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Flood Retention on Private Land

Climate change increases the frequency and intensity of future flood events, leading to higher costs of flood damages and increasing the public demand for protective measures.
Traditional flood protection measures, mainly based on grey infrastructure (i.e. dikes, dams, etc), are not sufficient to cope with dynamic flood risk alone. Nature-based solutions such as Natural Water Retention Measures (NWRM) are promising options to mitigate flood risks as a complement to grey infrastructure.
The challenge is to consider multifunctional land uses, which enable temporary flood retention and flood storage on private land without restricting the provision of other ecosystem services.

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Flood Retention on Private Land

The reconciliation of flood risk management and land management is needed. Since all NWRM primarily need to be implemented on private land the consideration of multiple aspects includes:

economic issues (e.g. how to compensate for or incentivize flood retention services);
property rights issues (e.g. how to allow temporary flood storage on private land);
issues of public participation (e.g. how to ensure the involvement of private landowners) as well as issues of public subsidies (e.g. how to integrate/mainstream flood retention in agricultural subsidies).

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LAND4FLOOD COST Action

LAND4FLOOD COST Action aims to address these different aspects and to establish a common knowledge base and channels of communication among scientists, regulators, landowners and other stakeholders in field.

Key Questions:

Which synergies can be identified between different land uses[...]?
How can the knowledge base [...] be strengthened and their importance communicated to different actors [...]?
How can land owners be encouraged to adapt land uses and land management strategies which allow for increased water retention capacity?
How can public and private stakeholders in urban and rural areas engage with each other to reduce flood damage [...]?

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LAND4FLOOD COST Action

Key Questions:

Which synergies can be identified between different land uses[...]?
How can the knowledge base [...] be strengthened and their importance communicated to different actors [...]?
How can land owners be encouraged to adapt land uses and land management strategies which allow for increased water retention capacity?
How can public and private stakeholders in urban and rural areas engage with each other to reduce flood damage [...]?

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TU-games

A TU-game is a pair (N,v) where N is the finite set of players, and v: 2^N⟶ℝ is the characteristic function of the game, which satisfies v(∅) = 0.
We interpret v(S) as the benefit that S can generate. We often refer to (N,v) simply as v.

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Imputations and the Core

Let v be a TU-game. An imputation of v is an x ∊ ℝ^N such that

∑_i_∊_N x_i = v(N)
x_i ≥ v(i) for all i∊N.

We denote by I(v) the set of imputations of v.

The core of v is the following set:

Core(v) = {x ∊ I(v) : ∑_i_∊_S x_i ≥ v(S) for all S ⊂ N}

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The model

There is a finite number of land owners on a biodiversity area.
Land owners can use their land for either forests or for other uses (up to 14 different land uses).
Forests help to conserve biodiversity.
Forest are also less affected by climate change.

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Biodiversity index

Forest generates:	Neighbor forests	Neighbor “other uses”
Better water quality	Positively affected	Positively affected
More fauna diversity
Well-being	Not affected
CO₂ absorption

Social benefit due to externalities computed as the addition of four factors:

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The model

Let N = {1,...,n} be a finite set of landowners on a biodiversity area.
Profit of having a forest is given by a vector f ∈ ℝ₊^N.
Profit of having other land use is given by a vector o ∈ ℝ₊^N.
The positive externality for land j ∈ N due to the presence of a forest on land i ∈ N \ {j} is given by a matrix E = (e_ij)_i,j_∈_N, where e_ij ≥ 0.

Model 1: It only affects other uses.
Model 2: it affects forests and other uses.

A biodiversity problem is a tuple B = (N, f,o,E) with the properties given above.

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Non-cooperative approach

Allowing agents to choose the use of their land may lead to inefficiencies. For example:

Prisoner's dilemma game with unique Nash equilibrium payoff allocation (1,1).

f₁ = 0

o₁= 1

f₂ = 0

o₂= 1

	o₂	f₂
o₁	(1,1)	(0,3)
f₁	(3,0)	(2,2)

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Cooperative approach (Model 1)

A biodiversity game is a cooperative game (N, v¹^B) generated by a biodiversity problem B, where the worth of a coalition S is given by the maximization problem:

v¹^B(S) = max_F⊆S Ψ¹(F, S, f , o, E)

where

We say that any set F ∊ arg max_F_⊆_N Ψ¹(F, N, f, o, E) is an optimal configuration. When unique, we denote it as F^B.

Profit for having a forest

Profit for other uses

Externalities

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Cooperative approach (Model 2)

A biodiversity game is a cooperative game (N, v²^B) generated by a biodiversity problem B, where the worth of a coalition S is given by the maximization problem:

v²^B(S) = max_F⊆S Ψ²(F, S, f , o, E)

where

We say that any set F ∊ arg max_F_⊆_N Ψ²(F, N, f, o, E) is an optimal configuration. When unique, we denote it as F^B.

Profit for having a forest

Profit for other uses

Externalities

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Example (Model 1)

Let B = (N,f,o,E) defined as

N = {1,2,3}
f = (1,1,2)
o = (2,1,1)
e₁₂ = e₁₃ = 2�e₂₃ = 1�e_ij = 0 otherwise.

v¹^B({1}) = 2, v¹^B({2}) = 1, v¹^B({3}) = 2�v¹^B({1,2}) = 4, v¹^B({1,3}) = 4, v¹^B({2,3}) = 3,�v¹^B(N) = 7.

In particular, considering all the agents, the optimal structure is that land 1 uses the land for forest, and lands 2 and 3 have other different uses.

f₁ = 1

o₁= 2

f₃ = 2

o₃= 1

f₂ = 1

o₂= 1

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Example (Model 2)

Let B = (N,f,o,E) defined as

N = {1,2,3}
f = (1,1,2)
o = (2,1,1)
e₁₂ = e₁₃ = 2�e₂₃ = 1�e_ij = 0 otherwise.

v²^B({1}) = 2, v²^B({2}) = 1, v²^B({3}) = 2,�v²^B({1,2}) = 4, v²^B({1,3}) = 5, v²^B({2,3}) = 4,�v²^B(N) = 9.

In particular, considering all the agents, the optimal structure is that each land be forest.

f₁ = 1

o₁= 2

f₃ = 2

o₃= 1

f₂ = 1

o₂= 1

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Cooperative approach

Definition

The core of a biodiversity problem B = (N,f,o,E) is the set of efficient payoff allocations that cannot be objected by any coalitions:

Core(B) = {x ∊ ℝ^N : ∑_i_∊_Nx_i = v^B(N), ∑_i_∊_Sx_i ≥ v^B(S) ∀S⊆N}.

A sharing rule is a function that assigns to each biodiversity problem B = (N,f,o,E) a payoff allocation ϕ(B) ∊ ℝ^N.

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Properties

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

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Properties

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

Proposition (Model 1) The core may be empty.

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Properties

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

Proposition (Model 1) The core may be empty.

o₁=1

f₅=1

v¹^B(N) = 3

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Properties

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

Proposition (Model 1) The core may be empty.

o₁=1

f₅=1

v¹^B(N) = 3

x₁ + x₂ + x₃ + x₄ ≥ 2

x₁ + x₂ + x₃ + x₅ ≥ 3

x₁ + x₂ + x₄ + x₅ ≥ 3 ⇒ 4∑_i_∊_Nx_i ≥ 13 ⇒ ∑_i_∊_Nx_i ≥ 3.25

x₁ + x₃ + x₄ + x₅ ≥ 3

x₂ + x₃ + x₄ + x₅ ≥ 2

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Properties

Efficiency Given B = (N,f,o,E),

∑_i_∈_Nϕ_i(B) = v^B(N).

Additivity Given B = (N,f,o,E), B′ = (N′,f′,o′,E′),

ϕ(B+B′) = ϕ(B) + ϕ(B′).

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Properties

Efficiency Given B = (N,f,o,E),

∑_i_∈_Nϕ_i(B) = v^B(N).

Additivity Given B = (N,f,o,E), B′ = (N′,f′,o′,E′),

ϕ(B+B′) = ϕ(B) + ϕ(B′).

Proposition

There is no Efficient rule satisfying Additivity.

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Properties

Efficiency Given B = (N,f,o,E),

∑_i_∈_Nϕ_i(B) = v^B(N).

Additivity Given B = (N,f,o,E), B′ = (N′,f′,o′,E′),

ϕ(B+B′) = ϕ(B) + ϕ(B′).

Proposition

There is no Efficient rule satisfying Additivity.

f₁ = 0

o₁= 1

f₂ = 0

o₂= 0

f₁ = 0

o₁= 1

f₂ = 0

o₂= 0

f₁ = 0

o₁= 2

f₂ = 0

o₂= 0

v^B(N) = 2

v^B’(N) = 3

v^B+B’(N) = 4

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Properties

Efficiency Given B = (N,f,o,E),

∑_i_∈_Nϕ_i(B) = v^B(N).

Additivity Given B = (N,f,o,E), B′ = (N′,f′,o′,E′),

ϕ(B+B′) = ϕ(B) + ϕ(B′).

Proposition

There is no Efficient rule satisfying Additivity.

f₁ = 0

o₁= 1

f₂ = 0

o₂= 0

f₁ = 0

o₁= 1

f₂ = 0

o₂= 0

f₁ = 0

o₁= 2

f₂ = 0

o₂= 0

v^B(N) = 2

v^B’(N) = 3

v^B+B’(N) = 4

Necessary condition for additivity:

v^B⁺^B′(N) = v^B(N) + v^B′(N)

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Properties

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with v^B+B’(N) = v^B(N) + v^B’(N):

ϕ(B + B′) = ϕ(B) + ϕ(B′).

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Properties

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with v^B+B’(N) = v^B(N) + v^B’(N):

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Proposition Two biodiversity problems B = (N,f,o,E), B′ = (N,f′,o′,E′) satisfy v^B+B’(N) = v^B(N) + v^B’(N) if and only if they share a common optimal configuration.

Moreover, this common optimal configuration is also optimal in B + B′.

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Properties

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with v^B+B’(N) = v^B(N) + v^B’(N):

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Proposition Two biodiversity problems B = (N,f,o,E), B′ = (N,f′,o′,E′) satisfy v^B+B’(N) = v^B(N) + v^B’(N) if and only if they share a common optimal configuration.

Moreover, this common optimal configuration is also optimal in B + B′.

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Properties

Efficiency Given B = (N,f,o,E), ∑_i_∈_Nϕ_i(B) = v^B(N).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

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Properties

Efficiency Given B = (N,f,o,E), ∑_i_∈_Nϕ_i(B) = v^B(N).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Individual rationality: Given B = (N,f,o,E),

ϕ_i(B) ≥ v^B({i}) = max{f_i,o_i}

for all i ∈ N.

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Properties

Efficiency Given B = (N,f,o,E), ∑_i_∈_Nϕ_i(B) = v^B(N).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Individual rationality: Given B = (N,f,o,E),

ϕ_i(B) ≥ v^B({i}) = max{f_i,o_i}

for all i ∈ N.

Proposition (Model 1) No efficient rule satisfies Maximal Additivity and Individual Rationality.

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Properties

Efficiency Given B = (N,f,o,E), ∑_i_∈_Nϕ_i(B) = v^B(N).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Individual rationality: Given B = (N,f,o,E),

ϕ_i(B) ≥ v^B({i}) = max{f_i,o_i}

for all i ∈ N.

Proposition (Model 1) No efficient rule satisfies Maximal Additivity and Individual Rationality.

f₁ = 1

o₁= .6

f₂ = .7

o₂= 1

f₁ = 1

o₁= 1

f₂ = 1

o₂= 1

f₁ = 0

o₁= .4

f₂ = .3

o₂= 0

ϕ₁ ≤ 1.7

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Properties

Efficiency Given B = (N,f,o,E), ∑_i_∈_Nϕ_i(B) = v^B(N).

Maximal Additivity Given B = (N,f,o,E), B′ = (N,f′,o′,E′) with a common optimal configuration:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

Individual rationality: Given B = (N,f,o,E),

ϕ_i(B) ≥ v^B({i}) = max{f_i,o_i}

for all i ∈ N.

Proposition (Model 1) No efficient rule satisfies Maximal Additivity and Individual Rationality.

f₁ = 1

o₁= .2

f₂ = .9

o₂= 1

f₁ = 1

o₁= 1

f₂ = 1

o₂= 1

f₁ = 0

o₁= .8

f₂ = .1

o₂= 0

ϕ₁ ≥ 1.8

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Two options (Model 1)

Weakening Individual Rationality
Weakening Maximal Additivity

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Two options (Model 1)

Weakening Individual Rationality
Weakening Maximal Additivity

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Weakening Individual Rationality (Model 1)

We consider the class of problems B with a unique optimal configuration F^B.

Minimum Rights Given B = (N,f,o,E),

ϕ_i(B) ≥ f_i for all i ∊ F^B and

ϕ_i(B) ≥ o_i for all i ∊ N \ F^B.

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Weakening Individual Rationality (Model 1)

We consider the class of problems B with a unique optimal configuration F^B.

Minimum Rights Given B = (N,f,o,E),

ϕ_i(B) ≥ f_i for all i ∊ F^B and

ϕ_i(B) ≥ o_i for all i ∊ N \ F^B.

Separability Given B = (N,f,o,E) and S ⊂ N such that e_ij = e_ji = 0 for all i∊S, j∊N\S,

ϕ(N,f,o,E) = ϕ(S,f_S,o_S,E_S)×ϕ(N\S,f_N\S,o_N\_S,E_N\S)

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Weakening Individual Rationality (Model 1)

Theorem 1.1

In the class of problems with a unique optimal configuration, a rule ϕ satisfies Efficiency, Maximal Additivity, Minimum Rights, Separability, and Anonymity iff there exists α ∊ [0,1] such that, for all B = (N,f,o,E) and i ∊ N,

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Two options (Model 1)

Weakening Individual Rationality
Weakening Maximal Additivity

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Weakening Maximal Additivity (Model 1)

We say that B = (N,f,o,E) and B′ = (N′,f′,o′,E′) are compatible if

N = N′,
they have a common optimal configuration, and
for each i∊N, f_i < o_i implies f_i′ ≤ o_i′ , and f_i > o_i implies f_i′ ≥ o_i′.

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Weakening Maximal Additivity (Model 1)

We say that B = (N,f,o,E) and B′ = (N′,f′,o′,E′) are compatible if

N = N′,
they have a common optimal configuration, and
for each i∊N, f_i < o_i implies f_i′ ≤ o_i′ , and f_i > o_i implies f_i′ ≥ o_i′.

Compatible Additivity Given B, B′ compatible:

ϕ(B + B′) = ϕ(B) + ϕ(B′).

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Weakening Maximal Additivity (Model 1)

Theorem 1.2

The equal surplus rule ψ is the only rule that satisfies Efficiency, Compatible Additivity, Individual Rationality, and Anonymity.

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Properties (Model 2)

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

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Properties (Model 2)

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

o₁=1

f₅=1

v²^B(N) = 6

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Properties (Model 2)

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

Proposition

For all B = (N,f,o,E), game (N,v²^B) is convex.

o₁=1

f₅=1

v²^B(N) = 6

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Properties (Model 2)

Core Selection Given B = (N,f,o,E),

ϕ(B) ∊ Core(B).

Proposition

For all B = (N,f,o,E), game (N,v²^B) is convex.

o₁=1

f₅=1

v²^B(N) = 6

Corollary

For all B = (N,f,o,E), Sh(N,v²^B) ∊ Core(B) ≠∅.

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Characterization (Model 2)

Theorem 2

There exists a unique rule that satisfies Core Selection and Maximal Additivity, and it is given by

𝜙_i*(B) = max{o_i, f_i + ∑_j_∊_N_\{_i_}e_ij}

for each B=(N,f,o,E) and i ∊ N.

Remark

This rule does not coincide with the Shapley value and it is much faster to compute.

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