MODELING DYNAMICS OF POST-DISASTER RECOVERY Ali NEJAT1and Ivan DAMNJANOVIC2
1Assistant Professor, Department of Construction Engineering and Engineering Technology, Texas Tech University, Box 43107, Lubbock, Texas 79409-3107, Email: ali.nejat@ttu.edu
2Assistant Professor, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, E-mail: idamnjanovic@civil.tamu.edu ABSTRACT
Natural disasters result in loss of lives, damage to existing facilities, and interruption of businesses. The losses are not instantaneous, but rather continue to occur until the community is restored to a functional socio-economic entity. Hence, it is essential that policy makers recognize this dynamic aspect of the incurred losses and make realistic plans to enhance recovery. However, this cannot take place without understanding how homeowners react to recovery signals. These signals can come in different ways: from policy makers showing their strong commitment to restore the community by providing financial support and/or restoration of lifeline infrastructure; or from the neighbors showing their willingness to reconstruct. The goal of this research is to develop a model that can account for neighbors’ dynamic interactions by incorporating their signals in a spatial domain. A multi-agent framework is used to capture emergent behavior such as formation of clusters. The results from the model confirm the important role of spatial externality in agents’ decision-making and the process of recovery. The results further highlight the significant impact of discount factor and the accuracy of the signals on the percentage of reconstruction. Finally, cluster formation was shown to be an emergent phenomenon during the recovery process.
INTRODUCTION
Advanced planning is an essential factor to enable communities to promptly recover from natural and human-induced catastrophes. Effective planning needs to incorporate the magnitude of economic losses incurred by catastrophes. Economic losses from both natural and anthropogenic disasters do not take place instantly; rather they are accumulated over the time of recovery (Chang and Miles, 2004).
Despite the substantial literature on post-disaster loss modeling such as Input-Output models, econometric models, and computable general equilibrium models, few studies have focused on the dynamics of the recovery process. The existing literature on modeling the recovery process can be grouped into five major categories: 1) Studies with a focus on recovery as a temporal process: This includes lagged expenditure models (Cole 1988, 1989) as well as recovery optimization by minimizing economic losses (Rose et al. 1997), 2) Studies founded on a conceptual recovery framework introduced by Haas et al. (1977) in which the recovery process was modeled as a four-stage sequential incident. This study was followed by case studies by Hogg (1980), Rubin and Popkin (1990), Berke et al. (1993), and Bolin (1993) which extensively questioned this four-stage sequential approach to recovery, its predictability, and argued that the order of the sequence can be different from what was suggested by Haas et al. (1977). These subsequent studies characterized recovery as an uncertain event affected by social disparities and decision-making, 3) Studies centered on disparities in recovery, which were pursued by a two-pronged effort. The first effort captured disparity in social classes among people (Hewitt (1997), Blaikie (1994)), where the second effort covered the recovery issues associated with disparities in businesses (Durkin (1984), Kroll et al. (1991), and Alesch and Holly (1998)), 4) Studies attempted to capture the effect of spatial externality on different aspects of disaster recovery. Among those the spatial impact of lifeline infrastructure was studied by Gordon et al. (1998), while Chang and Miles (2004) proposed an object modeling technique to capture interactions between industry sectors and community planning, and finally 5) Studies to determine the key performance measures and indicators to capture the different aspects of the recovery process. These include psychological or perceptional measures related to stress and frustration, to more objective indicators such as regaining income, employment, household assets, and household amenities (Bates 1993, 1994; Bolin and Bolton, 1983; Bolin and Trainer, 1978; Peacock et al., 1987).
While these efforts capture the different aspects of the problem, a spatial-behavioral model at the decision-making level (agents’ level) is still missing. The objective of this paper is to develop a theoretical model that can capture the behavior of residents/owners in the presence of other reconstructing neighbors which is defined as the existing spatial externality. Using this model we investigate the emergent behavior of cluster formation in a post-disaster market-driven residential reconstruction. The scope of this paper is limited to modeling behavior of the owners of single-family owner-occupied residences affected by a disaster. Even though this assumption constrains the real life application of the developed model, it allows for the assessment of the effects of spatial externality on the recovery process. Note that this model does not attempt to fully explain a very complex post-disaster reconstruction process, nor to capture heterogeneity in owners’ behavior, but to
provide a theoretical foundation for investigating the phenomenon of spatial cluster formation and decision-making under uncertainty. This emergent behavior in the spatial domain of interactions is captured using multi-agent system simulation. To assess the model’s capacity of depicting spatial clusters of reconstructed properties, the null hypothesis of having a random rate of cluster formation was formulated and tested.
The paper is organized as follows. The next section presents the theoretical framework for assessment of the value of reconstructed properties based on the signals from the neighborhood. Neighborhood is defined as a set of properties within a certain distance to each household. For this research this distance has been set to one block for each property owner. This is followed by a description of the agents’ interaction game that defines agents’ preferences among two options: reconstruct now or wait-and-see. A multi-agent system simulation model is presented next and used in analysis of cluster formation. Finally, the last section presents major findings and suggests topics deserving future attention.
VALUE OF RECONSTRUCTION
Faced with property damage and the partial or even complete destruction of neighborhoods, the owners often question whether to reconstruct the property immediately, or to wait and collect more information about the future value of such action. This new information comes from signals from the other owners. If there is observed value in reconstruction (e.g. property values are restored as the community is being fully rebuilt), the owner will rebuild as well; otherwise, the owner may wait until the next time period to observe the reconstruction value. In this paper, the owner’s option to relocate is not explicitly considered; the owner is assumed to have relocated if he does not rebuild at any time in the simulation. This concept of “do-now” or “wait-and-see” has been extensively utilized in multiple applications where uncertainty is resolved sequentially.
Given that the value of neighboring reconstruction has a direct impact on the future value of yet-to-be reconstructed property, it is essential for the owners to update their beliefs accordingly. This externality, the effect of decision making of a set of property owners on the rest of the owners without considering their interests, normally leads to a free-rider effect in which some owners prefer to wait and observe the state of the world while other owners rebuild, to reduce their risk and add value for their property. However, waiting may not always be the optimum strategy as the owners have limited resources for reconstruction that are decreasing as the owners wait to reconstruct and the benefits arising from the rebuilt properties are foregone.
In this research, much like Hendricks and Porter (1996), we assume that the agents (i.e. residents) are rational agents seeking to maximize their utility. Assume that homeowner y makes the reconstruction decision in a neighborhood of N homeowners. Homeowners’ future property values (e.g. homeowner y’s (xy) and that of the N-1 neighbors ݔ , i=1,…,N-1), are assumed to be random variables from a lognormal distribution with geometric mean exp(ߙሻ and precision (inverse of the variance) . Therefore it can be concluded that the log of the future property values would be random draws from a normal distribution with mean ߙ and precision ߚ. Without loss of generality, precision ߚ is normalized to 1. It is assumed that
homeowner y’s reconstruction cost is cy and the reconstruction period is limited to time T. Hence, the net present value (NPV) of homeowner y’s utility at time t can be expressed as:
NPV U y t( ( ), )tU y( )t xycy (1)
Where t represents the discount factor for time period t
and U y( ) denotes homeowner y s' utility from reconstruction. Homeowner y s'
belief about the mean of future property values of all homeowners ( ) together with its neighbors’ beliefs about its actual future property value are updated through future market appraisal information known as signals. For all homeowners, signals ( ,s ii 1,...,N) are considered to be normally distributed with mean xi, and precision i where.
Signals represent a belief about the future property values. Therefore, homeowner y starts with its initial belief about the value of reconstruction in the neighborhood. This initial belief can be updated based on the signals (i.e. revealed property values from neighborhood). When the signals are observed, homeowner y
updates its belief about the mean of future property values in the neighborhood. This further results in updating the belief of the other homeowners in the neighborhood about xy. This process continues until homeowner yreconstructs, or decides to sell
at its fair market value or abandon the property and leave. Here, it is critical to understand how signals coming from different agents (e.g. neighbors reconstructing their houses) affect homeowners y s' perception of the value of reconstruction
(sy xy).
The belief-updating model used in this subsection is built upon previous studies that have investigated sequential decision-making and information updating (e.g. the “free-rider” problem, clock game, war of attrition, predator – prey waiting game, etc.). More specifically, the model extends the Hendricks and Porter (1996) study on the effect of timing on exploratory drilling, and develops a Bayesian value model to account for new information and signals. The signals are revealed sequentially as homeowners reconstruct (si xi). Following Bayes rule and ignoring prior beliefs, it
can be shown that homeowners’ belief about the mean of future property values in the neighborhood is a random draw from a normal distribution with the following parameters: 1 1 1 [ (1 ) ] , (1 ) N i i i N i i i i s
(2)Additionally, homeowners’ beliefs about x ii, 1,..., (e.g. )N xy conditional on the observed signals from the neighborhood can be shown to be a random draw from a normal distribution with the following parameters:
2 ( 1) , 1 ( 1) 1 y y y y y x x y y s (3)
Now if some (e.g. h number) of the homeowners reconstruct, homeowners’ beliefs regarding the mean of the future property values in the neighborhood () will
change respectively. The new value will depend on: 1) number of homeowners that reconstructed and have a revealed future value ( ,..., )x1 xh ; and 2) the remainder of the signals (N h )which, ignoring prior beliefs (using a non-informative prior), is a normal random variable with parameters shown in Equations 4 (Hendricks and Porter 1996). In these equations x denotes the average revealed future property value and N represents the total number of homeowners in the neighborhood.
1 1 1 [ (1 ) ] , (1 ) N i i i N h i h h i h i h i hx s h
(4)As shown in Equations 4, the new mean is a weighted mean of average revealed future property values and the sum of the remaining signals from properties on which reconstruction has not been started yet.
Therefore, homeowner y starts with its initial belief about the value of reconstruction in the neighborhood. This initial belief is updated based on the signals and revealed property values from neighboring property homeowners. When the signals are observed, homeowner y updates its belief about the mean of the future property values in the neighborhood. This will as well result in updating the belief of the other homeowners in the neighborhood about xy. This process continues until homeowner y either reconstructs, or sticks to the do-nothing strategy.
THE GAME: RECONSTRUCT NOW OR WAIT-AND-SEE
Owners’ behavior can be characterized from a game-theoretic approach based on the fact that the decision to reconstruct can also depend on the neighbors. In other words, the neighbors play a game with two strategies: wait, observe the signals, and reconstruct only if there is a sufficient revealed value to do so; or reconstruct immediately, without waiting for the others.
In cases where the net value from waiting exceeds that of immediate reconstruction, the game is referred to as war of attrition where a follower may end up with a higher payoff than a pioneer. On this premise, a game-theoretic model is developed to account for spatial interactions (e.g. the decision to reconstruct depends also on the neighbors and their decisions). To illustrate the concept, a situation with only two neighbors (neighbor i and j) is considered in which ( ), and ( )t t represent the mean and precision of future property values at the time of consideration. The logic behind a two-neighbor case consideration is two-fold: 1) its simplicity, and 2) the fact that the behavioral analysis for the multiple neighbor cases is not significantly different from the equilibrium solution for the two-neighbor case
(Hendricks and Porter 1996). The expected payoff from immediate reconstruction for homeowner i considering no prior reconstruction can be shown as:
EVI i[ | ( ), ( )] t t
f x( | ( ), ( )) ( )i t t U i dxi (5)Where EVI i t[ , | ( ), ( )] t t denotes the expected value from immediate reconstruction for homeowner i at time t given the current state of information about future property values in the neighborhood, ( | ( ), ( ))f xi t t represents the normal probability density function of xi, and Ui represents the gained utility for owner i at time t. On the other hand, if homeowner i waits and observe its neighbor’s (homeowner j) action, the state of information about the mean of future property values in the neighborhood changes to a new normal distribution with the following parameters (Hendricks and Porter 1996):
-1 -1 [ ( ) ( ) ( - (1 ) )] 1 ( ) , ( ) ( ) [ ( ) (1 ) ] 1 j j j j j j t t x s t t t t t t (6)
As shown in Equation 6, homeowner i’s belief about (t t) is a function of homeowner j’s property value which has not been revealed yet. It can be shown that
(t t)
has a normal distribution with mean ( )t and precision t t2(1j) and therefore the expected payoff of waiting for homeowner i at time t which can be denoted as:
2
[ | ( ), ( )] max[0, ( | , ( ))] ( ; ( ), ( ) ( ) (1 j))
EVW i t t
EVI i t t f t t t d (7)Where EVW i[ | ( ), ( )] t t denotes the expected value from waiting for homeowner i at time t given the current state of information about future property values in the neighborhood, (t t) and EVI i( | , ( t t))denotes the expected value from immediate reconstruction for homeowner i at time t t.
Game Solution
Based on these assumptions, the game between the homeowners (homeowners i and j) can be defined as a war of attrition where homeowners have two pure strategies, 1) starting the reconstruction, and 2) waiting for neighbors to reconstruct first and deciding accordingly. In the case where pure strategies do not result in equilibrium or homeowners are not determined about their reconstruction decisions, mixed strategies can solve the problem.
Mixed strategies are formed by assigning probabilities to pure strategies. For this game, the mixed strategy equilibrium can be expressed by the probability of reconstruction at each time period conditional on no prior reconstruction. The solution of this game can be found using backward induction. In the last period (T), homeowner i will start reconstruction if reconstruction has a positive expected value
occurred, homeowner i has two options. If it chooses to reconstruct then its expected payoff of immediate reconstruction (EPI) would be the same as the last period. This is shown in Equation 8.
[ , ( 1), ( 1)] max[0, [ , ( 1), ( 1)]
EPI i T T EVI i T T (8)
If homeowner i chooses to wait, its expected payoff from waiting (EPW)
depends on the probability of its neighbor (homeowner j) reconstructing (P R T( j| 1)), where Rj denotes the action of reconstruction for homeowner j.
Consequently homeowner i’s payoff can be shown as:
( | 1) [ , ( 1), ( 1)] [ | ( 1), ( 1)] (1 ( | 1)) max(0, ( | ( 1), ( 1)]) j j P R T EVW i T T EPW i T T P R T EVI i T T (9)
The first part of the formulation shown in Equation 9 indicates the state in which homeowner j reconstructs and homeowner i updates its belief accordingly, while the next part refers to the state in which homeowner j does not reconstruct and homeowner i reconstructs if the payoff is positive. Homeowner i would be indifferent between immediate reconstruction and waiting if the payoffs are the same. Equating equations 8 and 9 leads to the equilibrium solution for the probability of reconstruction at each time period:
(1 ) max[0, [ | ( 1), ( 1)]
( | 1) [ | ( 1), ( 1)] max[0, [ | ( 1), ( 1)] j EVI i T T P R T EVW i T T EVI i T T (10)Considering no prior reconstruction, the same reasoning can be applied to other reconstruction periods. Hence, the probability computed in Equation 10 is the mixed strategy solution for the formulated game.
MULTIAGENT SYSTEM SIMULATION MODEL
This section presents a multi-agent system simulation which is based on the aforementioned game theoretic, agent’s behavior model. For the purpose of simulation a residential neighborhood in College Station, Texas was selected. Figure 1 shows the workspace and the spatial configuration of the property owners. The simulation process was performed using 4 different modules. Module 1 was designed to import the exact location of the homeowners to the multi-agent system framework and define homeowners as agents, Module 2 was to assign agents their properties, module 3 was to simulate the interaction of agents with each other and finally module 4 was to detect clusters among the property owners who have reconstructed.
As shown in Figure 1, homeowners are presented by pins inside Google Earth map and are detected as squares inside the imported map. Homeowners differ from each other based on their property values, and their properties’ level of post disaster
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hesitant in starting reconstruction and prefer to wait and observe other neighbors’ action to guarantee a positive net value for reconstruction.
Sensitivity to Discount Factor
The next characteristic of the model is its sensitivity to the level of discounting. The results show that given a fixed CoV, reconstruction rate increases as discount factor (DF) increases. This is due to the fact that discount rate is one of the driving parameters in the process of decision making. Applying a high discount rate to the present value calculations assigns more weight to the current payoffs compared to payoffs anticipated in the future. Therefore one expects to observe an increase in the number of reconstructing owners by increasing the discount rate and vice versa. It is assumed that all the homeowners have the same discount rate.
Conclusions
The results from simulation model confirm that spatial externalities play an important role in agents’ decision-making and can greatly impact the recovery process. The results further highlight the significant impact of discount factor and the accuracy of the signals on the percentage of reconstruction. Finally, cluster formation was shown to be an emergent phenomenon during the recovery process. Also the model assumed that reconstruction could peak out at less than 100 percent. Therefore if the property-owners cannot secure a positive payoff from reconstruction, they will not reconstruct but effectively leave the location eventually. The issues deserving further attention are those which might have implications for policy holders. These include setting constraints for availability of funding in regards to reconstruction of residential units and transportation infrastructure, limiting the number of insured homeowners, optimizing the methods of resource allocation, incorporating the dynamics of the transportation infrastructure and others.
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