Coupled economic-coastline modeling with suckers and free riders
Zachary C. Williams,
Center for Marine Science, University of North Carolina Wilmington, Wilmington, North Carolina, USA
Department of Physics and Physical Oceanography, University of North Carolina Wilmington, Wilmington, North Carolina, USA
Corresponding author: Z. C. Williams, Department of Physics and Physical Oceanography, University of North Carolina Wilmington, 601 South College Rd., Wilmington, NC 28403, USA. (firstname.lastname@example.org)
 Shoreline erosion is a natural trend along most sandy coastlines. Humans often respond to shoreline erosion with beach nourishment to maintain coastal property values. Locally extending the shoreline through nourishment alters alongshore sediment transport and changes shoreline dynamics in adjacent coastal regions. If left unmanaged, sandy coastlines can have spatially complex or simple patterns of erosion due to the relationship of large-scale morphology and the local wave climate. Using a numerical model that simulates spatially decentralized and locally optimal nourishment decisions characteristic of much of U.S. East Coast beach management, we find that human erosion intervention does not simply reflect the alongshore erosion pattern. Spatial interactions generate feedbacks in economic and physical variables that lead to widespread emergence of “free riders” and “suckers” with subsequent inequality in the alongshore distribution of property value. Along cuspate coastlines, such as those found along the U.S. Southeast Coast, these long-term property value differences span an order of magnitude. Results imply that spatially decentralized management of nourishment can lead to property values that are divorced from spatial erosion signals; this management approach is unlikely to be optimal.
 Humans move more surface mass than all natural processes combined [Hooke, 2000]. We are considered the most significant geomorphic force on the planet and have even prompted the naming of a new epoch, the Anthropocene [Crutzen and Stoermer, 2000]. Surface processes that transport mass can become coupled with dynamics associated with human agency. Coupling arises as humans and their institutions attempt to alter surface processes, and then these altered surface processes influence future human interactions with the environment.
 Coupling between humans and landscape processes is often strongest on intermediate time scales (years to decades) [Werner and McNamara, 2007]. On these scales, many landscape processes are governed by spatial dynamics such as linear diffusion, nonlinear diffusion, and advection [Martin and Church, 1997; Kirkby, 1987; Dietrich et al., 2003]. Human decisions, however, are often focused on time and neglect the role that spatial processes play in influencing the coupled human-landscape system. As humans alter local regions of a landscape, those alterations proceed to influence other locations through natural spatial dynamics. The altered dynamics in nearby regions can then impact human intervention in those locations, which eventually can influence the original human intervention. This spatiotemporal coupling can give rise to unanticipated emergent behavior [Haff, 2003; Werner and McNamara, 2007; Parker and Munroe, 2007; Brock and Xepapadeas, 2010; Moritz et al., 2004]. Spatial feedbacks also have potential to give rise to strategic interactions that have been explored using game theoretic models in the experimental psychology literature [Kerr, 1983]. “Sucker” is a term related to our analysis of coupled human coastline dynamics and was originally proposed by Kerr . Sucker behavior occurs in small groups of people and involves one group member inadvertently carrying a disproportionate amount of burden which the entire group was originally tasked with, meanwhile the rest of the group “free rides” on the sucker's effort. Along a coastline, coastal processes move sand from one location to another, so that when one coastal community locally alters the coastline and coastal processes, other communities are affected. Our work explores how future shoreline stabilization in one community could depend on current stabilization decisions in neighboring communities. These interactions could potentially give rise to coastal community sucker and free rider type behavior.
 Many coastal systems have a strongly coupled human component. Humans manage the coastline system either with hard stabilization such as rockwalls and jetties or through soft stabilization, namely beach nourishment. Our work here focuses on beach nourishment, rebuilding an eroded beach by placing sand on the shoreface. Numerous recent studies have explored feedbacks in the dynamics of the coupled human-coastline system. In a coupled barrier island resort model, dunes on undeveloped portions of the barrier provide protection that justifies resort development, and once development occurs, artificial protection justifies further resort development. This feedback leads to heavily protected barrier island resorts that filter low-energy storm damage at the cost of enhancing larger and less frequent damage [McNamara and Werner, 2008]. Along a cuspate coastline similar in morphology to the North Carolina coastline, a model in which communities constantly stabilize the coastline predicts that local stabilization not only affects the nearby shoreline but also those on the order of tens of kilometers away, through both alongshore propagating effects and instantaneous long-range changes to rates of alongshore sediment transport [Slott et al., 2010; Ells and Murray, 2012]. Model experiments on a straight coastline where communities project optimal nourishment intervals based on perceived erosion show an instability due to the timing of alongshore sediment transport delivery in one's own nourishment cycle, and this instability can lead to temporally chaotic beach width [Lazarus et al., 2011].
Smith et al.  suggested an important feedback whereby as the cost of nourishment sand rises, so does a community's locally optimal rate of nourishment. This unexpected feedback exists for some but not all regions of the optimal nourishment interval parameter space. Considering that sand is a limited resource, an increasing cost of sand as the available offshore nourishment sand dwindles could provide a positive feedback resulting in unsustainable nourishment [Smith et al., 2009]. It is unclear how this and other feedbacks will manifest along a spatially complex, dynamic coastline. How does the practice of spatially decentralized management that is locally optimal (optimal from the perspective of one community without regard for effects on others, i.e., spatially myopic nourishment) influence patterns of nourishments along a spatially explicit coastline? What are the effects of spatiotemporal feedbacks on property values? To what degree does large-scale coastal morphology influence these feedbacks? We explore these questions with a numerical model that couples locally optimal nourishment decision making with a model of coastal dynamics. Model experiments focus on straight and cuspate plan view coastal morphologies as found along the U.S. East Coast, as these are geomorphically interesting, data rich, and societally relevant case studies.
2.1 Coastal Model
 We use a coastline change model put forth previously in the literature [Ashton et al., 2001; Ashton and Murray, 2006a]. The plan view model domain (Figure 1a) is discretized into a cellular grid with individual cells of size 1 × 1 km. Fluxes of sediment, Qs, between cells are calculated according to an alongshore sediment transport equation
where Hb is the breaking wave height, k1 is an empirical constant reflecting local geology, and ϕb − θ is the angle of the breaking wave crest relative to the coast (Figure 1a). Mass is conserved in the model domain and sand is assumed to spread evenly over, or eroded evenly from, the active shoreface up to a depth, D. This assumption leads to a conservation of sediment equation
 Equation ((2)) states that the time rate of change in cross-shore shoreline position, η, depends on alongshore gradients in Qs. Input offshore waves are refracted and shoaled over shore-parallel contours. On large scales (relative to the cross-shore extent of the shoreface), depth contours tend to approximately parallel the shoreline out to the shoreface depth (D) [e.g., Ells and Murray, 2012]. Thus, the assumption of shore parallel contours means that “offshore” wave inputs represent conditions at the offshore extent of the shoreface. Wave refraction and shoaling occur until the point of breaking according to Airy wave theory, and once a wave has broken due to shallow depths, equation ((1)) is used to determine sediment transport at shoreline cell boundaries. The model geometry is referred to as a “one-line” approach since the shoreline can be specified by a single line. A critical assumption underlying this approach is that alongshore transported sand spreads evenly along the entire active shoreface (gray area in Figure 1b) allowing for a simplified model that bypasses the details of cross-shore dynamics that can be important on time scales that are shorter than that required for sand to spread across the entire shoreface [van den Berg et al., 2012]. On longer time scales (years to decades), large-scale shoreline evolution can be investigated by tracking alongshore sediment transport [e.g., Ashton and Murray, 2006a, 2006b].
 The model is forced with a distribution of offshore wave angles, which vary from one set of experiments to another, depending on the simulated coastal configuration. In model experiments involving a cuspate coastline, a new offshore wave angle is chosen each model day from a four-bin probability distribution function (PDF) while the wave height and period are fixed for all wave angles. The weight of each bin from the PDF represents the probability that the corresponding range of wave angles is randomly chosen, which can be related to the relative influence on alongshore transport from waves in each angle bin, based on observational data [Ashton and Murray, 2006b]. For a cuspate coastline similar to the North Carolina coast, we use a PDF (Figure 2) derived from 20 years of hourly data collected by the Army Corp of Engineer's Wave Information Studies Station 509 (WIS 509). WIS 509 is located on edge of the continental shelf perpendicularly offshore of Cape Fear, North Carolina. Details on how we create a PDF from wave data can be found in Ashton and Murray [2006b] and Slott et al. . Model experiments on straight coastline morphologies are forced with zero degree offshore water wave angles (shore-normal approach) every day. (The resulting diffusion of coastline shape that drives the results described below also results from any wave climate dominated by “low-angle” waves [e.g., Ashton and Murray, 2006a]; we chose shore normal waves only for simplicity.) We provide an additional source of linear erosion in the model by assigning a mean erosion rate that can represent shoreline retreat from sea level rise (or could represent alongshore-transport gradients on larger scales). For simulations on straight coastlines we also impose a random shock to the shoreline position representing stochasticity in shoreline position due to faster time scale processes such as daily changes in wave height and storms. We draw shocks from a mean-zero normal distribution and vary the standard deviation to explore sensitivity to magnitude of the stochastic component and the implications of increasing stochasticity. Along straight coastlines, this additional erosion signal can be represented mathematically as
where γ represents a mean erosion rate and the term on right side, N(0, σC), is a variable erosion component chosen randomly from a normal distribution with a mean of zero and standard deviation of σC. Although the original model formulation bypasses faster time scales processes, we effectively reintroduce some of their effects in the N(0, σC) term. While it is generally accepted that shoreline erosion occurs faster than accretion, we allow similar magnitudes of erosion and accretion to occur with the same frequency since there is no generally accepted form to the distribution and because our primary concern is long-time shoreline change.
 A cuspate coastline domain was generated by running the physical model described above with the WIS 509 wave climate parameterization for 10,000 years allowing an initially straight coastline to self-organize to a configuration with aspect ratios approximately equal to those calculated for the cusps located along North and South Carolina. For details and explanation as to how large-scale coastal morphology self-organizes, refer to Ashton et al.  and Ashton and Murray [2006a]. After this domain was generated, the coast was left to evolve for an additional 20 years while tracking the daily position of every shoreline cell. From this, the standard deviation in daily shoreline change was obtained at each cell. We define the alongshore average of the standard deviation in cuspate shoreline change at each cell as σc.
2.2 Economic Model
 The economic model is based on a capital accumulation framework developed for a single nourishing community [Smith et al., 2009]. Nourishment in practice is done periodically rather than continuously. This pattern reflects the high fixed costs of planning projects and mobilizing dredges and other equipment [Smith et al., 2009]. Beach managers choose the nourishment interval, T, which is the length of time between consecutive nourishments. A model community determines the optimal time to nourish the beach by choosing a nourishment interval that maximizes the present value net benefits from a series of nourishments. Net benefits (NB) from a single nourishment event are
where B(T) is a benefits function that represents the discounted flow of benefits accumulated over T and C(T) is the cost of nourishment. The cost comprises fixed and variable nourishment costs:
where cf is the fixed cost associated with equipment, environmental impact analyses, and permits. The second term in equation ((5)) is the variable cost, and it depends on the current cost of sand per cubic meter (ϕ) and the amount of sand needed to nourish the beach out to x0, the maximum cross-shore location. Benefits from beach nourishment, capitalized by the coastal real estate market, are derived from a hedonic pricing model, which decomposes the price of coastal property into constituent parts: property characteristics (year built, building materials quality, etc.), location specific characteristics (proximity to parks, crime rates, etc.), and environmental characteristics (beach width) [Bin et al., 2008; Gopalakrishnan et al., 2011; Landry and Hindsley, 2011]. The benefits function takes the form
where α is the base property value that includes all attributes excluding beach width (x(t)) and β is the hedonic price of beach width. Both property and locational characteristics are captured in the base property value α, while beach width (environmental characteristics) is captured in the x(t)β term. Equation ((6)) shows that beach width is a fundamental a component of property value. Our previous empirical work controlled for all of the features that also affect differences in home values: square footage, lot size, unit type (single-family, duplex, or condominium), construction type, number of bedrooms, number of bathrooms, and regional effects (to account for features like school district) [Gopalakrishnan et al., 2011]. In this analysis, we use the earlier empirical results to isolate the contribution of beach width to property value and represent the average value of all other characteristics through a single parameter reflecting baseline value (α). The sale price of a house is the stock of value representing the total value of amenities and services provided over the lifetime of the property. If the owner were to rent out the house, the rental income would be a flow of value over time. If the owner occupies the house, she or he receives a similar flow of value. In real estate markets, sale prices are equivalent to adding up the flows of value over the life of the property, where future flows are discounted. When an environmental amenity such as the width of the beach is an important component of the value of a house, flows of value need to account for how the amenity changes over time and the discount rate, δ, serves to convert a stock value into a flow benefit.
 In our model, each community along the coast determines an optimal nourishment frequency based on projections for future erosion rates. The model community assumes that beach width change can be attributed to two phenomena: exponential relaxation and linear shoreline retreat. Following nourishment, beach erosion is enhanced because the new cross-shore profile is out of equilibrium, steeper than the profile that would produce no net long-term cross-shore transport [Dean and Dalrymple, 2002]. The community assumes that erosion from relaxation decreases exponentially with a time scale θ [Dean, 2002; Smith et al., 2009]. The shoreline is assumed to retreat linearly with rate γ, leading to a projected beach width that evolves according to
where μ is the fraction of beach that experiences exponential erosion and (1 − μ) is the fraction of beach that experiences linear erosion; μ must lie in the range 0 ≤ μ ≤ 1. The beach width will change through time and based on the community's projection of beach dynamics (equation ((7))) the community chooses the interval between nourishments that maximizes the net benefits associated with all future nourishments. The sum of all net benefits is the total present value of the coastal property,
with each Ti being the time since the beginning of management. If erosion dynamics are stationary, then equation ((8)) can be rewritten as an infinite geometric series,
 Here the infinite-horizon problem is an analytical convenience and can be considered a close approximation to a finite-horizon problem with a long planning horizon (>50 years). The optimal nourishment interval T* can now be recovered through the following maximization:
 It should be noted that while T* is locally optimal, it may not be societally optimal as the loss of ecosystem services is not included in the cost function (equation ((5))), and effects on neighboring communities are ignored [Smith et al., 2009]. Solving equation ((10)) for a range of sand costs (ϕ) and shoreline retreat rates (γ), we generate an optimal nourishment interval (T*) parameter space (Figure 3). In doing this, we hold all other economic and physical parameters constant: exponential relaxation time scale is θ = 0.05, hedonic price of beach width is β = 0.25, discount rate is δ = 0.06, fixed cost of nourishment is cf = 10, and the base property value is α = 200.
 Nourishment grade sediment is a finite resource and may become increasingly scarce or hard to access [Cleary et al., 2004]. We model the effect of limited nourishment stock by allowing the sand cost (ϕ) to adjust according to the level of nourishment sand available. In the model, the cost of sand increases linearly with a diminishing sediment reservoir of volume R according to
where c0 is the baseline sand cost, cm is the maximum sand cost, and λ is the rate at which the sand cost increases per unit volume of sand removed from the reservoir [McNamara et al., 2011]. The initial costs of nourishment sand used in the simulations are of the same order of magnitude as those found along the U.S. Coast [Gopalakrishnan et al., 2011]. The issue of stock effects is fundamental and ubiquitous in the study of renewable resource economics (e.g., fisheries) and nonrenewable resource economics (e.g., fossil fuel and other mineral extraction). In all of the theoretical and empirical literature, cost increases as the stock of resources decreases, and debates for any particular resource are really only about how large or small the effect is. Given the empirical evidence of dwindling sand resources near coastlines that are engaging in a lot of nourishment, this is a natural feedback to explore. Whether we are seeing stock effects already in the empirical data is difficult to say and would be very difficult to isolate. We certainly have evidence of feedbacks between property values and the geomorphology through coastal management [Gopalakrishnan et al., 2011]. Because sand cost is such an important parameter in this empirical relationship, it is essential to understand how the dynamics of this coupled system would evolve if these costs were to increase as theory predicts and as anecdotal evidence at least suggests.
2.3 Coupling Economic Dynamics to Coastal Processes
 Historical nourishment along the U. S. East Coast has occurred over a wide range of spatial scales [Valverde et al., 1999, Program for the Study of Developed Shorelines]. In order to explore the general consequence of locally optimal nourishment along a coast, economic decisions are made at the resolution of the physical model, which corresponds to some of the smaller historical alongshore scales. Specifically, nourishment decisions are made over 1 km alongshore stretches of coastline. At each shoreline cell in the model domain, the shoreline retreat rate (γ) is projected forward by averaging the τ previous years of shoreline change at each cell. τ is a time scale for averaging previous shoreline change and is varied in model simulations. Communities essentially assume that a beach will erode in the future at a rate similar to the past, and the time scale τ determines how far into the past they consider. A separate sediment reservoir (and therefore a unique sand cost) is associated with each shoreline cell. Knowing both the erosion rate and sand cost at each shoreline cell, the locally optimal time to nourish is found using Figure 3. As time steps forward, the erosion rate is updated. When the time since the last nourishment equals the optimal nourishment interval T*, nourishment takes place, and the shoreline cell is extended by setting the beach width to x0. If the beach width is greater than x0 at the time of nourishment, no change occurs at the cell, and the sediment reservoir and sand cost remain unchanged. A shoreline cell will not be nourished if the projected erosion rate is negative.
 The size of the reservoir for a single shoreline cell is determined such that after 150 years of drawing from the reservoir to hold the shoreline in place against 1.5 m yr−1 of erosion, the sand cost will go from the initial cost (c0) to the maximum cost (cm) [McNamara et al., 2011]. Through all simulations, the initial cost and maximum cost are set to 0.5 $ m−3 and 30 $ m−3, respectively. In some simulations, a string of 100 shoreline cells is connected to a common reservoir such that when one cell nourishes, the cost of sand rises for all cells within the string. For these simulations, the size of the common reservoir is the individual reservoir size multiplied by 100. The sand cost increase parameter λ is re-calculated such that if all cells constantly draw on the reservoir to mitigate 150 years of 1.5 m yr−1 of shoreline retreat, the cost will reach the maximum cost.
 All simulations are run for 150 model years. The first 40 years are characterized by transient behavior as the system settles toward its steady state or attractor. This 40 year transient period indicates an intrinsic time scale of the coupled economic coastline system; it is the time required for the fully coupled system with its inherent feedbacks to evolve to its longer-term attractor. Therefore, our analysis uses results from 40 to 150 years. Initial model experiments incorporate a straight plan view coastal morphology forced with zero degree incident waves, wave height of 1.7 m, and 10 s wave period. This describes a straight beach where cross-shore perturbations diffuse through alongshore sediment transport. The overall shoreline retreat rate was set to 1.5 m yr−1 and is within predicted limits of shoreline retreat rates due to sea level rise [Zhang et al., 2004; Moore et al., 2010].
 We find that spatial interactions lead to significant alongshore variability in the total present value of coastal property (v). We use the standard deviation as a measure of variability in v. Variability in v is largest for experiments when the standard deviation in forced erosion rate equals zero (σc = 0). Variability in v decreases with increasing σC and is a minimum for the highest value of σc (Figures 4 and 5). We find that the high variability in v (Figure 4a) is negatively correlated with the high variability in the alongshore distribution of total nourishments (Figure 6, black line). The coefficient of correlation between these alongshore signals is −0.85, indicating that when nourishment frequency is high, v is low, and vice versa. Likewise, when the variability is lowest in v (Figure 4d), the alongshore distribution of total nourishments is also less variable (Figure 6, gray line).
 The alongshore variability found in both v and in total nourishments is the result of feedbacks between the economic and physical processes. An erosion feedback on straight coastal morphologies with low angle wave climates occurs because as a cell nourishes, it experiences a higher erosion rate from sand being lost to neighboring cells through the gradient in coastline position that is created after nourishment. Thus, the community in that cell forecasts a higher erosion rate. This forecast makes the nourishing cell more likely to nourish sooner since it is optimal according to Figure 3 to decrease the nourishment interval with increasing erosion rates and associated forecasts. Additionally, neighbor cells see decreases in their erosion rates and forecast lower erosion, prompting them to decrease their nourishment rate (Figure 3). Another feedback exists between the cost of sand and optimal nourishment rates for part of the parameter space. As the cost of sand increases from declining sediment resources, the locally optimal nourishment interval decreases. This feedback causes an already nourishing cell to be likely to nourish more and more frequently, thereby enhancing the spatial variability in v. Cells move rightward in the optimal nourishment parameter space (Figure 3) as erosion rate feedbacks take hold and upward as the cost of sand rises. Cells adjacent to frequent nourishers move left (decreasing erosion rate) in the optimal nourishment space and occasionally up corresponding to infrequent nourishments.
 A partial phase space representation with axes consisting of nourishment effort and erosion rate conveniently captures model behavior that would otherwise be difficult to convey due to the high dimensionality of the system. Nourishment effort is the cumulative volume of sand used at a cell for nourishment divided by the time passed since the beginning of the model run. A phase space density plot is obtained by summing the number of cells occupying a given region of the partial phase space at any time. Model experiments corresponding to no variation in erosion forcing (σc = 0) and a 10 year erosion averaging time scale (τ) yield five frequently occupied regions of the phase space (Figure 7a). This grouping of regions is indicative of the geometry of the attractor governing the behavior of the coupled system. Area I of Figure 7a, where nourishment effort and erosion rate are highest, corresponds to cells with negative total present value (v). Negative v is possible not just in our modeling but, as observed for the case of the East Coast of the U.S., a large share of the cost of nourishment can be subsidized by the federal government. These cells become locked into frequent nourishment and high sand costs. For these cells, spatially myopic nourishment benefits neighboring cells while it devastates the local property value. For this reason, we refer to them as “suckers.” If a cell is not a sucker, then it occupies any of the other regions either exclusively or for some part of its trajectory in the partial phase space. The counterpart to the sucker cell is the “free rider” cell, which is characterized by low nourishment frequency, a low erosion rate, and therefore low nourishment effort. Free riders are so named because they rely on the efforts of other cells to contribute benefits to them for free. Free rider cells begin in the vicinity of area II and move to area III if they nourish. If a free rider nourishes, it will never return to II and instead move between areas III and IV. Not every cell is a free rider for an entire simulation. There are a small number of cells that transition from free rider to sucker behavior. Area V is an unstable region occupied by cells that could transition to being a sucker. For example, a cell could begin near II, pass through III and IV en route to V. At area V, a cell may move back to IV or the erosion feedbacks may force it to I. Keeping all variables in the previous example the same while increasing the standard deviation erosion forcing to 2 × σc changes the dynamics of the system (Figure 7b). There is now only one distinguishable region, area I. The partial phase space is contracted horizontally and vertically so that cells can no longer be described as sucker or free rider. The large random changes to the shoreline (from increased σc) cause cells to forecast more variable erosion rates with the additional consequence of preventing erosion signals due to nourishment from appearing in the erosion rate forecasts. The less a nourishment alters neighboring (or a cell's own) erosion projections, the less the erosion feedback is able to take hold. The increased randomness in projected erosion rates leads cells to collectively obscure any vertical distinctions in the phase space density, while the lack of long term suckers and free riders leads to the horizontal focusing.
 We explore the erosion averaging time scale (τ) for values of 10 and 30 years. In Figure 5, increasing τ reduces the alongshore variability in total present value (v). When τ is large, a cell is less likely to be influenced by recent trends in beach width. This effect occurs because it takes a longer trend in shoreline change to alter the forecasted erosion rate. Similarly, cells with shorter τ are more influenced by short trends and stochastic variations and are more susceptible to changing the forecasted erosion rate if a neighboring cell nourishes or vice versa. The partial phase space density representation for the longer τ (30 years) and zero standard deviation in erosion forcing (σc = 0) (Figure 7c) shows a similar structure to the results with a shorter τ (Figure 7a). While there are distinct areas in Figure 7c (I, II, III), their closeness means that free rider and sucker behaviors do not occur with the same magnitude as in Figure 7a. Distinct regions are observable because adjacent nourishments still show up in a cell's erosion calculation, but the 30 year τ means a signal due to neighboring beaches makes less of a contribution to the erosion forecast, leading to smaller, but still distinct, moves in the phase space. As in simulations employing 10 year τ (Figures 7a and 7b), increasing the standard deviation of erosion forcing from 0 × σc to 2 × σc causes the coupled system to a occupy a single region (I) in the partial phase space (Figure 7d).
 We also explore whether a spatially forced pattern of erosion can appear in the alongshore distribution of total present value (v). Along a straight coastline, we impart a spatial erosion signal that consists of 100 km stretches of coast with of a mean erosion rate of 1.5 m yr−1 that are interrupted by 20 km stretches with a mean erosion rate of 3 m yr−1. The standard deviation in erosion forcing is set to zero (σc = 0). From visual inspection of v we find that the spatial pattern is less easily recognized when τ = 10 and more pronounced when τ = 30 (Figures 8a and 8b). When τ is large (Figure 8b), cells within the increased erosion region are better able to lock into the long-term trend of erosion compared to the simulation with smaller τ (Figure 8a). As in previous results, increasing τ leads to decreased alongshore variability in the v.
 We also compare the effect of a common reservoir on the total present value (v) for coastlines with and without a spatial pattern of erosion. Reservoir commons reduce alongshore variability in v for a straight coastline (Figure 8c) and enhance the signal shredding effect of the spatially forced erosion signal (Figure 8d). Any alongshore variation of sand cost is eliminated because each cell is tied to the same reservoir. Therefore, the sand cost feedback is inhibited in common reservoir simulations. Individual cells still become caught in the erosion feedback, but with a sand cost that increases more slowly. With each individual withdrawal from the reservoir, a runaway nourishing cell will not have nourishment costs as high as those in individual reservoir simulations, thereby reducing the negative troughs in v. Comparing the straight coastline simulation with a reservoir commons (Figure 8c) to the initial straight coast simulations (Figure 4a), we note the emergence of areas of highly reduced alongshore variability in v (Figure 8c). This is due to the absence of the sand cost feedback such that reservoir commons have the ability to further shred spatially forced patterns of erosion. When we force a straight coastline full of cells connected through reservoir commons with a spatial pattern of erosion, the spatial erosion pattern is no longer easily distinguished in property value. This enhanced signal shredding effect occurs for two reasons. The cost of sand and therefore the cost of nourishments no longer strongly reflect which cells nourish more frequently since the sand cost is uniform. Also, inhibiting the sand cost feedback leads to regions of highly reduced alongshore variability as observed in Figure 8c.
 A final set of model simulations are performed to explore the role of emergent cuspate features in human coastline interactions. Prior to the first of these simulations, we obtained the mean (Figure 9b) and standard deviation in erosion rate at each shoreline cell along a cuspate coastline during a 20 year simulation without nourishment that was forced by the wave climate shown in Figure 2 and with 1.5 m yr−1 of shoreline retreat. For the first simulation, we take these cell by cell erosion statistics and apply them cell by cell to a straight coast. The incident wave angle is set to zero along the straight coast. Forcing a straight coastline with cuspate erosion statistics serves as a basis for comparison to coupling along the cuspate coastline. The resulting total present value (v) along the straight coastline (Figure 9c, black line) reflects the mean erosion rate (Figure 9b) where higher erosion rates correspond to more negative v due to more frequent nourishment and regions with less erosion correspond to higher v. The variability in v is of the same magnitude as found in straight coast model simulations where the standard deviation in erosion forcing was equal to σcs (Figure 4c).
 The second model simulation takes the cuspate coastline forced by the WIS 509 derived wave climate depicted in Figure 2 along with a 1.5 m yr−1 mean erosion rate and zero standard deviation. The resulting distribution in total present value (v) (Figure 9c, red line) along a cuspate coastline is different from the straight coastline forced with cuspate spatial erosion statistics (Figure 9c, black line). The property value along the cuspate coast is characterized by large and negative v at cape tips (Figure 9a, region I) that are 1–2 orders of magnitude more negative than v on a straight coastline for the corresponding regions. Within cusp bays (Figure 9a, region II) where the wave climate is diffusional due to wave shadowing Ashton and Murray [2006b], the values for v are 1 order of magnitude more negative than predicted by the straight coast simulation.
 Along a cuspate coastline without human intervention, there are characteristic patterns of erosion and accretion over long time scales due to gradients in sediment transport that reinforce the emergent cuspate morphology (Figure 9b). On short time scales (months), local stretches of coastline can temporarily come to disequilibrium with the long-term wave climate as a result of stochastic variations in local wave climate and in wave shadowing due to the large cuspate features. If left unmanaged, these shoreline states quickly return to equilibrium. Figure 10a shows the shoreline position through time of three adjacent cells within the red box of Figure 9a. Solid lines correspond to the cuspate coast simulation with nourishment intervention and dashed lines correspond to simulations without nourishment intervention. The green lines (solid and dashed) correspond to the alongshore cell positioned 558 km from the left edge of the model domain (Figure 9a) and is where we find a large negative total present value (v) of −2390 × 1000 $ house−1 and is indicated by a green dot in Figure 9c. The red lines correspond with the cell downdrift to cell 558 (green lines) and is 559 km from left edge of the model domain while and the blue lines are for the cell updrift of cell 558 and is 557 km from the left edge of the model domain. v at the updrift (557 km) and downdrift (559 km) cells are 515 × 1000 $ house−1 and 530 × 1000 $ house−1, respectively. In the simulation without nourishment (Figures 10a, dashed lines; and 10b), gradients in sediment transport force the unstable convex seaward configuration of cells to be concave seaward. Figure 10b reflects this adjustment where cell 558 km (green lines) experiences a temporarily higher rate of erosion until the equilibrium coastal configuration is reached at which point all cells experience the same erosion rate oscillating about the rate of sea level rise. Human intervention prevents the shoreline from returning to equilibrium. In Figure 10a (solid lines), the green cell experiences the higher erosion rate associated with the shoreline returning to equilibrium and nourishes. The nourishment of the cell returns the local configuration back to the disequilibrium configuration. This response increases the forecasted erosion rate of the green cell, causing it to nourish sooner while driving up the cost of sand (Figure 10c). In this example, natural cuspate coastline dynamics intensify the erosion feedbacks leading to suckers previously observed along straight coastlines.
 The total present value (v) of cells near the cape tips (Figure 9a, region I) reflects dynamics associated with physical location. To the left of cape tips where the erosion rate is highest, v is low or negative. To the right of cape tips where the erosion rate is low, little nourishment is necessary and v is high. The large negative values at cape tips do not appear in the simulation where a straight coast is forced with the cuspate erosion pattern. This relatively modest effect on values occurs because along a straight coast when a cell is eroded, sand is transported in from both sides by adjacent cells due to diffusive processes. Along a cuspate coastline, the area immediately left of cape tip cells does not benefit by such transport from neighbors because the effective diffusivity is low [Ashton and Murray, 2006b].
 Beach nourishment that counteracts coastal erosion is an internationally practiced form of coastline management, and while model experiments presented here involve straight and cuspate morphologies characteristic of the U.S. East Coast, our results are applicable to other coastlines where communities myopically engage in nourishment. For example, France, Denmark, and the United Kingdom are three such countries that have engaged in beach nourishment with all or partial funding supplied at the local level [Hamm et al., 2002]. We explore two coastline types characteristic of the U.S. East Coast by necessity; parameters used to generate optimal nourishment intervals come from data collected along the North Carolina coast [Gopalakrishnan et al., 2011] and are not available for any other coastal region in the world.
 A limitation of the physical model is that it abstracts from the behavior of individual storms and other short time scale processes and collapses these processes into the standard deviation in erosion forcing (σc). A key economic model limitation is that it abstracts from regional market forces that influence property values independently of the physical system; to develop insights about the coupled model, we focus only on value changes associated with beach width. Broader market forces undoubtedly contribute to what communities decide are locally optimal nourishment strategies; however, it is well established that communities inform future nourishment based on location specific histories of shoreline change [Psuty and Douglas, 2002]. We do not explore the possibility of stopping nourishment as it is beyond the scope of this paper. We also do not explore the scenario where the cost of nourishment is partially or fully supported by subsidies that are then removed. In reality government subsidization would vary over time; however, here we assume the commitment stays the same in order to isolate the effects of the nourishment feedback itself. Lastly, erosion occurs for reasons other than sea level rise, e.g., alongshore gradients in Qs on scales larger than the model domain. The purpose of this model is to explore nourishment under conditions of shoreline retreat, and we do not explore scenarios of no erosion (for which shoreline stabilization would not be necessary).
 These model experiments are not quantitative predictions for specific coastal locations and are instead simulations exploring the general features of long time scale emergent patterns of human-coastline behavior. The coastline model contains a representation of the dynamics that drive large-scale coastline change. Previous work has shown large-scale shoreline change along sandy wave-dominated coasts is strongly related to the local wave climate, wave shadowing, and alongshore sediment transport. The zero degree incident wave climate we use to force straight coastlines represents one possible “low angle” wave climate that leads to diffusion of the coastline shape through gradients in alongshore sediment transport. There are many low angle wave climate configurations that cause large scale diffusion of the coastline shape. However, our aim is to explore generally the effect of the diffusional dynamic of alongshore sediment transport and not to simulate a particular straight coastline and we therefore choose a symmetric and purely diffusive wave climate. Additional simulations (not shown here) show results for straight coastlines do not change when using a distribution of low angle waves versus the simpler choice of zero degree wave angle forcing. The coastline model leads to coasts that share some of the main characteristics with coastlines observed in nature [Ashton and Murray, 2006a, 2006b]. Likewise, the economic framework for beach nourishment decisions [Smith et al., 2009] matches data collected along the North Carolina coast [Gopalakrishnan et al., 2011].
 The term free rider is used in economics and social sciences to describe the inaction of an individual or institution which allows that individual or institution to receive benefits derived from the actions of another. In the social sciences, the individual who exerts more effort due to the existence of the free rider is fittingly named the “sucker” [Kerr, 1983]. The “sucker effect” empirically found in Kerr  refers to individuals in small groups who lose motivation because they do not want to be suckers; they may otherwise be willing to contribute to the group, but they see free riders and react by free riding themselves. In essence, the sucker effect is a negative feedback. In our case, there is no “sucker effect” because sucker cells react to free riding cells by becoming even bigger suckers (a positive feedback). Along sandy coastlines, the combination of erosion feedbacks and the sand cost feedback can lead to an alternating pattern of suckers and free riders that persists. The peaks in the total present value of Figure 4a correspond to free riders and the troughs correspond to suckers. As rates of shoreline retreat increase, this process leads to greater alongshore variability in the total present value.
 It would be natural to suppose that property situated in a high-erosion location would have a lower value and vice versa. This would be a situation where property value reflects coastal erosion. Our works suggests the counter intuitive result that spatial signals of property value and spatial patterns of erosion become separated such that one could not predict how property value is distributed along a coastline even if they had information of historical “pre-human” erosion patterns. Negative total present value suggests that the true economic value of the property is negative. Despite a positive sale price, this could occur because the value of the property is being subsidized by contributions from the federal government, typically through beach nourishment in our case study region. The implication is that the property would become worthless were the federal government to stop paying for nourishment.
 The positive feedback that creates suckers and free riders has two economic preconditions: spatial myopia and the presence of mixed private/public benefits. Without spatial myopia, prospective sucker cells would realize how their actions would influence actions of their neighbors and, in turn, feedback on themselves. We expect a spatially extended view to sever the positive feedback that generates patterns of suckers and free riders. The mixing of private and public benefits is more subtle. Many studies of collective action focus on individual contributions that benefit the group as a whole [Ostrom et al., 1994], but nourishment provides benefits to individual cells and the group of cells separately (nourishment is an impure public good in the language of economics). Even if a cell's nourishment actions contribute large benefits to its neighbors through sand transfer, the cell may still find it locally optimal to nourish because it retains enough sand to justify the cost based on private benefits alone. In that sense, sucker cells are still behaving in their own best interests but are not as well off as they would be if they were able to coordinate with the free riders.
 The extent of the feedback depends on other key parameters. Model simulations (not shown here) show the sucker feedback is robust up to cells sizes (7 km) greater than the historical average alongshore extent of U.S. East Coast nourishments projects (2–5 km). In simulations (not shown here) with larger cell sizes (beyond 7 km), edge effects become an important dynamic but this is beyond the scope of our paper and is not discussed further. Increased variation in erosion forcing reduces the degree to which neighboring nourishments affect a cell's erosion calculation, therefore inhibiting the feedback that gives rise to the alternating pattern of suckers and free riders. The prevalence of suckers is also reduced when management decisions are made with a longer view of past shoreline change, since projected erosion rates become less sensitive to neighboring nourishments. The shorter view of past shoreline change also has the effect of shredding spatial patterns of erosion, an effect that becomes more pronounced when stretches of coastline cells are tied together through a common sand reservoir. Recent research in geomorphology shows that sediment transport can shred environmental signals [Jerolmack and Paola, 2010]. In our case, coupling geomorphology to human behavior can also shred signals. Model simulations along a cuspate coastline predict the occurrence of suckers and free riders in cusp bays while cells located just to the left and right of cape tips are controlled by physical location. In cuspate bays, the feedbacks due to human coastline coupling interact with transient undulations in shoreline position leading to the presence of more suckers, which can be as costly as cells located at the heavily eroding cape tip locations.
 Regions along the U.S. East coast where communities are in close proximity to each other and are not unified in their nourishment efforts are most susceptible to a future dominated by suckers and free riders. Historical nourishment data are highly fragmented, making the town-by-town analysis necessary for a direct comparison to our model difficult. We also note that nourishment has become widespread only since the 1960s and our simulations suggest an approximately 40 year period before sucker/free rider behavior begins. Nonetheless, it is possible we are already beginning to see early signs of sucker/free rider patterns in data from The Program for the Study of Developed Shorelines (PSDS Beach Nourishment Database). Along the New Jersey coast, Avalon is a borough situated between the city of Sea Isle and another borough, Stone Harbor. Since 1987, Avalon has nourished their beaches six times while Sea Isle nourished once in 1987 and Stone Harbor nourished once in 2003. At another area in New Jersey, the township of Long Beach has engaged in nourishment 15 times since 1956, while the flanking communities of Surf City and Beach Haven together have nourished 7 times with their most recent nourishment in 1963. Although these instances only reflect differences in nourishment behavior, our model simulations suggest subsequent corresponding deviations in property value if such practices persist. Data from PSDS also reveal interesting nourishment behavior in North Carolina that is consistent with suckers and free riders. The adjacent coastal towns Wrightsville Beach and Carolina Beach are approximately equal in alongshore extent but display very different nourishment history. Since the 1950s when nourishment began, the cumulative volume of sand used by Carolina Beach is more than twice that used by Wrightsville Beach. This same dynamic is observed in Ocean City and Atlantic Beach, New Jersey. Nourishment began in the 1940s, and at the present time, Ocean City has nourished 2.5 times the total volume of sand compared to Atlantic City.
 As sand resources decline, the cost of sand will continue to rise and coastal towns may become more prone to a prevalence of sucker and free rider communities. As this happens, the value of coastal property in sucker locations could eventually destabilize real estate markets and lead to future abandonment. The looming threat of reduced government assistance for beach nourishment could accelerate this process. In this scenario, who wins and loses is ultimately unrelated to physical environmental forcing along straight coastlines. Along cuspate coastlines, the same is true and potentially magnified along cuspate bays. However, at cape tips, winners and losers are strongly tied to location and physical forcing. Given these results, future management may find it beneficial to assess regional wave climates, if nourishments are subject to low angle waves that tend to smooth the coastline, the best strategy may be to coordinate beach nourishment aiming to minimize losses occurring from gradients in shoreline position. Along coastal stretches where alongshore transport's smoothing effect is less pronounced and erosion signals are dominated by the evolution of large-scale coastline shape, such as cape tips along the North Carolina coast, it may be more beneficial to enact policies that reflect the local environmental response to wave climates.