# Accounting for Ecological Choices through Individual Commitments in Collective Actions: The Kovenant Model

Massimo Cervesato, Mathieu Guigourez

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Massimo Cervesato, Mathieu Guigourez. Accounting for Ecological Choices through Individual Commitments in Collective Actions: The Kovenant Model. 2025. halshs-05450446

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CES

Centre d'Économie de la Sorbonne
UMR 8174

Accounting for Ecological Choices through Individual Commitments in Collective Actions: The Kovenant Model

Massimo CERVESATO, Mathieu GUIGOUREZ

2026.01
# Accounting for Ecological Choices through Individual Commitments in Collective Actions: The Kovenant Model

Massimo Cervesato *† and Mathieu Guigourez ‡§

December 19th 2025

## Abstract

This paper introduces the *Kovenant model*, a formal framework for understanding ecological choices as resulting from norm-guided individual commitments that emerge in fragmented collective contexts. Rather than primarily seeking to optimise outcomes or induce cooperation through incentives, the model represents how agents act as if an ecological covenant were in place, thereby reshaping their own decision-making structure. The model captures key behavioural features such as over-investment or crowding-out effects, showing that these are not irrational deviations, but responses to the absence of well-defined shared normative expectations. The Kovenant model offers, thus, theoretical ground for explaining ecological behaviours in emerging social norms.

**Keywords:** Commitments; Collective Action; Ecological Behaviours; Common-Pool Resources; As if reasoning

## 1 Introduction

It is now a well-established fact that climate change poses a pressing political, economic, and ethical challenge. Individuals are caught in a social dilemma regarding the climate crisis, which can be modelled using a Common-Pool Resources (CPR) game. They are driven to overproduce and overconsume, thereby contributing to the neglect of the crisis. Economic literature has emphasised solutions to this collective dilemma, traditionally relying on externally imposed sanctions or incentive-based mechanisms to steer behaviour in the desired direction (Ostrom et al., 1999). Instead of fostering such solutions, this article develops a formal model of ecological commitment — based on a CPR game —

<footnote>*CONTACT: massimo.cervesato@univ-paris1.fr</footnote>

<footnote>†Université Paris 1 Panthéon-Sorbonne, Centre d'Économie de la Sorbonne, France</footnote>

<footnote>‡CONTACT: mathieu.guigourez@univ-paris1.fr</footnote>

<footnote>§Université Paris 1 Panthéon-Sorbonne, Centre d'Économie de la Sorbonne, France</footnote>
to account for the voluntary reductions in individual carbon emissions. To illustrate such behaviours, consider an individual—whom we will refer to as *Dan* throughout the article — who decides to become vegetarian in response to the climate emergency. This choice is (1) personally costly, (2) marginally ineffective in isolation, and (3) difficult to explain within the standard rational choice framework. How can this phenomenon be explained?

We propose a redefinition of what we call an *ecological covenant* — a normative framework in which agents see themselves as bound by a shared ecological commitment, even in the absence of external enforcement. By committing to this covenant, agents do not aim primarily at maximising expected payoffs (such as global harm reduction), but rather contribute to *something else*. We argue that this *something else* is the enactment of a social and ecological covenant: agents act as if such a covenant were already in place, and by doing so, help bring it into fruition. Instead of prescribing the precise actions individuals ought to undertake, *commitment* to the covenant is associated with a particular understanding of ecological problems and their resolution—that is, with a specific structure of each agent's payoff function. We thus develop a covenantal framework for understanding the emergence of moral and social norms in ecological collective actions. In this way, the central claim of this paper is that individual actions in favour of ecology, although often dismissed as marginal or irrational, can be understood as the seeds of a broader normative structure: the *covenant* reshapes the individual's valuation expectations, i.e. the individual's payoff function, and stabilises new forms of collective action. Our *Kovenant model* demonstrates that agents, when acting as if such a shared covenant existed, are indeed driven to contribute to a more desirable social-ecological equilibrium, albeit sometimes in an inefficient manner. Crucially, this normative structure does not rely on pre-specified ethical rules (as in Kantian optimisation) nor on the aggregation of preferences (as in utilitarian logic), but emerges from within the game as a product of commitment-based reasoning.

To establish such a result, we develop a CPR game in which the collective production function is represented by what we term the *social state of the world*—that is, the institutions and modes of production through which human societies sustain economic activities. This is directly tied to a *state of the living*, which includes natural resources, biodiversity, and ecosystems (our CPR), all of which are being exhausted in order to fuel these activities. In this game, agents choose whether to reduce their ecological impact based on the structure of their payoff function. When the structure of the function changes, the game itself is transformed, leading to different equilibrium outcomes.

Our framework builds explicitly on two foundational literatures. On the one side, it draws from Elinor Ostrom's work on CPR governance, particularly the role of informal norms, self-regulation, and community-based rule-making (Ostrom et al., 1992, 1994; Ostrom, 1998; Ostrom et al., 1999; Ostrom,
2010, 2014). On the other, it draws from Amartya Sen's account of commitments, understood as deliberate shifts in motivation that can override standard preference-based reasoning (Sen, 1973, 1977, 1985, 2005). In this article, we analyse a CPR game through a Senian lens, treating commitments as endogenous determinants of individual behaviour, and we interpret ecological commitments as if they constituted a new form of covenant in the Ostromian sense — one that emerges through agents' reciprocal engagement with the community (Ostrom et al., 1994). Two complementary bodies of literature further inform our framework. First, recent work on how climate policies can induce changes in preferences provides important insights into the plasticity of individual motivations in ecological contexts (Dasgupta et al., 2016; Mattauch and Hepburn, 2016; Mattauch et al., 2022). Second, our model resonates with ongoing efforts to establish a renewed ecological and social contract, which we explicitly reinterpret in covenantal terms (Serres, 2009; Engel, 2010; Combet and Pottier, 2024).

Furthermore, addressing the climate crisis meaningfully requires rethinking the normative criteria through which climate and public policies are evaluated. Traditional benchmarks such as Paretian optimality — even when refined through formal frameworks like Kantian equilibrium — remain inadequate. They are ill-suited to account for ongoing coordination failures and offer limited insights into how agents might engage with collective action in the absence of institutional enforcement. Instead of outcome-based criteria, the Kovenant model responds to this limitation by providing a process-based framework in which individual ecological commitments can give rise to new normative structures—covenants—that guide behaviour and shape shared expectations. In doing so, it paves the way for a new approach to individual and collective responsibility, not as a derivative of optimal outcomes, but as a condition for the emergence of collective action.

The remainder of the article proceeds as follows. Section 2 defines the analytical framework of the *Kovenant model*, treating ecological commitments as shifts in moral stance rather than fixed behavioural prescriptions. Section 3 presents the formal model, introducing our CPR game basis and the resulting shifts in the individual decision-making process. Section 4 derives the resulting equilibria, showing that under conditions of strategic uncertainty, such commitments lead to over-investment—revealing the tension between unilateral commitment and collective coordination. Section 5 compares our approach with Paretian optimality and Kantian equilibria, highlighting key conceptual departures. Section 6 concludes by reflecting on the broader implications of covenantal reasoning for climate ethics, institutional design, and public policy.
## 2 The Framework: An ‘As-if’ Covenantal Reasoning

According to Ostrom et al. (1999, p. 278), CPR situations arise under two conditions: (1) it is extremely costly—physically or institutionally—to exclude an agent from benefiting from the resource, and (2) individual consumption diminishes the resource available to others. Both of these conditions map closely onto the structure of the global environmental crisis: first, it is nearly impossible to exclude any agent from drawing on environmental resources (broadly understood); second, each agent’s depleting actions incrementally degrade those shared resources, bringing the system closer to irreversible tipping points. While the scale of the problem may seem too extensive to be appropriately modelled using a CPR framework, Ostrom herself noted that effective responses to climate challenges may be bolstered by smaller-scale, decentralised commitments. As she writes, “[b]uilding a strong commitment to find ways of reducing individual emissions is an important element for coping with this problem” (Ostrom, 2014, p. 97). Besides, such commitments are often more viable within polycentric governance structures composed of interlinked local units (Ostrom, 2010).

In this article, we treat the depleting collective resource as the state of the living — that is, the biosphere and its regenerative capacities — which is progressively compromised by carbon emissions. More precisely, carbon emissions are conceptualised as a depleting shared stock of atmospheric stability. For instance, if 450ppm is taken as a tipping threshold (as implied by the Paris Agreement target of limiting warming well below +2°C), then each additional unit of carbon emission constitutes a withdrawal from a finite global “carbon budget.” From this perspective, the climate crisis can be understood as a CPR game: it is extremely costly to exclude any agent from emitting, and each action contributes — however marginally — to the cumulative depletion of a shared resource.

### 2.1 A Definition of Commitments

In modelling ecological behaviour, we adopt the concept of *commitment* in the sense developed by Amartya Sen. Sen's work provides one of the earliest and most sustained critiques of standard rational choice theory, particularly in contexts—such as climate-related action—where individuals often act against narrowly self-interested incentives (Sen, 1973, 1977, 1985, 2004, 2005). For Sen (1977, p. 329), commitment refers to forms of agency that reflect a broader array of motivations — including ethical duties, moral norms, and rule-following behaviour — that need not align with the maximisation of personal welfare.

A classic example of such behaviour is Sen (1973, p. 254)'s illustration of returning glass bottles for recycling. In this scenario, an individual may value environmental sustainability but is disinclined to incur the personal inconvenience of returning bottles. The situation mimics the prisoner's dilemma:
each person would prefer that others act (so the environment is protected) while they themselves do nothing. Consequently, even when all agents care about the environment, nobody takes action. This dilemma does not arise from ignorance or indifference, but from the logic of strategic individualism implied by the situation in which the agents find themselves.

Commitment, for Sen, offers a way out of such dilemmas. Committed agents act as if they had different preferences — ones that reflect obligations or imagined goals — even when this entails personal cost. As he writes: “if people are ready to act (individualistically) on the basis of some ‘as if’ — more cohesive—orderings, then they can do better than acting individualistically in direct pursuit of their real goals” (Sen, 2004, p. 214). This as if reasoning can involve both internalised norms and the imagined adoption of others’ goals or collective purposes (Pettit, 2007, p. 33). Commitments, in this light, are not reducible to altruism or rule-deontology, as they involve a “rational recognition of rules associated with social membership” (Pauer-Studer, 2007, p. 75).

In our model, ecological commitments are understood as agents acting as if they recognise the desirability of an alternative social state of the world — one that is normatively correlated with a specific and sustainable state of the living. Therefore, we do not formulate explicit rules or institutional imperatives as such.<fnref n="1" /> Rather, this type of commitment reflects a change in the agent’s decision structure, shaped by their interpretation of others’ actions and communications as if guided by the same covenantal logic. Thus, we understand ecological commitment as an agent’s willingness to adopt an implicit normative order, even when such a covenant is not formally instituted.

## 2.2 A Definition of Covenants

As argued by Lewis and Petersen (2023, pp. 853–856), there is a strong theoretical convergence between Sen’s account of commitment and Ostrom’s work on collective action and rationality. This connection rests on a shared emphasis on social identity and non-instrumental motivation. As Lewis and Petersen (2023, p. 856) put it, “[i]n identifying with a group, therefore, individuals understand that they have accepted an obligation to adhere to the relevant rules even when doing so involves them departing from their individual (self)goals, making a commitment that motivates their subsequent actions”. In other words, social identification and normative commitment are co-constitutive: to commit is also to see oneself as embedded in a rule-bound community. In Ostrom’s framework, such commitments are not abstract moral dispositions but emerge through concrete processes of communication and shared normative expectations (Ostrom et al., 1994). Covenants, in this sense, are

<footnote n="1">¹In a recent theorisation of a renewed ecological social contract, Combet and Pottier (2024, p.187) define it as implicit rules which aim at “enabling socio-economic functioning [that are] compatible with planetary boundaries.” Their conception of an ecological contract is thus to reshape social imaginaries of what a “transition” stands for; they do not elaborate laws or imperatives upon which each agents ought to behave (Combet and Pottier, 2024, p.189).</footnote>
the institutional forms that agents construct to navigate collective action problems in the absence of centralised authority (Ostrom et al., 1992; Ostrom, 1993, 2005).

It is important to distinguish covenants from (social) contracts. Although contracts typically require enforceable obligations and external guarantees, covenants are grounded in mutual promises and normative recognition among participants (Ostrom et al., 1992). However, a covenant is more than a shared norm: it is what enables the existence and effectiveness of norms and, in a more general way, valuation expectations.<fnref n="2" /> A covenant is what collectively shapes the decision-making structures of individuals.

In our model, the renewed ecological covenant is understood as shared normative expectations towards a more sustainable *state of the living* that do not require a formal institution. Consequently, commitments to this covenant are conceived as a shift in the decision-making process *as if* the individual valued the covenant itself. Our framework is thus explicitly *fictionalist*: we are not concerned with the realism of the agent's motives, but with the structure of reasoning that treats the ecological covenant *as if* it were already binding.<fnref n="3" />

## 2.3 A Fictional Framework: The As If Covenant

Our goal is not to offer a detailed explanation of the origins of ecological commitments — whether psychological, evolutionary, or cultural.<fnref n="4" /> Rather, we aim to capture the shift in the agent's decision structure once such a commitment is in place. From an epistemological perspective, “theoretical fictions are first developed with the aim of making knowledge possible, and then in order to give direction to actions” (Schmutz, 2006, p. 595). On this view, fictions are not empirical hypotheses to be tested, but conceptual tools that articulate epistemic, legal, heuristic, or ethical ideals. From these, one may derive universalising norms or action-guiding frameworks, without assuming their literal truth. Practical fictions, unlike theoretical postulates, do not aim to perfectly describe the world or uncover causal mechanisms. Their role is normative and motivational: they provide orientation for action even when their content is not empirically grounded (Vaihinger, 1931). In this way, adopting a fictional framework allows us to think beyond cooperation and efficiency as the primary evaluative standards in collective action theory. While the hypothesis aims to describe and predict, “fiction seeks to invent”, *i.e.* to construct new normative possibilities that have yet to be institutionalised or even socially recognised (Schmutz, 2006, p. 524).

<footnote n="2">²In Ostrom (2005, p. 174)'s own formalisation, such expectations are represented through a “delta parameter” added to an agent's payoff function: a value that reflects the moral or emotional costs and benefits associated with norm compliance.</footnote>

<footnote n="3">³By “fictionalist,” we mean that our model does not require direct access to agents’ actual motives or beliefs.</footnote>

<footnote n="4">⁴For psycho-evolutionary approaches to the emergence of commitment behaviour, see Simon (1985, 1990).</footnote>
Our model uses this fictional lens to reinterpret ecological behaviours: not as expressions of altru-
ism or external incentives, but as normative stances anchored in imagined social membership. This
fictional, or as if covenant, thus, serves as both a conceptual tool and a motivational device: it allows
agents to act as if they were already part of a shared ecological order, even when such a structure
remains institutionally unrealised.

3 The Kovenant Model

In this section we introduce the Kovenant model, specifying the CPR game basis and the consequences of the *as if* ecological covenant.

3.1 A Common-pool Resources Game Basis

Before examining how agents become committed to the *as if* ecological covenant, we first model
the social dilemma they face using a CPR framework: even when agents acknowledge the ecological
consequences of their polluting choices, they remain driven to overproduce and overconsume, thereby
exacerbating the crisis.

Let $N \gg 1$ denote the number of agents. They are ordinary individuals confronted with a collective action dilemma tied to the climate crisis as it manifests in everyday life. The relevant horizon is long, and although no one expects an imminent collapse, the cumulative impact of individual actions remains significant. For each issue related to the climate crisis, each agent $i \in \{1, \dots, N\}$ chooses a level of investment or consumption $x_i \in \mathbb{R}_+$ of a good contributing to this problem. For instance, Dan chooses to eat a certain amount of meat each day during his lunch break at work. The vector $\mathbf{x} \in (\mathbb{R}_+)^N$ represents the agents' choices regarding the issue under consideration.

What ultimately matters in this context is the aggregate contribution of these choices to a social state of the world. Individual action, such as Dan's meat consumption, not only satisfies an immediate need but also reinforces economic, cultural, and institutional structures that sustain high-carbon production. Consequently, we assume that individual choices are positively correlated with carbon emissions: higher aggregate investment or consumption further degrades the state of the living—that is, the ecological system sustaining life.

The aggregate of individual choices thus generates a social state via a production function $F$ :
$\mathbb{R}_+ \to \mathbb{R}$, which maps total investment or consumption into a social outcome: $F(\sum_{j=1}^N x_j)$. We adopt a broad interpretation of $F$, as illustrated in our previous paragraph on Dan's choice to eat meat. It encompasses not only economic output but also cultural and social institutions, including imag-
inaries surrounding nature and its value. Accordingly, we assume $F$ is $C^\infty$, strictly concave, and single-peaked, capturing diminishing returns and the possibility of overproduction and depletion at the collective level.<fnref n="5" />

Production is then distributed across individuals via $D_i : (\mathbb{R}_+)^N \to [0, 1]$, which maps investment and consumption choices into agent $i$'s share of the social state. We assume $x_i$ increases $D_i$, but leave the functional form unspecified for now. Since production is fully allocated, $\sum_{j=1}^N D_j(\mathbf{x}) = 1$.

Let $C_i : \mathbb{R}_+ \to \mathbb{R}_+$ denote the cost associated with agent $i$'s choice (e.g., monetary, time, or energetic loss, etc.).

The payoff function $U_i : (\mathbb{R}_+)^N \to \mathbb{R}$ for each agent $i$ is then:

$$U_i(\mathbf{x}) = D_i(\mathbf{x})F\left(\sum_{j=1}^{N} x_j\right) - C_i(x_i) \quad (1)$$

In such a social dilemma, agents are driven to overconsume and overproduce (Ostrom et al., 1992, 1994). Moreover, individuals like Dan face substantial epistemic uncertainty in this context. First, they do not know the precise impact of their action on the social state (i.e. the shape of $F$), nor the ecological implications of that state (i.e. how much $F$ is exhausting the state of the living).<fnref n="6" /> For instance, what specific ecological benefit, if any, can be causally attributed to Dan's substitution of meat consumption with soy-based alternatives?<fnref n="7" /> Second, they cannot precisely evaluate the personal benefit of their contribution, since the form of $D_i$ is unknown. Even if substituting soy for meat yields positive effects for society and the environment, the specific benefit accruing to Dan remains opaque.

Yet individuals — such as Dan, who becomes vegetarian — continue to make committed ecological choices. This raises a twofold challenge: for agents, who lack the information required to evaluate the real impact of their actions; and for theorists, who must explain why such agents nonetheless undertake costly ecological behaviours.

Our answer is that Dan's decision is structured by a prior commitment. By adopting an *as if* reasoning, he effectively transforms his payoff structure: he behaves as if a normative ecological covenant were operative, thereby re-evaluating both social production and distributive outcomes. In other words, such commitments reconfigure the agent's interpretation of $F$ and $D_i$, giving shape to an *as if* payoff function.

<footnote n="5">⁵We shall get back to the particular trend of $F$ in the next subsection.</footnote>

<footnote n="6">⁶Such a hypothesis has also been used to justify critics of individual responsibility to curb one's carbon emissions, see for instance Sinnott-Armstrong (2005); Kingston and Sinnott-Armstrong (2018).</footnote>

<footnote n="7">⁷For a related discussion on the complex relations between agricultural production and carbon impact, see Norwood et al. (2015).</footnote>
## 3.2 The *As if* Ecological Covenant: A Shift in the Decision-making Structure

### A Shift in Attitude Toward Ecological Issues

Initially, the agent treats the world as one in which nature is intact, viewing their actions as contributing to its degradation. Yet, upon recognising that environmental degradation is already underway, the reference point shifts accordingly: (1) their existing behaviour is now understood as contributing to the production of a polluted social state, and (2) the question becomes not how to preserve a pristine environment, but how to intervene in an ongoing process of deterioration. For instance, once Dan acknowledges environmental degradation, meat consumption can no longer be seen as neutral but as participating in an entrenched system of ecological harm.

Ecological commitment thus does not merely aim to minimise one's impact in order to restore a former social state — one that is, in fact, irretrievably lost. It is about producing something else, namely a qualitatively different social state. The committed agent reframes the problem so as to align their action with a redefined normative orientation, effectively transforming the structure of their decision-making.<fnref n="8" />

Let $n \gg 1$ denote the number of committed agents, with $n \le N$. Let $y_i \in \mathbb{R}^+$ be the ecological effort of a committed agent $i$.<fnref n="9" /> If $y_i = 0$, the agent does not modify their behaviour; if $y_i > 0$, they act positively in response to environmental concerns. For instance, Dan reducing his meat consumption implies $y_{\text{Dan}} > 0$. The behaviour of non-committed agents is treated as fixed and exogenous, contributing in no positive way.<fnref n="10" />

This reframing can be represented as a straightforward translation. Let $x_i^{\text{ref}}$ denote the total level of agents' investment and consumption prior to agent $i$'s commitment. Then, $y_i = x_i^{\text{ref}} - x_i$. Accordingly, let $c_i > 0$ be the constant marginal cost of ecological effort, so that $C_i(y_i) = c_i y_i$.

Moreover, the committed agent reinterprets the collective production function as $F^{\text{as if}}: \mathbb{R}_+ \to \mathbb{R}$, which remains $C^\infty$, strictly concave, and single-peaked. Its origin lies on the decreasing segment of the single-peaked concave function $F$, reflecting that we are already operating under climate crisis conditions. To produce positively something else, the committed agent now considers how their ecological effort allows them to deviate from the present trend.

<footnote n="8">⁸Given the one-dimensional structure of our model, any attempt to capture this transformation is necessarily reductive. However, it provides a fruitful basis for future multidimensional extensions.</footnote>

<footnote n="9">⁹We assume for simplicity that committed agents are indexed first in the list, i.e., those for which $i \in \{1, \dots, n\}$.</footnote>

<footnote n="10">¹⁰In other words, once commitment occurs, the agent $i$ no longer interacts strategically with all the agents: the actions of non-committed agents are assumed to remain constant.</footnote>
To clarify the underlying intuition, we present a simplified formal translation. Let $F^{\text{ref}} = F(\sum_{j=1}^{N} x_j^{\text{ref}})$ be the reference output before commitment. The revised collective production function can then be defined as: $F^{\text{as if}}(\sum_{j=1}^{n} y_j) = F(\sum_{j=1}^{N} x_j^{\text{ref}} - \sum_{j=1}^{n} y_j) - F^{\text{ref}} = F(\sum_{j=1}^{N} x_j) - F^{\text{ref}}$.

Therefore, an ecological commitment changes an agent's attitude towards ecological issues. However, the underlying uncertainty persists: agents still lack sufficient information to evaluate their payoff. We argue that the committed agent reasons as if an ecological covenant provided a clearer outline of the production and distribution functions within their payoff.

## Shaping the Production Function

The as if covenantal reasoning involves two elements:

1. The agent values a new social state $K_i$, correlated with a sustainable state of the living, as if it were real and attainable;

2. The agent behaves as if there existed a threshold level of collective effort, $a_i$, sufficient to achieve it, i.e., whenever $\sum_{j=1}^{n} y_j \ge a_i$.

Allowing each agent to define $K_i$ and $a_i$ would introduce excessive relativism. Commitments, however, are not grounded in purely subjective judgements: they are informed by shared knowledge, social norms, and collective imaginaries, such as those articulated in IPCC assessments or internationally recognised climate targets. We therefore posit the existence of $K$ and $a$ such that, for all $i$, $K_i \approx K$, and $a_i \approx a$.

The value of the as if covenant is assumed to exceed each agent's cost of contributing to its realisation, i.e., $\forall i, K > c_i a$ — hereafter referred to as (A1). It is further assumed to greatly exceed its average cost, insofar as it constitutes a substantial collective benefit, i.e., $K \gg \bar{c}a$, where $\bar{c}$ denotes the mean individual cost — hereafter referred to as (A2).

Through his commitment, Dan understands his choice as contributing to a covenant valued at $K$, binding a number of equally committed individuals who must reduce their meat consumption by at least $a$.

In this way, ecological commitment does not come down to specifying $y_i$, but rather to adopting a normative frame $(K, a)$ that defines a broader collective production:

$$F^{\text{as if}}(\sum_{j=1}^{n} y_j) = K * \mathbf{1}_a(\sum_{j=1}^{n} y_j), \quad \text{with} \quad \mathbf{1}_a = \mathbf{1}_{\sum_{j=1}^{n} y_j \ge a}. \qquad (2)$$
## Shaping the Distribution Function

By acting as if an ecological covenant exists, each committed agent implicitly associates a personal reward with its realisation. We assume these benefits are distributed in proportion to the agent's ecological effort. Formally, assuming there exists $k$ such that $y_k > 0$, the as if share attributed to agent $i$ is:

$$D_i^{\text{as if}}(\mathbf{y}) = \frac{y_i}{\sum_{j=1}^{n} y_j} \qquad (3)$$

This proportional rule, while not unique, captures the rivalry characteristic of CPR settings, where benefits depend on relative rather than absolute contributions. An agent is rewarded only to the extent that they contribute.<fnref n="11" />

## Shaping the Payoff Function

As a result of the agent's as if covenantal reasoning, assuming there exists $k$ such that $y_k > 0$, the as if payoff function is:

$$U_i^{\text{as if}}(\mathbf{y}) = \frac{y_i}{\sum_{j=1}^{n} y_j} K \cdot \mathbf{1}_a \left( \sum_{j=1}^{n} y_j \right) - c_i y_i \qquad (4)$$

In what follows, we derive the equilibrium resulting from it.

# 4 Analytical Results: Ecological Choices as Over-investments

## 4.1 Collective Optimum

Suppose all committed agents cooperate to maximise the total collective payoff. The resulting aggregate utility is:

$$U_i^{\text{as if}}(\mathbf{y}) = \sum_{i=1}^{n} U_i^{\text{as if}}(\mathbf{y}) = K \cdot \mathbf{1}_a \left( \sum_{j=1}^{n} y_j \right) - \sum_{i=1}^{n} c_i y_i \qquad (5)$$

Since agents act as if the covenant were in place, their objective is to realise its benefits $K$ at the minimal collective cost. Hence, the collective optimum is achieved when:

$$\sum_{j=1}^{n} y_j = a \qquad (6)$$

Moreover, to minimise cost, only agents with the lowest marginal cost $c_{\min}$ should contribute. Let

<footnote n="11">¹¹Importantly, non-committed agents receive no direct benefits from the covenant. Although they may experience marginal indirect gains (e.g., through an improved social state), full participation requires normative alignment. Only agents who adopt the norms and values encoded in $K$ and act accordingly can access its rewards. Hence, the model remains consistent with a CPR framework rather than with a standard public-good setting.</footnote>
$I_{min}$ denote the subset of agents with $c_i = c_{min}$. The optimal effort profile $y^{opti}$ is then:

$$ egin{cases} y_i^{opti} = 0, & \text{for all } i \notin I_{min}, \\ \sum_{j \in I_{min}} y_j^{opti} = a. \end{cases} \tag{7} $$

The exact distribution of effort among agents in $I_{min}$ is indeterminate, as it does not affect the total outcome. The corresponding optimal collective payoff is thus:

$$ U^{\text{as if, opti}} = K - c_{min}a \tag{8} $$

which is strictly positive under (A1).

## Two key observations follow

First, this result does not imply that ecological problems are resolved. Agents act only to reach the threshold $a$, doing what appears necessary to make a more sustainable world, but no more. Given the uncertainty linking individual efforts to an ecological impact, $F^{\text{as if}}(a) = K$ cannot be assured; more likely, $F^{\text{as if}}(a) < K$. By acting as if the covenant were real, agents risk underestimating the actual effort required to achieve the desired outcome. For example, if Dan reduces his meat consumption to twice per week, it may appear *reasonable* provided that others adopt the same practice.<fnref n="12" /> However, 1) the link remains uncertain, and 2) if some do not follow, he may inadvertently sustain the status quo — supporting the meat industry — while bearing the ineffective cost of change. This underlines a key issue explored in Section 4.2: individual strategies actually involve over-investment, such as complete abstention from meat.

Second, the cooperative optimum raises normative concerns. It implies a division in which only the lowest-cost agents act and receive rewards, while others remain inactive. Although economically efficient, this outcome is ethically problematic. It mirrors real-world inequalities in ecological transition: affluent groups able to afford eco-friendly practices receive disproportionate moral and economic credit, whereas marginalised populations are excluded, undermining the inclusiveness and legitimacy of ecological action.

We now derive the Nash equilibrium to illustrate how the covenantal commitment framework can generate over-investment.

<footnote n="12">¹²On the question of what constitutes a *reasonable* reduction in emissions from a climate ethics perspective, see for instance Hiller (2011); Johnson (2011); Baatz (2014); Pellegrino (2018).</footnote>
## 4.2 Nash Equilibrium and Over-investment

Up to relabeling, we may assume that agents are ordered such that the sequence $(c_j)_{1\le j\le n}$ is increasing. Given this ordering, there exists a unique Nash equilibrium<fnref n="13" /> if and only if

$$K \ge a \frac{\sum_{j=1}^{m} c_j}{m-1}, \qquad (9)$$

and the corresponding equilibrium profile $y^{\text{Nash}}$ is

$$y_i^{\text{Nash}} = \begin{cases} \frac{(m-1)K}{\sum_{j=1}^{m} c_j} \left( 1 - \frac{(m-1)c_i}{\sum_{j=1}^{m} c_j} \right), & i \in M, \\ 0, & i \notin M, \end{cases} \qquad (10)$$

where $m \ge 2$ is the unique integer such that $M = \{1, \dots, m\}$ satisfies

$$\begin{cases} c_i < A_m & \text{for all } i \in M, \\ c_i \ge A_m & \text{for all } i \notin M, \end{cases} \quad \text{with} \quad A_m := \frac{\sum_{j=1}^{m} c_j}{m-1}. \qquad (11)$$

In what follows, we refer to the condition stated in equation (9) as (C1).

The total equilibrium effort is therefore

$$\sum_{j=1}^{n} y_{j}^{\text{Nash}} = \frac{(m-1)K}{\sum_{j=1}^{m} c_{j}}, \qquad (12)$$

and exceeds the covenant threshold $a$ if and only if $K \ge a \frac{\sum_{j=1}^{m} c_j}{m-1}$.

The individual payoff is

$$U_i^{\text{as if, Nash}} = \begin{cases} K \left( 1 - \frac{(m-1)c_i}{\sum_{j=1}^{m} c_j} \right)^2, & i \in M, \\ 0, & i \notin M, \end{cases} \qquad (13)$$

and the total payoff, which must necessarily be positive, is

$$U^{\text{as if, Nash}} = \sum_{i=1}^{n} U_i^{\text{as if, Nash}} = K \left( 2 - m + (m-1)^2 \frac{\sum_{j=1}^{m} c_j^2}{\left(\sum_{j=1}^{m} c_j\right)^2} \right). \qquad (14)$$

<footnote n="13">¹³See Appendix B for the proof.</footnote>
Three key observations follow

First, under our assumptions, (C1) is always satisfied, and hence a unique Nash equilibrium exists.
Indeed, two cases must be distinguished:

• If $m < n$, then we know that $c_{max} \ge \frac{\sum_{j=1}^{m} c_j}{m-1}$. By (A1), we have $K > c_{max}a$, and therefore
$K > a \frac{\sum_{j=1}^{m} c_j}{m-1}$.

• If $m = n$, then (C1) can be rewritten as $K \ge a \frac{n\bar{c}}{n-1}$. Since $n \gg 1$, we have $\frac{n}{n-1} \approx 1$. Moreover, by (A2), we have $K \gg \bar{c}a$. Thus we necessarily obtain $K > a\bar{c}$.

As shown, in both cases, (C1) is strictly satisfied, implying $\sum_{j=1}^{n} y_j^{\text{Nash}} > a$. Yet the Pareto-efficient effort profiles are precisely those $\mathbf{y}$ satisfying $\sum_{j=1}^{n} y_j = a$.<fnref n="14" /> Hence, the Nash equilibrium is not Pareto-efficient and $U^{\text{as if, Nash}} < U^{\text{as if, opti}}$. This outcome aligns with standard CPR dynamics, where lack of coordination leads to over-investment. Here, over-investment stems not from direct competition but from individual commitment absent institutionalised cooperation. Dan, acting on his own, may adopt more stringent ecological practices (e.g., full abstention from meat) than collectively necessary to reach $K$.

Second, the result relies on a narrow distribution of marginal costs: any agent $i$ with $c_i \ge \frac{\sum_{j=1}^{m} c_j}{m-1}$ will not contribute. In other words, in unequal settings, agents with the highest costs rationally withdraw despite recognising the *as if* covenant. This is not free-riding<fnref n="15" />, since non-contributors receive no direct benefit. Rather, it reflects a ‘crowding-out’ effect: structural inequality (*e.g.*, poverty, lack of autonomy, etc.) constrains participation even among the willing. High-cost agents could never enjoy the covenant’s benefits and therefore rationally deviate, reducing or abstaining from ecological effort whenever their marginal impact is smaller than that of more powerful actors.

Third, the preceding point shows that the Nash equilibrium depends on agents having broadly comparable living conditions. Since only agents with similar marginal costs contribute, the equilibrium presupposes relatively homogeneous communities. In such contexts, it is indeed reasonable for each agent to expect others to be equally committed, which reinforces mutual commitment and induces them to adopt equivalent strategies, acting as if the covenant were already in place.

<footnote n="14">¹⁴See Appendix C for the proof.</footnote>

<footnote n="15">¹⁵While our model captures over-investment, it abstracts from certain forms of free-riding. In practice, agents may signal their commitment without incurring any real cost. Dan, for instance, may join a vegetarian association without altering his habits—thereby gaining symbolic credit at no cost. However, modelling such behaviour lies beyond the scope of the present analysis.</footnote>
## 4.3 Summary of the Symmetric-case Results for the Kovenant Model

The table below summarizes the results for the symmetric case, i.e., $\forall i, c_i = c$, and under the assumption that $\frac{n-1}{n} \approx 1$. In this setting, we have $m = n$ and, as shown in the previous subsection, (C1) is strictly satisfied and thus $\sum_{j=1}^{n} y_j^{\text{Nash}} > a = \sum_{j=1}^{n} y_j^{\text{opti}}$. It follows that the Nash equilibrium is not Pareto-efficient and $U^{\text{as if, Nash}} < U^{\text{as if, opti}}$.

<table><caption>Table 1: Symmetric-case Results for the Kovenant Model</caption><thead><tr><th></th><th>y<sub>i</sub></th><th>&sum;<sub>j=1</sub><sup>n</sup> y<sub>j</sub></th><th>U<sup>as if(y)</sup></th><th>Pareto-efficient</th></tr></thead><tbody><tr><td>Collective optimum</td><td>[0; a]</td><td>a</td><td>K &minus; ca</td><td>Yes</td></tr><tr><td>Nash equilibrium</td><td><sup>K</sup>&frasl;<sub>nc</sub></td><td><sup>K</sup>&frasl;<sub>c</sub></td><td><sup>K</sup>&frasl;<sub>n</sub></td><td>No</td></tr></tbody></table>

Hence, our model shows that individual ecological actions can be accounted for without assuming a collective strategy. The resulting outcome, however, is not Pareto-efficient. Agents who commit themselves by treating the collective action problem as if a covenant existed tend to over-invest in their ecological choices, reproducing dynamics typical of classic CPR settings. By attempting to address a collective problem individually, agents are inherently unable to resolve it unilaterally. For instance, all agents could simply reduce meat consumption to twice per week. Yet once Dan commits, he is unilaterally driven to adopt a fully vegetarian lifestyle.

# 5 Discussion

In the following section, we discuss the main differences between our approach and the traditional economic solutions to the management of common resources.

## 5.1 Analysing Ecological Behaviours: Kantian vs. Covenantal Reasonings

In the context of CPR dilemmas, Roemer's Kantian optimisation offers a compelling alternative to the Nash equilibrium (Roemer, 2010, 2015, 2019; Braham and Hees, 2020). It aims to provide a micro-foundation for cooperation by proposing a mode of reasoning that resolves social dilemmas without relying on altruism or external enforcement. Kantian optimisation is defined by a universalisation criterion: an agent adjusts their behaviour only if they would endorse everyone making the same adjustment. In this formulation, the Kantian equilibrium is necessarily Pareto-efficient. Importantly, it remains inherently individualistic, as agents achieve a collectively optimal outcome not
through altruism, communication, or external enforcement, but through individual counterfactual rea-
soning grounded in trust. This naturally raises the question: if Kantian optimisation already offers
Pareto-efficient, individually grounded solutions to CPR dilemmas, why is the Kovenant model needed?

As noted, Kantian optimisation relies on *common knowledge* of the game’s symmetry and of play-
ers’ reasoning capacities. Cooperation emerges from the mutual trust that others will adopt the same
universalising logic (Roemer, 2019, p. 20). Cooperation thus requires neither *ethical* nor *social* concern
for others (Roemer, 2015, p. 46). Instead, Kantian optimisation is a fully normative framework, aimed
at providing a Pareto-efficient solution to the tragedy of the commons (Roemer, 2010, p. 18). Its
structure is explicitly teleological: the aim of resolving the social dilemma determines the normative
rule for cooperation. Hence, *unilateral* strategies (i.e., strategies that do not rely on common trust or
symmetry) lie outside its purview. Similarly, it has little to say about social norms or the endogenous
formation of expectations and preferences.

By contrast, the Kovenant model does not aim to provide a normative solution to climate-related
collective action problems. It does not seek to derive individual duties from the collective require-
ment to avert ecological catastrophe. Instead, our approach begins with the real-world observation
that agents often commit themselves to ecological action in fragmented, uncertain, and non-optimal
contexts. They act without guarantees of reciprocity, without clear expectations of success, and often
in ways that exceed what would be considered rational under either standard rational choice frame-
work or Kantian optimisation. The Kovenant model accounts for this by emphasising the formation
of normative expectations, *e.g.* norms that shape agents’ decision-making structures.

In this sense, our model reverses the Kantian logic. While Kantian equilibrium assumes trust
and rational universalisation—prescribing what rational cooperation should look like—the Kovenant
model describes how ecological behaviours emerge in practice, often despite structural disincentives
and without institutional coordination. It is a framework for understanding unilateral commitment
not as a deviation from rationality, but as a meaningful form of ecological agency grounded in social
processes. Hence, if Kantian optimisation provides a prescriptive model for achieving efficient cooper-
ation, the Kovenant model offers an interpretive framework for understanding the forms of ecological
over-investment and fragmentation observed today.

5.2 Moving Beyond Paretian Optimality to Address the Climate Crisis

More generally, a central normative shift advanced by the Kovenant model is that ecological com-
mitments should not be assessed primarily through the lens of social optimality (top-down), but rather
understood as emergent, norm-guided responses to collective problems under institutional and epis-
temic uncertainty (bottom-up). Standard economic models often adopt a teleological perspective:
they identify desirable social outcomes, and then work backward to define the individual strategies or
incentive structures that could lead to those outcomes. This typically translates into incentive-based
'carrot-and-stick' frameworks (Ostrom et al., 1992, 1999), or into Kantian optimisation. While these
models may perform well under idealised conditions, they struggle to account for the fractured, often
over-invested behaviours observed in practice, that is, behaviours that are neither coordinated nor
strictly optimal, yet persist.

The Kovenant model introduces an alternative: agents adopt ecological commitments based on a
shift in their *perception* of the problem they face. By reasoning *as if* an ecological covenant were
in place, they reconfigure their own decision-making structure, not to achieve social efficiency, but
to affirm their responsibility within a deteriorating collective context. This is why the Kovenant
model shows that ecological commitments still give rise to persistent coordination problems: without
mechanisms for collective deliberation, agents often either fail to act (due to crowding-out effects) or
over-invest beyond what is socially required.<fnref n="16" /> These are not behavioural anomalies, but predictable
consequences of fragmented norm adherence and of the various institutional structures in which agents
operate.

The idea that ecological behaviour is embedded in social contexts and influenced by evolving
norms is also being explored by an increasing body of research in environmental economics — and,
more broadly, in the economic literature (see, *inter alia*, Bowles, 1998, 2008, 2016; Dasgupta et al.,
2016; Ulph and Ulph, 2021; Mattauch et al., 2022). For instance, Dasgupta et al. (2016) show that
consumption choices adjust through social interaction and peer influence, while Ulph and Ulph (2021)
demonstrate that agents derive utility from norm compliance, thereby limiting the effectiveness of
Pigouvian taxation when behaviours are socially embedded. These contributions underscore a grow-
ing consensus: policies that neglect the normative and relational dimensions of ecological action risk
overstating the efficacy of incentive-based mechanisms.

While our approach stems from the same diagnosis, it represents a significant normative depar-
ture. Rather than assessing ecological action against a benchmark of global efficiency or cooperative
success, we treat it as a way of giving form and meaning to one's position in a fragmented and in-
terdependent world. The model, therefore, shifts the evaluative focus: *from* outcome-based criteria,
*to* process-based reasoning. This shift allows us to account for unilateral, norm-guided actions that

<footnote n="16">¹⁶Given the model’s unidimensional structure, such failures are likely to be even more pronounced in real-world settings, where ecological issues are inherently multidimensional.</footnote>
remain opaque within standard incentive-based or optimality-driven frameworks. In this way, we respond to recent calls for a renewed ecological social contract (Combet and Pottier, 2024), interpreting it in a covenantalist way that prioritises norm emergence over optimisation. It also raises the question of institutionalisation: how can such norm-driven actions scale and stabilise into shared frameworks for decision-making, beyond isolated acts of commitment? The Kovenant model provides a formal apparatus for beginning to address this issue.

Consequently, our model does not prescribe a solution to the climate crisis. Rather, it offers a conceptual framework for understanding how agents act *meaningfully* in a context where neither institutions, nor incentives, nor equilibria can guarantee that their actions will be efficient. This, we argue, is a precondition for the formation of ecological institutions, and a necessary complement to models of rationality and collective agency.

# 6 Conclusion

Our Kovenant model specifies new accounts of unilateral ecological commitments. By conceptualising the agent's decision *as if* they were bound by an emergent ecological covenant (*K*), we show how ecological actions can arise even in the absence of pre-specified imperatives, incentives, or external enforcement. As such, the existence of *K* is not introduced *ex ante* by the economist or the social planner, but is only derived from an *as if* reasoning from agents deciding to commit themselves. Once established, *K* justifies new ecological behaviours to be meaningful and may, thus, lead to the emergence of a new social state of the world (*i.e.*, institutions and collective production) and a new specific state of the living (*i.e.*, sustaining biodiversity and biotopes). However, because the game takes the form of a CPR structure, agents seek solutions through unilateral action only, leaving them exposed to persistent coordination failures — such as inefficient over-investment and crowding-out effect.

The relevance of the Kovenant model lies in starting from a process-based reasoning as it enables us to: 1) conceptualise how normative expectations emerge from individual commitments; 2) explain agents' actions by embedding these expectations within a covenant; and 3) reorient the focus of public policies toward the institutionalisation of covenants themselves as a means to overcome the coordination traps of fragmented, individualistic strategies. Framing and sustaining the emergence of covenants may, therefore, be essential to achieve any large-scale, durable collective action in response to ecological challenges.
# Appendix

## A – Lemma

Let $(c_j)_{1 \le j \le n} \subset \mathbb{R}_+^*$ be an increasing sequence. Then there exists a unique $m \ge 2$ such that $M = \{1, \dots, m\}$ satisfies

$$ \begin{cases} c_i < A_m & \text{for all } i \in M, \\ c_i \ge A_m & \text{for all } i \notin M, \end{cases} \quad \text{where} \quad A_m := \frac{\sum_{j=1}^{m} c_j}{m-1}. $$

*Proof. Reformulation.* Define the function

$$ \phi : \mathbb{N}_{\ge 2} \to \mathbb{R}, \quad \phi(k) := \sum_{j=1}^{k} c_j - (k-1)c_k. $$

Also define

$$ A_k := \frac{\sum_{j=1}^{k} c_j}{k-1}, \quad \text{for all } k \ge 2. $$

By construction, the desired conditions are equivalent to the existence of a unique $m \ge 2$ such that

$$ c_m < A_m \le c_{m+1}. $$

Observe that

$$ \phi(k) = (k-1)(A_k - c_k), \quad \text{for all } k \ge 2. $$

Hence, for all $k \ge 2$:

* $A_k > c_k$ if and only if $\phi(k) > 0$,

* $A_k \le c_{k+1}$ if and only if $\phi(k+1) \le 0$, since $\phi(k+1) = k(A_{k+1} - c_{k+1}) = (k-1)(A_k - c_k)$.

Thus, the conditions are equivalent to the existence of a unique $m \ge 2$ such that

$$ \phi(m) > 0 \quad \text{and} \quad \phi(m+1) \le 0. $$

**Uniqueness.** Note that for all $k \ge 2$,

$$ \phi(k+1) - \phi(k) = (k-1)(c_k - c_{k+1}) \le 0, $$

because $(c_j)_j$ is increasing. Therefore, $\phi(k)$ is a decreasing function of $k$, which guarantees uniqueness.

**Existence.** We have $\phi(2) = c_1 > 0$. Two cases arise:
* If $\phi(n) > 0$, then $m = n$ satisfies the conditions by default.

* If $\phi(n) \le 0$, then by monotonicity of $\phi$, there exists at least one $m \ge 2$ such that $\phi(m) > 0$ and $\phi(m+1) \le 0$.

□

## B – Nash Equilibrium

Up to relabeling, we may assume that the agents are ordered so that the sequence $(c_j)_{1\le j\le n}$ is increasing. By Lemma A, there exists a unique $m \ge 2$ such that $M = \{1, \dots, m\}$ satisfies

$$ \begin{cases} c_i < A_m & \text{for all } i \in M, \\ c_i \ge A_m & \text{for all } i \notin M, \end{cases} \quad \text{with} \quad A_m := \frac{\sum_{j=1}^{m} c_j}{m-1}. $$

We show that there exists a unique Nash equilibrium if and only if

$$ K \ge a \frac{\sum_{j=1}^{m} c_j}{m-1}, $$

and the corresponding profile $y^{\text{Nash}}$ is given by

$$ y_i^{\text{Nash}} = \begin{cases} \frac{(m-1)K}{\sum_{j=1}^{m} c_j} \left( 1 - (m-1) \frac{c_i}{\sum_{j=1}^{m} c_j} \right), & i \in M, \\ 0, & i \notin M, \end{cases} $$

*Proof. Case 1: $\sum_{j=1}^{n} y_j < a$.*

Since $K > c_i a$ for all $i$ by assumption (A1), each agent can benefit from deviating unilaterally. Hence, no such profile can be a Nash equilibrium.

**Case 2:** $\sum_{j=1}^{n} y_j \ge a$.

Let $S_{i-} := \sum_{j \ne i} y_j$ for each agent $i$. The first-order condition for agent $i$ is

$$ \frac{\partial U_i}{\partial y_i}(y_i) = \frac{S_{i-}}{(S_{i-} + y_i)^2} K - c_i. $$

This derivative is decreasing in $y_i$, and at $y_i = 0$ it equals $\frac{K}{S_{i-}} - c_i$. Therefore, a non-contributor $i$ has no incentive to deviate if and only if

$$ S_{i-} \ge \frac{K}{c_i}. $$

A profile with a single contributor $i_0$ and $y_{i_0} > 0$ cannot be a Nash equilibrium. Indeed, if $y_{i_0} > a$, agent $i_0$ would reduce his contribution to $a$. Furthermore, if $y_{i_0} = a$, since $K > c_j a = c_j S_{j-}$ for all
j by assumption (A1), each other agent $j \neq i_0$ would have an incentive to contribute, contradicting
equilibrium.

Multiple contributors. Let $L \subset \{1, \dots, n\}$ with $|L| = l \ge 2$ and define

$$
y_i = \begin{cases} > 0, & i \in L, \\ 0, & i \notin L. \end{cases}
$$

For $i \in L$, the best response is

$$
y_i^* = \sqrt{\frac{S_{i-}K}{c_i}} - S_{i-}.
$$

Defining $S^* := \sum_{j=1}^{n} y_j^* = \sum_{j \in L} y_j^*$, we obtain

$$
y_i^* = S^* \left( 1 - \frac{S^* c_i}{K} \right), \quad \forall i \in L.
$$

Summing over $i \in L$ gives

$$
S^* = (l-1) \frac{K}{\sum_{j \in L} c_j}.
$$

Hence for all $i \in L$,

$$
y_i^* = \frac{(l-1)K}{\sum_{j \in L} c_j} \left( 1 - \frac{(l-1)c_i}{\sum_{j \in L} c_j} \right).
$$

For this profile to be a Nash equilibrium, it is necessary that

$$
\forall i \in L, y_i^* > 0 \iff \forall i \in L, \frac{\sum_{j \in L} c_j}{l-1} > c_i,
$$

and

$$
\forall i \notin L, y_i^* = 0 \iff \forall i \notin L, \frac{K}{c_i} \le S_{i-}^* \iff \forall i \notin L, \frac{\sum_{j \in L} c_j}{l-1} \le c_i.
$$

Since the costs are increasing, by defining $A_l := \frac{\sum_{j=1}^l c_j}{l-1}$, the two conditions above are equivalent to

$$
\begin{cases}
\forall i \in L, & A_l > c_i, \\
\forall i \notin L, & A_l \le c_i.
\end{cases}
$$

By Lemma A, there exists a unique solution $m$ with $M = \{1, \dots, m\}$. For $l=m$, the profile defined
above is thus a Nash equilibrium. Moreover,

$$
\sum_{j=1}^{n} y_j^* = \frac{(m-1)K}{\sum_{j=1}^{m} c_j} \geq a \quad \Leftrightarrow \quad K \geq a \frac{\sum_{j=1}^{m} c_j}{m-1}.
$$
Conversely, defining $M = \{1, \dots, m\}$ with the same solution $m$, and defining the profile

$$y_i = \begin{cases} \frac{(m-1)K}{\sum_{j=1}^{m} c_j} \left( 1 - (m-1) \frac{c_i}{\sum_{j=1}^{m} c_j} \right), & i \in M, \\ 0, & i \notin M, \end{cases}$$

with $K \ge a \frac{\sum_{j=1}^{m} c_j}{m-1}$, we trivially obtain a Nash equilibrium.

□

## C – Pareto-Efficient Solutions

We show that the Pareto-efficient effort profiles are precisely those $\mathbf{y}$ such that $\sum_{j=1}^{n} y_j = a$.

*Proof. Case 1:* Let $\mathbf{y}$ be an effort profile such that $\sum_{j=1}^{n} y_j < a$.

Then, for all $i$,

$$U_i^{\text{as if}}(\mathbf{y}) = -c_i y_i \le 0$$

Since $U^{\text{as if}}$ is only defined when some agent exerts positive effort, there must exist at least one agent $i_0$ with $y_{i_0} > 0$.

Let $0 < \lambda < 1$ and define $\mathbf{z}$ by $z_{i_0} = \lambda y_{i_0}$, and $z_i = y_i$ for all $i \ne i_0$. Then,

$$U_{i_0}^{\text{as if}}(\mathbf{z}) > U_{i_0}^{\text{as if}}(\mathbf{y}), \quad \text{and} \quad U_i^{\text{as if}}(\mathbf{z}) = U_i^{\text{as if}}(\mathbf{y}) \quad \forall i \ne i_0.$$

Thus $\mathbf{y}$ is not Pareto-efficient.

*Case 2:* Let $\mathbf{y}$ be an effort profile such that $\sum_{j=1}^{n} y_j = a$.

Then, for all $i$,

$$U_i^{\text{as if}}(\mathbf{y}) = \frac{y_i}{\sum_{j=1}^{n} y_j} K \cdot \mathbf{1}_a \left( \sum_{j=1}^{n} y_j \right) - c_i y_i = \left( \frac{K}{a} - c_i \right) y_i \ge 0,$$

since $K > c_i a$ for all $i$, according to (A1).

Suppose there exists an alternative profile $\mathbf{z}$ that Pareto-dominates $\mathbf{y}$.

* If $\sum_{j=1}^{n} z_j = a$:

There exists some $i_0$ such that $U_{i_0}^{\text{as if}}(\mathbf{z}) > U_{i_0}^{\text{as if}}(\mathbf{y})$ and for all $i \ne i_0$, $U_i^{\text{as if}}(\mathbf{z}) \ge U_i^{\text{as if}}(\mathbf{y})$.

This implies

$$\left(\frac{K}{a} - c_{i_0}\right) z_{i_0} > \left(\frac{K}{a} - c_{i_0}\right) y_{i_0}, \quad \text{and} \quad \left(\frac{K}{a} - c_i\right) z_i \ge \left(\frac{K}{a} - c_i\right) y_i \quad \forall i \ne i_0.$$

By (A1), it follows that

$$z_{i_0} > y_{i_0}, \quad \text{and} \quad z_i \ge y_i \quad \forall i \ne i_0,$$
which implies $\sum_{j=1}^{n} z_j > \sum_{j=1}^{n} y_j = a$, a contradiction.

* If $\sum_{j=1}^{n} z_j > a$:

Let $\lambda = \frac{a}{\sum_{j=1}^{n} z_j} < 1$ and define $\mathbf{v}$ by $v_i = \lambda z_i$ for all $i$. Then, for all $i$,

$$U_i^{\text{as if}}(\mathbf{v}) = \left( \frac{K}{\sum_{j=1}^{n} z_j} - \lambda c_i \right) z_i \geq U_i^{\text{as if}}(\mathbf{z}) = \left( \frac{K}{\sum_{j=1}^{n} z_j} - c_i \right) z_i,$$

with a strict inequality for some $i_0$ satisfying $z_{i_0} > 0$, which follows from $\sum_{j=1}^{n} z_j > a$.

Hence, $\mathbf{v}$ Pareto-dominates $\mathbf{z}$. Since $\mathbf{z}$ Pareto-dominates $\mathbf{y}$ by assumption, transitivity implies that $\mathbf{v}$ Pareto-dominates $\mathbf{y}$.

Yet, by construction, $\sum_{j=1}^{n} v_j = a$, which is a contradiction as established in the previous subcase.

* If $\sum_{j=1}^{n} z_j < a$:

For all $i$, $U_i^{\text{as if}}(\mathbf{z}) \leq 0$.

Since $\sum_{j=1}^{n} y_j = a$, there exists at least one agent $i_0$ with $U_{i_0}^{\text{as if}}(\mathbf{y}) > 0$.

Hence $\mathbf{z}$ does not Pareto-dominates $\mathbf{y}$.

Therefore, $\mathbf{y}$ is Pareto-efficient.

**Case 3:** Let $\mathbf{y}$ be an effort profile such that $\sum_{j=1}^{n} y_j > a$.

As established above, any such profile is Pareto-dominated by a rescaled profile with total effort $a$, and is therefore not Pareto-efficient.

□
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