Abstract
This article examines how the spatial configuration of forest ownership influences risk-mitigating behavior of public and private forestland owners over time. We use a game theoretic framework to examine how the public landowner’s investment in fuel management influences, and is influenced by, decisions made by private forestland owners. We find that spatial configuration and location affect the timing and amount of fuel treatment on the landscape. There is less investment in fuel management on landscapes characterized by fragmented ownerships. We also find that the type of strategic interaction between landowners depends critically on the shape of the wildfire damage function. (JEL Q23, Q24)
I. Introduction
Wildfire is an intrinsic ecosystem process throughout much of the western United States (Pyne, Andrews, and Laven 1996); however, the number of catastrophic, uncharacteristically severe, and uncontrollable fires increased dramatically in recent years. Every year, in spite of the millions of dollars federal agencies spend on fire suppression, thousands of homes are destroyed by wildfire.1 Increasing numbers of landowners are choosing to live in the wildland-urban interface (WUI)— the area where humans and their development meet or intermix with wildland fuel (Federal Register 2001)—because of the amenities provided by the natural, forested setting, both on and off their own land (Garber-Yonts 2004). This trend has added to the complexity and immediacy of the wildfire problem. While little can be done to prevent wildfires from occurring, landowners can undertake actions to reduce the amount of damage caused by fire. Wildfire responds to changes in the amount and configuration of forest fuels, therefore a forest can be managed through the use of various hazardous fuel reduction treatments to minimize wildfire risk (Agee and Skinner 2005; Amacher, Malik, and Haight 2005; Graham et al. 1999; Hirsch and Pen- gelly 1999; Pollet and Omi 2002; Yoder 2004).
Because fire moves through a landscape, the spatial pattern of fuel treatments can influence the landscape pattern of wildfire damage risk. Positive spatial externalities created by fuel treatments have been documented (Hann and Strohm 2003) and found to be especially significant in the case of large wildfires (Finney 2001; Gill and Bradstock 1998). Good management on the individual unit, which limits the accumulation of forest fuels, decreases the risk of fire damage on neighboring forests, whereas poor management on the individual unit increases the risk of fire damage on neighboring forests. The growing number of landowners living in the WUI consider the state of neighboring forests when making decisions about investment in fuel treatment (Brenkert-Smith, Champ, and Flores 2006; Monroe and Nelson 2004). Hence the pattern of fuel treatment on the landscape depends, increasingly, on the pattern of private and public landownership and, in particular, the degree of ownership fragmentation and the pattern of values across the landscape. To understand the risk of fire damage on a landscape, then, we must first understand how landowner risk-mitigating decisions interact.
In previous studies, Butry and Donovan (2008), Konoshima, Montgomery, et al. (2008), and Konoshima, Albers, et al. (2010) explicitly account for the spatial externalities associated with fuel treatments and use stochastic optimization models to determine the optimal spatial allocation of fuel treatment effort for a single landowner or decision maker. Crowley et al. (2009) explore the effect of spatial fuel treatment externalities on optimal fuel management when there are two adjacent private landowners.
We extend this literature by modeling fuel treatment decisions across a range of landownership patterns. We explore how these different spatial patterns of ownership affect the optimality of landowners’ fuel treatment decisions when (1) fuel treatment on one site affects fire risk in adjacent sites, (2) the values at risk involve spatial externalities so that fire damage on one site affects the value of adjacent sites, and (3) landowners behave strategically, knowing that their fuel treatment choices may influence the level of fuel treatment on adjacent sites. We examine these issues in three steps. First, we model the effect of the spatial externalities associated with fuel treatment for a range of ownership patterns by defining two landowners, which we call public and private, with objective functions that are functionally equivalent (e.g., private owners care only about private values generated on private land, and public owners care only about public values generated on public land). We specify fire damage to be linear in fuel stock so that the deviations in the outcomes from social optimality are due only to the fuel treatment externality. Second, we examine the effect of heterogeneous objective functions by adding amenity values generated on public land to the private objective function so that there are externalities associated with the values at risk. Finally, we examine strategic behavior by specifying nonlinear damage functions: one with diminishing marginal returns to treatment and one with increasing marginal returns to treatment. Because the marginal benefit of fuel treatment depends on the overall level of fuel treatment on the landscape, landowners respond strategically to induce more or less treatment on neighbors’ land.
To explore the interaction between owners on various spatial patterns of ownership, we develop a spatially explicit game theoretic model. We simulate a game between public land managers and private landowners for a range of stylized, but typical, ownership patterns and settings, and compare the pattern of fuel management that emerges from the game to the socially optimal fuel treatment pattern. Our modeling framework is described in Section II, along with our approach to modeling fire and fuels. The base case, where landowners’ values at risk on each land unit are equal and fire damage is linear in fuel stock, and a set of scenarios that allow ownership objectives to differ and allow owners to display strategic behavior are described in Section III; and the resulting outcomes are described in Section IV. We conclude with a discussion of the characteristics of the landscape ownership pattern, landowners, and damage function that determine when multiple landowners generate a socially preferred amount and location of fuel treatment on a landscape, and when they do not. Public agencies can use these scenarios to assess the likelihood of adequate fuel treatment in different real-world settings and the likely response of private landowners’ fuel treatment patterns to changes in public fuel treatment decisions.
II. Modeling Framework
We approach the wildfire problem using a spatially explicit, dynamic, game-theoretic model. The game is set on a forested landscape with mixed ownership: public and private. Similar in spirit to Albers (1996), Swallow and Wear (1993), and Swallow, Talukdar, and Wear (1997), all spatial interactions between adjacent units are considered within a dynamic choice setting. In the model, spatial interactions include the effect that fuel load on one unit has on neighboring units’ fire risk, in addition to the public good and amenity values generated by other landowners’ units in the landscape. These effects are included by their explicit consideration in landowners’ fuel management decisions. The setting is dynamic because individuals are forward-looking and recognize that optimal decisions in future periods will depend on whether fire has occurred in the interim. Unlike the models of sole ownership of Albers (1996), Swallow and Wear (1993), and Swallow, Talukdar, and Wear (1997), and similar to the land conservation framework of Albers, Ando, and Batz (2008) but in a dynamic setting, we examine these decisions on a landscape with mixed ownership and owners with different objectives.
Fire, Fuels, and Suppression
Fuel treatment affects the behavior of a forest fire in at least two ways. First, reducing fuels slows the spread of fire through treated areas. A slower spread rate reduces the probability of fire arrival at sites beyond the treated areas. Second, reducing fuels reduces the intensity of the fire and, hence, the severity of damage when a treated area does burn.2 Investments that reduce the probability that a damaging event occurs are referred to as self-protection, while investments that reduce the severity of damage when a damaging event occurs are referred to as self-insurance (Ehrlich and Becker 1972). In our analysis, we treat fuel treatment as self-insurance. That is, landowners take the probability of a fire occurring on their property as independent of the fuel stock on their property, as in Amacher, Malik, and Haight (2005, 2006). That means that when landowners choose to treat fuels, they seek to reduce damage to their own and nearby properties when fire arrives, but they neglect the effect on the spread of fire across the landscape. In contrast, Konoshima, Montgomery, et al. (2008) explicitly model fire spread from different ignition points but neglect the effect of fuel treatment on fire severity, so that when a fire burns a unit, it is a total loss regardless of fuels. While we believe our assumption for this analysis is reasonable for individual landowners,3 we recognize that the socially optimal pattern of fuel treatment should be selected on the basis of both the self-protection and self-insurance aspects of fuel treatment, and we consider the implications of this omission in the concluding discussion.
Although fire affects the entire landscape, its particular effect on each unit depends on the prefire fuel stock both on the unit and on neighboring units. The differential impact of damages arises because, when fuel stocks are high, fires burn more intensely and cause more damage, but when fuel stocks are low, the opposite is true. To illustrate, a high-intensity fire might kill all the standing trees and completely destroy structures, thereby eliminating all amenity values and destroying private property values, while a low-intensity fire might kill only the low shrubs and debris, causing little damage to existing structures and leaving amenity values largely intact. While fire can provide ecological benefits, because our focus is the WUI where damage tends to outweigh the ecological benefits, we do not consider this possibility.
Once a fire starts, we assume that an exogenously determined level of full force suppression occurs. This assumption is a reasonable representation of current fire management—especially in the WUI, where protection of structures is a dominant concern— because in state and federal land management agencies, fire suppression and fuel treatment decisions are made by different groups of people, at different points in time, and using different budgets (Hesseln 2001; US GAO 2005). In contrast, Crowley et al. (2009) focus their analysis on the trade-off between fuel treatment and suppression and, as a result, find that the greatest inefficiencies in fire management are caused by free-riding on public provision of fire suppression effort.
Values at Risk and Fuel Stocks
Let there be i = (1,. .., N) land units owned by either a private individual or a public agency and t =(0,...,T) time periods. Land units are located in the fire-prone WUI and contain flammable forest fuels. In what follows, we use scalars, indexed by unit and time, to represent actions taken and values generated on an individual unit i at time t, and vectors, indexed only by time, to represent the full set of actions taken or values generated on the landscape at time t. For example, at the beginning of each time period t, public and private owners simultaneously choose whether to treat fuels on each unit i to minimize expected loss from fire damage plus cost of fuel treatment. Given the decision to treat, we assume the amount of fuel removed on unit i at time t, xit, is the amount required to reduce fuel stock to a “safe” standard. This generally depends on local standards and may involve removing flammable plants, pruning low branches to raise crown height, thinning tree stands to separate trunks to safe distances, and so on (e.g., see Sunriver Owners Association 2003). The configuration of fuel treatment actions on the landscape at time t is represented by the vector xt =(x1t, x2t,...,xNt).
In this model, fire affects fuel stocks, sit, private structure value, hit, private amenity value, ait, and public goods value, pit, on each unit i at time t. The configuration of fuel stocks, private structure values, private amenity values, and public goods values on the landscape are st, ht, at, and pt, respectively. Structure value, hit, is simply the value of the physical structure and is generated and consumed on the same unit i. Private amenity value, ait, is generated on unit i but may be consumed by private landowners on multiple neighboring units. The value of amenities generated on unit i is capitalized into the market property value of each of the units on which it is consumed. For example, consider a scenic vista of unit i from unit j . The market value of unit j will be enhanced by the view, but a fire on unit i might destroy that value. Therefore the fuel management decisions on i affect the property value of j. Public goods values, pit, such as biodiversity, ecosystem function, and carbon sequestration, generated on unit i benefit society at large and are not capitalized into private property values.
We assume that public landowners consider only public goods values and that private landowners consider only those values that they privately enjoy (e.g., private structure and amenity values) when making fuel management decisions. Private structure values are generated on private land. We assume, further, that public goods values and private amenity values are generated on public land. We recognize that this assignment of values is a simplification; private landowners may consider public goods values on their property in the interest of good land stewardship and may also manage for private amenity values, recognizing the value to themselves of doing so; public landowners surely worry about how their decisions might influence private market values on adjacent properties in the interest of being good neighbors. However, this simplification allows us to focus on the primary distinction between public and private management objectives (management for public goods vs. management for private values) and define three distinct combinations of production/consumption that will influence the direction of the externality: public/society at large, public/private, and private/private. Within this framework, a public objective that omits private values can contribute to inefficient outcomes, just as the private objective that omits public goods can.
Damage Function
Landowners choose to treat fuels, or not, at the beginning of each time period. After the decision is applied, fire may or may not occur. If there is no fire, posttreatment fuel stock on unit i evolves according to the following equation:
[1]where G is a growth function. Structure, amenity, and public goods values are unchanged so that hi,t +1 = hit, ai,t+1 = ait, and pi,t+1 = pit. We do not consider timber values here, which would grow over time in the absence of fire.
If there is a fire, we represent its impact using a damage function that gives the proportion of k lost in unit i as a function of the configuration of posttreatment fuel stock on the landscape, 
, where k = s, h, a, p indicates what is damaged. In this study, we assume that fire acts similarly on all four variables. This relationship holds when the effect of fire is, in general, closely correlated with fire intensity (e.g., flame length) and if fire intensity depends on fuels. However, there are settings where this assumption may not hold. For example, it may be possible that a fire of moderate flame length improves wildlife habitat (a public goods value) while destroying scenic value (an amenity value). We don’t explore this possibility here, but keep the door open for future research. This assumption allows us to specify one damage function, 
, so that after a fire, fuels are
[2]and structures, amenities, and public goods are
To begin, we assume that the damage function for unit i is a linear function of the posttreatment fuel stock on every unit, so that
[3]The weight, ωij, represents the contribution of unit j’s posttreatment fuel stock, (sjt- xjt), to fire damage on unit i. The bigger the weight assigned to unit j, the more important that unit’s fuel stock in determining fire damage on unit i. This damage function is deterministic, an increasing function of the posttreatment, prefire fuel stock, bounded by 0 and 1, and continuous. We use a spatial weighting scheme of first-order, common boundary, rook contiguity (Anselin 1988) to represent the cross-boundary externalities inherent in the fuel treatment problem. This weighting scheme implies that land units that share common boundaries affect each other’s fire risk, while land units that contain common vertices or share no boundaries do not.4
After establishing the base case using a linear damage function, we then model two variations of the damage function: nonlinear with increasing marginal returns to fuel treatment and nonlinear with decreasing marginal returns to fuel treatment (Figure 1). These nonlinear variations represent two distinct real-world settings and allow for strategic behavior on the part of the landowners, because their fuel treatment choices influence the marginal returns to fuel treatment by their neighbors and, hence, their treatment decisions. In some settings, such as those characterized by extreme drought conditions, large expanses of beetle-killed trees, or forests where stand-replacing fires are typical, only by removing large amounts of hazardous fuels are landowners able to significantly reduce fire damage risk. This type of setting can be represented by a damage function that exhibits increasing returns to fuel treatment. However, in settings where vegetation is adapted to low-to-moderate intensity fire, such as the Ponderosa pine forests of eastern Oregon and Washington, much fire damage can be avoided by removing a small amount of hazardous fuel. This setting can be represented by a damage function that exhibits decreasing returns to fuel treatment.
![Damage Functions Showing Proportion of Value Loss for Unit i as a Function of Fuel Stock at the Time of Fire for (a) Equation [3], Linear Damage Function; (b) Equation [5], Nonlinear Damage Function Reflecting Decreasing Marginal Returns to Treatment; and (c) Equation [4], Nonlinear Damage Function Reflecting Increasing Marginal Returns to Treatment](https://le.uwpress.org/content/wple/88/3/496/F1.large.jpg?width=800&height=600&carousel=1)
Damage Functions Showing Proportion of Value Loss for Unit i as a Function of Fuel Stock at the Time of Fire for (a) Equation [3], Linear Damage Function; (b) Equation [5], Nonlinear Damage Function Reflecting Decreasing Marginal Returns to Treatment; and (c) Equation [4], Nonlinear Damage Function Reflecting Increasing Marginal Returns to Treatment
In this study, we specify two nonlinear specifications of the damage functions in equations [4] and [5] to represent nonconstant returns to fuel treatment.5 For increasing returns to fuel treatment, we use
[4]where 
from equation [3], and κ is a constant that determines the curvature of the function and which we set at 1,000, giving an upper limit of D i = 0.95.6 For decreasing returns to fuel treatment, we use
[5]where η is set at 0.95, giving a lower limit of 
.
Loss Function
Because public and private owners have different values at risk, the specification of the equation describing total loss from fire depends on whether the unit is publicly or privately owned. In this study, we model the private landowners as a single coordinated decision maker, internalizing fuel stock externalities among private owners, in order to focus on the interaction between public and private owners. In practice, private landowners often make fuel management decisions as a coordinated group of individuals in a housing development, as a homeowners’ group, or informally as a group of neighbors (Brenkert-Smith, Champ, and Flores 2006; Monroe and Nelson 2004). While this type of coordinated behavior is widespread throughout eastern Oregon including in Sunriver,7 Sisters, and Bend, such coordination may be less likely or more costly on landscapes with more-fragmented ownership patterns, and the implications of this possibility will be discussed in Section V. Our ongoing research further explores private-private interactions and the costs of coordination, which we discuss briefly below, but we maintain an assumption of costless coordination among private landowners in this study to maintain a focus on the public-private landowner interactions. Fuel treatment decisions on publicly owned units are made by a single public decision maker.
Total loss due to fire damage in time period t on privately owned units is given by equation [6], and loss on publicly owned units is given by equation [7]:
[6]
[7]where vt is aggregate private property value at the beginning of the period, which is determined by previous decisions, and vt +1 is aggregate private property value at the end of the period, which depends on the vectors of end-of-period (that is, postfire) amenity and structure values that, in turn, depend on the configuration of posttreatment fuel stock on the landscape. Likewise, Ptand Pt+1 are aggregate public good values at the beginning and the end of the period, respectively, where Pt +1 depends on the vector of end-of-period public goods values generated on each unit, which, again, depends on the configuration of posttreatment fuel stock on the landscape.
The Game
When landowners are forward-looking and consider how decisions in the current time period affect future decisions and outcomes, the game becomes dynamic. Specifically, our game is dynamic because decision makers recognize that their fuel treatment decisions in future periods will depend on whether fires occur in intervening periods and on their current period actions. Each landowner chooses the fuel treatment pattern that minimizes the discounted sum of expected fuel reduction costs and loss from fire, subject to the fuel stock growth and damage functions. Landowners will undertake fuel treatment only if they determine that the benefit—the expected reduction in fire damage—is greater than the cost.
The uncertainty in each period of the game is from fire and is completely resolved at the end of each period. There are two states of the world in each period; “fire” occurs with frequency ρ, and “no fire” occurs with frequency 1 - ρ. Let Vit(st) denote the expected present value of the aggregate cost plus loss on unit i, β ∈ [0,1] denote the discount factor, and C(xit, sit) be the cost of reducing fuel stock by xit on unit i when the current fuel stock is sit. The stochastic dynamic programming problem (or Bellman equation) is given by equation [8] for privately owned units and by equation [9] for publicly owned units for all periods t =1,...,T:
[8]
[9]Equations [8] and [9] add current loss due to fire, which occurs with probability ρ, and fuel treatment cost to the present value of future losses and treatment costs, both of which are functions of values at risk, fuel stock on the individual unit and neighboring units, and the probability of fire. A player’s optimal strategy 
is the choice of fuel treatment in time t that minimizes expected loss plus cost, given that the other player is also making fuel treatment choices to minimize expected loss plus cost.
In repeated or multiperiod games, the past can matter for two reasons: either because the players believe past behavior influences future behavior, or because past decisions affect the future environment in which the game is played (Fudenberg and Triole 1992). We assume that the past matters only through its effect on the environment in which the game is played. Past decisions about fuel removal influence the fuel stock (or “state variable”) in future periods. For this reason we use “Markov” or “state space” equilibrium strategies. A Markov perfect equilibrium (MPE) is a profile of Markov strategies that yields a Nash equilibrium (NE) in every time period, or “subgame” (Fudenberg and Triole 1992). The concept of MPE is a refinement of the more general NE with several advantages over the NE concept: MPE reduces the number of possible equilibria in dynamic games, thereby improving the predictive power of the model; by allowing only the state variable to affect strategic behavior, the impact of state variables on outcomes is made clear; and, finally, Markov models can be easily simulated (Maskin and Tirole 2001).
Socially Optimal Fire Risk Management
Socially optimal fire risk management is the landscape-level fuel treatment pattern that is best for society as a whole. The socially optimal fuel treatment pattern is found by solving the “social planner’s” problem in which fuel treatment decisions are made to protect all values, both public and private, in all periods t =1,...,T:
[10]The social planner’s optimal decision xi∗t( s t) is the choice of fuel treatment on all public and private units in time t that minimizes the sum of public and private expected total costs.
III. Base Case and Scenario Parameters
To explore the role of ownership pattern on fuel treatment, we solve the model described above for a two-period time horizon (t =0, 1) on a set of 10 three-by-three grids of ownership patterns, or “types,” that characterize the western U.S. landscape (Figure 2).8 While these types may appear stylized (Figure 3), the grid pattern they depict is quite common in the western United States.9 Ownership fragmentation, as determined by the number of public-private adjacencies, is greatest for the checkerboard pattern and least for the adjoining pattern. We chose parameter values that reflect relative values found in a variety of landscapes rather than those that represent one setting. In those choices, and sensitivity analysis across those values, we implicitly characterize the ecological and landscape scale. For example, a simulation in which one unit’s fuel load creates a large spatial externality represents either a small landscape scale or an ecosystem with fast-moving fires. To focus on the interactions within the grid of landowners, we control for edges by assuming that adjoining units outside the three-by-three focal grid receive the same treatment as their neighbor within the focal grid, except in the case of the checkerboard ownership pattern, where we assume that the checkerboard continues beyond the focal grid.
Ownership Patterns in 3-by-3 Grid
Ownership Map for Harney County, Oregon (Includes U.S. Forest Service, Bureau of Land Management, State of Oregon, Burns Paiute Reservation, and Private Land Ownerships)
Fuel treatment decisions are made and applied at the beginning of each time period t =0 and t = 1. Fire can then occur in each time period after the fuel treatment decision has been applied. Finally, the fuel stock and values are updated to the beginning of the next period. The fuel treatment decision in the second and final time period, t = 1, is made to minimize expected loss plus fuel treatment cost given the expectation of fire in that period.
The parameters we use for the fuel stock growth rate, posttreatment fuel stock, and probability of fire on the landscape are reasonable estimates for eastern Cascade forests dominated by Ponderosa pine (Agee and Lolley 2006; Everett et al. 2000). Fuel stock doubles each period if it is not burned; equation [1] becomes st+1= G(st,xt)=2(st- xt). We assume fuel treatment cost is positive and the fuel management technologies are the same for all owners. Fuel treatment cost on an individual unit i in period t is given by C(xit) = 1 + 0.5xit. We used these parameters in the base case and in each scenario. A summary of the parameters is provided in Table 1.10
Parameter Values for the Base Case Specification
The weighting scheme reflects the characteristics of the landscape (e.g., slope, dominant wind direction) and captures how fuel stock on the individual and neighboring units matters in determining fire damage. Fire behavior across a full boundary is more significant than fire behavior across a point or vertex, so we use a rook contiguity weighting scheme for our grid landscape, a common assumption in fire behavior models (Li et al. 2008). In the first-order common boundary rook contiguity weighting scheme, the characteristics of an individual unit and its four adjacent units contribute to the damage on the individual unit. In our base case, the impact of the four common boundary units and the individual unit i matter equally in determining the damage on unit i. We model three additional weighting parameterizations within this rook contiguity weighting scheme. One sets weights ωij = 0 for i ≠j to represent the case of no fuel treatment externalities; only the fuels on individual unit i determine the damages on i. The other two parameter scenarios give equal weight to the individual unit and a single neighboring unit; one gives weight to the neighboring unit to the right and the other to the neighboring unit above. These represent scenarios where the fuel stock on the upwind or downslope unit is significant in determining damage on the individual unit, and depending on the underlying ownership pattern, they lead to different outcomes.
In our base case, public and private landowners generate the same per unit value (set to 400), public good values are generated only on public units, private structure values exist on every privately owned unit, and there are no private amenity values. This starting point, where the two types of landowners have objective functions that are functionally equivalent, allows an exploration of the basic behaviors that arise from the spatial externality between adjacent units for 10 ownership patterns. We then model spatial externalities and differing owner objectives by setting amenity values generated on each public unit to 400 per unit. Amenity values, which appear in the private loss function, and the cost parameters were chosen for given fire occurrence probability, ρ, so that landowners undertake some positive level of fuel treatment.11
IV. Results
We use a numerical solution method, programmed in Matlab (Mathworks 2007), to solve for the equilibrium to the game represented by equations [8] and [9] and for the socially optimal outcome represented by equation [10] in the following four scenarios for each of the spatial weighting schemes: (1) a simple base case with linear damage functions, public goods values on public land, and structure values on private land, results shown in Figure 4; (2) linear damage functions with public goods and structure values plus private amenity values generated on public land, results shown in Figure 5; (3) nonlinear damage function with increasing marginal returns to fuel treatment and values as in the base case, results shown in Figure 6; and (4) nonlinear damage function with decreasing marginal returns to fuel treatment and values as in the base case, results shown in Figure 7. The public and private landowners’ objective functions, while qualitatively distinct, are quantitatively unique only for Scenario (2), where there are private amenity values on public land. For each scenario and ownership pattern, there is a unique pure strategy NE.12
Base Case Treatment Pattern Outcome in Each Period with and without Fire
Treatment Pattern in Each Period with and without Fire and Outcome from the Game for All Ownership Patterns When There Are Private Amenity Values on Public Units
Treatment Pattern in Each Period with and without Fire and Outcome from the Game from All Ownership Patterns When There Are Increasing Returns to Fuel Treatment
Treatment Pattern in Each Period with and without Fire and Outcome from the Game from All Ownership Patterns When There Are Decreasing Returns to Fuel Treatment
For each case, we show a measure of the difference between the socially optimal outcome and the outcome of the game between private and public landowners (Table 2). The departure from the social optimum is calculated as the difference between total expected loss from the game and from the social optimum, as a percent of the total expected loss from the socially optimal treatment pattern:
Departure from Social Optimum for the Base Case, Private Amenity Value on Public Land, and Increasing and Decreasing Returns to Fuel Treatment
[11]Base Case Results
For the base case, the damage function is linear (equation [3]), public and private values are equal, public values are generated only on public land, and private values are generated only on private land (amenity values, at, are zero in equation [6]). Figure 413 depicts the two sets of results shown for each ownership pattern: one for the treatment decision resulting from the game between two forward-looking landowners (equations [8] and [9]) and one for the treatment decisions resulting from the social planner’s optimization14 (equation [10]). Units marked with a “Y” are treated in the given period and those marked with an “N” are not treated. Because these values imply that public and private decisions are symmetric, (e.g., the treatment patterns of public and private owners “mirror” one another), we need only 5 of the 10 possible ownership patterns in Figure 4, where public (private) units are shaded and private (public) are not. Outcomes of the game for the remaining 5 ownership patterns are identical to those described in the figures, except that the owners are reversed.
The base case results demonstrate that (1) the outcome from the game between forward-looking landowners is not socially optimal, (2) the degree of the inefficiency arising from the spatial fuel load externality increases with ownership fragmentation, and (3) fuel treatment decisions by each owner type depend on ownership pattern.
In general, the outcome of the game involves less fuel treatment than is socially optimal. Because of the spatial externality, this free-riding result occurs even though landowners have perfect information about fire risk. The greatest departure from optimal as measured in equation [11] (and the least amount of fuel treatment) occurs on the landscapes with the most fragmented ownership pattern, with the most public-private adjacencies. To illustrate the point, compare the outcome of the game with the private corridor ownership pattern (Figure 4 “corridor” pattern) to the outcome from the game with the isolated ownership pattern (Figure 4 “isolated” pattern). In both cases, the center unit’s value is the same, fuel treatment cost is the same, and fuel treatment decisions on the surrounding units are the same. When the private center unit has public land all around it, as it does in the isolated unit pattern, it is not treated in the second period regardless of whether fire occurs. When that private center unit has adjacent private units on either side, as it does in the corridor pattern, it’s treatment depends on whether a fire occurs (e.g., it is treated if no fire occurs). This treatment difference arises because, in the corridor pattern, the positive externality from reducing forest fuels on the center unit provides additional protection to the two neighboring private units and increases the total benefit of treating that unit to private owners.
In this base case, fuel treatment decisions are driven by the spatial externality and the ownership pattern rather than the game’s interaction. With the linear damage function, the marginal benefit of additional fuel treatment to the landowner is constant for parcels with an equal number of same-owner adjacencies. That is, the calculus of the fuel treatment decision does not depend on fuel treatment decisions by neighboring landowners. Given the current period’s fuel stock, each landowner has a dominant strategy and there is a unique NE. Therefore, the inefficiency that results from the game is not a result of strategic interaction, nor is it a result of differing landowner objectives. It is a result of a coordination failure among public and private landowners stemming from their inability to internalize the external benefits of fuel treatment. To emphasize that the inefficiencies here arise solely from the spatial externality, we set ωij = 0 for i≠ j, eliminating the spatial externalities associated with fuel treatment and making the model aspatial, and we resolved the model. As a result, with constant returns to fuel treatment, the outcomes of the game are optimal, that is, they match the outcomes of the social planner’s problem.
Private Amenity Value on Public Land
When private amenity value is generated on public units, private landowners will want to protect amenity values on the public units where they are generated because the value accrues to them. For example, in a survey of Florida and Minnesota residents regarding defensible space for fire protection, individuals reported the value of wildlife habitat and recreation opportunities on public land surrounding their homes as a motivation for risk-reducing actions (Brenkert-Smith, Champ, and Flores 2006; Monroe and Nelson 2004). However, because only the public landowner can carry out fuel treatment on public units, there exists the potential for a divergence in the public and private landowners’ desired level of fuel treatment on those units. Nonetheless, because of the spatial fuel load externality, private owners can reduce risk to amenity values generated on public units by treating fuel on private units that are adjacent to those units.
As previously described, we model amenity value by adding to the value generated on public units (amenity values, at, are positive in equation [6]) and, as a result, the socially optimal outcome targets fuel treatment on the high-value public units, which leads to more fuel treatment overall than results from the game. Compared to the base case, in this game, private landowners invest more in fuel treatment for all ownership patterns because total private value on the landscape is greater. But their ability to reduce risk to the amenity values is limited by the extent of the spatial externality and, hence, the ownership pattern influences the private landowner’s pattern of fuel treatment. For half of the ownership patterns, the game’s outcome relative to the social optimum is worse than that of the base case (Table 2) because of the private landowner’s inability to directly protect values on public units. However, for the other half of ownership patterns, the outcome of the game is actually closer to the socially optimal treatment pattern than is the base case. These five ownership patterns include isolated private, private extending into public, public extending into private, and the two checkerboard patterns. Because these ownership patterns have more public-private adjacencies, they offer more opportunities for private landowners to protect amenity values through treatment of adjacent private units than the other ownership patterns, as shown in Figure 5.15
Returns to Fuel Treatment
In the base case, the relationship between weighted fuel stock and fire damage (i.e., the damage function) is linear (equation [3]), and the marginal benefit of fuel treatment is constant at all levels of weighted fuel stock. In this section, we explore two nonlinear relationships between weighted fuel stock and fire damage. First we look at the case where fire damage increases with weighted fuel stock at a decreasing rate and there are increasing marginal returns to fuel treatment, such as in a severe drought region as described above. Second, we look at the case where fire damage increases with weighted fuel stock at an increasing rate and there are decreasing marginal returns to fuel treatment, such as in an area that is adapted to frequent low-intensity fires.16 These different ecological/fire settings create distinct strategic behavior among landowners.
Increasing Returns to Fuel Treatment
When the damage function is nonlinear with globally increasing marginal returns to fuel treatment (equation [4]), the game between the public and private landowners becomes strategic because one owner’s decision can alter the marginal returns to treatment and, hence, the treatment decision on adjacent properties. Knowing that, the owner may choose treatment levels with the intent of inducing adjacent owners to treat more or less than they would have otherwise treated. In this case, landowners no longer have a dominant strategy that depends only on fuel stock and ownership pattern. Instead, the optimal fuel treatment decision depends on the other landowner’s treatment and the marginal benefit associated with that treatment level. With increasing marginal returns to fuel treatment, when one landowner treats a unit, the marginal benefit of treating fuels on neighboring units increases via the spatial weighting matrix for those units. Hence, the more one landowner treats, the more likely the other landowner will choose to treat, and vice versa. This interaction between the two landowners causes them both to perform more fuel treatment than they would without the other landowner, which leads to a higher total amount of fuel treatment on the landscape.
Compared to the base case, the outcome from the game with increasing marginal returns to fuel treatment is closer to the socially optimal treatment pattern for all ownership patterns except isolated (Table 2). The outcome of the game is identical to the socially optimal treatment pattern for the adjoining, corridor, and extension ownership patterns (Figure 6).17 Only for the two most fragmented ownership patterns—isolated and checkerboard—does the game lead to inefficiently low levels of fuel treatment. In general, where ownership is more fragmented, because landowners are unable to capture the positive spatial externalities from fuel treatment, it is more likely that the treatment cost exceeds the treatment benefit and, hence, less likely that there will be fuel treatment on the landscape.
Decreasing Returns to Fuel Treatment
As with increasing returns to fuel treatment (equation [5]), when there are decreasing marginal returns to fuel treatment, the game between the public and private landowners becomes strategic, as the players no longer have a dominant strategy that depends only on fuel conditions and each landowner’s optimal fuel treatment choice depends on the other’s choice. Because the marginal benefit of additional fuel treatment decreases as weighted fuel stock decreases, the more (less) fuel treatment on publicly owned units the lower (greater) the marginal benefit of fuel treatment on private units, and vice versa. This interaction leads to lower amounts of fuel treatment on the landscape than would arise if landowners’ actions had no impact on their neighbors’ weighted fuel stocks, and, compared to the base case, the outcome of the game with decreasing returns to fuel treatment is further from the social optimum for all ownership patterns (Table 2). It is the smaller marginal value to fuel treatment with this damage function shape that leads to lower levels of fuel treatment than in the base case.
Compared to the socially optimal fuel treatment pattern with decreasing returns, the game results in inefficiently low levels of fuel treatment for all ownership patterns (Figure 7).18 As each landowner attempts to free ride on the fuel treatment of the other, the result is that both perform less fuel treatment than they would without the other landowner, leading to a lower total amount of fuel treatment on the landscape. As in the base case and the increasing returns case, the outcome is furthest from the social optimum for the most fragmented ownership pattern, checkerboard.
Alternative Spatial Externality Weighting Schemes
We also solved the base case model using the three alternative sets of parameter values for the spatial weighting described above. As described above, when we set ωij = 0 for i≠ j, there are no spatial externalities associated with fuel treatment, and the model becomes aspatial so that the outcomes of the game are optimal. Relative to our base case weighting parameters, when we gave weight only to the unit to the right or above the individual unit, we observed a tendency to treat units on the right side and on the upper portion of the landscape, respectively. But the results are otherwise completely consistent with the results for the base case described above. See Busby (2008) for a detailed description of the results for these alternative parameter values.
V. Discussion and Concluding Remarks
With increasing numbers of private landowners in the WUI, efforts to prevent damage from wildfires have increased in importance and complexity. Fire damage is related to the total amount of fuel treatment in a landscape, but that total amount results from the uncoordinated actions of a landscape of landowners. Because fuel treatment creates a spatial externality, we developed a game to determine the patterns of fuel treatment and damages that result from different patterns of land ownership. Although simplified, this stylized game- theoretic model provides basic insights that are not site-specific and can be applied across a range of settings. In applying this structure to any specific case, the analyst must modify our assumptions about all landowners of a particular type acting in concert. homogeneity of values across same-owner space, and the relationship between fuel treatment and the pattern of fire damage risk, among others. Still, adding these complexities does not overturn the basic insights developed from exploring this simpler model. In addition, an advantage of the modeling framework used here is that it forms a platform for numerous extensions, such as our ongoing evaluation of regulatory, market insurance, and land management policies to improve risk management in a spatially explicit setting.
The primary objective of this study is to examine how spatial configuration and location affect fire risk management on a landscape with mixed ownership. Results from the model indicate that ownership pattern and location do influence individual fire risk management decisions and outcomes. Three major conclusions from this research are (1) ownership fragmentation increases the inefficiency of fire risk management; (2) off-site amenity values decrease, and in some cases may offset, that inefficiency; and (3) the nature of strategic interaction among landowners depends on the relationship between fuel stock and fire damage severity, for example, the shape of the damage function and whether there are increasing or decreasing returns to fuel treatment. In some cases, government agencies can use information about land ownership patterns to recognize where inefficiencies exist and improve outcomes by prioritizing fuel treatment in those areas. We note that by ignoring the effect of fuel treatment on fire spread, we underestimate the benefit of fuel treatment in the socially optimal fire risk management problem. To the extent that the probability of fire reaching individual units on the landscape is reduced by fuel treatment on the landscape, and that individual landowners ignore that effect, the inefficiencies we identify in our model are understated.19
Because reducing forest fuel generates positive spatial externalities, landowners are unable to capture the full benefit of fuel treatment on a single unit. A landscape in which ownership is highly fragmented—as measured here by the adjacencies between private and public owners—generally leads to inefficiently low levels of fuel treatment because of that spatial externality. However, the inefficiencies created by fragmentation depend on the direction and the degree of spatial interdependence between units in terms of the impact of on-site fuel treatment to off-site damages. Low levels of spatial interdependence between land units generate only a small difference between the socially optimal fuel treatment pattern and the outcome of the game between forward-looking landowners. With the growth of the WUI, fragmented ownership landscapes are becoming increasingly common (Garber-Yonts 2004; Stewart et al. 2005). The analysis here signals the failure of highly fragmented ownership landscapes to provide adequate levels of fuel treatment, further increasing the risk of social losses due to wildfire.
In a more optimistic finding, greater ownership fragmentation leads to better outcomes when private landowners value aspects of public land, or vice versa. When landowners receive values, such as amenity values like scenic views and recreation access, from other parcels, fragmentation and ownership adjacencies allow owners to partially protect those off-site values through treatment on their individual units due to the positive externality. In recent years, amenity migrants, or landowners who move to the WUI to enjoy the wilderness setting both on and off their own land, have caused an increase in the amount of private landholdings in the WUI (Garber-Yonts 2004). Through the lens of our model, where this type of private landowner abounds, higher levels of land ownership fragmentation can offset the inefficiencies associated with uncoordinated fuel treatments across the landscape.
In order to focus on the interactions of private and public landowners, we assume that private landowners coordinate their fuel treatment decisions. In some settings, such as fire-prone eastern Oregon, homeowners associations and other social networks encourage this kind of coordination. However, in other settings, ownership fragmentation, the sheer number of landowners, and the dispersion of properties create prohibitively high coordination costs. In such settings with private landowners acting independently, our modeling framework suggests that the lack of consideration of the positive spatial externalities of treatment would lead to low levels of fuel treatment and move the outcome of the game further from the social optimum. Shafran (2008), focusing almost exclusively on private-private interactions in the absence of coordination, conducted an econometric analysis of homeowners’ decision to manage fuels to create defensible space around their homes in the WUI that supports this notion, which we also are exploring in ongoing research. Our current analysis emphasizes the inefficiencies that arise even when private landowners perfectly coordinate their spatial fuel treatment decisions; but the role of collective action through homeowners associations in achieving such coordination has potential importance for reducing spatial inefficiencies across a multilandowner landscape.
Returning to our focus on private-public ownership interactions, cross-ownership externalities are the sole source of inefficiency in this model when there are constant returns to fuel treatment. However, when the relation between fuel stock and damage is nonlinear, landowners behave strategically and the direction of that strategic action depends on whether returns to fuel treatment exhibit increasing or decreasing returns to total fuel treatment. With decreasing marginal returns, landowners spend as little as possible on fuel treatment as they attempt to free ride on their neighbor’s effort. Here the best responses for each landowner reflect the motivation for free-riding; as one landowner’s fuel treatment increases, the response of the neighboring landowner is to decrease fuel treatment. In general, the owner with less value on the landscape is able to perform low levels of fuel treatment and free ride because the owner with more value on the landscape protects that value by choosing a high level of fuel treatment. In contrast, with increasing marginal returns, the opposite strategic behavior emerges. Here landowners employ fuel treatment on additional units to increase the marginal benefit of fuel treatment to all landowners, thereby inducing higher levels of treatment by their neighbors. Because the direction of the strategic behavior in terms of amounts of fuel treatment depends on the shape of the function describing returns to fuel treatment, determining that shape for a particular area informs expectations about the level of fuel treatment on the landscape and the response of private landowners to changes in public fuel treatment decisions. For example, in settings where strategic landowner interaction is determined by increasing returns to fuel treatment, increasing fuel treatment on public land may play a pivotal role by creating additional incentive for fuel treatment on private land. On landscapes where private landowners might otherwise have done nothing, this public initiative might lead to large areas of fuel treatment on private land, answering public land managers’ calls for more fuel treatment (Stephens and Ruth 2005), better protected public values, and potentially significant suppression cost savings.
In addition to strategic behavior in quantities, in this spatially explicit landscape, landowners also behave strategically in the location of their fuel treatment. In the equilibria to this game, landowners are willing to treat units with lower on-site values at risk than untreated units when the treated unit’s location induces the other landowner to treat more units through changes in the marginal benefit of treatment on neighboring units. The configuration of land ownership determines the opportunity for landowners to behave in this spatially strategic manner, with more opportunities created in fragmented landscapes like the checkerboard. In addition, although not spatially strategic in a sense of game theory, landowners here also make fuel treatment location decisions that use the positive externality of fuel treatment to protect amenity values generated on neighboring land units. In this case, the size of the off-site values that the landowner sees combines with the configuration of land ownership to determine the opportunity for landowners to make such spatial decisions, with more spatial decisions in landscapes with fragmented ownership pattern.
Our stylized framework helps public landowners determine the likely impact on total fuel treatment in a landscape as a function of the landscape ownership pattern, private perspectives on public land values, and the shape of the damage function. In further analysis, we will extend the public landowner’s choices beyond levels and location of fuel treatment to include the role of regulation, liability, insurance, and disaster relief, which all interact with the spatial configuration of landowners in ways similar to this paper’s exploration of fuel treatment alone. Several studies suggest that private landowners perceive the benefits of fuel treatment as greatest when forest fuels are reduced on both their property and that of neighbors, which implies that public fuel treatment may induce more private fuel treatment, as in our case of increasing returns to fuel treatment (Brenkert-Smith, Champ, and Flores 2006; Agee and Skinner 2005). Many fast-growing WUI areas report that the growing populations come from amenity migration, which might imply that private landowners will perform high levels of fuel treatment in order to protect values on nearby public land. Where these private perspectives hold, our analysis signals that public landowners can perform lower levels of fuel treatment and allow private landowners to carry more of the cost of risk-mitigating activities. They can also place their fuel treatment in locations where it induces more private actions and/or where private landowners hold alternative views or damage functions and have incentives to free ride on public fuel treatment.
Acknowledgments
This research was made possible by financial support from the U.S. Forest Service’s Pacific Northwest Research Station and Western Wildland Environmental Threat Assessment Center. The views described here are those of the authors alone and do not represent those of the U.S. Forest Service.
Footnotes
The authors are, respectively, adjunct assistant professor, Department of Forest Resources and Environmental Conservation, Virginia Tech, Blacksburg, and senior scientist, Department of Environmental Sciences, University of Virginia, Charlottesville; professor, Applied Economics/FES, Oregon State University, Corvallis; and professor, Department of Forest Engineering, Resources, and Management, Oregon State University, Corvallis.
↵1 See the National Interagency Fire Center fire statistics, available at www.nifc.gov.
↵2 Within the wildfire management literature, fire severity is a measure of damage and fire intensity is a measure of fire temperature and heat generation (Sousa 1984). Here our focus is on wildfire damage to human and natural resources.
↵3 This is similar to assuming that individuals are pricetakers in a market in which price is determined by the joint actions of all individuals. Our landowners take the probability of fire reaching their property as given even though it is influenced by the joint actions of landowners on the landscape. Further, there is some evidence that fuel treatment by public agencies in the WUI acts more to reduce damage when fire occurs than to prevent fire from reaching the WUI, while fuel treatment outside the WUI, where fuel accumulation is the highest, may act to reduce the probability of fire reaching the WUI (Ager, Vaillant, and Finney 2010).
↵4 In some fire regimes, a vertex of contiguity might be enough to spread fire, and future work could examine the patterns that derive from a nonzero weight on all land units with queen contiguity. Similarly, moving away from a grid landscape to a landscape of hexagonal units would obviate the problem of fire spread across a corner point.
↵5 Other specifications are possible. We tried several (for example, the natural log of a linear transformation of weighted fuel stock greater than 1 for increasing returns to treatment) and obtained the same qualitative results as we obtained with these specifications, because the changes in outcomes relative to linear were driven by the presence of increasing or decreasing returns to scale rather than the particular form they took.
↵6 In our spatially constrained setting, we assume globally increasing returns; this would not be realistic in an unconstrained setting, where the corresponding assumption would be one of locally increasing returns. See Baland and Platteau (1997) for a description of other examples of increasing returns in resource management.
↵7 For a description of the Sun River homeowners’ fuels management plan, see www.sunriverowners.org/Environmental-Services-Department-%E2%80%A2-541.593.152214345413934.htm.
↵8 In Figure 2, gray = private represents “private adjoining public, private corridor, isolated private center unit, private extending into public, and private checkerboard”; gray = public represents “public adjoining private, public corridor, isolated public center unit, public extending into private, and public checkerboard.”
↵9 In fact, these landscape patterns were identified from land ownership maps from Beaverhead County, Montana; Deschutes County, Oregon; Harney County, Oregon; Lemhi County, Idaho; Clearwater County, Idaho; and Catron County, New Mexico.
↵10 A sensitivity analysis of the base case parameters was conducted and results were consistent with the model’s predictions. With a higher beginning fuel stock, we observed more fuel treatment in both time periods. Compared to less- fragmented ownership patterns, a more-fragmented landscape required more significant increases in the probability of fire for landowners to treat every parcel. However, only a slight decrease in the probability of fire was sufficient for landowners to choose zero fuel treatment. More-fragmented landscapes were found to be more sensitive to increases in fuel treatment costs.
↵11 Realistic values could be selected that result in treatment of no units or of all units regardless of the spatial pattern of ownership. Both cases can be observed in the real world. However, we are interested in examining how treatment choices are affected by the spatial configuration of ownership and so we chose parameter values that narrow the range to that at which the margin is close and behavior changes when the ownership pattern changes.
↵12 We do not consider the possibility of mixed strategy NE because public and private landowners do not appear to follow such probabilistic or randomized management plans. This type of behavior is politically risky for a public landowner, responsible for the thoughtful management of public lands. And private landowners, acting as a single coordinated decision maker, might prefer predictability in their hazard mitigation strategy.
↵13 In Figure 4, gray = private represents “private adjoining public, private corridor, isolated private center unit, private extending into public, and private checkerboard”; gray = public represents “public adjoining private, public corridor, isolated public center unit, public extending into private, and public checkerboard.”
↵14 The socially optimal fuel treatment pattern is the same for all ownership patterns except the checkerboard pattern. This difference is due to differing assumptions about management decisions outside the nine-by-nine grid landscape.
↵15 In Figure 5, gray = private represents “private adjoining public, private corridor, isolated private center unit, private extending into public, and private checkerboard.”
↵16 Direct comparisons between base case treatment patterns and treatment patterns for the two cases with nonlinear damage functions are not possible because of differences in the parameters that describe each of the three damage functions. Due to our chosen parameters, for all levels of weighted fuel stock, fire damage is greater for the case with increasing marginal returns and less for the case with decreasing marginal returns than with the base case’s linear damage function. However, direct comparisons can be made between the socially optimal treatment pattern when the damage function is nonlinear and the outcome of the game when the damage function is nonlinear. Further, the strategic interaction between the landowners driving the results described in this section is independent of the chosen parameters.
↵17 In Figure 6, gray = private represents “private adjoining public, private corridor, isolated private center unit, private extending into public, and private checkerboard.”
↵18 In Figure 7, gray = private represents “private adjoining public, private corridor, isolated private center unit, private extending into public, and private checkerboard.”
↵19 To fully model the effects of spatial pattern of ownership on both the self-insurance and the self-protection aspects of fuel treatment would require explicitly modeling the spread of fire across the landscape from different ignition points for each ownership pattern and fuel treatment pattern. That complicates the computational aspect of game substantially and would, we think, necessitate using approximate solution methods.
