Hi My topic is The Interplay between anti-smuggling measures taken by two collaborative
countries.
Let’s have a brief review.
The goal of this project is to find the Nash Equilibrium Strategy of two countries troubled by
smuggling. The smuggling that we are talking about here is transnational goods smuggling, which
is the most common case in all kinds of smugglings.
The model established in this project is a simple model that contains two players, country H and
country F. In this simple model, I assume them to move simultaneously.
Now I will introduce the model established in this project.
Besides two countries H and F, I assume there is a group of smugglers called T, and once T
succeed in a smuggling, there would be certain damage to H and F, I denote them as TH and TF.
Here Mh and MF represents the level of anti-smuggling measures taken by H and F. Intuitively,
We know that TH and TF could be reduced by anti-smuggling measures, which means they have
the following first-order partials.
The expression in (1) and (2) indicates that level of anti-smuggling measures would decrease
damages at a diminishing rate. Also level of anti-smuggling measures is public good which means
anti-smuggling measures taken by either H or F would benefit the other country as well. As a
public good, anti-smuggling measures imply strong free-rider incentives.
Next I try to formulate the damage function which is denoted by TH and TF.
First of all, there are several constrains on TH and TF as I list here.
H
F
a. Each has two variables m and m .
b. Must satisfy expression (1) and (2) ,which means level of anti-smuggling measures
would decrease damages at a diminishing rate.
c.
They should Reveal the property of free-rider incentives.
d.
T H , T F should have similar expressions.
e.
m H would be more important in T H
f.
m F would be more important in T F
The following damage function captures these properties:
T H  T (1  m H )(1  m F )  T (1  m H );
T F  T (1  m H )(1  m F )   T (1  m F );
We can see that both have two varibles, both satisfy expression (1) and (2) , also we could
find that these two expressions could reveal the property of public good which means TF benefits
from a big value of MH and TH benefits from a big value of MF. Here αand βare two important
parameters in this model. I regard αT and βT as an extra damage to country H and F. In fact,
αand βallow us to describe the differerence of the damage to country H and country F. For
example, considering the smuggling between Mexico and America, assume America to be H and
Mexico to be F, then we could set αto be big and set βto be relatively small.
H
If country H faces a constant marginal cost c for its defensive measures, then its total loss
V H is given by TH plus the cost of defensive resource times the defensive levels taken by H,
which is showed here
V H  T H  c H mH  T (1  m H )(1  m F )  T (1  m H )  c H m H
Similarly, we would have the expression of VH
V F  T F  c F mF  T (1  mH )(1  mF )   T (1  m F )  c F m F
Now we already have the total loss function and what we need is the Nash Equilibrium.
H*
F*
H
F
Denote m and m be the best response respectively to V , V .
H*
F*
In order to find the Nash Equilibrium, we want to get the intersection of m and m .
Following are some graphs of different value of  ,  .
There are four possible cases.
As we can see in figure 1, there is only one NE existed that is (𝑚𝐹∗ , 𝑚𝐻∗ )=(1,0)
In case 2, there are three NE existed
In case 3, there is only one NE.
In case 4, there is only one NE.
So the total Nash Equilibrium could be written as following.
Now assume the marginal cost of defensive resource CH and CF is known by both countries,
and we assume them to be a given value, then αand βwould be the key factors that influence
the Nash Equilibrium. So the last question would be how to quantify α and β.
Let’s see an extreme case,  ,  <<1, which means the damage to these two countries
could be regarded as equal. Then let’s look back into the Nash Equilibriums.
In case i),
we could see that

cH
cF
F
,which means c <<T. Thus Country F would choose to
, 
T
T
take a full defensive level which would make T
F
 T (1  mH )(1  mF )   T (1  mF ) =0 and
Country H , on the other hand, would like to take a free ride. The final loss of these two countries
would be
V H  T ,V F  c F
In case ii),

cH
cF
F
H
, which means c and c couldn’t be ignored. In this case, there would be
, 
T
T
three NE as listed.
In case iii), we can see that CH is far less than T and CF is far less than T. In this case, the Nash
Equilibrium would be (1,1) which means both countries would prefer to take a full level of
anti-smuggling measures. And their total loss would be
V F = c F , V H = c H , which is just the
cost of defensive resources.
In case iv).
This case is symmetric to case I and we could find that ( m
F*
,m
H*
)= ( 0,1) is also symmetric to
that in case i). ( V , V )=(  T , c )
F
H
H
As we could see in these four cases, this model represents an intention that a country with very
low cost of defensive resource would like to take a full level of anti-smuggling measures and let
the other one take the free-ride. On the other hand , when the cost is not that low, there would
be three Nash Equilibriums. Again, these conclusions are under the extreme cases that α，βfar
less than 1.
So the future work is to find a way to quantify αand β, and thus this model could be applied
into the real problem.