting to V i I that are best to be vaccinated on the side I according to his utility density function.The same will be followed by counselor C A after C B has expressed his preferential choices at each B-labeled point p i .
Accordingly, the counselor in charge at each point p i , regardless of being labeled A or B, ends up vaccinating exclusively at one of the sides.Nonetheless, they would benefit from vaccination on both sides.Since both utility density functions assume nonzero values along the entire simplex, each counselor must account for a benefit coupled with the other´s choice.In the example above we understand that, at that particular point p i , counselor C A has chosen side II and the best sub-region to vaccinate the V i II individuals at that side.This choice is foreseen after evaluating the total benefit from u A composed of: (i) the amount obtained from u A at a sub-region of II comprising V i II individuals that have been chosen according to her utility function u A , and (ii) the coupled benefit that corresponds to the amount obtained from u A at a sub-region of I comprising V i I individuals that have been chosen by suggestion of C B , based on his utility density u B .She concluded that the sum of (i) and (ii) is greater than the amount she would have obtained if she has chosen to vaccinate on the side I and received the coupled benefit from side II.For this, one must assume that both counselors know each other´s utility density functions.
To extend (i) and (ii) to arbitrary choices, it will be useful to define for Γ ∈ {I, II} and η ∈ {A, B}, the interval Ω Γ η (p i , t), as the sub-region on the side Γ of the simplex with respect to point p i where counselor η evaluates the maximum benefit from u η at time t.According to this, counselor C A looks for the largest between the total benefit , are defined as: and If U I A (p i , t) U II A (p i , t) she decides for side I, otherwise she decides for side II.
Likewise, to decide which side to vaccinate at each B-labeled point p i , according to his utility function, counselor C B looks for the largest between the total benefit U I B (p i , t) and U II B (p i , t), defined as: and If U II B (p i , t) U I B (p i , t) he decides for the side II, otherwise he decides for the side I.
The example discussed above corresponds to the case for which the pre-evaluation of the benefit by C A , at the We finally observe that even though side I has no individuals to be vaccinated at the end-point p 0 at y p0 = 0, the counselor in charge there might have two options: either to let the other vaccinate on side II using the entire amount V of doses, or to vaccinate the V individuals on side II.For example, if the point p 0 is A-labeled then counselor C A will still be in charge to decide about her preferential side based on the largest between On the contrary, if the end-point p 0 is B-labeled then counselor C B would select the side based on the largest between .Therefore, the counselor in charge at p 0 will oneself prefer to indicate the individuals to be vaccinated, and this would happen on side II, instead of leaving vaccination up to the other counselor.Using similar arguments, one finds that for p 1 at y p1 = 1 either one of the counselors would choose side I. Therefore, whoever counselor at p 0 , would choose the right side whereas whoever counselor at p 1 ,would choose the left side.These conclusions assure that the conditions under which Sperner's Lemma holds are fully satisfied by the simplex defined above.
Finally, after the two counselors have expressed their preferential sides at each of the corresponding points p i and the simplex looks like that sketched in Figure 1 it allows one, through simple visual inspection, to list all pairs of consecutive points, referred here generically as (p L , p R ), such that the counselor at the point on the left p L has expressed a preference to vaccinate on one of the sides say on side II, while the counselor at the point on the right p R has expressed a preference to vaccinate on the opposite side, i.e. side I.
The existence of at least one such pair of points is ensured by Sperner's Lemma.Accordingly, for sufficient large partition d, an internal point p * of the interval defined by any of these pairs will approximate a position at which the preferred sides of the two counselors are opposite to one another.The choice of any of those points p * (if more than one) identifying opposite preferred sides for each counselor to allocate the available vaccine doses, characterizes an approximate envy-free division for which either or In order to carry on this strategy until all susceptible individuals in the population have the opportunity to get their doses, it is assumed that the procedure described above is repeated at each time t when a new batch containing V doses becomes available.For simplicity, we consider the unrealistic case for which V does not change along the entire process.On each of these occasions, the simplex must be re-scaled and the utility densities attributed benefits Φ η (t): U Ω(t´) η (16) for each of the two counselors η = A, B, and the mean: The outcomes obtained for all selected countries exhibited a similar pattern (results not shown).
Figure 6 depicts the time average of the differences (absolute values) between the contributions to the cumulative benefits of the two counselors, evaluated for all the countries listed: Figure 7 shows the corresponding results for time average Φ of Φ(t) (17): These include the results for the minimize-benefit which is worth considering here precisely because it offers a lower bound to set a scale that allows one comparing outcomes, as we discuss next.Figure 8 merges the results for the averages of cumulative benefits ∆Φ and Φ using the considered strategies and combinations of utilities listed above extended for 236 countries (not specified).

Discussion and concluding remarks
The realistic case addressed here is that of deciding about strategies for allocating vaccine doses that become available to a community at a certain frequency but in very limited quantities.In the example used, we consider two guidelines to drive allocation.The first focuses on the direct benefit of decreasing the severity of the symptoms.The second focuses on the indirect benefit of decreasing transmission.We approach this situation by representing each of these guidelines as the priority of a qualified counselor in scoring the entire population ordered by age.Assuming that full protection of an individual is achieved after a single dose, the challenge is to select the group of individuals to be vaccinated every time a new batch becomes available to balance these two contributions.The difficulty relies on the fact that, in general, the amount by which a given vaccinated individual contributes to each of the benefits differ from each other.On the contrary, if both benefits were of the same magnitude, any strategy would result in a balanced condition.We claim that the strategy based on an envy-free division for dose allocation, as outlined above, offers a suitable and efficient choice to achieve such a balance in unpaired cases, as exemplified by the considered utilities.Our approach adapts the constructive analysis of the classic cake-cutting division problem [9] to conceive distributing doses optimizing the benefits envisaged by each counselor, which include the benefits coupled with the other´s choice.
Consistent with this, the results in Figure 3 of a case study certify that under the envy-free strategy, the increments to the benefit acquired at each time by each counselor remain very close together until vaccination is completed.Such results contrast with those obtained through oldest-first and maximize-benefit for which the increments to the two benefits differ considerably from each other across time.By adopting any of these two strategies, the selected regions of the simplex for doses allocation, either Ω = Ω oldest or Ω = Ω max along which u A (y) and u B (y) may assume very different values, leads to unbalanced U Ω(t) A and U . This is also the case with the random procedure.Although each of the benefits accumulates to the unity at the same time, the way that this is accomplished and the effects on the achievements of the two counselors can differ considerably.In this respect, the comparative results in Figure 5 offer information about the efficiency with which the benefits evolve under different strategies.This can be better seen by interpreting cumulative data as the positions in time of the two particles in the space of benefits driven, each of them, by the corresponding counselor.Extending the analysis for the population age distributions of the selected countries, as shown, Figure 6 depicts the average distance kept between these two particles in each case, until reaching their common final position simultaneously.Large values indicate that on average, one of the particles reached positions close to the goal considerably faster than the other.That is, for such strategies, the two benefits evolve out of sync over a considerably large period.This is the case for maximize-benefit and oldest-first in these examples.On the contrary, the positions of the two particles under the envy-free remain very close together at each instant through the entire time interval, suggesting that in addition to offering a way to promote a balance between acquired benefits, the strategy offers also a way to balance the instantaneous rates at which this happens.This is important for practical purposes since the intervals between consecutive batches may be very large, especially during the initial vaccination.The effects of a time delay between the achievements of each of the two benefits may have devastating consequences for the community.The random choice procedure offers balanced rates, on average, although the instant rates differ considerably since the sizes of the increments to the benefits alternate unbalanced.
Keeping with this kinematic interpretation, data in Figure 7 refer to the time averages Φ of the positions of the center of mass of the two particles achieved through the diverse strategies, and for all of the selected countries.
The maximize-benefit presents the largest average value, as expected.Even though the partial benefits, i.e. the ones envisaged by each counselor, evolve at different rates in this case, both of them reach large values within relatively short times.Envy-free is also efficient in accumulating benefits almost as fast as the maximize-benefit does.Apparently, in all cases, the oldest-first and the random choice promote the worst results among all the considered procedures, except for a strategy introduced here named minimize-benefit.Under this, one looks for regions of the simplex that minimize the total benefit at each time.Although very implausible to be adopted in practice, this strategy is useful to consider in the present analysis since it provides a lower bound to compare the efficiency of the diverse strategies investigated.Accordingly, the average benefit accumulated upon envy-free is much closer to the quantity accumulated by the maximize-benefit than that accumulated by the minimize-benefit (Figure 7).Random choice accumulates benefits at an average rate between the maximize and minimize-benefit strategies.
The oldest-first spreads its contributions along the interval showing a strong dependence on the population age distribution.
Figure 8 depicts the results for (a) the time averages of the instantaneous difference ∆U and (b) the time averages of the cumulative difference ∆Φ, both against the average cumulative benefit Φ.Each point from a total of 236 composing a colored set, corresponds to the population distribution of a given country (not identified).The emphasis is given to the different combinations of utilities and strategies employed.In all cases, the results are in line with the behavior depicted in Figures 4, 6, and 7, for the utility pair named Default.The envy-free strategy is unique in achieving the smallest differences ∆U and ∆Φ among all strategies and in producing total benefit at a rate that, on average, is the closest to that achieved by maximize-benefit.Although the maximize-benefit (and in parallel, the minimize-benefit) approaches the results for the envy-free regarding the cumulative difference ∆Φ, the dispersion of data increases considerably in these cases.A surprising outcome from the study in Figure 8 is that the envy-free reveals a tendency to minimize the dispersion of the distributions for both ∆Φ and Φ when compared to the corresponding results achieved by the other strategies.For all pairs of utilities chosen, the remaining strategies produced large dispersion either for ∆Φ or Φ, or for both.Collectively, these results indicate that among all of the considered strategies the envy-free promotes a good balance between the benefits envisaged by the two counselors over time, and also that this happens at similar and relatively high rates at initial times resulting in fast accumulation of benefits.In addition, it is the strategy that tends to equalize the benefits of vaccination among diverse countries, which is desirable within a scenario of a pandemic.We thus believe that the proposed strategy fulfills the requirements stated in the introduction since it maintains the balance in agreement to different measures comprising a) the amount of benefit acquired at each time by each counselor, b) the efficiency of the process given the speed with which the cumulative benefit approaches limiting values, and c) the tendency to equalize the effects achieved by distinct population age distributions.
We have assumed throughout that the only mechanism by which individuals are removed from the simplex is through vaccination.We do not account for varying vaccine efficacy or deaths, whether caused by the disease or by any other reason across the vaccination period.Once the two counselors provide the utilities, we predict the fraction of individuals from each age group, and at each time, that should be vaccinated to guarantee the balance.The results in Figure 9 exemplify in the case of the U.S. the kind of outcome provided by each of the four combinations of utilities, as indicated.In all cases, individuals of 65+ and those comprised within 15 − 24 years old are indicated to be prioritized across the initial batches.
The effects of vaccine efficacy have been considered in previous studies using optimization algorithms [15], [16] in connection to the evolution of age-stratified population models.In particular, an SEIR (susceptible, exposed, infectious, recovered) model dynamics has been considered for analyzing different scenarios for the choice of the age groups at the initial period of vaccination [17].Given the proposal developed here, it might be interesting to conceive a vaccination plan based on an interplay between the dynamic of the envy-free and that of the SEIR model.Such a protocol would be able to minimize morbidity while balancing benefits.
Finally, it should emphasize that the model offered here is not in any possible way restricted to the specific guidelines addressed above.These have been selected as references to explain and illustrate the practice of the method.Any other guideline could have been chosen to drive the allocation of available units.In addition, because Sperner's Lemma can be extended to more dimensions [9], [10], this opens the possibility to extend the constructive strategy described above to approach more realistic situations in which there are more than two guidelines defining priorities [3].We believe that this offers an attractive perspective to resolve such complex problems, with the help of careful and skilled counselors.
where ∆s j (t) = s F j (t) − s I j (t) is the size of the interval s j (t).The remaining intervals [ζ I j (t + 1), ζ F j (t + 1)] for all j > 1 are set as: ζ F j (t + 1) = ζ I j (t + 1) + ∆s j (t)r(t; t + 1) Z(t + 1) defines the simplex that will be considered in the next iteration, at the time t + 1.The three steps described above are iterated up to the vaccination is completed.

About the choice of parameter d
Sperner's Lemma guarantees that the envy-free strategy will always find at least one pair of points (p L , p R ) enclosing an envy-free point p * at the end of each iteration time.However, if the considered number d of divisions of the simplex is too small, the average between these two points may not be a good approximation to p * , as we have assumed.In this case, changing d may lead to oscillations in the value of p * and, most probably, in the quantities derived from it.Improving the approximation by increasing d is expected to reduce such oscillations as p * approaches its actual value.This, however, implies adding considerable computational costs to the numerical procedure.To achieve a compromise between mathematical accuracy and computational performance in this case, we examine how the change in d directly affects the temporal averages of the quantities shown in figures 4, 6, and 7 using as an example, the population age distribution from the U.S..

Other strategies
The other strategies considered to simulate the dynamics of vaccination, in particular the maximize benefit and the oldest-first, introduce changes into the Decision Step described above.
The maximize-benefit, is based on the choice of the region Ω(t) max in the simplex which is of the size of the total fraction v(t) = V /N (t) of individuals to be vaccinated at each time t, such that it maximizes the total benefit, accounting for both counselors according to the prescription in (14) for Ω(t) = Ω(t) max .The iterating procedure follows then the same removal and re-scaling steps as for the envy-free.
The implementation of the oldest-first strategy consists in allocating the total fraction v(t) = V /N (t) of vaccine doses available at the time t to the oldest fraction of the population present at that time.The resulting benefit is evaluated according to (14) for Ω(t) = Ω(t) oldest .The iterating procedure follows then the same removal and re-scaling steps as for the envy-free.

y
jI < y jF , comprises a number [(y jF − y jI ) N ] of individuals, where the notation [z] indicates the integer part of the real number z.
think of two different priority criteria suggested by two counselors C A and C B expressing different opinions about how one should rank the population in the community to guide this choice.The utility density functions u A (y) and u B (y), y ∈ [0, 1] conceived, respectively, by C A and C B assume nonnegative real values and represent a measure of the relevance for vaccinating the individuals ordered according to some rule.To present the methodology, we choose to order the individuals by their age, although this does not exclude any other possibility.Such an ordered list of individuals mapped into the interval [0, 1] defines the 1-simplex as detailed above.We aim to present a fair division strategy of an envy-free class through which the choice of the V individuals at each time balances the perspectives of the two counselors in the best possible way.The proposal offers an approximate solution extending

Figure 4
suggests a measure ∆U(15) for this imbalance averaged over time.The results for the diverse strategies are depicted for each of the selected countries.It shows that ∆U approaches null values through the envy-free strategy.Relatively large values are obtained by applying the other two strategies and also by choosing the regions at random.
Figure S2(a) shows the average temporal behavior of the difference of the benefits (absolute values) ((15)) due to the contributions of the two counselors obtained through the envy-free strategy, as d increases.Figure S2(b) shows the corresponding behavior of the cumulative differences (18), and Figure S2(c) shows the mean cumulative benefit (19).As expected, the amounts oscillate around the mean until stabilizing at a certain value of d that is not the same for the different quantities analyzed.We proceed into the whole numerical calculation presented above choosing d = 100 which seems suitable to ensure convergence of the results in all cases.

Figure 9 Re
Figure 9 Since at p 0 the values reached by u A at Ω II A (p 0 , t) are higher than or at least equal to the values reached by u A at Ω II B (p 0 , t) then U II A U I A .Similarly, since the values reached by u B at Ω II B (p 0 , t) are higher than or at least equal to the values reached by u B at Ω II A (p 0 , t) then U II