Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 2, Number 1, March 2021, pp.10-21 https://doi.org/10.5206/mase/10093
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR
ADVERSE EVENTS
MYLES NAHIRNIAK, MONICA COJOCARU, AND TANGI MIGOT
Abstract. Natural disasters occur across the globe, resulting in billions of dollars of damage each
year. Effective preparation before a disaster can help to minimize damages, economic impact, and loss
of human life. This paper uses a game theory framework to set up a leader-followers model for resource
distribution to several geographic zones before an adverse event. The researchers model population
members who may choose to prepare in advance of an event by acquiring supplies, whereas others
may wait until the last minute. Failure to prepare in advance could result in a significant loss due
to the chance that supplies may no longer be available. Numerical simulations are run to determine
how the leader should distribute supplies to maximize the preparedness of the overall population. It
was found that population size is a significant factor for supply distribution, but the behaviour of
individuals within a zone is also important. Much of the current resource allocation research focuses
on the logistics and economics of supply distribution, but this paper demonstrates that social aspects
should also be considered.
1. Introduction
Throughout 2019 in the United States alone, there were 14 natural disasters each causing 1 billion
USD or more of damage [13]. One example of such a disaster is hurricane Dorian, which in August and
September affected several Caribbean nations, most notably the Bahamas, as well as the Eastern United
States and provinces in Atlantic Canada. Effective preparation in advance of a disaster is necessary to
mitigate damages, financial costs, and loss of human life.
A review by Altay and Green [1] identified a lack of cooperation between humanitarian agencies.
As a response, Galindo and Batta [5] recommended modelling a leader agency that is responsible for
overall coordination. Muggy and Heier Stamm [10] agreed that cooperative models are lacking, and
determined that game theory is an appropriate tool for determining an optimal allocation of resources.
Disaster relief is divided into four stages, known as mitigation, preparedness, response, and recovery
[8]. Our motivation for this paper is to use a bilevel game to model the distribution of supplies among
various geographical zones in preparation for a disaster, while also considering how members of the
population choose to prepare. Many existing environmental and disaster relief models which use a
game theory framework consider cases where relief agencies or suppliers either compete or cooperate.
It is known that two parties cooperating to implement an environmental project is more effective than
each party working independently [3].
Previous studies have examined socio-cognitive reasons that members of a population may or may
not prepare for adverse events, however these factors have not been included in mathematical models.
Even with advance knowledge, many individuals will choose not to prepare for a natural disaster [15].
The level of preparation amongst a population can depend on the ease of acquiring supplies, as well
their ease of implementation [14]. Another issue is the public’s compliance with recommended safety
Received by the editors 20 October 2020; accepted 12 January 201; published online 22 January 2021.
2000 Mathematics Subject Classification. 91A40, 91B32, 91A65.
10
https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/10093
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 11
measures put forward by authorites; some population members may not have trust in the authority
while others believe the adverse event will be less severe than predicted [2]. Finally, Lopes [7] determined
that individuals overestimate their level of preparedness compared to their actual level of preparedness.
In our paper, we posit that social aspects are significant, and incorporate these factors in our model.
There is much literature with the focus on optimizing the cost and transport of supplies. Nagurney
and Flores [12] develop an equilibrium model to ensure consistent flow of supplies under cost constraints.
Similar research was conducted to specifically analyze and reduce the transportation costs of supplies to
different sites [6]. These models, however, ignore the problem of how to effectively allocate resources to
multiple players. Nagurney et al. [11] adapted a generalized Nash game model for post-disaster relief,
focusing on the distribution of funds. Their model assumes multiple organizations that have all decided
in advance the amount of relief to be provided at each point.
In contrast, we propose an alternative approach. In our model, we consider how resources should
be allocated to multiple geographic zones in advance of a disaster, to ensure that the maximum overall
population is prepared. Additionally, we consider the human behaviour aspect of how a population
prepares in advance of a disaster. We model the situation as a leader-followers game in which an
overseeing party is responsible for distributing supplies among several geographical zones. In each zone,
the population is divided into two groups: one who chooses to acquire supplies in advance of a disaster
and the other who waits until the last minute, knowing there is a risk that supplies will run out. We
consider two other important factors in our analysis: first, we differentiate the populations in each
zones not only by their size (total number of inhabitants per zone), but also by their cost of acquiring
preparedness supplies in advance. The cost here should not be regarded as a dollar amount, but rather
as a personal cost, in time, money and effort to acquire appropriate supplies. Second, we differentiate
the zones by assuming different probabilities of running out of distributed supplies and that costs of
acquiring supplies may be reduced (by incentives) to nudge individuals to consider supplies in advance,
rather than last minute. Reductions in personal costs would be for instance: convenience of locations
for supplies distribution centers, distribution of supplies ”at residence” for people of higher risk (such
as senior citizens or long-term care homes), information campaigns, and ”drive-through” or delivery
options, etc.
This paper provides an overview of the model, and sets up the problems for the followers and the
leader. We provide two types of analyses, based on parameter sensitivity analysis techniques. To begin
the analysis in Section 3, we first establish a “base case” scenario of the followers problem, then we
complete a sensitivity analysis on their possible best strategies. We continue in Section 4 with a further
sensitivity analysis of the leader’s best decision based on models parameters and leader’s constraints.
In section 5 we link the probability of running out of supplies in each zone to the choices of individuals
in that zone. We re-evaluate the model under this new assumption, and contrast the results with our
previous analyses. We conclude that population density and advance incentives are the most important
factors for a leader to consider when allocating supplies.
2. Game theoretic resource allocation model in diverse populations
In this section, we develop a model for the distribution of resources among several geographical zones.
We formulate the model as a leader-followers game. Each zone’s followers is described by a symmetric
two-player bimatrix game. The leader’s problem is formulated as an optimization problem, given the
behaviour of the players in each zone. We will explain the parameters and assumptions used in the
model below.
Consider a leader to be an authority designated to distribute supplies among N zones, labelled as
n = 1, . . . ,N. This could represent the government, or a charitable organization providing disaster
12 M. NAHIRNIAK, M. COJOCARU, AND T. MIGOT
relief. For some total quantity of supplies, Qtot, denote the amount distributed to each of the zones as
Qn, and any leftover supplies not distributed to be Qremain.
The total supplies are Qtot =
∑N
n=1 Qn + Qremain. The fraction of total supplies provided to a given
zone is wn =
Qn
Qtot
which implies
∑N
n=1 wn ≤ 1.
Represent the population of each zone by Pn. The population spread among the zones is given by
pn =
Pn∑
N
n=1
Pn
, where pn is the fraction of the total population in zone n.
2.1. Followers’ game. We now assume that the players in each of the zones play the same game.
Their two pure strategies are represented by: 1 = ”stocking up on supplies before an adverse event”,
or 2 = ”waiting until the last minute to purchase supplies”. Advance purchasing may be viewed as
carrying a lower personal cost to a player, whereas last minute search has a risk that supplies may have
run out and the player may not be able to acquire needed supplies. We model the players’ choices as
a symmetric bimatrix game in each zone, where the two players share the same matrix, A, of payoffs
given by
yn 1 −yn
xn
1 −xn
(
Tn −Cn Tn −γCn
Tn −γCn Tn − (qnLn + (1 −qn)2Cn)
)
.
In the above bimatrix, xn represents the frequency that player 1 in zone n plays strategy 1 = ”supply
in advance”. Because each player may only choose between one of two strategies, the value 1 − xn is
the frequency that player 1 in zone n plays strategy 2 = ”supply last minute”. Similarly, yn and 1−yn
are the frequencies that player 2 in zone n plays strategies 1 and 2, respectively.
The “budget” available to each player in zone n is denoted Tn, and the cost of acquiring supplies is
Cn. There is a probability of running out of supplies, qn, which leads to the potential for either a large
loss, Ln, or for acquiring supplies at a larger personal cost. We assume that at the last minute, the
remaining supplies require double the effort/cost to be acquired. There is an incentive factor for the cost
of supplies if individuals from a single zone supply in advance, denoted by γ. With no incentive (γ = 0),
individuals pay full price; a high value of γ provides an incentive to supply in advance, thus lowering the
effective cost of supplies. This can signify, for example, that the time they spent for acquiring supplies
is very short, or that supply distribution centres were conveniently located and well-stocked.
The parameters in the model, Tn,Cn,Ln,qn, and γ are normalized to be between 0 and 1. We assume
further that a player must have enough funds to purchase supplies if desired, and the potential loss is
greater than the cost of supplies, so 0 ≤ Cn < Ln ≤ Tn. The expected payoff En for a player in zone n
is given by
En(xn,yn) = (xn, 1 −xn)A(yn, 1 −yn)T .
A strategy (x∗n,y
∗
n) in zone n is a Nash equilibrium if x
∗
n is the best response to y
∗
n and vice versa, that
is
En(x
∗
n,y
∗
n) ≥ En(x
∗
n,yn), ∀yn ∈ [0, 1],
and since the game is symmetric, then x∗n = y
∗
n. Direct computation allows us to determine the
equilibrium points for the game. We find that the mixed equilibrium in the zone n is given by:
x∗n =
Cnγ + 2Cnqn − 2Cn −Lnqn
2Cnγ + 2Cnqn − 3Cn − 2qn
.
We verify that 0 ≤ x∗n ≤ 1 in Section 3.
2.2. Leader’s problem. The leader’s goal is to supply as much of the population as possible, in
advance of an event. This is done by optimizing the allocation of supplies to each zone and choosing
appropriate incentives in the distribution process that would amount to a lesser personal cost Cn for a
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 13
player to supply in advance. We assume that the probability of running out of supplies qn is different
for each zone, depending on the behaviour of the population in that zone and the amount of supplies,
wn, distributed.
The leader’s problem then is given by:
max
(w,γ,x)
θL :=
N∑
n=1
pnxn
s.t. (w1, ...,wN,γ) ∈ [0, 1]N+1,
0 ≤
N∑
n=1
wn ≤ 1,
xn ∈ arg max
xn
{En : 0 ≤ xn ≤ 1},∀n.
(2.1)
3. Sensitivity analysis of followers’ optimal strategies in a 2-zone allocation
problem with fixed costs and losses for followers
For ease of presentation, let us consider the leader’s problem (2.1), with N = 2 (i.e. for two zones).
The leader’s function for two zones now becomes:
θL(w1,w2,γ,x
∗) = p1x
∗
1(p1,w1) + (1 −p1)x
∗
2(p2,w2).
using the solution points from the followers’ games:
x∗1 =
C1γ + 2C1q1 − 2C1 −L1q1
2C1γ + 2C1q1 − 3C1 − 2q1
x∗2 =
C2γ + 2C2q2 − 2C2 −L2q2
2C2γ + 2C2q2 − 3C2 − 2q2
.
To analyze the leader’s problem, we assume here an equal population distribution between the two
zones (p1 = p2 = 0.5) and an equitable distribution of resources of w1 = w1 = 0.5, then the leader’s
function becomes θL = 0.5x
∗
1 + 0.5x
∗
2. The leader’s function is maximized when the followers’ best
strategy is maximal. Thus we will strive to analyze the optimal strategy values for the followers.
3.1. Base case. To better understand the problem, we now define a base case scenario, that is: L1 = L2
and C1 = C2, with C1 < L1. We then first compute the followers’ best strategy x
∗
1 depending on possible
values of γ,q1.
We draw the evolution of x∗1 as a function of γ and q1 in the base case where C1 = C2 = 0.5 and
L1 = L2 = 0.75 in Figure 1. We see that, indeed, the values of the follower’s best strategy lie in [0, 1].
Assuming an equal population distribution between the two zones (p1 = p2 = 0.5) and an equitable
distribution of resources of w1 = w1 = 0.5, then the leader’s function θL is maximized when the followers
best strategy is x∗1 = x
∗
2 = 1 given in Figure 1. We can exactly compute it to be θL =
∑2
n=1 0.5x
∗
n with
a maximal value of θL = 1 when γ = 1, qn = 0, n = {1, 2} (i.e. all will supply in advance if incentive γ
becomes maximal (γ = 1) and there is no possibility of running out of supplies (q1 = q2 = 0).
3.2. Deviating from the base case. In this subsection we start to differentiate between the two
zones, in the way we setup the follower’s input parameters Cn,Ln, n = {1, 2}. We look at the following
two scenarios:
• C2 = 1.5C1 and L1 = L2 = 0.75, where cost in zone 2 is 1.5 times higher than in zone 1, but
losses are comparable;
• C2 = C1 and L2 = L1 + 0.15, where losses in zone 2 are 20% higher than in zone 1, but costs
are comparable.
14 M. NAHIRNIAK, M. COJOCARU, AND T. MIGOT
Figure 1. Evolution of x∗n as a function of γ and qn in the base case where Cn = 0.5
and Ln = 0.75, n = {1, 2}.
In these cases, the plots of the followers’ best strategies are given in Figure 2 and 3 as: We see that
Figure 2. Evolution of x∗n as a function of γ and qn where C2 = 1.5C1 and L1 = L2 =
0.75, n = {1, 2}.
the leaders’ function, under the base case scenario conditions (p1 = p2 = 0.5 and w1 = w2 = 0.5), is
maximized whenever x∗n values are maximal for both populations. In both Figure 2 and Figure 3 we see
that the leader’s function is maximized in the best of circumstances, i.e., all will supply in advance if
incentive γ becomes maximal (γ = 1) and there is no possibility of running out of supplies (q1 = q2 = 0).
What is interesting to look at is the least favourable scenario, i.e.: the case where there are no incentives
(γ = 0) and there is a certainty of running out of supplies (q1 = q2 = 1). Here, when losses are the same
between zones (Figure 2), individuals in the zone with higher supply cost will not supply in advance at
all; on the other hand, when costs are the same but losses differ (Figure 3) then both groups will supply
in advance in some measure. This will lead to higher values for the leader’s function in the latter case.
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 15
Figure 3. Evolution of x∗n as a function of γ and qn where C1 = C2 = 0.5 and
L2 = L1 + 0.15, n = {1, 2}.
4. Sensitivity analysis of leader’s function in a 2-zone allocation problem with fixed
costs and losses for followers
We are now interested to examine a case where costs, as well as losses, are differentiated between
zones, but are fixed throughout the analyses. We then present a full sensitivity analysis of the followers’
game in Table 1, where the varying parameters are presented in Table 2.
C1 0.5 C2 0.75
L1 0.75 L2 0.9
Table 1. Table listing the fixed values of input parameters used in this case
γ Discounting factor [0, 1]
qn Prob. supplies run out in zone n [0, 1]
wn Supplies fraction to zone n [0, 1]
p1 Population density in zone 1 [0, 1]
p2 Population density in zone 2 p1 = 1 −p1
Table 2. Table listing the parameters we vary in the sensitivity analyses below in this
problem. Here n ∈{1, 2}.
All simulations are conducted by selecting 500 randomly distributed points (γ,w1,w2,p1) ∈ [0, 1]4
that satisfy the conditions of the problem, and then computing the corresponding values for the leader’s
objective function. We note that there is interdependence between the probability of running out of
supplies in a zone, qn, its population density pn, and the allocation of resources wn. Specifically, with
higher population density in one zone, this probability may increase; on the other hand, with higher
allocation of supplies, this probability may decrease.
Let us consider that the probability qn is proportional to pn and inversely proportional to wn. To
ensure that 0 ≤ qn ≤ 1, we choose q1 = min(1, p1√w1 −p1) and q2 = min(1,
1−p1√
w2
− (1 −p1)). We show
16 M. NAHIRNIAK, M. COJOCARU, AND T. MIGOT
the plot of this functional dependency in Figure 4 below, and note that qn saturates at 1 for large
populations if supplies are not sufficiently allocated.
Figure 4. Plot of qn as a function of pn and wn, n = {1, 2}.
We use the following subcases to present our results:
(1) We can observe the effect of varying the incentive factor, γ.
(2) We consider how the population distribution pn between zones affects the allocation weights
w1,w2.
(3) Finally, we distribute all parameters freely to determine the optimal values of the leader’s
function.
Case 1.
Using a 30-70% population split, we show two scenarios: Figure 5 shows the effect of γ while freely
distributing supplies (left panel), and with all supplies distributed using w1 + w2 = 1 (right panel).
Figure 5. Three parameters are freely distributed: (w1,w2 ≤ 1−w1,γ) while p1 = 0.3,
q1 =
p1√
w1
−p1 and q2 = 1−p1√w2 − (1 −p1). In the right panel we depict the case where
supplies are exhausted, i.e. w1 + w2 = 1
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 17
The incentive factor has a strong effect for large γ, and it can be seen that the objective function
is much higher towards the top of the plot in the yellow zone (where γ = 1). This indicates that
incentivizing the purchase of supplies at a lower effective cost increases the value of the leader’s objective
function and allows more of the population to prepare in advance. Additionally, below γ = 0.6, there is
very little variation in the objective function values. Hence values of γ ≥ 0.6 affect supply distribution.
Case 2. To observe the effect of varying population density between the two zones, we consider
two scenarios: Figure 7 shows the effect of p1 while freely distributing supplies, as well as distributing
all supplies using w1 + w2 = 1. The highest objective values occur where the proportion of supplies
distributed to a zone is comparable to the fraction of the population in that zone. For the purposes of
illustration, a constant value of γ = 0.5 was used in both simulations, but the same conclusions can be
drawn for any other value of γ ∈ [0, 1].
Figure 6. Two parameters are freely distributed: (w1 and p1) while γ = 0.5, q1 =
p1√
w1
−p1 and q2 = 1−p1√w2 − (1 −p1). In the left panel, w2 is freely distributed with the
constraint w1 + w2 ≤ 1, whereas the right panel depicts the case where supplies are
exhausted, with w1 + w2 = 1.
Case 3. Here the parameters (w1,w2,p1,γ) are freely distributed, with p2 = 1 −p1, and results are
shown in Figure 8. The weights and population size are plotted along the axes, with γ being represented
by the size of the marker at each point.
The optimal points occur in the top right (p1 = 1 and w1 = 1) and lower left (p1 = 0 and w1 = 0)
regions of Figure 8a. These correspond to the extreme cases where all of the population is in a single
zone, and all supplies distributed to that zone, suggesting that higher weights are beneficial to the zone
with higher population density. For mixed population distributions, the objective function is optimal
when all supplies are distributed, i. e., w1 + w2 = 1 We also notice that higher values of γ increase
the objective function, as large-sized points (high γ) close to small-sized points (low γ) have higher
objective values. With all supplies distributed, as shown in Figure 8b, higher γ values are preferred.
18 M. NAHIRNIAK, M. COJOCARU, AND T. MIGOT
Figure 7. Two parameters are freely distributed: (w1 and p1) while γ = 0.5, q1 =
p1√
w1
−p1 and q2 = 1−p1√w2 − (1 −p1). In the left panel, w2 is freely distributed with the
constraint w1 + w2 ≤ 1, whereas the right panel depicts the case where supplies are
exhausted, with w1 + w2 = 1.
Figure 8. Three parameters are freely distributed (w1,p1,γ) with p2 = 1 −p1, q1 =
min ( p1√
w1
−p1, 1), and q2 = min ( 1−p1√w2 − (1 −p1), 1). In the left panel, w2 is freely
distributed with the constraint w1 + w2 ≤ 1. The size of each point corresponds to the
value of γ. The right panel shows the case where w1 + w2 = 1.
5. Interplay of supplies between zones
In this section, we further assume that the probability of running out of supplies depends in general
on the supply strategies in both zones. We can write functions for q1 and q2 in terms of the parameters
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 19
x1,x2,w1, and w2, noting w1 + w2 = 1. Let
q1 =
x21
w1
+
x1x2
2w1w2
−
x1
2w1
and q2 =
x22
w22
+
2x1x2
5w1w2
−
x2
2w2
+
1
10
By inserting the above expressions for q1 and q2 into the followers’ games and making the choice
γ = 0.5, the payoffs in each zone are:
E1 = x1(−
x1
4
+
3
4
) + (1 −x1)(
3x1
4
+ (
x21
4w21
+
x1x2
8w1w2
−
x1
8w1
)(1 −x1))
and
E2 = x2(−
3x2
8
+
5
8
) + (1 −x2)(
5x2
8
+ (−
11
25
+
3x22
5w22
+
6x1x2
25w1w2
−
3x2
10w2
)(1 −x2)).
The computations are a two-step process. First we distribute w1 ∈ [0, 1] with a step size of 0.05 and
using the constraint w1 +w2 = 1, then solve the followers’ game for each pair of values. It is worth noting
that different choices of γ significantly modify the followers’ payoffs. Using the equilibrium points x∗1
and x∗2 from these games, we can simultaneously solve for the values of q1 and q2 in the above formulae.
Once q1 and q2 are known, we again solve the followers’ game using those probabilities to gain x1 and
x2. This finally allows us to optimize the leader’s function to determine the optimal supply distribution
between the two zones. The solutions to the game are given in Figure 9.
Figure 9. Solutions x∗1 (left panel) and x
∗
2 (right panel) of the game with γ = 0.5,
varying supply allocation and population between zones
.
To observe the effects of relative population sizes, we additionally freely distribute the parameter p1,
with the constraint p1+p2 = 1, before solving the followers’ game. Similarly, to determine the impact
of the incentive factor, we can vary γ before solving the followers’ game. This leads to the results in
Figure 10.
20 M. NAHIRNIAK, M. COJOCARU, AND T. MIGOT
Figure 10. Leader’s function with γ ∈{0, 0.5, 1}, varying supply allocation and pop-
ulation between zones.
.
It turns out that γ has a large effect on the leader’s function, and in fact, the leader’s values are
optimal at the points where γ = 0. This is in contrast to the previous models from sections 3 and
4 where high γ values were preferred. This shows that with the interplay between both zones, larger
objective function values are found if no incentive is provided. In examining population size, there is
a strong correlation between the size of the population in a zone and the optimal weighting of supplies
distributed to that zone. When one zone’s population is double the other’s or greater (i.e., when
p1 ≥ 2p2), there is a bias towards distributing most of the supplies to the larger zone. If both zones
are of similar size, the leader has flexibility as to the weighting, with minimal impact on the objective
function.
6. Discussion and conclusion
This research examines the problem of resource allocation to multiple geographical zones to prepare
for an adverse event using a bilevel game-theoretic approach. The focus is to incorporate the choices
of the population on whether or not to prepare in advance, as well as zonal population densities and
probabilities of running out of supplies. We use a leader-followers problem to optimise the supplies
allocation to each zone for the leader, in order to supply as much of the population in advance as
possible. The main factors impacting the leader’s decision and allocation schemes are zonal population
densities, incentivizing the acquisition of supplies in advance by population groups (for instance groups
“at-risk”), and the probabilities of allocated supplies to be exhausted.
In the case of 2 zones with equal populations, the leader’s function is optimal for values of γ ap-
proaching 1, as in Figures 1, 2, and 3. This indicates that the leader should incentivize the purchase
of supplies in order to maximize the objective function. With the additional assumption that available
supplies depend on the population size of each zone, again, high γ values are preferred. The functional
form of the probability of running out of supplies is highly important: In the situation where supply
availability is also dependent on the supply strategies in each zone, γ = 0 is the best choice as in Figure
10. With the interdependence of both zones, neither zone should receive an incentive.
In the model from section 3, the result is straightforward that larger zones receive more supplies. As
soon as the probability of running out of supplies takes a functional form, the optimal supply distribution
is less clear, as is evident from Figure 8. In general, supplying the larger zone is still preferred. For the
A LEADER-FOLLOWERS GAME OF EMERGENCY PREPAREDNESS FOR ADVERSE EVENTS 21
case in Section 5 with q1 and q2 based on the strategies of each player, if the zones are comparably sized,
there is a large region in Figure 10 in which the objective function is roughly constant. This suggests
that the leader has freedom in how to distribute supplies to the zones, without significantly affecting
how much of the population supplies in advance. However, if the population sizes are dramatically
different, then the best policy is to supply the largest zone.
Future research will focus on the case where the populations in the different zones are subject to
interdependent constraints [4, 9].
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Corresponding author, Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E.,
Guelph, ON, N1G 2W1
E-mail address: mnahirni@uoguelph.ca
Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E., Guelph, ON, N1G 2W1
E-mail address: mcojocar@uoguelph.ca
Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E., Guelph, ON, N1G 2W1
Current address: Department of Mathematics and Industrial Engineering, Polytechnique Montral, Montral, Canada
E-mail address: tangi.migot@gmail.com
https://doi.org/10.1007/978-3-319-97442-2/
https://www.ncdc.noaa.gov/billions/
https://doi.org/10.17226/983
1. Introduction
2. Game theoretic resource allocation model in diverse populations
2.1. Followers' game
2.2. Leader's problem
3. Sensitivity analysis of followers' optimal strategies in a 2-zone allocation problem with fixed costs and losses for followers
3.1. Base case
3.2. Deviating from the base case
4. Sensitivity analysis of leader's function in a 2-zone allocation problem with fixed costs and losses for followers
5. Interplay of supplies between zones
6. Discussion and conclusion
References