ClimateChangeNegotiations

Princeton University
Program in
Applied and Computational Mathematics
Homophily in Climate Change Negotiations:
A Probabilistic SIR Model
Author:
Kashyap Rajagopal
Advisors:
Vitor Vasconcelos, Simon Levin
9 May 2014

Abstract
The Kyoto Protocol was an international treaty signed by various industrialized countries committed to
reducing greenhouse gas emissions (GHGE). In 2005, it went into effect but soon flatly failed without
participation from the United States and low compliance. Climate change negotiations continue as countries
around the world seek to halt the progression of global warming. At the same time, they are naturally
incented to free ride on the contributions of other countries. Free riders will defect; whereas others will
cooperate and contribute funds to a common trust that is dedicated to mitigating climate change by pooling
resources. The behavior of different countries in response to the possibility of an international agreement
can be modeled as a dynamical system. Consider first a generic model of an international trust which
demands contributions from each country. These countries are influenced by one another significantly, and
their mutual influence is captured in the ’homophily’ parameter. Country decisions at each time step will
be motivated by the number of cooperators and defectors at that time-step. These changing circumstances
will influence how many countries eventually accede to the terms of the global treaty. I capture the national
behavior of countries in this negotiations game using a novel approach: a modified two-strain SIR model,
whose parameters reflect homophilic probabilities of switching strategies. In putting such a model to work,
it seems progression towards cooperation is strongly sensitive to the share of initial endowments required as
deposit to the global trust.
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1 Introduction
1.1 The Problem of Climate Change
Over the past century, the increased emission of greenhouse gases such as carbon dioxide, methane, and
nitrous oxide have been cause for serious concern among global leaders. Unfettered coal mining has resulted
in anthropogenic (human-induced) global warming through ozone depletion. Often times, the public is under
the misunderstanding that global warming may represent one of many periods of warming that have occurred
previously in the history of the Earth. However, the detection of increased temperatures above the ocean
surface posits strong evidence that global warming is primarily attributed to exogenous, human factors.
Indeed, there is a consensus among 97 percent of scientists that global warming has been on the rise due
to human activity [2]. As a result, average temperatures have increased by around two degrees Fahrenheit.
Without efforts to abate global warming, the United States Environment Protection Agency projects this
trend will continue- with temperatures climbing as high as 10 degrees over the next hundred years [1].
Temperature increases may seem slight at first glance, but these poorly understood climatic fluctuations
can dramatically alter — and disrupt — existing ecosystemic networks and regimes. The protective atmo-
spheric layer of ozone that shields the Earth against UV radiation is depleted by greenhouse gas emissions
(GHGE). Without it, many ecological systems are thrown into flux as the mixture of flora and fauna is
altered over time. Moreover, carbon dioxide traps sunlight within the Earth’s atmosphere, which hampers
the reflection of radiation. These implications suggest a long-term possibility of climate change reducing
social weflare. However, of late scientists have been drawing public attention to the sharper disruptions that
are also possible. Above threshold temperatures, large glacial ice caps in the North Pole can melt more
quickly than we can control. Rising coastal sea levels can lead to widespread flooding in lower lying areas.
As another example of a sharp, ecosystemic disruption - consider the thermohaline circulation that charac-
terizes ocean currents across the Atlantic Ocean. This system is responsible for the Gulf Stream, sending
warmer currents eastward. It is responsible for the phenomenon in which we see European cities at a higher
latitude than New York experiencing the same weather, on average. If thermohaline circulation is disrupted,
the warmer currents will not reach European cities, and the continent as a whole may see cooling. Between
global warming and pockets of cooling, the effects of climate change are not perfectly understood. What is
without doubt, however, is that climate change can alter global ecosystem regimes in not only a gradual way,
but can also have catastrophic implications in the event of glacial melting or thermal current destabilization.
It is useful to frame these climatic upheavals in terms of the economic implications of climate change.
1.2 Climate Change Mitigation and Negotiations
There are various methods of mitigating climate change. For such a large problem, it is no surprise that
successful mitigation rests on joint, macro-level cooperation.
Over the past couple decades, countries have come to realize the importance of working together towards
halting the progression of rising greenhouse gas emissions. The Kyoto Protocol of 1997 was the first major
step taken by myriad countries to outline an agenda for reducing their carbon emissions.
Climate change negotiations reflect a classic tragedy of the commons situation. There are many countries
who are interested in contributing funds to tackle the problem, but no country wants to invest in an effort
that will not ultimately succeed — especially when fellow countries are shirking responsibility and actively
deciding not to contribute. In short, it is unclear in this Prisoner’s Dilemma whether a country can trust its
peers to chip in, if it makes any leap forward with its own contribution to the pot.
2 The Modeling of Public Goods Games
2.1 Economic Game Theory
In formulating a mathematical simulation for policy-making, Ostrom (2009)[4] prescribes a useful framework
for development. She highlights the importance of using a mutlifactorial approach that captures the way in
which decisions play out over various horizontal levels- for example, at the level of the individual, the country,
the zone, etc. as opposed to a singular construction. For the purposes of modeling climate change, I will
2

introduce a model that captures two types of problems country leaders face: (i) the regional environmental
problem that they face by virtue of being a participant in a group that is exposed to this regional problem,
and (ii) a localized problem that afflicts their country. By posing countries with this two-level problem, I
can better capture the effects of failing to meet certain mitigation thresholds through their game theoretic
payoffs.
2.2 Evolutionary Game Theory
Vasconcelos et al (2014) describe an evolutionary game theory model for capturing the dynamics of countries
engaged in climate change negotiations. I will adapt this model in Section 7 when incorporating a concept of
evolutionary fitness in the SIR model I introduce. Vasconcelos notes that poor and rich country cooperators
and defectors play a series of games at each timestep to gauge one another’s fitness. In our model, this series
of games is not necessary — one will be sufficient. Furthermore, the distinction between rich and poor will
not be important to our generic case.
The benefits of evolutionary game theory is that it can better capture a series of games over time rather
than a non-temporal utility maximization method.
2.3 SIR Models in Social Behavior
The term ‘SIR Model’ refers to a classic system of differential equations that is typically used to model
infectious diseases. S refers to a susceptible class of the population. This group is susceptible to acquiring
the infection from I, the infectious group that harbors the bug. Finally, with enough treatment and time,
members of I are passed along to the R group, or recovered/removed class. The SIR Model is founded on
network theory and the theory of differential equations. So at each timestep, the number of individuals in
each of these three classes changes due to movement that is modelled bya system of differential equations.
The simplified, classic model is presented below:
dS
dt
= −βSI
dI
dt
= βSI − γI
dR
dt
= γI
Here, the γ term tells us what fraction of the members of I are successfully treated and move into the
recovered class — it is the removal rate. The classic model SIR links the rate of transferrence from S to I
in a given timestep to the number of possible interactions between members of each group. For example, a
population consisting of many S and I members makes for a high SI value — this value represents the total
possible interactions between susceptible and infected individuals who can spread the disease. As S and I
change with time, this SI value does as well. This correlation factor is represented in the simplified model
by β. We can conceive of two probabilities then that jointly give rise to this classic β term— the probability
of a given interaction occuring, and the probability of infection over the course of that interaction. In that
sense, β captures the rate of disease spread or what is classically called the contact rate [3].
Scholars have used various SIR-type models with appropriate modifications. The most relevant for our
use is the two-strain SIR model shown below.
3

dS
dt
= −β1SI − β2SI
dI1
dt
= β1SI − γ1I
dI2
dt
= β2SI − γ2I
dR
dt
= γ1I1 + γ2I2
The two-strain model is typically used for modelling two strains of a disease, and an individual can only
be afflicted with one such version. In the model shown above, immunity from one confers immunity to the
other. A version of this two-strain model will be adapted in our problem.
A final note about relevant background literature — one of the more recent trends in theoretical ecology
and the practical use of SIR has been in modeling the spread of social behaviors, memes, and cultural
diffusion.
3 Development of the Climate Change SIR Model
We have seen multi-strain SIR models used to model economic behavior in prior studies. However, my
application of SIR to climate change negotiations would be a novel addition to the literature. To develop
this model, we first consider an overall population of Z countries that can participate in climate negotiations.
They are comprised of four different population classes:
• S: Neutral countries which have yet to agree to GHGE reduction and whose GHGE levels are poor
• I1: Cooperators — Treaty signatories (countries which accept terms) who have pledged to conribute
to an international trust and are taking efforts to reduce annual GHGEs
• I2: Defectors — Countries that cannot/refuse/show no indication of cooperating in near future due
to low resources or moral hazard; GHGE levels of these defectors are steeply worsening.
• R: Countries from I1 that have fulfilled their promise and annual commitment. These are Role
Models.
These population classes refer to a standard cooperative treaty that establishes an international trust that
is dedicated to mitigating climate change effects. Countries by default are neutral towards the treaty. They
continue GHGE emissions at a moderate rate. With a certain probability, they may become cooperators
that take active efforts to be a cooperator, that works at a regional level towards solving the climate change
problem. Or, they may defect and contribute towards exacerbating the climate change problem. The
cooperators set benchmarks for themselves, The below set of differential equations expresses how movement
between the various classes occurs.
dS
dt
= −β1SI1 − β2SI2 + α1I1I2 + α2I1I2 + κI2R
dI1
dt
= β1SI1 − α1I1I2 − γI1
dI2
dt
= β2SI2 − α2I1I2 − κI2R
dR
dt
= γI1
Notice that the total number of countries Z remains constant (sum the differential equations). In this
model, the following parameters are used:
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• β1: the influnce of signatories on neutral countries to sign the treaty.
• β2: the influnce of defectors on neutral countries to join these other defectors.
• α1: the influence of defectors on signatories to renege and take minimal effort towards GHGE mitiga-
tion.
• α2: the influence of signatories on defectors to reconsider their position and join the neutral contingent.
• γ: The rate at which cooperators become role models by virtue of fulfilling their GHGE mitigation
promise.
• κ: The rate at which defectors are influenced by role models to reconsider their position.
Figure 1: Negotiations Model: Bold lines indicate interaction and movement; Dashed lines indicate interac-
tion alone
In Figure 1 you can better visualize how the population classes interact. Blue indicates movement to
cooperators/defectors caused by interaction with the same. Red indicates movement to neutral caused by
interaction between cooperators and defectors (interactions are dotted). Green indicates movement to role
models.
4 Kappa Parameter Variation and Differential Analysis
4.1 Effect of Guilt: Kappa Variation
Recent work by Avinash Dixit and Simon Levin has sought to introduce a prosociality component to economic
game theory. By relating the utility of one group to the benefit of other individuals, it is possible to
skew the Nash equilibrium accordingly. Differential systems do not lend themselves to questions of utility.
Nevertheless, I capture a pseudo-prosociality component in the κ parameter. In this system, κ measures
the guilt and exogenous altruism felt by the defectors of I2 towards the role models who have fulfilled their
promise. To bring focus to this aspect of homophily, I have varied the parameter kappa to find any threshold
outbreaks for which the population as a whole converges on an equilibrium of total cooperation.
As a base case, I have made the following assumptions about the other parameters:
• β1: .03
• β2: .03, since effect of each class on neutral countries is equivalent
• α1: .02, marginal impact on signatories is less than impact on neutral, more malleable countries.
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• α2: .04, anti-climate change is twice as likely to join neutral countries since sanctions/negative effects
from abstaining are significant
• γ: .1, reflects 10 years of effort to meet GHGE goals after signing treaty
With around 200 countries in the world, we assume initial conditions are such that 170 countries are
neutral at the start, 10 are treaty signatories, and 20 are low-resource countries actively defecting. Each
timestep is a year, and under these assumptions, results were found for simulations varying κ over a course
of 50 years. These are shown in Figure 2.
Figure 2: It seems a prosocial guilt value of 0.10 is the threshold for equilibration towards cooperative
accomplishment
To get a better understanding of sensitivity of the model to values of κ, the sensitivity plot shows the
amount of time it takes for the number of role models to hit 80 percent of the population (160 countries)—call
this time τ— as we increase the value of κ. This is shown in Figure 3.
We can see from the figure that τ decreases sharply at a threshold point. After this, the effects of
increasing κ are minimal at best.
4.2 Differential Analysis
Clearly, there is a spike in role models that results from increasing κ — and this pheneomenon is not gradual.
It signals a threshold. The presence of a threshold around this value of κ = 0.06 begs the question of whether
this precise value could have been predicted.
To check, we can run some analysis a la classical SIR inspection. At the equilibrium point, all of the four
differential equations must be equal to zero. In fact, one is redundant since our population remains constant
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Figure 3: A decrease in τ is visible as we gradually increase κ
over time. We focus on the ﬁnal three equations. Notice in the ﬁnal equation for R, we require I1 = 0 at all
equilibria. Then notice our equation for I2:
dI2
dt
= β2SI2 − α2I1I2 − κI2R
dI2
dt
= β2SI2 − κI2R
So either I2 = 0 as well, leaving degrees of freedom for S and R, or—
dI2
dt
= β2S − κR = 0
β2S = κR
S =
κ
β2
R
I2 = Z − (R +
κ
β2
R)
This leaves only one degree of freedom for R. We can see how this coincides with the top-right panel of
Figure 2. There, you see a stable equilibrium where R = 12. We predict S = κ
β2
R = .05
.03 R = 20. This is
borne out by 20 neutrals. Further, I2 is what remains- 167.
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It would be nice to know the conditions that determine stability of this point with positive I2, as that
would tell us more about the threshold level of kappa. Notice though what happens with the Jacobian using
the expressions for S, I2, and R.
JS,I2,R =


a b c
d e f
0 0 0


So stability cannot be determined.
Note bene - a limitation to this model is that it does not account for economic benefits and costs that
accrue to cooperators and defectors. In the next section, I will develop a refined Markovian model that
addresses these defects from an evolutionary game theoretic perspective.
5 Incorporation of Probabilistic Homophily
5.1 Defining Transition Probabilities
The degree to which a country is influenced by another is a measure of its homophily. Until this point, we
have treated homophily as a parameterized constant effect of group interactions. The number of network
interactions between two groups increases the chance of switching due to behavioral influence. As such,
we have assigned arbitrary parameters in the SIR Model. Our goal here is to make the probability at
each timestep more “evidence-based” by incorporating relative fitness. Specifically, we will more precisely
determine a value for α1, α2, β1 and β2 at each timestep of the differential process. We can accomplish this
goal by drawing on evolutionary game theory. Through more rigorous mathematical manipulations—akin to
the approach used by Vasconcelos et al (2014) —- it is possible to have countries imitate one another based
on their relative fitness.
First, however, it is important that we adjust the model developed in the previous sections in order to
make our goal more tractable.
1. We will now consider an infinite population of countries divided into the same four classes: S, I1, I2,
R. S and R serve as sinks in this model, with a finite population I1 + I2 at the center.
2. We define S in this section as S
S+I1+I2+R from the previous section; and so forth for the other classes.
Each variable represents a proportion of the total population of countries.
At a given timestep, each country in each class has a chance of switching into another one. These are
transition probabilities that are dependent on the current proportion of individuals there are in each group.
As an example —- the transition probability can be defined for an individual in group I1 at timestep t. It
is the probability of switching to group S at timestep t + 1, given parameters S, I1, I2 at timestep t. This
probability is denoted as follows:
T
(I1→S)
(S,I1,I2)
This transition probability accurately reflects the probability of a member in I1 switching into S. It
is precisely what we want to capture with parameter alpha1. As mentioned previously, this transition
probability can then be expressed as some function F of relative fitness.
α1 = T
(I1→S)
(S,I1,I2) = F(fI1
− fI2
)
A homophilic country will look to gauge the fitness of employing its own strategy, the fitness of employing
the alternative strategy given the same conditions — and based on the difference in fitness, will switch
strategies with a certain transition probability in the next timestep. Fitness measures the typical, weighted
advantage conferred by employing a given strategy. It is a common term in theoretical ecology and has
direct bearing on this discussion.
In the example above, α1 will increase if the country in class I1 has greater incentive to switch to S.
This occurs if fI1 is lower than fI2 . Equal fitness makes the country indifferent to switching. A lower
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relative fitness increases transition probability, and vice versa for a higher relative fitness. Therefore, we
need a function F such that increasing values of fI1 −fI2 signify a lower probability of I1 members switching
strategies, and a higher probability of I2 members switching strategies. We use the following:
F(x) = (1 + ehx
)−1
This equation allows for homophilic imitation. The parameter h is a measure of the sensitivity of the
transition probabilities to fitness differences. Figure 2 demonstrates this sensitivity for different values of
h.
Figure 4: Homophily parameter h captures sensitivity of transition probability to relative fitness
The challenge now is to develop the idea of fitness. How can the fitness of agents in each class be
determined the players (countries) at each timestep so that they may modify their transition probabilities
accordingly? The model description requires an alteration with Ostrom’s message in mind. Previously,
movement between classes took place at each timestep. Now, at each timestep, we consider that cooperators
are taking efforts to mitigate GHGEs. Their efforts go towards solving two types of problems- a local problem
and a zonal or regional problem. To capture the effects of their strategy of cooperation, we will consider the
games they play at each timestep.
Before movement takes place, the countries in I1 and I2 are randomly placed into groups of size n, and
all these groups play an economic game wherein the objective is to solve a regional climate change problem
(e.g. pollution). Each country has its own local climate change problem, as well, and reaping local benefits
depend on solving this zonal problem. Each country starts off with a local endowment c, and they have the
option of contributing a fraction b of c to their regional trust. Indeed, contribution of this fraction is the
strategy taken by agents in class I1. The defectors in class I2 will contribute nothing. If by the end of the
9

game i ci > M, where M is a threshold amount of contribution, a regional catastrophe is averted. The
countries can enjoy their local endowments. If this threshold is not met, however, the the regional problem
is not solved, and so the local problem is not mitigated. In that timestep (or that year, for our purposes),
this given group of n countries failed to solve their regional problem and with risk r their work will be for
naught. So they will try next year, after gauging the payoffs of all the Z countries globally, to see how they
may improve their strategy. The cooperators of I1 and defectors of I2 will receive payoffs according to this
schematic:
Figure 5: The payoff matrix for I1 and I2 in the negotiation game
One may wonder why the R and S groups are not playing this game. The R countries are ones in I1
whose cooperative efforts were, by chance, sufficient at the local level to ensure their durability in the long
run. They have no impact on climate change negotiations because they have played their part. The countries
in S have no bearing on solving the problem because they have negligible effect on climate change— they
neither augment nor assuage the problem. Just as R is a sink, S is a source (and sink) for the cooperators
and defectors. We can arbitrarily assume the fitness of S and R are zero for the purposes of the game
depicted above.
Assume that in a group of n countries, there are j1 cooperators and j2 = n − j1 defectors. Now we
can write the payoff of each defector in a group of j1 cooperators using the Heaviside function based on the
schematic:
PI2
(j1) = b(H(j1 − M) + (1 − r)(1 − H(j1 − M)))
where
H(n) =
1, n ≥ 0
0, n < 0
A cooperator will always lose bc, so payoff for a cooperator:
PI1
(j1) = b(H(j1 − M) + (1 − r)(1 − H(j1 − M))) − bc
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With these payoffs defined, we can finally develop an expression for fI1 and fI2 . Consider that in their
group of n players, cooperator in question might find itself in a group with any number j1 of cooperators
(including itself). Because only the number of cooperators, who all behave identically, affects the outcome,
we can express the fitness as the weighted sum of all the possible payoffs for the cooperator:
fI1 =
games
QI1 (games)PI1 (games)
where Q is the probability of each game, and PI1
is the payoff to the cooperator in that given game. Each
game only depends on the number of cooperators j1 based on the payoff described- since all other parameters
b, r, and M are set. This payoff is already given. Then accordingly, we write the binomial probability of
each j1, given at least one member of I1 is in the group, as QI1 (j1) — see below:
QI1
(j1) =
n − 1
j1 − 1
(
I1
I1 + I2
)j1−1
(
I2
I1 + I2
)n−j1
fI1
=
n
j1=1
n − 1
j1 − 1
(
I1
I1 + I2
)j1−1
(
I2
I1 + I2
)n−j1
× b(H(j1 − M) + (1 − r)(1 − H(j1 − M))) − bc
We can apply a similar reasoning to find the expression for fI2
. We just keep in mind that the only
condition is that we are selecting from a group of n − 1 playing the game (apart from the defector), and
there may be as few as 0 cooperators or as many as n − 1 cooperators:
fI2
=
games
QI2
(games)PI2
(games)
fI2
=
n−1
j1=0
n − 1
j1
(
I1
I1 + I2
)j1
(
I2
I1 + I2
)n−j1−1
× b(H(j1 − M) + (1 − r)(1 − H(j1 − M)))
Just as we defined α1, we can define α2:
α2 = T
(I2→S)
(S,I1,I2) = F(fI2 − fI1 )
It is a property of the function we defined, that:
F(−x) = 1 − F(x)
Therefore,
α2 = T
(I2→S)
(S,I1,I2) = 1 − α1
Given that we defined parameter neutral fitness parameter fS = 0, we can define a few of the other
parameters, as well.
β1 = T
(S→I1)
(S,I1,I2) = F(fS − fI1
) = F(−fI1
) = 1 − F(fI1
)
β2 = T
(S→I2)
(S,I1,I2) = F(fS − fI2
) = F(−fI2
) = 1 − F(fI2
)
Since we will leave guilt parameter κ as an adjustable parameter, we have only γ pending. Here we
refer to the probability that the signatory wound up in a group which met its threshold. In that event, the
cooperator has not only solved the regional problem, but because of its efforts, solved its local problem. The
defector’s lack of effort did nothing to solve the problem, but also handicapped it in solving its local goals—
so it will most certainly not reach R. We can characterize the probability the cooperator ends up in such a
successful group:
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γ =
n
j1=M
n − 1
j1 − 1
(
I1
I1 + I2
)j1−1
(
I2
I1 + I2
)n−j1
This concludes the set-up of the model. The next section will show how simplification can make for
a more tractable parameterization that can abet the practical functionality of my current two-strain SIR
model.
5.2 Analysis and Simplification
Our goal is to simplify the expression for transition probability. The first step would be rewriting the sum
for fI1
such that it has the same limits as fI2
:
fI1 =
n
j1=1
n − 1
j1 − 1
(
I1
I1 + I2
)j1−1
(
I2
I1 + I2
)n−j1
× b(H(j1 − M) + (1 − r)(1 − H(j1 − M))) − bc
fI1
=
n−1
j=0
n − 1
j
(
I1
I1 + I2
)j
(
I2
I1 + I2
)n−j−1
× b(H(j + 1 − M) + (1 − r)(1 − H(j + 1 − M))) − bc
fI1
=
n−1
j=0
n − 1
j
(
I1
I1 + I2
)j
(
I2
I1 + I2
)n−j−1
× PI1
(j + 1)
Note that the probability portion of this is now the same expression as we have for QI2
. I’ll simplify the
expression for payoffs.
PI2
(j1) = b(H + (1 − r)(1 − H))
PI2 (j1) = b(H + 1 − H − r + rH)
PI2 (j1) = b(1 − r(1 − H(j1 − M)))
PI2 (j1) = b − br(1 − H(j1 − M))
Can we simplify the parenthetical?
H(j1 − M) =
1, j1 − M ≥ 0
0, j1 − M < 0
−H(j1 − M) =
−1, j1 − M ≥ 0
0, j1 − M < 0
1 − H(j1 − M) =
0, j1 − M ≥ 0
1, j1 − M < 0
1 − H(j1 − M) =
0, M − j1 ≤ 0
1, M − j1 > 0
1 − H(j1 − M) = H(M − j1 − 1)
So:
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PI2 (j1) = b − br(H(M − j1 − 1))
PI1 (j1) = PI1 (j + 1) = b − bc − br(H(M − j − 2))
We can now substitute these in our expressions for fI1
and fI2
. In doing so, let’s also simplify the
binomial expression:
fI1 =
n−1
j=0
n − 1
j
(
(I1)j
(I2)n−j−1
(I1 + I2)n−1
× PI1 (j + 1)
fI2 =
n−1
j1=0
n − 1
j1
(
(I1)j1
(I2)n−j1−1
(I1 + I2)n−1
× PI2 (j1)
Let’s look at the payoff cases now.
PI2
(j1) =
b − br, j1 ≤ M − 1
b, j1 > M − 1
PI1
(j + 1) =
b − bc − br, j ≤ M − 2
b − bc, j > M − 2
So then, we can find the term fI1
− fI2
by looking at the difference in payoffs, since the binomial terms
are equivalent:
PI1
− PI2
=



−bc, j ≥ M
br − bc, j = M − 1
−bc, j = M − 2
−bc, j ≤ M − 3
PI1
− PI2
= −bc + brδj(M+1)
fI1
− fI2
=
n−1
j1=0
n − 1
j1
(I1)j1
(I2)n−j1−1
(I1 + I2)n−1
(−bc + brδj1(M−1))
fI1 − fI2 = −bc + br
n−1
j1=0
n − 1
j1
Ij1
1 In−j1−1
2
(I1 + I2)n−1
δj1(M−1)
fI1
− fI2
= −bc + br
n − 1
M − 1
IM−1
1 In−M
2
(I1 + I2)n−1
We can use this simplified difference to compute the transition probability α1.
6 Conclusions and Future Direction
It is my sincere hope that future scholars will be able to use these findings to model homophilic interactions
using SIR techniques. By simplifying the expressions for transition probabilities, it should be possible to
acquire a more nuanced view of how negotiations may play out. Tweaking parameters such as the initial
endowment, cost of contribution c, and threshold level for catastrophe aversion M may yield some interesting
results. It will be good to carry out stability analysis on the more complicated development of this model.
This is beyond the scope of this paper, however.
I have laid the groundwork for policy-makers to rely on applied mathematical techniques to better develop
and craft significant decisions regarding climate change. As established in the first section, we are well past
13

the point of describing climate change as a problem of the future— the future is now. The steps global
actors take in the next 50 years will have a profound inﬂuence on the direction our planet takes and whether
we can leave something tangible behind for our children. Aligning incentives appropriately for long-term
success depends on modelling how that success can be brought about given present conditions. Applied
mathematics and computational analysis will be crucial to getting at the answers we need in order to avert
GHGE-induced climatic catastrophe.
7 Computational Appendix
7.1 Figure 1 Materials
Deﬁning the SIR Climate Negotiations Model
% Climate Change Negotiations- Multistrain SIR Model
function dx =climate(t,x, betaone, betatwo, alphaone, alphatwo,...
gamma, kappa)
%INITIALIZING:
dx=zeros(4,1);
%DYNAMIC EQUATIONS:
dx(1)= -betaone*x(1)*x(2) - betatwo*x(1)*x(3)+ alphaone*x(2)*x(3)...
+ alphatwo*x(2)*x(3) + kappa*x(3)*x(4);
dx(2)= betaone*x(1)*x(2) - alphaone*x(2)*x(3) - x(2)*gamma;
dx(3)= betatwo*x(1)*x(3) - alphatwo*x(2)*x(3) - kappa*x(3)*x(4);
dx(4) = x(2)*gamma ;
end
Prosociality Variation
%%DEMONSTRATED EFFECTS OF PROSOCIALITY
%Kappa of .0
clear
betaone= .03;
betatwo = .03;
alphaone= .02;
alphatwo= .04;
gamma= .1;
kappa= 0;
[t,x]=ode45(@(t,x) climate(t,x, betaone, betatwo, alphaone, alphatwo,...
gamma, kappa),[0 50],[170 10 20 0]);
subplot(2,2,1)
14

plot(t,x(:,1),'-g','LineWidth',1.5)
hold on
plot(t,x(:,2),'-b','LineWidth',1.5)
hold on
plot(t,x(:,3),'-r','LineWidth',1.5)
hold on
plot(t,x(:,4),'-m','LineWidth',1.5)
legend('Neutral','Signatories', 'Defectors', 'Role Models')
xlabel('Time')
ylabel('Population Size')
ylim([0 220])
title('No Prosociality')
hold off
shg
%Kappa of .05
clear
betaone= .03;
betatwo = .03;
alphaone= .02;
alphatwo= .04;
gamma= .1;
kappa= .05;
gamma, kappa),[0 50],[170 10 20 0]);
subplot(2,2,2)
hold on
hold on
hold on
xlabel('Time')
ylim([0 220])
title('Prosociality = .05')
hold off
shg
%Kappa of .1
clear
betaone= .03;
betatwo = .03;
alphaone= .02;
alphatwo= .04;
gamma= .1;
kappa= .1;
gamma, kappa),[0 50],[170 10 20 0]);
subplot(2,2,3)
hold on
hold on
hold on
15

xlabel('Time')
ylim([0 220])
hold off
shg
%Kappa of .15
clear
betaone= .03;
betatwo = .03;
alphaone= .02;
alphatwo= .04;
gamma= .1;
kappa= .15;
gamma, kappa),[0 50],[170 10 20 0]);
subplot(2,2,4)
hold on
hold on
hold on
xlabel('Time')
ylim([0 220])
hold off
shg
Tau Sensitivity to Kappa
%%DEMONSTRATED EFFECTS OF KAPPA SENSITIVITY OF TAU
clear
betaone= .03;
betatwo = .03;
alphaone= .02;
alphatwo= .04;
gamma= .1;
y = 200*ones(1, 100);
index = 1;
for kappa = (.01:.01:1)
[t,x]=ode45(@(t,x) Climate(t,x, betaone, betatwo, alphaone, alphatwo,...
gamma, kappa),[0 50],[170 10 20 0]);
if x(end) > 160
i=1;
y(index)=200;
while x(i,4) < 160
i=i+1;
16

y(index) = t(i);
end
end
index = index + 1;
end
plot((.01:.01:1), y, '-b', 'Linewidth', 1.5)
xlabel('Value of Kappa')
ylabel('Tau')
ylim([16 20])
xlim([.05 1.00])
shg
Homophily Deﬁnition
function y = Homophily(x, beta)
y = (1 + exp(beta*x)).ˆ-1
end
Variation of Homophily Parameters
%Higher values of beta increase sensitivity of transition probability
clear
x = -10:.1:10;
subplot(2,2,1)
beta = .5;
plot(x, Homophily(x, beta))
xlabel('Relative Fitness ')
ylabel('Transition Probability')
title('h = 0.5')
subplot(2,2,2)
beta = 1;
title('h= 1.0')
subplot(2,2,3)
beta = 2;
title('h = 2.0')
subplot(2,2,4)
beta = 5;
title('h = 5.0')
17

References
[1] United States Environment Protection Agency Climate Change: Basic Information.
http://www.epa.gov/climatechange/basics
[2] Scientiﬁc Consensus on anthropogenic climate change
http://www.sciencedaily.com/releases/2013/05/130515203048.htm
[3] Hastings, Alan. Population Biology: Concepts and Models Springer.
[4] Ostrom, Elinor. (2009). A General Framework for Analyzing Sustainability of Social-Ecological Systems
Science 325 (5939): 419–422. (http://www.sciencemag.org/content/325/5939/419.full.pdf)
[5] Ostrom, Elinor (1990). Governing the Commons: The Evolution of Institutions for Collective Action.
Cambridge University Press.
[6] Poteete, Janssen, and Elinor Ostrom (2010). Working Together: Collective Action, the Commons, and
Multiple Methods in Practice. Princeton University Press.
[7] Dixit, A., Governance Institutions and Economic Activity. American Economic Review 2009, 99 1, 5-24
[8] Santos, F.C., Pacheco, J.M. (2011) Risk of collective failure provides an escape from the tragedy of the
commons. Proc Natl Acad Sci USA 108 (26) 10421-10425
18

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