2200 MODELING DYNAMICS OF POST-DISASTER RECOVERY Al NEJAT 1 and Ivan DAMNJANOVIC 2 1 Assstant Professor, Department of Constructon Engneerng and Engneerng Technology, Texas Tech Unversty, Box 43107, Lubbock, Texas 79409-3107, Emal: al.nejat@ttu.edu 2 Assstant Professor, Zachry Department of Cvl Engneerng, Texas A&M Unversty, College Staton, TX 77843-3136, E-mal: damnjanovc@cvl.tamu.edu ABSTRACT Natural dsasters result n loss of lves, damage to exstng facltes, and nterrupton of busnesses. The losses are not nstantaneous, but rather contnue to occur untl the communty s restored to a functonal soco-economc entty. Hence, t s essental that polcy makers recognze ths dynamc aspect of the ncurred losses and make realstc plans to enhance recovery. However, ths cannot take place wthout understandng how homeowners react to recovery sgnals. These sgnals can come n dfferent ways: from polcy makers showng ther strong commtment to restore the communty by provdng fnancal support and/or restoraton of lfelne nfrastructure; or from the neghbors showng ther wllngness to reconstruct. The goal of ths research s to develop a model that can account for neghbors dynamc nteractons by ncorporatng ther sgnals n a spatal doman. A mult-agent framework s used to capture emergent behavor such as formaton of clusters. The results from the model confrm the mportant role of spatal externalty n agents decson-makng and the process of recovery. The results further hghlght the sgnfcant mpact of dscount factor and the accuracy of the sgnals on the percentage of reconstructon. Fnally, cluster formaton was shown to be an emergent phenomenon durng the recovery process.
2201 INTRODUCTION Advanced plannng s an essental factor to enable communtes to promptly recover from natural and human-nduced catastrophes. Effectve plannng needs to ncorporate the magntude of economc losses ncurred by catastrophes. Economc losses from both natural and anthropogenc dsasters do not take place nstantly; rather they are accumulated over the tme of recovery (Chang and Mles, 2004). Despte the substantal lterature on post-dsaster loss modelng such as Input- Output models, econometrc models, and computable general equlbrum models, few studes have focused on the dynamcs of the recovery process. The exstng lterature on modelng the recovery process can be grouped nto fve major categores: 1) Studes wth a focus on recovery as a temporal process: Ths ncludes lagged expendture models (Cole 1988, 1989) as well as recovery optmzaton by mnmzng economc losses (Rose et al. 1997), 2) Studes founded on a conceptual recovery framework ntroduced by Haas et al. (1977) n whch the recovery process was modeled as a four-stage sequental ncdent. Ths study was followed by case studes by Hogg (1980), Rubn and Popkn (1990), Berke et al. (1993), and Boln (1993) whch extensvely questoned ths four-stage sequental approach to recovery, ts predctablty, and argued that the order of the sequence can be dfferent from what was suggested by Haas et al. (1977). These subsequent studes characterzed recovery as an uncertan event affected by socal dspartes and decson-makng, 3) Studes centered on dspartes n recovery, whch were pursued by a two-pronged effort. The frst effort captured dsparty n socal classes among people (Hewtt (1997), Blake (1994)), where the second effort covered the recovery ssues assocated wth dspartes n busnesses (Durkn (1984), Kroll et al. (1991), and Alesch and Holly (1998)), 4) Studes attempted to capture the effect of spatal externalty on dfferent aspects of dsaster recovery. Among those the spatal mpact of lfelne nfrastructure was studed by Gordon et al. (1998), whle Chang and Mles (2004) proposed an object modelng technque to capture nteractons between ndustry sectors and communty plannng, and fnally 5) Studes to determne the key performance measures and ndcators to capture the dfferent aspects of the recovery process. These nclude psychologcal or perceptonal measures related to stress and frustraton, to more objectve ndcators such as reganng ncome, employment, household assets, and household amentes (Bates 1993, 1994; Boln and Bolton, 1983; Boln and Traner, 1978; Peacock et al., 1987). Whle these efforts capture the dfferent aspects of the problem, a spatalbehavoral model at the decson-makng level (agents level) s stll mssng. The objectve of ths paper s to develop a theoretcal model that can capture the behavor of resdents/owners n the presence of other reconstructng neghbors whch s defned as the exstng spatal externalty. Usng ths model we nvestgate the emergent behavor of cluster formaton n a post-dsaster market-drven resdental reconstructon. The scope of ths paper s lmted to modelng behavor of the owners of sngle-famly owner-occuped resdences affected by a dsaster. Even though ths assumpton constrans the real lfe applcaton of the developed model, t allows for the assessment of the effects of spatal externalty on the recovery process. Note that ths model does not attempt to fully explan a very complex post-dsaster reconstructon process, nor to capture heterogenety n owners behavor, but to
2202 provde a theoretcal foundaton for nvestgatng the phenomenon of spatal cluster formaton and decson-makng under uncertanty. Ths emergent behavor n the spatal doman of nteractons s captured usng mult-agent system smulaton. To assess the model s capacty of depctng spatal clusters of reconstructed propertes, the null hypothess of havng a random rate of cluster formaton was formulated and tested. The paper s organzed as follows. The next secton presents the theoretcal framework for assessment of the value of reconstructed propertes based on the sgnals from the neghborhood. Neghborhood s defned as a set of propertes wthn a certan dstance to each household. For ths research ths dstance has been set to one block for each property owner. Ths s followed by a descrpton of the agents nteracton game that defnes agents preferences among two optons: reconstruct now or wat-and-see. A mult-agent system smulaton model s presented next and used n analyss of cluster formaton. Fnally, the last secton presents major fndngs and suggests topcs deservng future attenton. VALUE OF RECONSTRUCTION Faced wth property damage and the partal or even complete destructon of neghborhoods, the owners often queston whether to reconstruct the property mmedately, or to wat and collect more nformaton about the future value of such acton. Ths new nformaton comes from sgnals from the other owners. If there s observed value n reconstructon (e.g. property values are restored as the communty s beng fully rebult), the owner wll rebuld as well; otherwse, the owner may wat untl the next tme perod to observe the reconstructon value. In ths paper, the owner s opton to relocate s not explctly consdered; the owner s assumed to have relocated f he does not rebuld at any tme n the smulaton. Ths concept of donow or wat-and-see has been extensvely utlzed n multple applcatons where uncertanty s resolved sequentally. Gven that the value of neghborng reconstructon has a drect mpact on the future value of yet-to-be reconstructed property, t s essental for the owners to update ther belefs accordngly. Ths externalty, the effect of decson makng of a set of property owners on the rest of the owners wthout consderng ther nterests, normally leads to a free-rder effect n whch some owners prefer to wat and observe the state of the world whle other owners rebuld, to reduce ther rsk and add value for ther property. However, watng may not always be the optmum strategy as the owners have lmted resources for reconstructon that are decreasng as the owners wat to reconstruct and the benefts arsng from the rebult propertes are foregone. In ths research, much lke Hendrcks and Porter (1996), we assume that the agents (.e. resdents) are ratonal agents seekng to maxmze ther utlty. Assume that homeowner y makes the reconstructon decson n a neghborhood of N homeowners. Homeowners future property values (e.g. homeowner y s (x y ) and that of the N-1 neghbors, =1,,N-1), are assumed to be random varables from a lognormal dstrbuton wth geometrc mean exp( and precson (nverse of the varance). Therefore t can be concluded that the log of the future property values would be random draws from a normal dstrbuton wth mean and precson. Wthout loss of generalty, precson s normalzed to 1. It s assumed that
2203 homeowner y s reconstructon cost s c y and the reconstructon perod s lmted to tme T. Hence, the net present value (NPV) of homeowner y s utlty at tme t can be expressed as: ( ( ), ) t t NPV U y t U ( y) xy c y (1) t Where represents the dscount factor for tme perod t and U( y ) denotes homeowner ys ' utlty from reconstructon. Homeowner ysbelef ' about the mean of future property values of all homeowners ( ) together wth ts neghbors belefs about ts actual future property value are updated through future market apprasal nformaton known as sgnals. For all homeowners, sgnals ( s, 1,..., N ) are consdered to be normally dstrbuted wth mean x, and precson where. Sgnals represent a belef about the future property values. Therefore, homeowner y starts wth ts ntal belef about the value of reconstructon n the neghborhood. Ths ntal belef can be updated based on the sgnals (.e. revealed property values from neghborhood). When the sgnals are observed, homeowner y updates ts belef about the mean of future property values n the neghborhood. Ths further results n updatng the belef of the other homeowners n the neghborhood about x y. Ths process contnues untl homeowner y reconstructs, or decdes to sell at ts far market value or abandon the property and leave. Here, t s crtcal to understand how sgnals comng from dfferent agents (e.g. neghbors reconstructng ther houses) affect homeowners ys ' percepton of the value of reconstructon ( sy xy ). The belef-updatng model used n ths subsecton s bult upon prevous studes that have nvestgated sequental decson-makng and nformaton updatng (e.g. the free-rder problem, clock game, war of attrton, predator prey watng game, etc.). More specfcally, the model extends the Hendrcks and Porter (1996) study on the effect of tmng on exploratory drllng, and develops a Bayesan value model to account for new nformaton and sgnals. The sgnals are revealed sequentally as homeowners reconstruct ( s x). Followng Bayes rule and gnorng pror belefs, t can be shown that homeowners belef about the mean of future property values n the neghborhood s a random draw from a normal dstrbuton wth the followng parameters: N 1 [ s(1 ) ] N 1, (2) (1 ) 1 Addtonally, homeowners belefs about x, 1,..., N(e.g. xy) condtonal on the observed sgnals from the neghborhood can be shown to be a random draw from a normal dstrbuton wth the followng parameters:
2204 xy 2 ysy ( y 1), x y 1 ( 1) 1 y y (3) Now f some (e.g. h number) of the homeowners reconstruct, homeowners belefs regardng the mean of the future property values n the neghborhood ( ) wll change respectvely. The new value wll depend on: 1) number of homeowners that reconstructed and have a revealed future value ( x1,..., x h) ; and 2) the remander of the sgnals ( N h) whch, gnorng pror belefs (usng a non-nformatve pror), s a normal random varable wth parameters shown n Equatons 4 (Hendrcks and Porter 1996). In these equatons x denotes the average revealed future property value and N represents the total number of homeowners n the neghborhood. N 1 [ s(1 ) ] N h1 h, h h h1 hx h (4) (1 ) As shown n Equatons 4, the new mean s a weghted mean of average revealed future property values and the sum of the remanng sgnals from propertes on whch reconstructon has not been started yet. Therefore, homeowner y starts wth ts ntal belef about the value of reconstructon n the neghborhood. Ths ntal belef s updated based on the sgnals and revealed property values from neghborng property homeowners. When the sgnals are observed, homeowner y updates ts belef about the mean of the future property values n the neghborhood. Ths wll as well result n updatng the belef of the other homeowners n the neghborhood about x y. Ths process contnues untl homeowner y ether reconstructs, or stcks to the do-nothng strategy. THE GAME: RECONSTRUCT NOW OR WAIT-AND-SEE Owners behavor can be characterzed from a game-theoretc approach based on the fact that the decson to reconstruct can also depend on the neghbors. In other words, the neghbors play a game wth two strateges: wat, observe the sgnals, and reconstruct only f there s a suffcent revealed value to do so; or reconstruct mmedately, wthout watng for the others. In cases where the net value from watng exceeds that of mmedate reconstructon, the game s referred to as war of attrton where a follower may end up wth a hgher payoff than a poneer. On ths premse, a game-theoretc model s developed to account for spatal nteractons (e.g. the decson to reconstruct depends also on the neghbors and ther decsons). To llustrate the concept, a stuaton wth only two neghbors (neghbor and j) s consdered n whch ( t), and ( t) represent the mean and precson of future property values at the tme of consderaton. The logc behnd a two-neghbor case consderaton s two-fold: 1) ts smplcty, and 2) the fact that the behavoral analyss for the multple neghbor cases s not sgnfcantly dfferent from the equlbrum soluton for the two-neghbor case
2205 (Hendrcks and Porter 1996). The expected payoff from mmedate reconstructon for homeowner consderng no pror reconstructon can be shown as: EVI[ (), t ()] t f ( x (), t ()) t U () dx (5) Where EVI[, t (), t ()] t denotes the expected value from mmedate reconstructon for homeowner at tme t gven the current state of nformaton about future property values n the neghborhood, f ( x ( t), ( t)) represents the normal probablty densty functon of x, and U represents the ganed utlty for owner at tme t. On the other hand, f homeowner wats and observe ts neghbor s (homeowner j) acton, the state of nformaton about the mean of future property values n the neghborhood changes to a new normal dstrbuton wth the followng parameters (Hendrcks and Porter 1996): -1 [ ( t) ( t) ( xj - sjj(1 j) )] 1-1 [ ( t) (1 j ) ] 1 ( tt), ( tt) ( t) j (6) As shown n Equaton 6, homeowner s belef about ( t t) s a functon of homeowner j s property value whch has not been revealed yet. It can be shown that ( t t) has a normal dstrbuton wth mean () t and precson 2 (1 ) t t j and therefore the expected payoff of watng for homeowner at tme t whch can be denoted as: 2 EVW[ ( t), ( t)] max[0, EVI (, ( t t))] f ( ; ( t), ( t) ( t) (1 j )) d (7) Where EVW[ (), t ()] t denotes the expected value from watng for homeowner at tme t gven the current state of nformaton about future property values n the neghborhood, ( t t) and EVI(, ( t t)) denotes the expected value from mmedate reconstructon for homeowner at tme t t. Game Soluton Based on these assumptons, the game between the homeowners (homeowners and j) can be defned as a war of attrton where homeowners have two pure strateges, 1) startng the reconstructon, and 2) watng for neghbors to reconstruct frst and decdng accordngly. In the case where pure strateges do not result n equlbrum or homeowners are not determned about ther reconstructon decsons, mxed strateges can solve the problem. Mxed strateges are formed by assgnng probabltes to pure strateges. For ths game, the mxed strategy equlbrum can be expressed by the probablty of reconstructon at each tme perod condtonal on no pror reconstructon. The soluton of ths game can be found usng backward nducton. In the last perod (T), homeowner wll start reconstructon f reconstructon has a postve expected value ( EVI( ( T ), ( T ) 0). In perod T 1, consderng that no pror reconstructon has
2206 occurred, homeowner has two optons. If t chooses to reconstruct then ts expected payoff of mmedate reconstructon ( EPI ) would be the same as the last perod. Ths s shown n Equaton 8. EPI[, ( T 1), ( T 1)] max[0, EVI[, ( T 1), ( T 1)] (8) If homeowner chooses to wat, ts expected payoff from watng ( EPW ) depends on the probablty of ts neghbor (homeowner j) reconstructng ( PR ( j T 1) ), where R j denotes the acton of reconstructon for homeowner j. Consequently homeowner s payoff can be shown as: P( R j T 1) EVW[, ( T 1), ( T 1)] EPW[ ( T 1), ( T 1)] (1 P( Rj T 1)) max(0, EVI( ( T 1), ( T 1)]) (9) The frst part of the formulaton shown n Equaton 9 ndcates the state n whch homeowner j reconstructs and homeowner updates ts belef accordngly, whle the next part refers to the state n whch homeowner j does not reconstruct and homeowner reconstructs f the payoff s postve. Homeowner would be ndfferent between mmedate reconstructon and watng f the payoffs are the same. Equatng equatons 8 and 9 leads to the equlbrum soluton for the probablty of reconstructon at each tme perod: (1 ) max[0, EVI[ ( T 1), ( T 1)] PR ( j T1) EVW[ ( T 1), ( T 1)] max[0, EVI[ ( T 1), ( T 1)] (10) Consderng no pror reconstructon, the same reasonng can be appled to other reconstructon perods. Hence, the probablty computed n Equaton 10 s the mxed strategy soluton for the formulated game. MULTIAGENT SYSTEM SIMULATION MODEL Ths secton presents a mult-agent system smulaton whch s based on the aforementoned game theoretc, agent s behavor model. For the purpose of smulaton a resdental neghborhood n College Staton, Texas was selected. Fgure 1 shows the workspace and the spatal confguraton of the property owners. The smulaton process was performed usng 4 dfferent modules. Module 1 was desgned to mport the exact locaton of the homeowners to the mult-agent system framework and defne homeowners as agents, Module 2 was to assgn agents ther propertes, module 3 was to smulate the nteracton of agents wth each other and fnally module 4 was to detect clusters among the property owners who have reconstructed. As shown n Fgure 1, homeowners are presented by pns nsde Google Earth map and are detected as squares nsde the mported map. Homeowners dffer from each other based on ther property values, and ther propertes level of post dsaster
2207 damage. For each homeowner a radus of one block was assumed to be ts neghborhood. Infra II & Agents Infra I Infra III Fgure1. The case study modeled n a Multagent System framework RESULTS AND CONCLUSIONS Hypothess Testng - Cluster Formaton The hypothess testng was formulated usng a mnmum rate of cluster- formaton. Ths mnmum rate was set to be 50 percent of reconstructed propertes. Hence, acceptng the alternate hypothess would ndcate that clusterng n reconstructon occurs more than 50 percent of the tme. Assumng normalty and unknown varance for the populaton, a t-statstc was used for the test: H, H :, T 0 : 0 1 0 0 X S / n 0, t0 t, n 1 (11) Where H s the nulll hypothess 0, 1 H s the alternatve hypothess, T s the test statstc, and t 0 s the rejecton crteron. Snce the t statstc s bgger than the rejecton crteron, the hypothess of havng a cluster formaton equal to 50% s rejected and the alternatve hypothess s accepted. Ths denotes that the rate of cluster formaton among reconstructed propertes s hgher than 50%. Table 1 shows the results. 0 T test for 0.5 Table 1. Results from t Test t P-value Low Interval 3.7438 2.375E-4 0.5452682 Hgh Interval 0.6460363 Based on the estmated p-values, t can be concluded havng a level of cluster formaton hgher than 50 percent s statstcally sgnfcant base on a 95% confdence level. Furthermore the confdence nterval for the mean les wthn the range of [0.5452, 0.646] ].
2208 Senstvty Analyss To examne the robustness of the model, a seres of senstvty analyses was performed. These analyses started wth the theoretcal model by testng ts reconstructon senstvty to a set of parameters, such as: 1) accuracy of sgnals through coeffcent of varaton; and 2) economc parameters, such as dscount factor. The base model was assumed to have a dscount factor of 0.9, coeffcent of varaton of 0.5, neghbor radus of 4, ntal value of $250K. To accomplsh the task, two values were consdered for each parameter. For dscount factor these values were 0.9 and 0.5, whereas for coeffcent of varaton thesee values were 0.5 and 2. The logc behnd these selectons was that each represents an extreme for that parameter. For example, the dscount factor of 0.5 ndcates a hghh dscountng whch s n contrast to the dscount factor of 0.9. On the other hand, the coeffcent of varaton of 2 denotes a hgh level of varance, whch s n contrast to the coeffcent of varaton 0.5. The results from the smulaton were depcted n Fgure 2. Explanaton of the results s separately shown for each parameter. Cumulatve Dstrbuton 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% DF=0.5-CofV=0.5 DF=0.5-CofV=2 DF=0.9-CofV=0.5 DF=0.9-CofV=2 0 0.040.08 0.12 0.16 0.2 0.24 0.28 0.32 0.366 0.4 0.44 0.48 Reconstructon Rate Fgure2. Senstvty analyss for reconstructonn rate Senstvty to Coeffcent of Varaton The frst feature of the model s ts senstvty to accuracy of sgnals through coeffcent of varaton (CofV). A hgher CofV ndcates that the sgnal s a poor ndcator of the true future value or n other words,, denotes a hgh varance assocated wth the value. The results show that gven a fxed dscount factor of 0.5, the reconstructon rate decreases as the level of uncertanty for the sgnals ncreases. Ths s due to the fact that homeowners decde based on ther observed sgnalss from ther surroundng neghbors. The accuracy of these sgnals s nversely proportonal to ther varance. Therefore wth ncreasng varance (uncertanty), owners become more
2209 hestant n startng reconstructon and prefer to wat and observe other neghbors acton to guarantee a postve net value for reconstructon. Senstvty to Dscount Factor The next characterstc of the model s ts senstvty to the level of dscountng. The results show that gven a fxed CoV, reconstructon rate ncreases as dscount factor (DF) ncreases. Ths s due to the fact that dscount rate s one of the drvng parameters n the process of decson makng. Applyng a hgh dscount rate to the present value calculatons assgns more weght to the current payoffs compared to payoffs antcpated n the future. Therefore one expects to observe an ncrease n the number of reconstructng owners by ncreasng the dscount rate and vce versa. It s assumed that all the homeowners have the same dscount rate. Conclusons The results from smulaton model confrm that spatal externaltes play an mportant role n agents decson-makng and can greatly mpact the recovery process. The results further hghlght the sgnfcant mpact of dscount factor and the accuracy of the sgnals on the percentage of reconstructon. Fnally, cluster formaton was shown to be an emergent phenomenon durng the recovery process. Also the model assumed that reconstructon could peak out at less than 100 percent. Therefore f the property-owners cannot secure a postve payoff from reconstructon, they wll not reconstruct but effectvely leave the locaton eventually. The ssues deservng further attenton are those whch mght have mplcatons for polcy holders. These nclude settng constrants for avalablty of fundng n regards to reconstructon of resdental unts and transportaton nfrastructure, lmtng the number of nsured homeowners, optmzng the methods of resource allocaton, ncorporatng the dynamcs of the transportaton nfrastructure and others. REFERENCES Alesch, D. J. and Holly J. N. (1998). Small busness falure, survval, and recovery: lessons from January 1994 Northrdge Earthquake. NEHRP Conference and Workshop on Research on the Northrdge, Calforna Earthquake, Rchmond, CA Bates, F. L., and Peacock, W. G. (1993). Lvng condtons, dsasters and development. Unversty of Georga Press, Athens, GA. Bates, F. L., and Pelanda, C. (1994). An ecologcal approach to dsasters. R.R. Dynes and K. J. Terney (eds.), Dsasters, collectve behavor, and socal organzaton. Unversty of Delaware Press, Newark, 149-159. Berke, P. R., Kartez, J. and Wenger D. (1993). Recovery after dsaster: Achevng sustanable development, mtgaton and equty. Dsasters, 17(2), 93-109 Blake, P., Cannon, T., Davs, I., and Wsner B. (1994). At rsk: natural hazards, people s vulnerablty, and dsasters. Routledge, New York.
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