The Interplay Between Anti-Smuggling
Measures by Two Collaborative Countries
By
Zhao Wu
Department of Industrial Engineering
University At Buffalo
Brief Review
Transnational Goods Smuggling
Simple Model:
Two Players
Move Simultaneously
Model Preliminaries
T
H
𝑇 𝐻 = 𝑇(𝑚𝐻 , 𝑚𝐹
F
𝑇 𝐹 = 𝑇(𝑚𝐹 , 𝑚𝐻
Model Preliminaries
𝜕𝑇 𝐻
𝜕𝑇 𝐻
< 0,
<0
𝐻
𝐹
𝜕𝑚
𝜕𝑚
(1)
𝜕𝑇 𝐹
𝜕𝑇 𝐹
< 0,
<0
𝐻
𝐹
𝜕𝑚
𝜕𝑚
(2)
Strong Free-Rider Incentives!
Formulate the Damage Function
The damage function must have following properties:
(a)Each has two variables 𝑚𝐻 and 𝑚𝐹 .
(b)Must satisfy expression (1) and (2) ,which means level of antismuggling measures would decrease damages at a diminishing
rate.
(c)Reveal the property of free-rider incentives.
(d)The two damage funtion 𝑇 𝐻 and 𝑇 𝐹 should have similar
expressions.
(e) 𝑚𝐻 would be more important in 𝑇 𝐻 .
(f ) 𝑚𝐹 would be more important in 𝑇 𝐹 .
Formulate the Damage Function
𝑇 𝐻 = 𝑇(1 − 𝑚𝐻 (1 − 𝑚𝐹 + 𝛼𝑇(1 − 𝑚𝐻 ;
𝑇 𝐹 = 𝑇(1 − 𝑚𝐻 (1 − 𝑚𝐹 + 𝛽𝑇(1 − 𝑚𝐹 ;
𝜶𝑻: extra damage other than the common
damage T to H
𝜷𝑻:extra damage other than the common
damage T to F.
The Total Loss
If country H faces a constant marginal cost for its defensive
measures, then its total loss is given by:
𝑉 𝐻 = 𝑇 𝐻 + 𝑐 𝐻 𝑚𝐻 = 𝑇(1 − 𝑚𝐻 (1 − 𝑚𝐹 + 𝛼𝑇(1 − 𝑚𝐻 +
𝑐 𝐻 𝑚𝐻
Similarly, if country F faces a constant marginal cost for its defensive
measures,
𝑉 𝐹 = 𝑇 𝐹 + 𝑐 𝐹 𝑚𝐹 = 𝑇(1 − 𝑚𝐻 (1 − 𝑚𝐹 + 𝛽𝑇(1 − 𝑚𝐹 + 𝑐 𝐹 𝑚𝐹
Nash Equilibrium
𝑚𝐻∗ = argmin(𝑉 𝐻 = argmin (𝑇 − 𝑇𝑚𝐻 − 𝑇𝑚𝐹 + 𝑇𝑚𝐻 𝑚𝐹 + 𝛼𝑇 −
𝑚𝐻∗ ∈(0,1
𝑚𝐻∗ ∈(0,1
𝑐𝐻
−
𝑇
𝑐𝐹
−
𝑇
Figure 1 . 1 + 𝛼
< 1,1 + 𝛽
>1
As shown in the graph, there is only one NE existed that is (𝑚𝐹∗ ,
𝑚𝐻∗ )=(1,0)
𝑐𝐻
𝑇
𝑐𝐹
𝑇
Figure 2 . 1 + 𝛼 − < 1,1 + 𝛽 − < 1.As shown in the graph,
there are three NE existed that is (𝑚𝐹∗ 𝑚𝐻∗ , )=(1,0) or (0,1) or
(1 + 𝛼 −
𝑐𝐻
𝑇
,1+𝛽−
𝑐𝐹
)
𝑇
Figure 3 . 1 + 𝛼
𝑐𝐻
−
𝑇
> 1,1 + 𝛽
𝑐𝐹
−
𝑇
> 1.As shown in the graph,
there is only one NE existed that is (𝑚𝐹∗ , 𝑚𝐻∗ )= (1, 1)
Figure 4 . 1 + 𝛼 −
𝑐𝐻
𝑇
> 1,1 + 𝛽
𝑐𝐹
−
𝑇
< 1.As shown in the graph,
there is only one NE existed that is (𝑚𝐹∗ , 𝑚𝐻∗ )= ( 0,1)
Analyze Nash Equilibrium
𝒄𝑯
i).If 𝜶 < , 𝜷
𝑻
𝐹∗
𝐻∗
𝒄𝑭
𝑻
>
(𝑚 , 𝑚 )=(1,0)
𝒄𝑯
𝒄𝑭
ii).If 𝜶 < , 𝜷 <
𝑻
𝑻
(𝑚𝐹∗ , 𝑚𝐻∗ )=(1,0) or
𝒄𝑯
> ,𝜷
𝑻
𝐻∗
iii).If 𝜶
(𝑚𝐹∗ , 𝑚
𝒄𝑭
𝑻
>
)= (1,1)
𝒄𝑯
𝒄𝑭
iv).If 𝜶 > , 𝜷 <
𝑻
𝑻
( 𝑚𝐹∗ , 𝑚𝐻∗ )= ( 0,1)
(0,1) or(1 + 𝛼 −
𝑐𝐻
𝑇
,1+𝛽
𝑐𝐹
− )
𝑇
Analyze Nash Equilibrium
Extreme Case of α and β：
α<<1，β<<1
𝑐𝐻
𝑐𝐹
i).𝛼 < 𝑇 , 𝛽 > 𝑇
(𝑚𝐹∗ , 𝑚𝐻∗ )=(1,0)
ii).𝜶 <
𝒄𝑯
,𝜷
𝑻
<
𝒄𝑭
𝑻
→ 𝑐 𝐹 <<T
→ 𝑇 𝐹 = 𝑇(1 − 𝑚𝐻 (1 − 𝑚𝐹 + 𝛽𝑇(1 − 𝑚𝐹 =0
𝑉 𝐻 = 𝛼𝑇, 𝑉 𝐹 = 𝑐 𝐹
Analyze Nash Equilibrium
Extreme Case of α and β：
α<<1，β<<1
𝒄𝑯
iii).𝜶 > , 𝜷
𝑻
𝐹∗
𝐻∗
(𝑚 , 𝑚 )=
𝒄𝑭
𝑻
> → 𝑐 𝐹 <<T,𝑐 𝐻 <<T
(1,1) → (𝑉 𝐹 ,𝑉 𝐻 )=(𝑐 𝐹 , 𝑐 𝐻 )
𝒄𝑯
𝒄𝑭
iv).𝜶 > 𝑻 , 𝜷 < 𝑻
( 𝑚𝐹∗ , 𝑚𝐻∗ )= ( 0,1)
(𝑉 𝐹 , 𝑉 𝐻 )=(𝛽𝑇, 𝑐 𝐻 )
Conclusion
1. Under the extreme case that 𝛼, 𝛽<<1, 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 the other one would like to
take a free-ride.
2. Under the extreme case that 𝛼, 𝛽<<1, this model represents that if the
cost of defensive resource of both countries is not that low then there
would be three NE.
Future Work
Find a Way to
Quantify 𝜶, 𝜷
Q&A