Social dilemmas and poor water quality in private water systems

Private water supply systems consisting of a domestic well and septic system are used throughout the world where households lack access to public water supply and sewers. In residential areas with high housing density, septic contamination of private wells is common and associated with multiple health concerns. This situation can give rise to social dilemmas, where individual costs dis-incentivize homeowners from investing in enhanced septic systems that would reduce well contamination and bring communal benefits. We combine a stylized game-theoretical model with a probabilistic groundwater model to char5 acterize how economic and hydrogeological conditions interact to produce misaligned incentives conducive to social dilemmas. The occurrence of social dilemmas depends on the relative costs of well contamination versus the cost of installing an enhanced septic treatment system, and the relative probabilities of cross-contamination versus self-contamination. The game reveals three three types of social dilemmas that occur in such systems, with each calling for distinct policy solutions. We demonstrate how the model can be applied to existing systems using a case study of St Joseph County, Indiana, where high nitrate contamination 10 rates have raised public health concerns. This analysis represents a step towards identifying alternative policy solutions for a problem that has remained difficult to address for decades.

Septic tanks represent one of the major threats to water quality in domestic wells and the high concentration of contaminants in septic leachate creates a variety health concerns (US EPA, 2005b;Katz et al., 2011). Septic contamination has been associated with endemic diarrheal illness in Wisconsin (Borchardt et al., 2003), viral discharge and transport exceeding typical well-drainfield setback distances (DeBorde et al., 1998), and multiple disease outbreaks in the United States and Canada (Craun et al., 1994;Beller et al., 1997). In addition to pathogens, septic systems also discharge high levels of nitrates, which can cause 40 acute health problems in infants (Knobeloch et al., 2000), and other emerging contaminants such as PFAS, flame retardants, and endocrine disruptors (Schaider et al., 2016), which also present long-term exposure risks (Bergman et al., 2013).
The issue of septic contamination has been widely recognized for at least half a century (Yates, 1985). While technological solutions (e.g., reverse osmosis) are available to treat water from contaminated wells, due to high costs such systems are often 2 Two-household contamination games 2.1 Expected utility and payout matrices We first consider a two-player static game with complete information, wherein the payout structure of both players is common knowledge. Each player chooses whether to upgrade their septic to an enhanced system with contaminant removal (E) at some 80 cost, or to keep a basic septic system (B) at no cost. The cost of upgrading, C σ , represents the difference between the enhanced septic system and the cost for a conventional system (e.g., as mandated by local regulations). If a player's well becomes contaminated, that player also incurs a cost, C x . The expected utility of an individual household i is therefore where σ i ∈ {0, 1} represents the choice of each household to maintain a basic septic system (0) or upgrade to an enhanced 85 system (1). The probability p i describes the likelihood of contamination of player i's well, which depends on the septic system decisions made by both players (σ i and σ j ) and on the groundwater flow direction. Lastly, the cost of contamination, C x , can either represent the individual cost of household water treatment to improve household water quality (which represents a lower bound on the cost, Yadav and Wall, 1998), or the cost of using contaminated water and associated consequences (Raucher, 1983). Note that no player would benefit from upgrading their system if doing so was more expensive the cost of 90 contamination. To avoid this trivial outcome, we consider only situations with C x larger than C σ . We evaluate this requirement and explore associated costs of nitrate contamination in greater detail in Sect. 4.
Because Eq. (1) depends on binary decisions taken by two interacting players, its outcomes can be represented as a 2 × 2 matrix. The rows and columns of the matrix represent the decision by players 1 and 2, respectively, on whether or not to upgrade. Each cell of the matrix is populated by a pair of normalized payouts representing the expected utility of each player 95 for each combination of decisions. Payouts from Eq. (1) are normalized so that the summed total payout across players for each cell ranges between 0 and 2. Two types of matrix cells are particularly noteworthy: -The Nash Equilibrium (NE) represents each player's best (i.e., utility maximizing) response to the other player's own utility-maximizing decision.
-The Social Optimum (SO) represent the combination of decision that maximizes the summed expected utility of both 100 players.
Whether or not the NE and SO decision outcomes overlap depends on the ratio of costs (i.e., the "cost ratio", C x /C σ ) and the groundwater configuration between households. We define a social dilemma as a situation where the SO outcome is different from at least one NE outcome. We proceed to enumerate the different configurations of NE and SO in the 2 × 2 games that emerge from Eq. (1)

Symmetric two-player games
If both players face identical costs (C x and C σ ) and contamination by (and of) each player is symmetric, four different game configurations emerge depending on the spatial configuration of the two players (Fig. 2). If neither player contaminates their own well or that of the other player, neither player wishes to upgrade to an enhanced septic system and the Nash Equilibrium 110 is identical to the Social Optimum (Fig. 2a). When both players contaminate only their own wells, they are both incentivized to upgrade their septic system and the Nash Equilibrium and Social Optimum solutions are again identical (Fig. 2b).
A social dilemma only occurs when each player's decision affects water quality in the other player's well. In the case where both players only contaminate the other player's well without contaminating their own well (Fig. 2c), the best response for each player would be not to upgrade, regardless of the decision of the other player. While neither player has anything to 115 gain from paying to upgrade their own septic, both players would benefit if the other player upgrades. This leads to a classic Prisoner's dilemma (see Kollock, 1998) where neither player upgrades, despite the fact that a situation where both players upgrading would improve their utility and is the social optimal outcome. The final situation, where players contaminate both their own and the other players' well ( Fig. 2d) gives rise to a second Nash Equilibrium where both players upgrade. In this case, upgrading is the best response for each player when the other player also decides to upgrade. However, not upgrading 120 will be the best response for each player if the other player decides not to upgrade. By upgrading in the latter case, the player would pay the full cost of upgrading while still being contaminated by the other player's septic system. Under these conditions, the benefits of upgrading are contingent on the other player doing the same (Fig. 2d). This scenario is known as a Stag-hunt game, where players face conflict between individual safety and social cooperation (Kollock, 1998). In the canonical Stag-Hunt game, two hunters independently decide whether to hunt a stag or a hare. A hare can be hunted by an individual hunter, while 125 an individual player deciding to hunt a stag will fail. The much more valuable stag can only be successfully hunted if both hunters decide to hunt it. The game has two Nash Equilibria, where either both players hunt the stag or both players hunt the hare. The second Nash Equilibrium (which, here, corresponds to a situation where neither household updates their septic) is not a Social Optimum and therefore creates a social dilemma.

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In addition to these symmetric games with complete information, households may have uncertainty with respect to groundwater behavior. We consider five different configurations of households (Fig. 3, I-V) and allow groundwater to flow west, east, or north, with corresponding probability P (W est), P (East), and P (N orth). Allowing flow towards the south would produce identical game structures as flow towards the north, and we exclude this unnecessary possibility. The expected utility can be calculated as the weighted average utility from the end-member scenarios. For instance, consider scenario II in an Asymmetric Dilemma (AD). In the latter dilemma, the social optimum differs from the Nash equilibrium which exhibits 140 asymmetric payouts favoring one player over the other.
To better understand how asymmetric dilemmas arise, consider configuration I as an example, where P (W est) = 0.2, P (East) = 0.8, and P (N orth) = 0 (i.e., marked by the +, •, and × in Fig. 3, I). If neither player upgrades, the player on the left faces low expected cost of contamination due to the low probability of flow to the west, whereas the player on the right faces a high expected cost of contamination due high probability of flow to the east. This creates an asymmetric dilemma 145 where the left household chooses not to upgrade in the Nash equilibrium, but would need to upgrade to reduce costs for the other player and achieve the Social Optimum. Multiple types of Asymmetric Dilemmas exist. In AD-i, one player upgrades in the Nash equilibrium because they have a high risk of self-contamination. However the social optimum requires that the second player upgrade because both have a modest risk of contamination from this player. In this situation the Nash equilibrium is (E, B) or (B, E) and the social optimum is (E, E). In AD-ii, neither player faces a high risk of self-contamination, but one 150 player faces a high risk of contamination from the other player. The Nash is (B, B) and the social optimal is (E, B) or (B, E).
Similarly in AD-iii, neither player has a high risk of self-contamination. However, the combined risk of contamination from either player means that both players must upgrade to achieve the Social Optimum. In this situation, the Nash is (B, B), and the social optimum is (E, E).
Generally, there tends to be greater likelihood of a social dilemma if the relative risk of contamination is high. This can 155 arise if the cost of contamination is high compared to the cost of upgrading (i.e. C x >> C σ , Fig. 3, right) or if the probability of contaminating the other player is high compared to the probability of self-contamination. Conversely, a high probability of Asymmetric Dilemma (AD) along with the potential benefits of cooperation, given as the difference in normalized payouts between the social optimal and the (least desirable) Nash equilibrium, averaged for both players. The main axes represent the probability of flow towards the west and east, respectively, with the remainder being the probability of flow towards the north (i.e., P (N orth) = 1−P (W est)−P (East)).
self-contamination circumvents social dilemmas by incentivizing the contaminating household to upgrade, as seen in scenario I (P (W est) → 1) and III (P (W est) → 1 or P (East) → 1).
Changing costs (either C x or C σ ) lead to transitions across different types of social dilemmas, indicated by the symbols 160 +, •, × in Fig. 3, with payouts shown in Fig. 4. For instance in configuration I, the social dilemma changes from AD-ii with (Fig. 4a). Conversely, in configuration III, the social dilemma changes from a Prisoner's Dilemma with C x = 1.5C σ (+) to AD-i with C x = 2C σ (•) to no dilemma with C x = 3C σ (×) (Fig. 4b).
7 https://doi.org/10.5194/hess-2021-312 Preprint. Discussion started: 15 June 2021 c Author(s) 2021. CC BY 4.0 License. Changes in the uncertainty associated with groundwater behavior can also lead to transitions across different types of social dilemmas. In particular, in configuration IV with C x = 3C σ , the social dilemma changes from AD-ii with P (East) = 0.2 165 (+) to PD with P (East) = 0.4 (•) to AD-iii with P (East) = 0.6 (×) (Fig. 4c). These dynamics have practical implications because each type of dilemma (PD, SH, and AD) is associated with a distinct set of appropriate policy responses, as discussed in Sect. 5.  The two-by-two games described above demonstrate the variety of social dilemmas that can exist within groundwater contamination games. We now translate the game to an N-player system in which each player faces the same decision (σ i ) and costs (C x and C σ ) as above, while also generating a more plausible representation of groundwater behavior and probabilities of contamination. game, the associated probability p i = P (X i ) is the union of probabilities that the well is contaminated by any septic system j: We write the probability of well i being contaminated by septic tank j as between the domestic well and septic system, and G represents the set of hydrogeological parameters that determine groundwater flow. The probability p ij therefore encapsulates the groundwater hydrology between i and j and the decision by j whether 180 or not to upgrade their septic system. This probability associated with each of the N players can therefore be determined by a groundwater model (Sect. 3.2).
We begin by considering the key economic trade-off that is encapsulated by the decision (and associated risks) of whether or not to upgrade a septic system. Utility within the game can be considered from the perspective of the individual household (whether or not they should upgrade) and the perspective of the community (whether or not everyone should upgrade). Because 185 the game is fully symmetric, we focus on the utility of an individual player given by Eq. (1), which also representative of the mean utility for households in the community. The reduction in probability of contamination for player i if everyone upgrades is given as positive. In contrast, the reduction in the probability of contamination of player i if he is the only one to upgrade is ∆p ii = is positive. Furthermore, a social dilemma occurs when In other words, all players would benefit from upgrading (right hand side), but any individual player will endure a decrease in utility if they are the only one to upgrade (left hand side). The occurrence of a social dilemma can be understood by plotting these quantities against each other (Fig. 5). In particular, high ∆p i and C x /Cσ increase the benefits of cooperation (everyone 195 upgrades) because the consequences of not upgrading are steep (high probability, and associated costs, of contamination).
Conversely, low ∆p ii and C x /Cσ increase the maximum loss of upgrading individually (i.e., the expected decrease in utility when no one else upgrades), because individual benefits of upgrading are small. Both the cost ratio C x /C σ and probability ratio ∆p ii /∆p i play an important role in determining whether or not a social dilemma occurs, and characterizing social dilemmas requires understanding both properties together (Fig. 5c).

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Cooperation is more likely to emerge when the maximum loss of upgrading (which occurs if a player is the only individual to upgrade) is low relative to the benefits of cooperation. This situation is represented by the • in Fig. 5c, which could occur when the cost of contamination is relatively high (C x >> C σ ) and there is a reasonable chance of self contamination (∆p ii /∆p i → 1) but not enough to convince an individual to upgrade without collective action (i.e., not in quadrant I). Conversely, cooperation is likely more difficult when the maximum loss of upgrading is high in comparison to the benefits of cooperation. This situation 205 is represented by the × in Fig. 5c, and would occur when the cost ratio (C x /C σ ) is just over 1 and the probability of self contamination is negligible (∆p ii → 0).

Groundwater model
Groundwater behavior determines the probability that any well i could be contaminated by leachate from a septic field j, p ij .
These probabilities and their unions, as required by Eq.
(3), could be determined via any groundwater model appropriate for of contamination reflects actual contamination rates. In the model, the probability of contamination p ij is equivalent to the probability that a particle from septic system j intersects well source area i (Fig. 6). The model encapsulates uncertainty in the mean horizontal direction of flow (θ, Fig. 6a) as well as uncertainty in the flow trajectory within the vertical plane (u z /u r , 220 Fig. 6b), given that the particle may pass over or underneath the well (Horn and Harter, 2009). Flow velocities in the θ-z plane are determined by the vertical and horizontal velocities, u z and u r , which are, in turn, determined by the vertical and horizontal fluxes. We assume steady state behavior, such that the vertical flux is given by average annual seepage to the water table, S. All water is assumed to eventually flow toward a linear sink (e.g., a gaining stream), so that the horizontal flux can be estimated as the rate of seepage multiplied by the uphill contributing area. When 225 considering a unit width of the aquifer, this rate is given by S × L, where L is the distance to the groundwater divide.
For parsimony and in the absence of additional information, we assume that uncertainty in the mean direction of groundwater flow can be represented by a uniform distribution, θ ∼ U (θ 1 , θ 2 ). Uncertainty in the θ-z trajectory of flow is determined by uncertainty in the distance to the groundwater divide, also parameterized as as uniform distribution, L ∼ U (L 1 , L 2 ). Wells are cased from the ground surface to depth z 1 and screened from z 1 to z 2 . Flow paths to the well occur only from groundwater 230 within the range of depth of the screened portion of the well, [z 1 , z 2 ]. The assumption of uniform uncertainty in flow direction has been applied in similar systems (Bremer and Harter, 2012), and the possibility that contamination can pass over or underneath a well has been demonstrated using MODFLOW (Horn and Harter, 2009). The groundwater model was implemented in an R package (Penny, 2021) that allows the calculation of both p ij and p i . As information is gained and the most critical components identified, more complex representations can readily be added, but starting with a parsimonious representation and 235 set of models has significant benefits with any probabilistic risk assessment (Bolster et al., 2009;de Barros et al., 2011). into Granger, but sewer connections are optional and residents must pay a fee for the connection (St. Joseph County, 2011).
New developments may be more likely to be connected to piped water and sewers (Sheckler, 2021), but the issue in existing homes remains. As yet, there is no clear solution to the problem. To explore the possibility of social dilemmas and potential policy solutions, we generate parameter estimates for the contamination game described above, and calibrate the model to observed prevalence of contamination in Centre Township and Granger CDP.

Model parameterization and calibration
Both the economic portion of the model and the groundwater portion must be parameterized. For the economic portion, a complete cost of contamination is difficult to obtain. The health consequences of nitrate contamination are most acute for infants who may develop methaemoglobinaemia (Knobeloch et al., 2000), whereas the long-term consequences of exposure for children and adults are potentially concerning but unclear (Ward et al., 2018). We therefore focused on the perceived cost houses. In particular, Guignet et al. (2016) used a hedonic analysis of nitrate well tests and real estate transactions in Lake County, Florida to determine that the typical reduction in home value was 2-6% for those homes with nitrate contamination over the EPA MCL. We consider this loss in home value to be a plausible estimate of the cost of contamination.
The cost of upgrading the septic system to reduce or prevent nitrate contamination has been documented for EPA approved  (Bayless and Arihood, 1996). Indiana regulations require that wells are installed at a depth of at least 25 ft (7.6 m) (Indiana State Department of Health, 2021), and the depth to the groundwater table is typically 5-15 ft (Bayless and Arihood, 1996). Given that wells reduce the water level locally, we assume a groundwater depth at the upper end of this range (15 ft, 4.6 m). We further assume that wells were screened down to the bottom of this aquifer layer (70 ft, 21.3 m) (Bayless and Arihood, 1996). We approximate the distance to the groundwater divide 275 as a uniform distribution characterized by the average width of the aquifer divided by the average height, giving an approximate ratio of 10-to-1 (Bayless and Arihood, 1996). Although these parameters are representative of the groundwater scenario, the uncertainty embedded in these estimates underscores the importance of calibrating the model to observed contamination probabilities.
We use the radius of well capture (r s ) as the only calibration parameter. This value represents the maximum lateral distance 280 from the well that the center of a septic plume could pass through to generate contamination within the well. Therefore, the value could potentially change for different regions of the aquifer and different thresholds of contamination. We calibrate the model in both Centre Township and Granger CDP for contamination thresholds of 5 ppm and 10 ppm (Fig. 7a).
In general, with a 10 ppm threshold, the probability of contamination is low (see figure 7a). With a 5 ppm threshold, the likelihood is much higher, and there is a clearer relationship between contamination rates and housing density. The issue of 285 social dilemmas depends on assumptions of the system. We specify four scenarios, representing different assumptions on the costs and likelihood of contamination. In all cases the cost of contamination is a fraction of the home value: i. Centre 5 ppm calibration, $4000 to upgrade septic, C x equals 5% of home value.
ii. Centre 10 ppm, $4000 to upgrade septic, C x equals 5% of home value.
iii. Granger 5 ppm, $8000 to upgrade septic, C x equals 4% of home value.
We calibrated r s for each scenario using observed data, allowing us to determine ∆p i and ∆p ii based on housing density.
We used three values housing density to evaluate the model: 1, 2, and 3 households per acre. We further calculated the cost of i. Centre > 5ppm ii. Centre > 10ppm iii. Granger > 5ppm iv. Granger > 10ppm Social dilemmas based on housing price and housing density (shown), and costs of contamination and septic upgrades (not shown, see text).
Cooperation benefits, ∆piCx/Cσ − 1, are shown as background shading (in blue) and the maximum loss of upgrading, 1 − ∆piiCx/Cσ, is shown as the fill color of example points (in red). Note that both are defined the same as in Fig. 5.
contamination for multiple values of home prices: $100k, 400k, and 700k. These can be mapped to the space ∆p i vs C x /C σ (Fig. 7c).

Social dilemmas
With a 10 ppm threshold, it is generally unlikely, except for more expensive houses (see Fig. 7c for ranges), that a social dilemma will occur under the assumptions in scenarios ii and iv. With a 5 ppm threshold and assumptions as shown, it is much more likely that a social dilemma would occur. In fact, in some cases everyone chooses to upgrade (no social dilemma). This represents an extreme case with a large-enough region of well capture and co-located wells and septic systems. If the direction 300 of groundwater flow were known, the septic would likely be placed downstream from the well such that ∆p ii would be reduced or not there is a Nash equilibrium that is Pareto inefficient, and the number of players that would have to upgrade to achieve a social optimal outcome (PD and AD games). In the Stag-Hunt game, if one player can be convinced to upgrade then the other player will be incentivized to upgrade as well.
and these situations could become social dilemmas. The 5 ppm threshold is below the EPA MCL for nitrates, and therefore C x /C σ could be overestimated. However, this ratio captures only the internalized costs to the homeowner and not additional public health or environmental externalities that could affect the socially optimal utility. This scenario is therefore illustrative of what could happen when such externalities are incorporated. Heterogeneity in groundwater contamination still plays a large 305 role. For instance, in some neighborhoods the rates of contamination were as high as 50% over 10 ppm, whereas the calibrated model estimated a contamination rate of about 12% (Fig. 7a, ii and iv). Thus the model is conservative in some cases and, if these differences could be resolved, then social dilemmas could be evaluated with greater precision. For instance, some wells tap into the deep aquifer, but only 10% of wells in the dataset came with well depth, therefore it is possible that wells in the shallow aquifer have higher rates of contamination. Other factors include localized abnormalities. For instance, septic design 310 and maintenance practices play a role in septic water quality and well contamination, and lakes exist in some neighborhoods that may be recharging and diluting groundwater.

Discussion
The results demonstrate that multiple types of social dilemmas can arise in the context of groundwater contamination of private water supply and sanitation systems. When considering two-player games with uncertainty in the mean direction of 315 groundwater flow, we identified five types of social dilemmas including a Prisoner's Dilemma, Stag-Hunt game, and three types of Asymmetric Dilemmas. The type of social dilemma can serve as a guideline to understanding barriers to cooperation and policy opportunities to shift from the Nash equilibrium to the Social Optimum (Fig. 8).
The Prisoner's dilemma is, in essence, a free-rider problem: a market failure that occurs when people might benefit from services without having to pay the associated cost (Fischbacher and Gachter, 2010). Here, the decision keep a basic system (B) 320 strictly dominates the decision to upgrade to an enhanced system (E), and each player benefits the most when only the other player upgrades. This allows the player in question to utilize the cleaner water associated with the other player's upgraded septic without having to pay for an upgrade themselves. A variety of market design approaches have been identified to address the free-rider problem in the context of public goods, including assurance contracts (Tabarrok, 1998), Coasian bargaining (Farrell, 1987) and incentives for cooperation (Balliet et al., 2011). The Nash equilibrium in the Prisoner's dilemma game is not Pareto 325 efficient, meaning that all players can benefit if everyone upgrades their septic system. The free-rider problem can therefore also be addressed through altruistic social norms (potentially combined with enforcement and sanctions) to support collective action (Kerr, 1992). In other words, despite the various obstacles to overcoming the Prisoner's Dilemma, there exist multiple governance approaches to address this challenge and initiate community-wide upgrades to enhanced septic systems, allowing everyone to benefit from cleaner groundwater and reduced contamination costs.

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In contrast, the Stag-Hunt game (also known as assurance game) is a problem of strategic uncertainty, where players might fail to achieve a socially-optimal equilibrium by not being able to coordinate their decision-making. Here, either player can only benefit from upgrading if the other player upgrades as well. Under these conditions, they will only endure the cost of upgrading if they are reasonably sure that the other player is also upgrading. Strategic uncertainty can be resolved through trust and reliable information (Jansson and Eriksson, 2015). In contrast with the Prisoner's Dilemma, where all players must 335 be incentivized to upgrade, the assurance game only requires any player to buy in (e.g., through targeted subsidies). The other player will reciprocate if she has sufficient assurance of the other player's buy in. As such, public programs that provide some measure of transparency or assurance that some households have upgraded their septic systems would encourage additional households to also upgrade.
In each of the Asymmetric dilemmas, the Nash equilibrium is Pareto optimal but one player receives outsize benefits at 340 the expense of the other player. A greater social optimum can be achieved by one player sacrificing some of their utility.
Asymmetric Dilemmas i and ii are akin to the Stag-Hunt game in that the social optimal is one upgrade away from the Nash equilibrium. However, unlike the Stag-Hunt game, achieving the social optimal requires convincing a specific player to upgrade. Any subsidy must therefore specifically targeted to the appropriate player, which requires reliable information on all players' utility parameters. Finally, Asymmetric Dilemma iii is partially similar to a Prisoner's dilemma because the social 345 optimal requires both players to upgrade. However, unlike the Prisoner's dilemma where both players increase their utility by achieving the Social Optimal, one player is worse off in the social optimal compared to the Nash Equilibrium. In this situation, targeted subsidies or side payments might be needed, in addition to social or market approaches to address potential free-rider problems.
Social dilemmas in N-player games can be understood by non-dimensionalizing the expected utility when everyone upgrades 350 (i.e., the cooperation benefits) and expected utility when only one player upgrades (i.e., the maximum loss of upgrading). Social dilemmas arise when both are positive. Situations akin to the Prisoner's dilemma are likely to arise when the cooperation benefits are modest and the maximum loss of upgrading is high. Dilemmas akin to a Stag-Hunt game are more likely when the cooperation benefits are high and the maximum loss of upgrading is low and p ii ∩p ij > 0, meaning that an individual's upgrade becomes more appealing after another player upgrades. The three asymmetric dilemmas only emerge in situations where 355 (known) specific players have undue influence on the contamination of other players' wells. We assume that the N-players are homogeneously located throughout the neighborhood and do not have perfect information on the flow direction of groundwater, and that all households face similar costs and emit comparable pollution loads. Under these conditions, asymmetric dilemmas would only emerge near the edges of the neighborhood and therefore, assuming the neighborhood is sufficiently large, only concern a small fraction of the households. Although both types of symmetric social dilemmas (Prisoner's Dilemma and Stag-360 Hunt) are possible, the Prisoner's dilemma situation is more likely because the case study in St. Joseph County suggests that our model potentially overestimates ∆p ii and the maximum loss of upgrading (Fig. 4). However, the specific benefits and costs of such upgrades are likely to be location-specific and depend on a variety of factors. While we show that such dilemmas are possible in St. Joseph County, additional groundwork would be needed to more clearly identify the type of situation that exists within the county, including the specific locations where housing and groundwater behavior give rise to such dilemmas.

Conclusions
Contamination of domestic wells by septic leachate is a public health concern in residential areas with high housing density. The issue has been particularly difficult to resolve for public health officials and residents alike. We develop a theoretical framework to understand how misaligned incentives can give rise to social dilemmas within such private water systems. A variety of social dilemmas can occur in such situations, depending on the groundwater hydrology, costs of various treatment technologies, and 370 configurations of households. These dilemmas include the classic Prisoner's Dilemma and Stag-Hunt game, as well as an Asymmetric Dilemma that favors some players over others. These dilemmas can occur even if all households face the same economic costs of contamination and treatment. As such, this manuscript provides a theoretical basis to identify when social dilemmas could be occurring and offers a new perspective through which to approach management of such systems. A better understanding of the challenges for any specific location could be achieved using more sophisticated groundwater and transport 375 models combined with location-specific estimates of the associated costs of contamination and septic upgrade. Such analyses would better resolve the details pertaining to social dilemmas, but the overarching framework would remain unchanged.
Code and data availability. Code for the two-player games and groundwater model is available in an R package (Penny, 2021) and can be accessed online at https://github.com/gopalpenny/nitratesgame. Household nitrates data are available from the corresponding public agencies.
Author contributions. G.P., D.B., and M.F.M. designed the research. G.P. conducted the analyses. G.P. and M.F.M. wrote the manuscript.