#### The Conceptual Framework of Robustly Fair Offsets

Our goal of offsetting is consistent with NNL in the sense that present loss is compensated by future gains, accounting for uncertainty and time lags in the development of these gains. We specify that the probability of incurring net loss must be small, thereby ensuring what we call “robustly fair offsets.” The uncertainty is a critical component when the aim is to avoid net loss due to unfavorable growth of conservation value at the restoration areas.

We assume three components of uncertainty. (1) Future value could be less than estimated, which could, e.g., represent the case that an area of forest develops fewer nesting holes than expected or that forest understory develops a community which is less species rich than expected. Outcome could be uncertain even when it is practically immediate, e.g., if compensation sites do not require restoration but the areas are poorly surveyed so that what is gained by the exchange is not accurately known. (2) Some feature of conservation value might completely fail to be established, e.g., a focal species may fail to colonize the area. (3) We also allow for the possibility that success and failure could be correlated between different restoration areas. The uncertainties in our analysis are most relevant where restoration action is applied at compensation areas. However, the proposed framework is equally applicable when compensation areas are such that they already hold substantial conservation value and some form of protection is applied rather than restoration action. In this case, uncertainties are smaller (or even zero), but the structure of the proposed calculations need not be changed.

We account for uncertainty by adopting a decision-theoretic approach to the calculation of offsets. If statistical models are available for the components above, one could use a statistical approach for identifying an offset ratio, which has, e.g., less than 5% chance of resulting in net loss. However, our formulation includes parameters, such as long-term success of restoration effort, for which it may be difficult to obtain reliable distributional information. In such a case, information-gap decision theory (Ben-Haim 2006; hereafter info-gap theory), which we employ here, provides a straightforward way of analyzing the influence of uncertainty on the offset ratio.

Time discounting (Carpenter et al. 2007) of the offset ratio is included because it is not fair to compensate immediate loss by hypothetical distant future gain. Presumably, the conversion of the development site would produce a relatively immediate economic return in the order of some percents per year. This revenue could plausibly be used for further environmentally harmful activity either directly or indirectly. On the other hand, conservation benefits arising from restoration effort may take a very long time to materialize fully, e.g., if one needs to wait for forest to grow. Consequently, we find it reasonable that the offset ratio should be calculated as a time-discounted weighted average across the planning frame. Omitting time discounting could place nature conservation efforts at an overall disadvantage.

These components have been noted in prior work: The outcome of restoration is often different from expected, for instance, due to existence of alternative equilibria and differences in ecological dynamics between degraded and less-impacted systems (Zedler & Callaway 1999; Folke et al. 2004; Suding et al. 2004; Hilderbrand et al. 2005). Following restoration, ecosystems can recover into different states from the same initial condition (Folke et al. 2004). Restoration action can fail despite the correct management action if, for instance, rainfall does not occur (Vesk & Dorrough 2006). Several authors note that there is uncertainty associated with the expected outcome of restoration (Cuperus et al. 2001; Bruggeman et al. 2005; Morris et al. 2006; Gibbons & Lindenmayer 2007) but do not explicitly account for it in their analyses. Keagy et al. (2005) investigate the feasibility of compensation for maintaining overall population abundance in the study area, when the compensation areas are of inferior quality compared to the lost habitat. Gibbons and Lindenmayer (2007) conclude that offsets will only contribute to NNL if (1) clearing is restricted to vegetation that is simplified enough so that its functions can be restored elsewhere; (2) any temporary loss in habitat between clearing and maturation of an offset does not represent significant risk to a species, population, or ecosystem process; and (3) offsets are substantial enough and they are complied to. HEA explicitly includes time discounting as an option (Dunford et al. 2004; Bruggeman et al. 2005). Morris et al. (2006) and Roach and Wade (2006) both mention that there is a time lag between impact and compensation, although they do not present methods that explicitly take that into account in analysis. Here we combine all these factors together into the same quantitative theoretical analysis.

#### Evaluating Offset Solutions Using an Uncertainty-Analytic Approach

We use info-gap theory (Ben-Haim 2006) to analyze the consequences of uncertainty for establishing a fair offset ratio. The main components of the info-gap theory are the goal (performance aspiration), the performance function, the nominal model, the uncertainty model, and the robustness function.

Our goal is to robustly achieve NNL. The nominal model is our best estimate for the expected conservation value in the development area and compensation areas (thick lines in Fig. 1). We indicate nominal models by and for conservation value at time *t *at the development area and compensation area *i*, respectively. The nominal model represents our best understanding of how conservation value will change in these areas over time. However, this information may be quite uncertain, which is modeled by the second central component of info-gap analysis, the uncertainty model (thin lines in Fig. 1). Note that instead of staying stable, conservation value at the development site could be declining, which would lead to smaller offset ratios.

The info-gap uncertainty model does not simply place bounds around the nominal estimate, as it might appear from Figure 1 because worst-case bounds are at best poorly known. Rather, the robustness of solution candidates are analyzed in terms of an uncertainty parameter, the horizon of uncertainty *α*. When this parameter is zero, it indicates full confidence in our nominal model and the nominal model is accepted as the true model. Higher values for *α*indicate less confidence in the nominal model: the true model is somewhere within an expanding bound around the nominal model. In our example of Figure 1, the uncertainty model is represented by the thin lines around the nominal model. When *α*= 0, the thick line is taken as the truth, and increasing *α*implies expanding bounds of possible outcome. Importantly, different areas and restoration actions could have different nominal estimates as well as different levels of uncertainty (often called error weights). For example, smallest error weights could be associated with a presently high-quality area that has been well surveyed. A relatively higher error would go for an area that is apparently valuable but is poorly surveyed. Highest error weights would be associated with areas where there is substantial lack of knowledge concerning the growth of conservation value there, e.g., as a consequence of trying out a completely new restoration technique. Technically, when evaluating a solution at any given level of *α*, the solution is evaluated according to the most adverse choice of the model inside the uncertainty bounds. However, since the horizon of uncertainty, *α*, is unknown, a solution is evaluated according to the greatest *α*up to which that solution yields adequate outcomes.

The aim of our uncertainty analytic approach is to identify solutions that are robust in the sense that they achieve our performance aspiration even when allowing for high uncertainty. In the typical info-gap formulation, the robustness of a solution, *α**, is the highest *α*at which it is guaranteed to meet the performance target (Fig. 2a). A solution is not robust if it may fail to achieve the goal even at low *α*, indicating that a small deviation from expected restoration outcome might miss the target of NNL.

Each offset candidate solution would be examined in terms of its performance under increasing uncertainty. This is illustrated in Figure 2. Assuming that offset candidates A, B, and C have equal cost, then A is the best option because it achieves goals while allowing for highest uncertainty (Fig. 2a). Candidate C is the second best option assuming nominal models are correct. However, candidate B is more robust to increasing uncertainty than C.

The robust optimal solution is the one solution that achieves the planners specified goals while allowing for highest possible errors in the nominal models. If only a few scenarios need to be compared, then solution performance and robustness can be evaluated for all candidates. If, however, the robust optimal solution needs to be identified from a large set of options (such as selecting 100 out of 1,000 sites), then some optimization method is needed. Below, we calculate the offset ratio that is sufficient for guaranteeing NNL while accounting for the modeled uncertainties (Fig. 2b).

#### A Simple Example of the Method

We illustrate the proposed method for the simple case where one unit area of land with relatively high conservation value is offset by a number of units of less valuable land that is restored. In this example, conservation value is treated as a one-dimensional construct. Table 1 gives a summary of symbols used in the equations.

Table 1. Explanation of symbols used. *t*_{p} | Length of planning period |

*β* | Reliability requirement, the probability of net loss should be less than (1 −*β*) |

*p* | Failure probability of restoration action at an area |

*ρ* | Correlation coefficient for failure of restoration action between areas |

*d* | Time discounting rate |

*α* | Info-gap robustness parameter, horizon of uncertainty |

| Best estimate for per unit area conservation value of the development site at time *t *(per unit area) |

| Best estimate for per unit area value of compensation area option *i *at time *t* |

*w*_{0}(*t*) | Size of error envelope (weight) of |

*w*_{i}(*t*) | Error weight of ; with restoration *w*_{i}(*t*) >> *w*_{0}(*t*) |

*N*_{method}(*α*, *t*) | Number of equal-sized offset areas needed according to an offset calculation using the method indicated by subscript, *N*_{simple}, *N*_{IG}, *N*_{prob}, *N*_{corr}, and *N*_{discounted}, for Equations 1, 2, 3, 5, and 7, respectively. This quantity depends on both *α*and *t *via Equation 2 |

Assuming that all conservation value of the high-quality development area will be lost following the land exchange, a naive solution using matching of mean expected utility for the offset ratio is as follows:

- (1)

where is the best estimate for the conservation value of the development area presently (at time 0) and is the best estimate for the final conservation value of the restoration area at the end of the planning period at time *t*_{p}. This is the ratio *A*/*B *in Figure 1. *N*_{simple} units of restoration land are eventually predicted to hold the same conservation value as the development area.

- (2)

Here, *w*_{0}(*t*) and *w*_{i}(*t*) are relative error weights for conservation value at the development area and compensation areas at time *t *in the future. For instance, these envelope functions may derive from statistical modeling and/or expert opinion. Because other experts may have yet other opinions, or differently framed questions may elicit different expert responses, the uncertainty envelopes are multiplied by the unknown horizon of uncertainty, *α*. In our example *w*_{0}(*t*) and *w*_{i}(*t*) were calculated as the difference between the nominal estimate and the hypothetical error bounds of Figure 1, indicating that at *α*= 1, the uncertainty envelope has expanded to the outer thin lines.

In the next level of sophistication, we allow for the possibility that conservation action in any one land unit could also fail altogether with a probability *p*. It is then logical to require that the even exchange would be achieved with a given reliability level *β*, say *β*= 0.95. The number of unit areas where conservation action would succeed, *N*_{S}, is now distributed binomially as *N*_{S}∼ Bin(*N*, *p*). To satisfy the reliability requirement, we need Prob[*N*_{S} < *N*_{IG}(*α*, *t*)] < (1 −*β*). Denoting by *N*_{prob}(*α*, *t*), the minimum number of unit areas needed, this number can be determined by finding smallest *N*_{prob}(*α*, *t*) > *N*_{IG}(*α*, *t*) for which

- (3)

Equation 3 assumes statistical independence in success of restoration effort between different sites when calculating *N*_{prob}(*α*, *t*). The assumption of independence is a strong one, and in general restoration, success between distinct restoration sites would be correlated to some degree (Fig. 3 illustrates effects of correlation). Ovaskainen and Hanski (2003) give a formula for the effective number of independent units, *N*_{eff}, when there is an uniform level of pairwise correlation, *ρ*, between *N*_{corr} sites,

- (4)

This equation essentially states that if the correlation is *ρ*, then there can be at most 1/*ρ*independent units irrespective of how many sites there are. Note that Equation 4 ignores higher-order correlations but, even so, it provides useful insight into the influence of correlation on the fair offset ratio.

Assuming *N*_{corr} correlated sites, we have only *N*_{eff} effective independent units, each of average size *S *=*N*_{corr}/*N*_{eff}. We then require that unit-size times the minimum number of units that succeed with reliability greater than *β*must be greater than *N*_{IG}(*α*, *t*). The number of effective units where conservation action would succeed, *N*_{S}, is now distributed *N*_{S}∼ Bin(*N*_{eff}, *p*). To satisfy the reliability requirement, we need Prob[*SN*_{S} < *N*_{IG}(*α*, *t*)] < (1 −*β*). The minimum number of real units needed for this relation to be true can be determined numerically by finding smallest *N*_{corr}(*α*, *t*), for which

- (5)

where *N*_{eff} comes from Equation 4 and *N*_{min} is the smallest number of units (out of *N*_{eff}) that succeed with a probability of at least *β*. *N*_{min} can be determined by inspecting the tail of the binomial distribution for the effective number of successful independent units. It is the largest number such that, out of *N*_{eff} units, at most *N*_{min}− 1 can fail with probability (1 −*β*) or less, which implies that *N*_{min} or more units will succeed with probability greater than *β*:

- (6)

Note that Equation 6 cannot always be satisfied. For example, with *ρ*= 0.25, there can be at most four effective independent units. Then, if the failure probability of a unit is 0.5, a 95% reliability can never be achieved because 0.5^{4}= 0.0625 > (1 − 0.95) meaning that the chance of all units failing is greater than the 5% allowed.

We add one final component, time discounting, to our analysis. A time-discounted offset ratio can be obtained simply as follows:

- (7)

in which *d *is the time-discounting coefficient and *N*_{method}(*α*, *t*) represents any of the offset ratios from Equations 1, 2, 3, or 5, where the offset calculations have been done at time *t *using given horizon of uncertainty *α*. For practical purposes, this means that the offset ratio is weighted most heavily by the early years when the quality of the restoration areas is worst.