Provision of Public Goods via Unilateral but Mutually Conditional Commitments—Mechanism, Equilibria, and Learning

Abstract

We propose a one-shot, non-cooperative mechanism that implements the core in a large class of public goods games. Players simultaneously choose conditional commitment functions, which are binding unilateral commitments that condition a player’s contribution on the contributions of others. We prove that the set of strong Nash equilibrium outcomes of this mechanism coincides exactly with the core of the underlying cooperative game. We further show that these core outcomes can be found via simple individual learning dynamics.

1. Introduction

Public goods are systematically underprovided due to free-riding incentives. While binding agreements and repeated games offer theoretical solutions, many real-world contexts lack strong enforcement mechanisms or sufficient repetition. This paper studies a third approach: unilateral but mutually conditional commitments.

Our contribution. We introduce a one-shot, non-cooperative mechanism where players simultaneously choose conditional commitment functions (CCFs) that bind their contributions to the contributions of others. Our main theoretical result establishes an exact equivalence: the set of strong Nash equilibrium outcomes of this mechanism coincides with the core of the underlying cooperative game. As a second contribution, we prove that simple individual learning dynamics converge to these core outcomes, demonstrating practical implementability.

This mechanism formalizes emerging real-world practices. The US National Popular Vote Interstate Compact (Bennett, 2001; Muller, 2007) exemplifies such conditional commitments. In it, federal states unilaterally commit to electoral reforms conditional on sufficient other states making similar commitments. Similarly, conditional Intended Nationally Determined Contributions under the Paris Agreement (Pauw et al., 2020) allow countries to pledge emission reductions contingent on others’ actions.

Relationship to the existing literature. Our work contributes to the core implementation literature by providing what appears to be the first one-shot mechanism implementing the core in public goods games—a contribution to the Nash program of justifying cooperative solution concepts through non-cooperative mechanisms. We extend the conditional commitment literature beyond the setting of Reischmann and Oechssler (2018) and Oechssler et al. (2022) to continuous action spaces with generic utility functions. While our mechanism shares mathematical similarities with supply function equilibria (Delgado & Moreno, 2004; Klemperer & Meyer, 1989), it achieves efficiency rather than merely replicating Nash outcomes. Also, although CCFs are similar to best-response functions in (quasi-)supermodular games (Milgrom & Shannon, 1994) in that they are non-decreasing, our mechanism is not a quasi-supermodular game in itself, as the best-response CCF to an opponent’s CCF is not necessarily a non-decreasing function of that CCF (see Appendix E for a counterexample).

The mechanism. Each player i submits a conditional commitment function, $c_i:A_{-i}\to A_i$, mapping others’ action profiles to their own commitment level. The mechanism selects the largest mutually feasible action profile $\ell \left(c\right)=sup\{a:a_i⩽c_i\left(a_{-i}\right)$ for all $i\}$. Unlike best-response functions that typically yield low contributions in public goods games, CCFs enable commitments to high contribution levels.

Our theoretical analysis covers a general class of costly positive externality problems where players prefer to contribute less themselves while benefiting from others’ contributions. We prove that strong Nash equilibria exist if and only if the core is nonempty and establish core nonemptiness for important special cases including separable utility functions and Cournot oligopolies.

The learning results show that when players iteratively update CCFs by choosing favorite points in others’ feasible sets and responding with indifference curve commitments, the process converges to core outcomes. This provides a constructive path to efficiency without central coordination.

Our mechanism also generalizes existing approaches: the approach of Gerber and Wichardt (2009) can be viewed as one that restricts players to a single conditional commitment (contribute optimally if all others do), while matching mechanisms (Buchholz et al., 2011) prescribe, rather than endogenize, the dependence of contributions on others’ actions.

This paper proceeds as follows: Section 2 defines the CCF mechanism for single public goods. Section 3 establishes the core equivalence result and proves core nonemptiness. Section 4 analyzes learning dynamics with applications to climate policy. Section 5 concludes the paper.

2. The Conditional Commitment Function Mechanism

In this section, we describe the CCF mechanism for a case where each player contributes only to one public good, whether the same or different ones, saving the technically more involved treatment of the case of more than one action dimension per player for the Appendix A, Appendix B and Appendix C.

We assume the following: All $N>1$ players, i, can supply any non-negative amount $q_i⩾0$ of some public good (which might or might not be the same good for all players). From everyone’s contributions, i derives utility $v_i\left(q\right)$, where $q=(q_1,\dots ,q_N)$ is the players’ joint supply profile. For convenience, we sometimes also write $(q_i,q_{-i})$ instead of q, where $q_{-i}=(q_1,\dots ,q_{i-1},q_{i+1},\dots ,q_N)$ is the other players’ supply profile. Each i has an optimal amount $q_i^0$ that maximizes $v_i(q_{-i},q_i)$ for all $q_{-i}$. In the domain where $q_i⩾q_i^0$ and all $q_j⩾q_j^0$, $j\ne i$, the function $v_i$ is weakly increasing in each $q_j$, $j\ne i$ and weakly decreasing in $q_i$.

If the players were to play the one-shot game of choosing $q_i$ directly and simultaneously, it would be a strictly dominant strategy to choose $q_i=q_i^0$. This unique strategic equilibrium would lead to individual and joint utilities of $u_i^0=v_i\left(q^0\right)$ and $U^0={\sum }_i{u}_i^0$. Let us call this the unconditional equilibrium.

As is well known, the socially optimal contributions and resulting joint utility are typically much larger than this. Consider the simple symmetric case with linear benefits and linear marginal costs, where $v_i\left(q\right)={\sum }_{j=1}^N{q}_j-q_i^2/2$. In that case, $q_i^0=1$ and $U^0=N(N-1/2)$. In contrast, the social optimum would have $q_i^{★}=N$ and total utility $U^{★}=N^3/2$. The quotient between $U^{★}$ and $U^0$, $N^2/(2N-1)$, is then basically proportional to the number of players N.

The classical challenge in this social dilemma is to find a mechanism that incentivizes players to supply more than the unconditional equilibrium. To focus on this, and using a little more general terminology, we introduce the notation $a_i=q_i-q_i^0$ and call $a_i$ the action of player i. Here, $a_i$ is represented by a single real number, while in the more general case addressed in the Appendix A, Appendix B and Appendix C, it can be an element of any partially ordered set. When dealing with utilities, we will also use the utility functions in terms of actions, $u_i\left(a\right)=v_i(q^0+a)$. Since the socially optimal contributions $q_i^{★}$ are above the equilibrium ones $q_i^0$, we restrict our focus to non-negative actions $a_i\in [0,\infty )$, so that our above assumptions imply that $u_i\left(a\right)$ is weakly decreasing in $a_i$ and weakly increasing in $a_{-i}$.

Now, the following CCF mechanism allows players to sustain the socially optimal levels $q^{★}$ in equilibrium. Instead of choosing $a_i$ directly, in the CCF mechanism, each player simultaneously submits a conditional commitment function (CCF), $c_i:{[0,\infty )}^{N-1}\to [0,\infty )$, that maps each possible action profile $a_{-i}$ of the others to a “limit” $c_i\left(a_{-i}\right)$ for the player. Only CCFs which are weakly increasing in each argument $a_j$ are allowed. The interpretation is that by submitting $c_i$, player i commits to accept the liability to realize any level $a_i⩽c_i\left(a_{-i}\right)$ if $a_{-i}$ are the liabilities of the others. From the profile $c=(c_1,\dots ,c_N)$ of submitted CCFs, the CCF mechanism determines the set of feasible action profiles:

$$\begin{array}{cc}\hfill F\left(c\right)& =\{a:a_i⩽c_i\left(a_{-i}\right)\mathrm{for}\mathrm{all}i\}.\hfill \end{array}$$

(1)

It then selects its largest element, $\ell \left(c\right)=supF\left(c\right)$, as the liabilities the players have to fulfill. In Lemma A2 in Appendix B.1, we show that $F\left(c\right)$ is nonempty and, indeed, that it has a largest element. In our one-dimensional case, it is simply given by

$$\begin{array}{cc}\hfill \ell {\left(c\right)}_i& =max\{a_i:a\in F\left(c\right)\}.\hfill \end{array}$$

(2)

So, in game-theoretic terms, the CCF mechanism is the one-shot simultaneous-move game form in which player i’s strategy (or “message”) space is the set of possible CCFs $c_i$, and the outcome is the action profile $\ell \left(c\right)$. In terms of the public good, this means that each player then supplies an amount of $q_i^0+\ell {\left(c\right)}_i$.1

Depending on the application context, the mechanism can be implemented in different ways. In a domestic context, a regulator might set up a central institution similar to a stock exchange that accepts CCF submissions rather than bids and offers and then announces resulting liabilities. In an international context, countries might pass domestic legislation that commits them to realizing their part of $\ell \left(c\right)$ based on a CCF they specify in that bill and the CCFs that other countries passing similar legislation state in their respective bills—somewhat similar to how the US National Popular Vote Interstate Compact works.

Note that players can choose any kind of function $c_i$ as their CCF as long as it is weakly increasing. Figure 1a shows a two-player example with continuous CCFs, in which case the outcome $\ell \left(c\right)$ is simply the uppermost intersection of $c_1$ and $c_2$. Two special types of CCFs turn out to be particularly interesting: simple step functions and indifference curves.

Step functions correspond to specific offers. A player could choose some combination of “target levels”, $q^t$, translate it into target actions, $a^t=q^t-q^0$, and submit as her CCF the simple step function with

$$\begin{array}{c}\hfill c_i\left(a_{-i}\right)=a_i^t\mathrm{if}{a}_j⩾a_j^t\mathrm{for}\mathrm{all}j\ne i,c_i\left(a_{-i}\right)=0\mathrm{otherwise}.\end{array}$$

(3)

She would thus make sure that she will contribute her target, $q_i^t$, only if all others contribute at least their targets, $q_j^t$, and she will otherwise contribute the unconditional equilibrium value. We will call this type of CCF a canonical CCF through $a^t$. Examples of canonical CCFs and resulting outcomes are seen in Figure 1b.

While it will turn out that canonical CCFs already suffice to construct strong equilibria leading to efficient outcomes, the learning rules that find these equilibria will also require a second special type of CCF that correspond to more general offers. If a player’s utility function, $u_i\left(a\right)$, is continuous in a, she could choose a “target utility”, $u_i^t$, and submit, as her CCF, the largest function that guarantees this utility level whenever possible:

$$\begin{array}{cc}\hfill c_i\left(a_{-i}\right)& =max\{a_i⩾0:u_i(a_i,a_{-i})⩾u_i^t\}\mathrm{if}{u}_i(0,a_{-i})⩾u_i^t,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill c_i\left(a_{-i}\right)& =0\mathrm{otherwise}.\hfill \end{array}$$

(5)

This way, she would act as a “satisficer” who offers to supply any amount strictly above the unconditional equilibrium if she gets at least her target utility. If $u_i^t$ is given by i’s utility in some reference profile, $a^t$, we call this type of CCF an indifference curve CCF through $a^t$. Examples of indifference curve CCFs and resulting outcomes can be seen in Figure 1c.

3. Correspondence Between Strong Equilibria and Core Action Profiles

Let us now study the strategic implications of the CCF mechanism. When considering only stability against single-player deviations, it turns out that there are very many equilibria, both efficient and inefficient ones:

Theorem 1.

For the class of public goods problems considered in the main text, an action profile a is the outcome of some pure-strategy Nash equilibrium of the CCF mechanism if and only if $u_i\left(a\right)⩾u_i^0$ for all players i. In particular, every Pareto-efficient combination of public good supplies, q, can be sustained in pure-strategy Nash equilibrium, and this requires canonical CCFs only.

This nonsurprising fact follows directly from the first part of Theorem A1 in Appendix B.2; see the formal proof there. The main reason for this result is the following. Assume all players weakly prefer a to the zero action profile and submit their canonical CCF through a. Then, any deviation by a single player i that changes the outcome either makes i supply more or makes all others’ actions go down to zero (i.e., to the unconditional equilibrium levels), but neither of these two things does player i prefer to a.

A more thorough analysis takes into account that several players may plan deviations together. For studying stability against such joint deviations, the proper game-theoretic solution concepts are coalition-proof Nash equilibrium (Bernheim et al., 1987) and strong Nash equilibrium (Aumann, 1959). We only discuss strong equilibrium here, as we show in Theorem A1 in Appendix B.2 that, in our context, both concepts are essentially equivalent. For a pure strategy equilibrium to be strong, no group (“coalition”) of players must have a joint deviation producing an outcome that all members of that group prefer (for subtleties regarding weak and strong preferences; see Appendix A).

To see that this makes a difference, note that any outcome a of a strong equilibrium must necessarily be Pareto-efficient. If not, there would be an alternative action profile $a^{\prime }$ that all players would prefer to a, and by changing everyone’s CCF to their canonical CCF through $a^{\prime }$, they could jointly produce that preferred outcome. However, characterising the whole set of action profiles that may result from some strong equilibrium in the CCF mechanism is not that trivial. We will see that it coincides with a certain version of the ‘core’ of the underlying public goods problem.

The basic idea of a core action profile is that no group of players can “on their own” produce a different outcome they all prefer by switching to a different combination of actions. (Note that while a strong equilibrium of the CCF mechanism is a profile of CCFs, a core action profile is a much simpler object: a profile of actions in the underlying public goods problem.) Formally, given utility functions $u_i$, we call an action profile a a core outcome if and only if there is no group G of players and action profile $a^{\prime }$ so that $a_j^{\prime }=0$ for all players j outside G and so that $u_i\left(a^{\prime }\right)>u_i\left(a\right)$ for all $i\in G$.2

Theorem 2.

For the class of public goods problems considered in the main text, an action profile a is the outcome of some strong Nash equilibrium of the CCF mechanism if and only if it is a core action profile. To sustain a core action profile in strong equilibrium, it suffices to use canonical CCFs. If all utility functions are continuous, it also suffices to use indifference curve CCFs. In particular, every strong equilibrium outcome is Pareto-efficient.

Again, this follows from Theorem A1 in Appendix B.2.3 The main reason for this result is that when all players submit their canonical or indifference curve CCF through a core action vector a, then each deviation by some group either makes every group member supply no less than in a, or makes all non-group members switch to action zero; but neither of these consequences do all group members prefer to a.

Note that the theorem does not imply that the core is nonempty or that there is any strong Nash equilibrium in the first place. In fact, it is well known that cores might be empty. Especially for public goods problems, however, Chander and Tulkens (1997) have shown that the core of public goods problems is often nonempty. In particular, one can easily see, with the help of the Bondareva-Shapley theorem, that the core is nonempty if utility is separable into costs and quasi-linear benefits of the form $u_i\left(a\right)=f_i\left({\sum }_j{\alpha }_{ij}{a}_j\right)-g_i\left(a_i\right)$, where $f_i$ are weakly increasing and weakly convex benefit functions, ${\alpha }_{ij}$ are nonnegative coefficients, and $g_i$ are arbitrary cost functions (see Lemma A1 in Appendix A.4 for a proof).

Another example where the core is nonempty is when utility has the quadratic form $u_i\left(a\right)=(\alpha -a_i)\left(3-{\sum }_j(\alpha -a_j)\right)$ with $\alpha =3/(N+1)$, where actions are constrained to $0⩽a_i⩽\alpha$. These utility functions arise when players are firms in a Cournot oligopoly with outputs $y_i$, uniform marginal production costs of unity, and demand function $Y\left(P\right)=4-P$, where P is the price. In that case, firm i’s output reduction with respect to the Cournot equilibrium output, which is $y^0=3/(N+1)=\alpha$, is the quantity $a_i=y^0-y_i⩾0$, which can be interpreted as that firm’s contribution to a “public” good among the firms (actually a club good). For $N=3$, Figure 2 shows the core of this problem as a light grey hexagon and depicts a canonical CCF for player 3 through some core point (large dot) as a dark grey step function. Together with the canonical CCFs of the other two players through that point, it forms a strong equilibrium that sustains that efficient outcome.

4. Finding Strong Equilibria via Simple Learning Rules

A strong equilibrium would not be worth much if players have no way to find it and converge to it. In this section, we will see how with the CCF mechanism, players can use a simple learning rule to achieve this, either asymptotically or even in only a small finite number of steps. We assume all utility functions $u_i$ are continuous in this section.

General learning rule. The learning rules we consider here are of the following simple type. Players start with all-zero CCFs, $c_i\left(a_{-i}\right)\equiv 0$, and then iteratively and repeatedly adjust their CCFs independently and without a need to communicate or negotiate other than indirectly by adjusting their individual CCFs. In each iteration, some player, i, (e.g., a randomly drawn one) identifies a favourite point (action profile) in the feasible set defined by the other players’ current CCFs. In other words, she picks some a for which $u_i\left(a\right)$ is largest among all a that fulfill the conditions $a_j⩽c_j\left(a_{-j}\right)$ for all $j\ne i$. Let us call the chosen point o, let $c_i^o$ be i’s canonical CCF through o, and let ${\tilde{c}}_i^o$ be i’s indifference curve CCF through o. She then replaces her current CCF by a new function $c_i$, which is between $c_i^o$ and ${\tilde{c}}_i^o$. In other words, she puts $c_i^o\left(a_{-i}\right)⩽c_i\left(a_{-i}\right)⩽{\tilde{c}}_i^o\left(a_{-i}\right)$ for all $a_{-i}$.

This can be interpreted as players treating others’ CCFs as a set of offers to which they adjust by making a set of counteroffers that lead to mutual improvement. By offering, at most, their indifference curve through their current favourite item on the others’ “offer menu”, they make sure the future outcomes can only be better than that point if they require the player to contribute positively.

For example, the updates could simply use convex combinations: pick a number $0⩽\lambda ⩽1$ and put $c_i=(1-\lambda )c_i^o+\lambda {\tilde{c}}_i^o$. Figure 3 shows the dynamics generated by this rule in a public goods problem typical for the literature on international environmental agreements (Barrett, 1994):

Example. Three emitting countries can reduce their greenhouse gas emissions by an amount $q_i$ with respect to some business-as-usual scenario, leading to total mitigation of $Q=q_1+q_2+q_3$ and welfare levels $v_i\left(q\right)={\beta }_{i}Q-q_i^2/2{\gamma }_i$, where $\beta =(1,1.5,2)$ are climate damage factors, and $\gamma =(2,1,1.5)$ are mitigation cost coefficients. In this case, the individually rational mitigation levels are $q_i^0={\beta }_i{\gamma }_i$, the ‘action’ in our terminology is additional mitigation, $a_i=q_i-{\beta }_i{\gamma }_i$, and utility functions in terms of actions are $u_i\left(a\right)={\beta }_i(a_1+a_2+a_3+6.5)-{(a_i+{\beta }_i{\gamma }_i)}^2/2{\gamma }_i$. The core of this problem is a curved surface whose boundary is shown in Figure 3 by thick blue lines. While all core points are Pareto-efficient, note that, here, the action profile that maximizes total welfare (the red dot) is not in the core, so it cannot be sustained in strong equilibrium, but only in simple Nash equilibrium via the CCF mechanism. Now, assume all three countries pass domestic laws containing some CCF and the commitment to realize the CCF mechanism outcome, and every year some country replaces their CCF as described above, using a random convex combination. Then, the resulting mitigation liabilities would evolve from zero to some core point, similar to the light colored lines in Figure 3, which are 100 different numerical simulations of this probabilistic learning process, each 30 years long. Only very few simulations have not well converged to the core within this time. These numerical results also suggest that the limit points cover the whole core.

While these numerical results are encouraging, it would be nice to have theoretical reason to expect such convergence. Our first result concerns the special case where players act modestly and simply use their indifference curve through o as their new CCF.

Theorem 3.

Assume players start from all-zero CCFs, and then in some random order, each player once replaces their CCF by their indifference curve through a favourite point on the others’ current CCFs. Then, after one such round of updates, the outcome of the CCF mechanism is in the core.

This is a special case of Theorem A2 in Appendix C.1.4 The thin blue lines and blue dots in Figure 3 show the results of this rule. It turns out that, in this example, the six different orders in which the players may update exactly lead to the six corners of the core.5 In Theorem A3 in Appendix C.1, we also give sufficient conditions for ending up in the core after only one round of updates, when players start from nonzero, rather than all-zero, CCFs and apply the same learning rule.

While the modest behaviour studied above leads quickly into the core, it appears not utterly realistic, since a player early in the line will later realize she could have gotten more than provided by the optimal feasible profile at the time she made her choice if she had offered to the later players less than what was given by her indifference curve. Our last result indicates that for other special cases of the general learning rule, it might be possible to prove eventual convergence to the core as well.

Theorem 4.

Assume players start with all-zero CCFs, and in each time step, a random player replaces her CCF by a random convex combination of her canonical and indifference curve CCFs through a favourite point on the others’ current CCFs. Then, if the resulting sequence of outcomes of the CCF mechanism converges to some action profile a, almost surely a is in the core.

This is a special case of Theorem A4 in Appendix C.2, which also contains two related conjectures.

5. Conclusions

This paper establishes two main theoretical results for public goods provision through conditional commitments. First, we prove an exact equivalence between strong Nash equilibria of our conditional commitment function (CCF) mechanism and the core of the underlying cooperative game. This provides a novel contribution to the Nash program by implementing the core through a one-shot, non-cooperative mechanism—the first such result for public goods games to our knowledge. Second, we demonstrate that simple individual learning dynamics converge to these core outcomes, offering a constructive path to efficiency without central coordination.

Our results have several implications for mechanism design theory. Unlike supply function mechanisms that generally cannot improve upon Nash equilibrium outcomes, the CCF mechanism sustains Pareto-efficient allocations in strong equilibrium. The mechanism’s success stems from allowing players to commit to contribution levels that exceed their best responses, transforming the strategic structure of the underlying game. This suggests broader potential for conditional-commitment-based mechanisms in addressing coordination failures.

The learning results are particularly striking given the general difficulty of achieving convergence to strong equilibria. While most learning dynamics in games struggle to reach even Nash equilibrium, our individual adjustment process converges to group-stable outcomes. This occurs because players treat others’ CCFs as offer menus and respond with counteroffers that improve all parties’ positions, creating a natural progression toward the core.

Several theoretical extensions merit investigation. First, characterizing which game classes beyond costly positive externality problems admit similar core implementation results is of great importance. Second, analyzing robustness to incomplete information about utility functions or commitment capabilities is also of interest. Third, exploring whether weaker commitment devices can achieve similar efficiency gains is worthy of investigation. The relationship between commitment strength and implementable outcomes represents a fundamental trade-off in mechanism design.

From a practical perspective, emerging institutions like the National Popular Vote Interstate Compact and conditional climate commitments demonstrate the real-world relevance of conditional commitment mechanisms. However, the theoretical insights—that unilateral commitments can solve collective action problems and that decentralized learning can discover efficient outcomes—have significance beyond any particular application.

Our analysis again confirms that the apparent tension between individual rationality and collective efficiency in public goods games can be overcome. When players can credibly commit to conditional strategies, efficient outcomes become individually and collectively rational. This shifts the focus from changing incentives to providing commitment technologies—a positive perspective for mechanism design in environments where traditional enforcement mechanisms are unavailable.