Beach Nourishment with Strategic Interaction between Adjacent Communities: A Game Theoretic Model

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Beach Nourishment with Strategic Interaction between Adjacent Communities: A Game Theoretic Model

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Title: Beach Nourishment with Strategic Interaction between Adjacent Communities: A Game Theoretic Model


1
Beach Nourishment with Strategic Interaction
between Adjacent Communities A Game Theoretic
Model
  • Camp Resources XV
  • August 7-8, 2008
  • Sathya Gopalakrishnan
  • Martin Smith
  • Nicholas School of the Environment Earth
    Sciences
  • Duke University

2
MOTIVATION
  • Dynamic Coastal Environment
  • Sea Level Rise
  • Changing Storm Patterns
  • Gradual Landward movement of shoreline
  • 90 Sandy beaches in the US face erosion

Source www.noaanews.noaa.gov/stories2006/s2622.
htm
(Leatherman, 1993)
3
MOTIVATION
  • Growing development in coastal towns that thrive
    on revenue from tourism
  • Policy intervention to protect property and
    infrastructure
  • Beach Nourishment Process of artificially
    re-building the beach by replacing lost sand

Source www.darylboyette.com
Source www.ncdot.org/doh/preconstruct/highway/pho
to/...
4
Factors influencing Coastal Change
Sea Level Rise ( x100)
earthtrends.wri.org
Alongshore Sediment Transport Max transport ( 45
degrees)
Human Manipulation
Photo credit Program of the Study of Developed
Shorelines, Western Carolina University
5
Simplified Dynamics of Beach Erosion and
Nourishment
  • Beach Nourishment
  • Process of artificially re-building the beach by
    replacing lost sand periodically with sand from
    off-shore locations

From Smith, Slott, and Murray, 2007 (in review)
6
Beach Nourishment analogous to Rotational Harvest
in Forestry
  • Benefits from maintaining beach width
    Recreational flow Storm Protection
  • High fixed cost of nourishment
  • Coastal dynamics influencing erosion rate
  • Rotational/Periodic Nourishment

Faustmann problem applied in reverse
7
THE MODEL Single Community
For each nourishment interval T
Net Benefits NB(T)
Benefits B(T)
Costs C(T)

-
Fixed cost
Variable cost
Discount rate
Width
Community chooses a series of nourishment
intervals
Infinite horizon problem (assuming stationarity)
Community chooses Optimal T
8
If there are two communities
  • How does strategic interaction between
    adjacent coastal communities affect volume and
    frequency of re-nourishment?
  • What are the policy implication of physical
    and economic interdependency?
  • Non-cooperative vs Cooperative solutions?
  • Can we estimate welfare effects?
  • Does North Carolina data support the model?

9
Spatial interaction between adjacent communities
Adapted (modified) from Smith, Slott, and Murray,
2007 (in review)
10
State Equations for Beach Width
Narrow Beach
Sediment Transfer from Beach B
Uniform Retreat
Exponential Retreat
Diffusion Parameter
Wide Beach
Portion experiencing exponential retreat In the
absence of Beach A
Slower erosion due to Beach A
Sediment Transfer to Beach A
11
Beach Width with No Spatial Interaction (K
0)10-year Nourishment Interval
Initial Width Beach A 100 Beach B 200
Baseline erosion 2 ft/yr Exponential retreat
rate for nourished beach 0.10 Portion of Be
ach A facing exponential retreat 0.35 Portio
n of Beach B facing exponential retreat 0.675

12
Beach Width with Spatial Interaction (K
0.5)10-year Nourishment Interval
Increase in K (0.5)
Slower in erosion at both A and B
Beach Width (Feet)
Change in curvature
Concave Function for A
Time (Years)
13
Beach Width with Higher Diffusion (K
1)Nourishment Interval at Wider Beach 10 years
Beach B nourishes every 10 years
Beach A experiences accretion
Beach Width (Feet)
Beach B
No Nourishment at A Positive externality
Beach A
Similar relation between rich and poor
communities?
Time(Years)
14
Strategic interaction in a 2-player game
  • PLAYERS Community A and Community B
    (Coastal Managers)
  • ACTION Nourishment Interval T ? 0,
    Tmax
  • PAYOFFS Present Value Net Benefits NB(T)
    f(x(T))
  • INFORMATION Full Information Game

Net Benefits for i
Beach Width at i
Tj
Ti
15
To solve for the optimal nourishment interval
  • Beach A Find Optimal TA given TB
  • Beach B Find Optimal TB given TA

TA f(TB)
As Reaction / Best Response Function
Bs Reaction / Best Response Function
TB f(TA)
Mutual Best Responses / Intersection of reaction
functions
NASH EQUILIBRIUM
16
Preliminary Results To solve for Optimal
Rotation
Kinky Reaction Functions
Not Unique Nash Equilibrium
Non-autonomous solution path?
17
Preliminary Results
  • Non-cooperative vs Cooperative solutions
  • Maximize of joint net benefits
  • Optimal Rotation Length
  • Increases at the Narrow Beach
  • Decreases at the Wider Beach
  • Joint Net Benefits under Cooperative solution
    greater than sum of Net Benefits under
    Competitive Solution

18
Next Steps
  • Non-autonomous solution path
  • Optimal Control Problem
  • Differential Game Optimal nourishment
  • Accounting for Fixed Costs?
  • No fixed costs - Bang-Bang Solution - Nourish
    every time period
  • Discrete Choice Dynamic Programming Problem
  • Binary Choice Variable at each time period
    Nourish / Dont
  • Two-stage model with endogenous initial width
  • Choice variables T, x0

19
Questions Suggestions
20
  • ADDITIONAL SLIDES

21
Reference Case
  • Simple Case - Starting with a Straight Shoreline
  • Choosing Initial Width x0(A) and x0(B) and
    mu(A) determines the portion of beach B facing
    exponential decay

22
State Equations for Beach Width
50-year horizon No Interaction
50-year horizon With Interaction
Slower Erosion in both beaches
23
Preliminary Analysis Sensitivity of Diffusion
Parameter
Optimal Nourishment Interval (T) varying K
Beach i assumes that Beach j will never nourish
(Tj 1000)
?
Beach A
K
TA
Large K ( 0.75) Narrow beach gains from sediment
transfer
Delays Nourishment
Optimal Nourishment Interval (Years)
?
Beach B
K
TB
High K creates incentive for wider beach to delay
nourishment and reduce gradient differential?
Diffusion Parameter (K)
24
Sensitivity to Diffusion Parameter
K 0.25
No Nash Equilibrium
25
State Equations with Nourishment
if
if
Width of beach i at time t depends on the
nourishment interval at beach j
26
The Resource Problem
Environmental Economics
  • Housing markets directly influenced by physical
    coastal processes
  • Coastal property values affected by erosion
  • Beach as a dynamic natural resource that
    generates value
  • Storm protection
  • Recreational flow

Value of beach reflected in property values

Optimal Beach Management
Natural Resource Economics
27
Beach as a Dynamic Natural Resource
  • Different from traditional resource economics
    problems
  • Economic Value of the resource derived from
    maintaining resource base (preventing erosion)
    rather than harvesting the resource
  • Analogous problem can be solved using tools from
    resource economics

Wide Beach
Storm Protection
Recreational Flow Value
28
State Equations for Beach Width
Wide Beach
Portion experiencing exponential retreat In the
absence of Beach A
Sediment Transfer to Beach A
Slower erosion due to Beach A
Transition Function