Game Theoretical Models of
Effort and Lobbying in a
Heterogeneous CPR Setting
Matthew Freeman
Louisiana State University/LA Sea Grant
Chris Anderson
University of Rhode Island
Effort & Lobbying
• Extraction from a common pool resource
(CPR), such as a fishery, can lead to socially
inefficient and undesirable outcomes.
• Through regulation of the CPR, users may
achieve a more profitable and socially efficient
outcome.
• However, assumptions made by traditional
CPR models provide an incomplete framework
to guide fishery policy.
Research Question
• Motivating Question
– As the assumptions that CPR users (1) are
homogeneous and (2) are unable to create, or
influence, management of the resource are
relaxed, what do theoretical models predict?
– In particular, we examine heterogeneity in the cost
function of CPR users and the influence of CPR
users on proposed regulation of the resource.
Outline of Model Presentations
• We present the game-theoretical models and
Nash equilibrium strategies so that we build
upon the preceding model by either
incorporating an additional element or
displaying a unique case.
• In doing so, we solve for an unregulated CPR,
several cases for a regulated CPR, and several
cases in which lobbying occurs in a regulated
CPR.
Theoretical Symbols & Equations
B( xi )  axi  bx
2
i
C( xi , X i )  (   X ) xi
NA + NB = N
X-i = total fishing effort all but the i th firm
x iA , x iB = effort of an individual from
Group A, from Group B
 A ,  B = externality cost experienced by
Group A fishermen, Group B fishermen
(1) Unregulated CPR Model
Model Details
• When δA>δB, the Nash equilibrium choice of individuals
in Group A is < that of individuals in Group B.
• Group B’s aggregate effort, however, could be >, <, or =
to that of Group A’s, dependent on the number of
individuals in each group.
• Interpreting δA>δB as the difference in the externality
cost experienced by small and large vessel operators,
we could expect large vessel operators to exert more
effort than small vessel operators in a CPR, based on
the Nash equilibrium predictions.
(2) Regulated Models
• Next, our model incorporates regulation to
limit the appropriation level of individuals,
and uses one of the allocation rules discussed
by Hackett (1992), whereby appropriation
rights are equally divided.
• For fisheries, this could take the form of equal
share quotas, or daily or weekly trip limits.
(2A) Regulated CPR Model –
Industry Social Optimum
Model Details
• In the solution, we observe in the
denominator both the number of individuals
in each group and the externality cost
experienced by them.
• As a result, if either an externality cost
increases or a group size increases, the socially
optimal regulation decreases.
(2B) Regulated CPR Model –
Nash Equilibrium
• Both groups of individuals have the
opportunity to choose their Nash equilibrium
effort levels when x  x , and both groups of
individuals would choose x when x  x .
• Thus, we are left to determine the effort levels
that occur when x < x < x .
*
iB
*
iA
*
iA
*
iB
Regulated CPR Model –
Nash Equilibrium
• However, there’s a level of effort where individuals
from Group A become bound by the constraint x ,
and must select the cap for an effort level.
Regulated CPR Model –
Nash Equilibrium
Regulated CPR Model –
Nash Equilibrium
• We have a unique solution wherein Group A
individuals are exceeding their unregulated
Nash equilibrium, since Group B individuals
are prevented from selecting their
unregulated NE.
• (The best response function of Group A
individuals has been altered due to the
constraint of the regulation on Group B
individuals.)
(2C) Regulated CPR Model –
Group Preferred Regulation
• This model becomes useful in our next step –
where CPR users lobby according to their
group preferred regulation.
(3) Regulated CPR Model –
Non-Cooperative Lobbying
• This model requires backward induction by
the CPR users: CPR users must first determine
the optimal final regulation and then
determine the level of lobbying required,
given the proposed regulation.
Regulated CPR Model –
Non-Cooperative Lobbying
• Lobbying expenditures to increase a proposed regulation are
as follows:
This is the
This helps to
group
determine the
preferred
“lower bound”
regulation that
for profitable
was
lobbying.
previously
identified.
• When δA > δB, we note (1) the preferred regulation for Group
A is smaller than for Group B and (2) the region about the
preferred regulation in which lobbying is unprofitable is
smaller for Group A than for Group B.
Regulated CPR Model –
Non-Cooperative Lobbying
• Based on where the ‘lower limit’ of profitable lobbying occurs,
one group’s NE may be to not spend any money lobbying & to
free-ride on the other group’s activities.
(3) Regulated CPR Model –
Non-Cooperative Lobbying
• To decrease a proposed regulation, we simply
examine only ‘negative’ lobbying
expenditures, whereas before we examined
‘positive’ lobbying expenditures in the
constraint.
Regulated CPR Model –
Non-Cooperative Lobbying
• Lobbying expenditures to decrease a proposed regulation are
as follows:
This sign
changed
compared to the
previous
equation, shows
the “upper
bound”.
• A boundary does exist where an individual would NOT want to
lobby using these equations. So, an individual would want to
compare the profit earned at his respective upper bound
(minus any lobbying expenditures) to that earned by selecting
the NE.
Regulated CPR Model –
Non-Cooperative Lobbying
• So, we’ve now identified the ranges wherein non-cooperative
lobbying could occur.
Regulated CPR Model –
Cooperative Lobbying
• Lobbying expenditures to increase a proposed regulation are
This now has
as follows:
group size in the
denominator,
which causes
the boundaries
to be smaller
compared to the
nc model.
• Below is a comparison of the cooperative & nc boundaries.
Regulated CPR Model –
Cooperative Lobbying
• Lobbying expenditures to decrease a proposed regulation are
as follows:
This sign
changed
compared to the
previous
equation, shows
the “upper
bound”.
• As before, a boundary exists where a group would NOT want
to lobby using these equations. A group would compare the
group profit earned at their respective upper bound (minus
the group’s lobbying expenditures) to that earned by selecting
the NE. This provides an interesting case where cooperative
behavior fails, mainly as a result of the cost of lobbying.
Summary
• (1) In an unregulated setting, individuals have no
incentive to deviate from the Nash equilibrium in our
model, (tragedy of the commons).
• (2) In a regulated setting, individuals may be bound
by the regulation and no longer able to extract from
the resource according to the unregulated Nash
equilibrium.
• (3) There is a range about a group’s social optimum
where lobbying is no longer profitable, which gets
smaller when groups act cooperatively.
Concluding Remarks
• Regarding resource sustainability, we observe
two positive outcomes with regards to
lobbying:
• (1) Individuals may lobby for a stronger
regulation on resource use.
• (2) If individuals act non-cooperatively, the
region increases in which lobbying is no longer
profitable; this may prove useful when
individuals would prefer a weaker regulation.
Questions?
We gratefully acknowledge the support for this research
provided by a NOAA/Sea Grant Joint Fellowship in Marine
Resource Economics and by RI Sea Grant.