Utility-based Multi-agent System with Spatial Interactions: The Case of Virtual Estate Development

Abstract

A multi-agent system is presented here. Agent decisions are based on economic partial utilities. It’s a new computational tool using cellular and agent-based models. Decision processes are represented through computational agents. Land-use and land cover changes are represented through a cellular model. It belongs to kind of models called Multi-Agent Systems model of Land-Use Cover Change. Decisions concern the estate development in the neighborhood of the agents. Space is discretized in heterogeneous cells where heterogeneous agents interact together (directly or via space). Based on partial utility probabilities, several stochastic simulations are achieved. These simulations are then summarized, replicated and a parameter sensitivity analysis provided. The main parameters influencing the decisions of an agent depend on his type (ecologist, etc.) and on the expected economic activities in his neighborhood. Little information is assumed to be known about the mechanisms determining the parameter values guiding the decisions of the agents, except their type. The whole system constitutes a general virtual model, mostly independent from initial conditions and real data. Sensitivity analysis are used to explore behavioral paths and influencing parameters. In summary, the main contribution of this paper consists in: (i) defining a simulation-based framework for modeling the decisions of many heterogeneous economic agents involved in spatial interactions; (ii) capturing the decision process using general utility-based equations; (iii) providing a sensitivity analysis of initial parameter values. An example of possible application investigating strategies of various categories of agents (institutions, landowners, homeowners, etc.) facing an intense residential development is provided. The framework consists of abstracting institutional characteristics of actual local Urbanism Plan procedures in France. This model constitutes a first step to be further specified.

Appendices

Appendix 1: Agent Classes—Utility and Decision Making

1.1 1.1: Mayor

At the beginning of the simulation a new type of Mayor is selected. Mayors can be more concerned by ecological or economic development. Three relative weights of partial utilities determine their sensibility to ³⁰ : environment ($\alpha _{EN}$), citizen welfare ($\alpha _{W}$) and tax revenue ($\alpha _{RF}$). $\alpha _{W}$ is fixed whatever Mayor type ( e.g. , $\alpha _{W}=0.2$),³¹ the two others weights determine the type of mayor: if $\alpha _{EN}\geqslant \alpha _{RF}$ Mayor is more sensitive to ecology and considered to be an ecologist ; if $\alpha _{EN}<\alpha _{RF}$ Mayor is more sensitive to economic development and considered to be concerned about economic development .

All partial utilities are calculated according to spatial constructions:

• The environmental utility ³²associated with a cell j:

  $$\begin{aligned} u_{EN}=ln\left( 1+Q_{j}\right) \end{aligned}$$

  (5)

where $Q_{j}$ is the environmental quality of a cell $j$.

$$\begin{aligned} Q_{j}=\left\{ \begin{array}{l} Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}\, {\text{ if }}\, Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}>0\\ 0\, {\text{ if }}\, Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}\leqslant 0 \end{array}\right. \end{aligned}$$

(6)

where $Q_{j0}$ is the initial environmental quality of cell $j$, which can be determined by various factors (for example, the vegetation type, the vegetation density, the vegetation type in a relevant neighborhood, etc.); $N$ is a fixed number of cells in the neighborhood of $j$; represents the environmental damages in cell $i$.

• Tax revenue utility associated with a cell $j$:

  $$\begin{aligned} u_{RF}=ln\left( 1+Z_{j}\right) \end{aligned}$$

  (7)

• Citizen welfare utility associated with a cell $j$ ³³ :

  $$\begin{aligned} u_{W}=ln\left( N+2-\rho \sum _{k=1}^{M}K_{k}\right) \end{aligned}$$

  (8)

$$\begin{aligned} { K_{k}=\frac{\sum _{l=1}^{X}K_{l}}{X};\, K_{l}=\left\{ \begin{array}{l} 1\, { \text{ if } \text{ protestation }}\\ 0\, {\text{ if } \text{ neutrality }}\\ -1\, {\text{ if } \text{ adhesion }} \end{array}\right. ;\,\rho =\left\{ \begin{array}{l} 1\, {\text{ if } \text{ the } \text{ cell } \text{ is } \text{ constructible }}\\ 1\, {\text{ if } \text{ the } \text{ cell } \text{ is } \text{ non } \text{ constructible }} \end{array}\right. } \end{aligned}$$

$K_{l}$ corresponds to the decision of a resident or a landowner for cells $l={1,{\ldots },X}$ after a mayor constructability proposal for a cell $j$ (when the agent is a resident $X=1$; when the agent is a landowner $X\geqslant 1$; in this case landowner’s final decision is the mean of the various decisions he takes for each of his cells); and $k={1,{\ldots },M}$ designed the set of agents (residents and landowners) in the neighborhood of $j$.

Environmental amenities and tax revenues attached to a cell depend on the cell activity. Tax revenues concern wild, residential, commercial and industrial cells as described below.

According to these partial utilities, the total utility function of Mayor for a cell $j$ is obtained:

$$\begin{aligned} U_{m}=\alpha _{EN}u_{EN}+\alpha _{RF}u_{RF}+\alpha _{W}u_{W} \end{aligned}$$

(9)

Decisions of Mayor are achieved in two steps:

1.1.1 Step 1—Constructability Proposal Decision

At the beginning of the simulation, Mayor decides and proposes (this is not a final decision) the constructability of wild cells. If $\beta$ the number of cells developed (cells which are not wild) surrounding the wild cell does not exceed a threshold $\theta$, the cell will not become constructable. The threshold $\theta$ is chosen at the beginning of the simulation. If $\beta >\theta$ the cell can become constructable. A probability proportional to the number of cells developed in the neighborhood ($\frac{\beta }{\theta }$) is used to determine if the cell is proposed for constructability.

Then, for cells potentially constructable, a new type of further potential activity is defined by a probability depending on the number of cells for each kind of activities in the neighborhood, as defined hereafter.³⁴

Let $N_{A}$; for $A={I,C,R,W}$; be the number of cells characterized by activity $A$ in the neighborhood and $f_{A}=\frac{N_{A}}{N}$, the proportion of cells characterized by activity $A$ in the neighborhood with $N=\sum _{A={I,C,R,W}}N_{A}$. We assume that the probability for the buyer to be an industrialist is positively correlated to the proportion of industrialists present in the neighborhood (for positive external effects expectation) and negatively correlated to the proportion of wild cells in the neighborhood: $p_{I}=f_{I}\left( 1-f_{W}\right)$. We assume that the probability that the buyer is a commercial is positively correlated to the proportion of residences and commerce present in the neighborhood: $p_{C}=f_{R}\times f_{C}$. We assume that the probability that a proposed selling cell is finally not sold and remains wild is positively correlated to the proportion of non-developed cells in the neighborhood and negatively correlated to the proportion of each kind of developed cells: $p_{W}=f_{W}\left( 1-f_{I}\right) \left( 1-f_{C}\right) \left( 1-f_{R}\right)$. Finally, the probability that the buyer is a resident is: $p_{R}=1-p_{I}-p_{C}-p_{W}$.

1.1.2 Step 2—Final Constructability Decision

Other economic agents (residents and landowners) are informed of Mayor’s development project. According to the constructability decision, the agent decides to protest or not. Decisions are then transmitted to Mayor who decides the final affectation of cells. The affectation depends on the utilities obtained in non constructable and constructable scenarios.

Let $U_{m}\left( cons\right)$ be Mayor’s utility if a wild cell becomes constructable and $U_{m}\left( \overline{cons}\right)$ Mayor’s utility in the opposite case:

$$\begin{aligned} U_{m}\left( cons\right) \!=\!\sum _{A={I;C;R;W}}p_{A} \left\{ \alpha _{EN}ln\left[ 1+Q_{j}\left( A\right) \right] +\alpha _{RF} ln\left[ 1+Z_{j}\left( A\right) \right] +\alpha _{W}u_{W}|_{\rho =1}\right\} \qquad \end{aligned}$$

(10)

$$\begin{aligned} U_{m}\left( \overline{cons}\right) \!=\!\alpha _{EN} ln\left[ 1+Q_{j}\left( W\right) \right] +\alpha _{RF} ln\left[ 1+Z_{j}\left( W\right) \right] +\alpha _{W}u_{W}|_{\rho =0}\qquad \end{aligned}$$

(11)

Where the environmental quality of the cell depending on the type of activity development is given by ³⁵ :

$$\begin{aligned} Q_{j}\left( A\right) =\left\{ \begin{array}{l} Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) \, if\, Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) >0\\ 0\, if\, Q_{j0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) \leqslant 0 \end{array}\right. ;\,\quad A=\left\{ I,C,R,W\right\} \nonumber \\ \end{aligned}$$

(12)

When a landowner decides to sell, a probability of buyer kind (residential, commercial or industrial ³⁶ ) is calculated based on the probability mechanism described below.

A wild cell proposed to constructability becomes finally constructable if:

$$\begin{aligned} U_{m}\left( cons\right) >U_{m}\left( \overline{cons}\right) \end{aligned}$$

(13)

Finally, Mayor informs every resident and landowner of the final decision.

1.2 1.2: Residents

Residents choose to protest or not after Mayor constructability decision. This decision is based on a utility function composed of two kinds of partial utilities. These partial utilities are respectively based on environmental preferences and on the value of their house:

• Environmental utility for a particular resident: $u_{E}=ln\left( 1+Q\right)$

Where $Q$ is the environmental quality of his or her house:

$$\begin{aligned} Q=\left\{ \begin{array}{l} Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}\, {\text{ if }}\, Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}>0\\ 0\, {\text{ if }}\, Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}\leqslant 0 \end{array}\right. \end{aligned}$$

(14)

• Wealth utility for a particular resident: $u_{F}=ln\left( 1+F\right)$

where $F$ is the value of his house ³⁷:

$$\begin{aligned} F=\left\{ \begin{array}{l} F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}\, {\text{ if }}\, F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}>0\\ 0\, {\text{ if }}\, F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}\leqslant 0 \end{array}\right. \end{aligned}$$

(15)

Coefficients $\delta$ and $\tau$ are constant weights relative to the different variables and $F$ is a constant parameter.

Residents can be ecologistsor property-value concernedaccording to the weight they give to the corresponding partial utilities . Let be $\alpha _{E}$ and$\alpha _{F}$ the respective weights a particular resident gives to the value of his house and to environmental preferences: if $\alpha _{F}\geqslant \alpha _{E}$ the resident is considered to be property-value concerned; if $\alpha _{F}<\alpha _{E}$ the resident is considered to be an ecologist.

Finally, the utility function of a particular resident is:

$$\begin{aligned} U_{v}=\alpha _{E}u_{E}+\alpha _{F}u_{F} \end{aligned}$$

(16)

The resident takes the decision to protest according to a potential change of the characteristic of a cell in his neighborhood. Let $U_{v}\left( cons\right)$ be the utility of the resident if a wild cell $j$ in the neighborhood becomes constructable and $U_{v}\left( \overline{cons}\right)$ if this cell remains non constructable.

Where:

$$\begin{aligned} U_{v}\left( cons\right) =\sum _{A={I,C,R,W}}p_{A}\left\{ \alpha _{E}ln\left[ 1+Q\left( A\right) \right] +\alpha _{F} ln\left[ 1+F\left( A\right) \right] \right\} \end{aligned}$$

(17)

With for $A={I;C;R;W}$:

$$\begin{aligned} Q\left( A\right)&= \left\{ \begin{array}{l} Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) \, if\, Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) >0\\ 0\, if\, Q_{0}-\sum \nolimits _{i=1}^{N}B_{i}-B_{j}\left( A\right) \leqslant 0 \end{array}\right. \\ F\left( A\right)&= \left\{ \begin{array}{l} F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}+\delta Z_{j}\left( A\right) \, if\, F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}+\delta Z_{j}\left( A\right) >0\\ 0\, if\, F_{0}+\tau Q+\delta \sum \nolimits _{i=1}^{N}Z_{i}+\delta Z_{j}\left( A\right) \leqslant 0 \end{array}\right. \end{aligned}$$

$$\begin{aligned} U_{v}\left( \overline{cons}\right) =\alpha _{EN} ln\left[ 1+Q\left( W\right) \right] +\alpha _{F}ln \left[ 1+F\left( W\right) \right] \end{aligned}$$

(18)

Every resident (and landowner) is characterized by a parameter $\varepsilon \in \mathbb{R }^{+}$, randomly attributed at the beginning of the simulation. This parameter reflects the cost attached for protesting or agreeing. Regarding $\varepsilon$, a resident protest if $U_{v}\left( cons\right) <\left( 1-\varepsilon \right) U_{v}\left( \overline{cons}\right)$; a resident is neutral if $\left( 1-\varepsilon \right) U_{v}\left( \overline{cons}\right) \leqslant U_{v}\left( cons\right) \leqslant \left( 1+\varepsilon \right) U_{v}\left( \overline{cons}\right)$; a resident agrees if $U_{v}\left( cons\right) >\left( 1+\varepsilon \right) U_{v}\left( \overline{cons}\right)$.

1.3 1.3: Landowners

Let be $U_{p}$ the utility function of a landowner with:

$$\begin{aligned} U_{p}=\alpha _{Q}u_{Q}+\alpha _{R}u_{R}+\alpha _{TR}u_{TR} \end{aligned}$$

(19)

(14) Where $\alpha _{Q}$, $\alpha _{R}$ and $\alpha _{TR}$ are, respectively, the weights the landowner gives to environmental quality, wealth and tradition ³⁸and $u_{Q}$, $u_{R}$ and $u_{TR}$ are landowners’ partial utilities for the same values.

Landowners can be: traditionalists , ecologists or land-value concerned depending on the higher weight of each kind of partial utility. Each partial utility represents preferences depending on the tradition (this partial utility is an increasing function of the time the cell is owned by the landowner), the environmental amenities and wealth regarding the potential value of his or her cell.

More precisely:

$$\begin{aligned} u_{Q}=ln\left( 1+Q\right) ;\, u_{R}=ln\left( 1+F\right) ;\, u_{TR}=\eta ln\left( 1+\frac{t_{pos}}{25}\right) \end{aligned}$$

(20)

where $t_{pos}$ is the time that cell belongs to landowner and $\eta =\left\{ \begin{array}{l} 0\, {\text{ if }}\, {\text{ the }}\, {\text{ cell }}\, {\text{ is }}\, {\text{ sold }}\\ 1\, {\text{ if }}\, {\text{ the }}\, {\text{ cell }}\, {\text{ is }}\, {\text{ not }}\, {\text{ sold }} \end{array}\right.$. $u_{TR}$ serve to capture the fact that the values that local residents hold for land may sensibly vary between long time residents and newcomers from more urban areas. ³⁹

Using this utility function, landowners take three decisions:

• 1. They choose or not to protest Mayor constructability decision. The mechanism used is the same than for residents.

• 2. When they agree the constructability decision they choose to sell or not their cells. They calculate the respective expected utility in both cases:

  $$\begin{aligned} U_{p}\left( sell\right)&= \sum _{A={I;C;R;W}}p_{A}\left\{ \alpha _{Q}ln\left[ 1+Q\left( A\right) \right] + \alpha _{R}ln\left[ 1+F\left( A\right) \right] \right\} \end{aligned}$$

  (21)
  $$\begin{aligned} U_{p}\left( \overline{sell}\right)&= \alpha _{Q}ln\left[ 1+Q \left( W\right) \right] +\alpha _{TR}ln\left( 1+\frac{t_{pos}}{25}\right) \end{aligned}$$

  (22)

• 3. The selling decision of the landowner is given by: Sell if $U_{p}\left( sell\right) >U_{p}\left( \overline{sell}\right)$; hold otherwise.

Appendix 2: Stepwise Regression

Appendix 3: Application of a Multinomial Logistic Regression to the Model

Let be “Adh” (for adhesion), “Neut” (for neutrality) and “Prot” (for protestation) the three dependent variable categories. One of these categories is choose to be the reference category. The reference category is indexed by 0 and the two others categories by $j={1;2}$.

Let be ⁴⁰ $X_{i}=\left( x_{1i}, \ldots ,x_{li}, \ldots ,x_{Li}\right)$ the observed outcome for the $i$th observation on the dependent variable and $\Omega _{j}=\left( \omega _{1j}, \ldots ,\omega _{lj}, \ldots ,\omega _{Lj}\right) ^{T}$the vector of the regression coefficients in the $j$th regression. The multinomial logistic model can be explained as:

$$\begin{aligned} P\left[ {\text{ Decision }}=j|X=X_{i}\right] =\frac{\exp \left( X_{i} \Omega _{j}\right) }{1+\sum _{j=1}^{2}\exp \left( X_{i}\Omega _{j}\right) };\, j=\left\{ 1,2\right\} \end{aligned}$$

(23)

$$\begin{aligned} P\left[ {\text{ Decision }}=0|X=X_{i}\right] =\frac{1}{1+\sum _{j=1}^ {2}\exp \left( X_{i}\Omega _{j}\right) };\, j=\left\{ 1,2\right\} \end{aligned}$$

(24)

Combining the two previous equations we obtain:

$$\begin{aligned} ln\frac{P\left[ {\text{ Decision }}=j|X=X_{i}\right] }{P\left[ {\text{ Decision }}=0| X=X_{i}\right] }=X_{i}\Omega _{j}=\sum _{l=1}^{L}x_{i}\omega _{j};\, j=\left\{ 1,2\right\} \end{aligned}$$

(25)

Let be ⁴¹ $x_{k}$ one of the $L$ continuous variables ⁴² and $x_{l0}$ a fixed quantity for. Using the previous equation we can write:

$$\begin{aligned} ln\frac{P_{j}}{P_{0}}\equiv ln\frac{P\left[ {\text{ Decision }}=j|x_{ik}=x_{k}+1\right] /P \left[ {\text{ Decision }}=0|x_{ik}=x_{k}+1\right] }{P\left[ {\text{ Decision }}= j|x_{ik}=x_{k}\right] /P\left[ {\text{ Decision }}=0|x_{ik}=x_{k}\right] } =\omega _{k};\, j=\left\{ 1,2\right\} \nonumber \\ \end{aligned}$$

(26)

So, $\varvec{\omega _{k}}$ can be interpreted as the impact of an increase of one unity of on the ratio between the probability that the decision is $j$ and the probability that the decision is 0; i.e. if $ln\frac{P_{j}}{P_{0}}>0$ an increase of $x_{k}$ makes more likely to have a decision of type $j$. Obviously opposite is true if $ln\frac{P_{j}}{P_{0}}>0$.

Appendix 4: Multinomial Logistic Regression Results