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Combating climate change with matching-commitment agreements

Abstract

Countries generally agree that global greenhouse gas emissions are too high, but prefer other countries reduce emissions rather than reducing their own. The Paris Agreement is intended to solve this collective action problem, but is likely insufficient. One proposed solution is a matching-commitment agreement, through which countries can change each other’s incentives by committing to conditional emissions reductions, before countries decide on their unconditional reductions. Here, we study matching-commitment agreements between two heterogeneous countries. We find that such agreements (1) incentivize both countries to make matching commitments that in turn incentivize efficient emissions reductions, (2) reduce emissions from those expected without an agreement, and (3) increase both countries’ welfare. Matching-commitment agreements are attractive because they do not require a central enforcing authority and only require countries to fulfil their promises; countries are left to choose their conditional and unconditional emissions reductions according to their own interests.

Introduction

Anthropogenic emissions of greenhouse gases (GHGs) have been identified as comprising a key factor in climate change1. In the Paris Agreement, countries overwhelmingly agree to limiting “the increase in the global average temperature to well below 2 °C […] and pursuing efforts to limit the temperature increase to 1.5 °C above pre-industrial levels”2. To do so, they must curb GHG emissions; yet according to recent assessments of present and future global GHG emissions, the Earth is still expected to warm by more than 2 °C3,4,5,6.

Achieving the necessary reduction in GHG emissions is difficult, primarily because it is an instance of Hardin’s classic “tragedy of the commons”7: each country benefits from reduced global emissions, but curbing its own emissions carries an economic cost. Countries thus prefer to free-ride on the emissions abatement (i.e., reduction of emissions) of others over reducing their own emissions8,9. The implied tragedy is that if countries pursue their own interests in this way, everyone will be worse off than if they had cooperated and acted in their collective interest.

To manage shared resources (or to resolve other collective action problems) within a single country, rules are typically agreed upon and enforced, making cooperation the rational choice by shaping individual incentives. In contrast, the Paris Agreement details countries’ voluntarily pledged emission reductions in the hope of averting a “climate tragedy”, but this agreement will likely fall short of its goal. For example, even if countries reduced their emissions by the amounts they pledged, the Earth would still be an estimated 3.2 °C warmer by 2100 (despite the agreement’s goal of limiting global warming to no more than 1.5 °C)10. The cause for this discrepancy between the Paris Agreement’s goal and individual countries’ abatement targets is that, because countries are sovereign, their participation must be voluntary. Thus, countries chose their own abatement targets, and had little incentive to make very ambitious pledges. Moreover, the abatement targets that countries have pledged are not legally binding, so countries are not incentivized to fulfill them; consequently some countries’ current policies are insufficient to fulfill even their modest pledges10. So, while, in principle, “prompt and substantial reductions in greenhouse gas emissions on a global scale” could limit warming to meet the 1.5 °C target set in Paris, at 2015 rates of emissions the global carbon budget for this goal will be exhausted already in 202111.

Seeking possible mechanisms for lowering global emissions, many researchers have turned to game theory (see ref. 12 for a review). Most of the resulting proposals involve establishing a central authority that can punish non-participation and noncompliance or otherwise redistribute money or resources (either directly or by approving sanctions imposed by countries)8,13,14,15,16. But enforcement is difficult to achieve in practice, because, as pointed out by Nordhaus8, it conflicts with the principles of countries’ sovereignty, equality and right to manage internal affairs without intervention, first established in the 1648 Treaty of Westphalia. Consequently, Nordhaus has suggested that international trade law be amended to allow sanctions against non-participants, while prohibiting retaliation by the latter8. This would require the comprehensive renegotiation of trade laws, which will be a difficult and lengthy process. Moreover, Barrett also stresses the importance of enforcement, but warns that even if countries agree to a treaty with strong enforcement mechanisms, they will likely do so at the expense of lowering their abatement targets “to ensure that punishments are not imposed”14. Hence, solutions that encourage countries to cooperate without encroaching on their sovereignty would be desirable, since the global carbon budget for 1.5 °C warming is quickly running out.

Barrett17,18 was perhaps the first to propose applying Guttman’s19,20 idea of matching contributions to lower GHG emissions and help mitigate climate change. More recently, Boadway et al.21 applied Guttman’s19,20 matching scheme to Finus’s22 Global Emission Game: every country’s emissions abatement is divided into an unconditional and a conditional abatement. Countries simultaneously declare matching factors, which define how much conditional abatement they will contribute for a unit of unconditional abatement contributed by each other country. (The literature typically refers to a country’s unconditional abatement as its flat—or direct—contribution, to its conditional abatement as its indirect contribution, and to the matching factors as matching rates. We have chosen our terminology in the hope that it is more transparent.) Assuming countries can commit to the matching factors they declare (see the section “Commitment” for a justification of this assumption), this program allows countries to alter the incentives of other countries without invoking a central authority.

Boadway et al’s21 inspirational analysis considers an arbitrary number of heterogeneous countries and arrives at an optimistic result. They show that a matching-commitment agreement can lead to new equilibrium emission levels that are locally Pareto efficient and unique and reduce emissions relative to the status quo. However, Boadway et al. consider the status quo emissions in the absence of an agreement to be exogenously determined, and their results require that at least one country has an incentive to unilaterally reduce emissions from this status quo even in the absence of an agreement. A more realistic scenario would be that countries follow their Nash equilibrium emissions strategy in the absence of an agreement, so that no country has an incentive to either increase or decrease their emissions6,23. In that case, Boadway et al.’s optimistic results disappear. In other words, in Boadway et al.’s analysis, without an agreement, at least one country’s emissions are irrationally high, even for its pure self-interest; matching-commitment agreements reduce emissions because they allow countries to respond to their incentives and thus lower their emissions. Thus, we do not know whether matching-commitment agreements can work to reduce emissions if countries are able to freely select emissions levels that align with their incentives in the absence of an agreement. Furthermore, Boadway et al. do not consider the question of global efficiency, i.e., whether the agreement improves global welfare, the payoff of all countries, or indeed any single country’s payoff, relative to the outcome in the absence of an agreement (i.e., either relative to a Nash equilibrium emissions profile, or to the non-Nash baseline Boadway et al. assume is selected in this case).

Against this background, this paper addresses two main questions: First, can a matching-commitment agreement incentivize countries to reduce their emissions relative to a baseline scenario without an agreement, and in which countries’ choices align with their incentives? Second, are countries better off with such an agreement than without? We build on Boadway et al.’s21 formulation, but require the baseline scenario—that is, countries’ emissions in the absence of an agreement for reducing them—to be a Nash equilibrium determined endogenously by the countries’ strategic decisions. For arbitrarily many countries that may differ in their economic and climate-related properties, and that choose their emissions independently, we find conditions guaranteeing that such a Nash equilibrium emissions profile exists and is unique—i.e., the baseline scenario is well-defined. Next, we analyze the effects of a matching-commitment agreement on the emissions of two such countries. We show that a matching-commitment agreement generates a new, unique equilibrium emissions profile. (More precisely, there are multiple subgame-perfect Nash equilibria—see “Model and methods” and ref. 24—which all yield identical emissions profiles.) At this equilibrium emissions profile, both countries’ emissions are lower and their payoffs are higher than at baseline, i.e., at the equilibrium without a matching-commitment agreement. Lastly, we show that the equilibrium emissions profile with matching-commitment agreements is locally Pareto efficient.

While Boadway et al.’s21 results broke new ground and motivated all that we have done, there are subtle problems with their arguments for both the existence and the uniqueness of the equilibrium, and hence the claims for the general case do not hold up (see Remark B.15 and Appendix C.11.6 for details). Indeed, even the two-country case is challenging for providing a rigorous proof of existence and uniqueness, which requires a different mathematical approach than that of Boadway et al. We introduce that method and proof in this paper for matching-commitment agreements between two countries, and show rigorously that the equilibrium (Eqs. (5) and (6)) was, in fact, correctly identified by Boadway et al. The n-country case remains an open problem.

Model and methods

Basic climate game (BCG)

We start by describing a model for the public goods problem of global GHG emissions between two countries (or unions thereof, e.g., the European Union), building on Boadway et al.’s21 model, which in turn is similar to one described by Finus (ref. 22, Ch. 9) and an example of Folmer and von Mouche’s25 transboundary pollution games. As is often the case for game-theoretical models of international cooperation, we focus on two countries, to retain mathematical tractability (see, e.g., refs. 26,27,28,29,30,31,32; we compare our analysis with the literature on n-player matching mechanisms in the section “Two vs. many countries”).

We consider two countries deciding on their emissions levels, $${e}_{i}\in {\mathbb{R}}\,(i=1,2)$$. We denote the emissions profile, i.e., the vector of countries’ emissions, by $${\boldsymbol{e}}=({e}_{1},{e}_{2})\in {{\mathbb{R}}}^{2}$$, and the total emissions by $$e={e}_{1}+{e}_{2}$$. Note that we allow countries to have negative net GHG emissions (e.g., by capturing carbon from the atmosphere), since this is assumed in many theoretical mitigation scenarios that meet the goals of the Paris Agreement33. We denote by $${B}_{i}({e}_{i})$$ and $${D}_{i}(e)$$ the benefit and damage, respectively, to country i (i = 1, 2) when its own emissions level is ei and the total emissions are $$e={e}_{1}+{e}_{2}$$. The benefit and damage functions are assumed to be increasing and twice differentiable, as well as decelerating and accelerating, respectively (see “Increasing damages from climate change”). Country i’s total payoff as a function of the emissions profile e is

$${\prod }_{i}({\boldsymbol{e}})={B}_{i}({e}_{i})-{D}_{i}(e)\mathrm{}.$$
(1)

Countries seek to maximize their own payoffs, and are indifferent to the other country’s payoff.

In our baseline scenario, which we call the basic climate game (BCG), representing the absence of any international agreement to curb emissions, countries independently and simultaneously choose their own emissions levels. When countries play the BCG and there is a unique Nash equilibrium, we view it as the baseline scenario, as it represents the predicted emissions profile for rational countries in the absence of any agreement to limit emissions. Unfortunately, and not surprisingly, the Nash equilibrium of the BCG does not maximize global welfare, i.e., the total payoff of both countries, ∏ = ∏1 + ∏2, and is hence not socially optimal. Indeed, we show in Appendix A.3 that the total emissions at a social optimum (SO), i.e., at an emissions profile that maximizes global welfare, must be lower than the total baseline emissions (Appendix A.3 discusses when an SO exists, which is in general not guaranteed by our assumptions). Moreover, since the baseline emissions profile is not Pareto-efficient, there are emissions profiles at which both countries’ payoffs are higher than at baseline (Appendix A.4). Thus, an international environmental agreement that results in lower emissions and higher national payoffs than baseline is desirable (but is still likely to be second-best from a global perspective).

Matching climate game (MCG)

One suggested approach to establishing an accord that incentivizes a reduction in GHG emissions is to use a matching-commitment scheme17,21. In this approach, country i may reduce its emissions relative to its baseline emissions level $${\bar{e}}_{i}$$ by an abatement level Ai ≥  (i = 1, 2), so that its actual emissions level is

$${e}_{i}={\bar{e}}_{i}-{A}_{i}\mathrm{}.$$
(2)

Country i’s abatement Ai is the sum of its unconditional abatement ai, and the conditional abatement that it performs in response to the other country’s unconditional abatement,

$${A}_{i}={a}_{i}+{m}_{i}{a}_{j},$$
(3)

where mi is the proportionality factor at which country i matches country j’s unconditional abatement and i, j = 1, 2, i ≠ j (that is, j is the nonfocal country). The countries then play the following game, henceforth referred to as the matching climate game (MCG) (see also Definition A.9):

• Stage I: Countries independently and simultaneously choose their (non-negative) matching factors, mi ≥ 0, to which they are subsequently committed.

• Stage II: Countries independently and simultaneously choose their unconditional abatement levels, ai, with full knowledge of the matching factors chosen in stage I.

Denoting the total baseline emissions and the total abatement by $$\bar{e}={\bar{e}}_{1}+{\bar{e}}_{2}$$ and $$A=\bar{e}-e$$, respectively, country i’s payoff becomes the nonlinear function

$${\prod }_{i}({\boldsymbol{e}})={B}_{i}({\bar{e}}_{i}-{A}_{i})-{D}_{i}(\bar{e}-A)\,{\rm{for}}\,i=1,2.$$
(4)

Note that the baseline emissions levels $${\bar{e}}_{1}$$ and $${\bar{e}}_{2}$$ must be specified to determine the country payoffs in the MCG. Below, we will take these to be the Nash equilibrium emissions of the BCG (which, under the conditions described in the section “Assumptions on the costs and benefits of emissions”, exists and is unique). These baseline emissions are thus endogenously determined in the sense that they are the rational levels of emissions in the absence of any climate agreement.

The process by which national abatements arise from the unconditional abatements and matching factors is visualized in Figs. 1 and 2.

Assumptions on the costs and benefits of emissions

In this paper, we aim to compare the equilibrium outcomes of the basic climate game (BCG) and the matching climate game (MCG). Since such a comparison is predicated on the existence of well-defined equilibrium outcomes for these two games, additional assumptions are needed to guarantee this. For example, in the BCG, it may be advantageous for countries to emit or abate without bound, in which case no Nash equilibrium exists. In the MCG, matching may incentivize a country to abate without bound, in which case the best-response functions for stage II (given matching factors chosen in stage I) need not even be well-defined.

In Appendix A.1, we identify a sufficient condition on the benefit and damage functions that guarantees the existence of a unique Nash equilibrium for the BCG; we call this condition bounded global emissions (BGE) (Definition A.2), because, roughly, it ensures that all rational emissions profiles lie inside a bounded set. In Appendix A.6, we show that under the assumptions of the BCG, the MCG’s stage-II best-response functions are well-defined if and only if (iff) for any country i, in the limit of negative infinite emissions, either the marginal benefits become infinite or the marginal damages vanish; we refer to this condition as bounded abatement with matching (BAM), because it guarantees that no matching factors can incentivize infinite abatement. Henceforth we assume the benefit and damage functions are such that BAM holds and that global emissions are bounded. It turns out that (at least for the case of two countries) the BAM condition is sufficient to ensure a unique equilibrium outcome for the MCG, in the sense described in our “Results” section. (Note that Boadway et al.21 do not assume the BAM condition, and their analysis does not address the possibility of the MCG’s stage-II best-response functions being undefined. Consequently, the MCG’s stage-II Nash equilibria in their analysis may not exist.)

While the model described above and the results that follow involve two countries only, note that for any number of countries, n ≥ 2, the BGE and BAM assumptions (resp.) ensure the existence and uniqueness of a Nash equilibrium for the BCG, and that the MCG’s stage-II best-response functions are well-defined. Regardless of the number of countries, these properties are necessary to sensibly ask whether and when the MCG yields equilibria that improve on the baseline emissions profile.

Solution concepts

The relevant solution concept for the basic climate game (BCG) is the Nash equilibrium, i.e., an emissions profile from which no country has an incentive to deviate unilaterally (see ref. 24, Ch. 4). The matching climate game (MCG), on the other hand, is a two-stage game in which a strategy for country i (i = 1, 2) specifies its choice of a matching factor, mi = μi in stage I, and of an unconditional abatement level for any combination of matching factors chosen by both countries in stage II, ai (m1, m2) (i = 1, 2). A natural choice of solution concept for the MCG is therefore the subgame-perfect equilibrium (SPE; see Ch. 7 of ref. 24), which yields Nash equilibria at every subgame (the MCG has infinitely many subgames: in addition to the MCG itself, for any choice of matching factors in stage I, the game played in stage II is a subgame of the MCG). For the MCG, an SPE must then satisfy two conditions: first, it must constitute a Nash equilibrium for the full MCG, and second, for any pair of matching factors (m1, m2), the abatements (a1 (m1, m2), a2 (m1, m2)) must constitute a Nash equilibrium of stage II. For any choice of matching factors played in stage I, an SPE selects abatements that are Nash equilibria for stage II, so this solution concept avoids Nash equilibria (for the full MCG game) that are sustained by incredible threats or other irrational behaviour in stage II. We focus on SPEs precisely because they avoid such irrational behaviour, and our main aim is to identify and compare SPEs of the MCG with the baseline emissions profile.

Results

The matching climate game has a unique equilibrium emissions profile

We first use backward induction to find subgame-perfect equilibria (SPEs) for the matching climate game (MCG; see the “Model and methods” section and Definition A.9) between two countries: We assume that given any choice of matching factors in stage I, countries may expect one another to choose unconditional abatement levels that are Nash equilibria given their choices of matching factors, (m1, m2); we then find which strategies will be played in stage I. A more detailed account of the process follows below, and the full details are given in Appendix B.

In Appendix B.1, we find the Nash equilibrium levels of unconditional abatement in stage II, given a choice of matching factors in stage I, which can be geometrically characterized as intersections of the stage-II best-response functions, $${ {\mathcal R} }_{1}\,{\rm{and}}\,{ {\mathcal R} }_{2}$$. The results of our analysis are summarized in Fig. 3. In particular, we find that there are four regions in the set of non-negative matching factors, in which the stage-II Nash equilibria are qualitatively different. These regions are delimited by the two stage-II delimiter curves, ϕ1 and ϕ2 (shown in Fig. 3a as the concave and convex curves, respectively), defined as follows: for any $${m}_{j}\ge 0$$, $${m}_{i}={\phi }_{i}({m}_{j})$$ is the matching factor at which the intercepts of the two countries’ stage-II best-response functions with the ai-axis are identical (Appendix C.7). Between ϕi and the aj-axis, country i’s stage-II best-response curve intercepts the ai-axis higher than does j’s.

The functions ϕ1 and ϕ2 intersect at two pairs of matching factors: (1) m1 = m2 = 0, for which the unique stage-II Nash equilibrium is no abatement (baseline), and (2) a pair of matching factors $$({m}_{1}^{{\rm{c}}},\,{m}_{2}^{{\rm{c}}})$$ that are reciprocal (satisfy $${m}_{1}^{{\rm{c}}},\,{m}_{2}^{{\rm{c}}}=1$$; since our model is a one-shot game, there is no future time at which countries have occasion to reciprocate in the sense of responding to one another’s actions), defined by Eqs. (5) and (6) below, for which the best-response functions overlap for positive unconditional abatements, yielding a continuum of stage-II Nash equilibria.

Furthermore, we find that for pairs of matching factors other than $$({m}_{1}^{{\rm{c}}},\,{m}_{2}^{{\rm{c}}})$$, the stage-II best-response functions cross at most once in the interior of the quadrant of positive unconditional abatements, i.e., $$\{({a}_{1},\,{a}_{2})|{a}_{1} > 0,\,{a}_{2} > 0\}$$. This implies that for $$({m}_{1},\,{m}_{2})\ne ({m}_{1}^{{\rm{c}}},\,{m}_{2}^{{\rm{c}}})$$ either one best-response function is above the other (upper-left and lower-right panels of Fig. 3b), or they cross at some point in the interior of that quadrant (upper-right and lower-left panels of Fig. 3b). Consequently, there are either one, two, or three stage-II Nash equilibria (Fig. 3b), and these cases can be distinguished by comparing the stage-II best-response functions’ intercepts with the a1- and a2-axes:

• On or between ϕ1 (resp. ϕ2) and the a2-axis (a1-axis), there is a stage-II Nash equilibrium at which only country 1 (2) abates unconditionally.

• Between the curves ϕ1 and ϕ2, there is a stage-II Nash equilibrium at which both countries abate unconditionally.

Next, for any pair of non-negative matching factors, $$({m}_{1},\,{m}_{2})\in {{\mathbb{R}}}_{\ge 0}^{2}$$, the payoffs ∏i(m1, m2) (i = 1, 2) can be determined assuming that a Nash equilibrium is played in stage II; we ascribe these payoffs to pairs of matching factors and obtain payoff functions ∏i(m1, m2) that we then use to find matching factors that are Nash equilibria of stage I (assuming Nash equilibrium unconditional abatements are played in stage II). These Nash equilibrium matching factors, along with the choice of Nash equilibria played in stage II that was used to calculate the stage I payoff functions, constitute SPEs.

However, the multiplicity of stage-II Nash equilibria for some pairs of matching factors introduces a complication, because the payoff to country i given (m1, m2) may then not be uniquely defined even under the assumption that only Nash equilibria are played in stage II (see top-right corner of Fig. 3b); one must then analyze an infinite family of continuous-strategy games, each with payoff functions ∏i(m1, m2) (i = 1, 2) determined by a particular choice of stage-II equilibria (for all pairs of matching factors for which multiple stage-II equilibria exist).

Despite this complication, it turns out that there is a unique SPE emissions profile. In particular, analyzing all possible pairs of payoff functions ∏i(m1, m2) (i = 1, 2) obtained assuming equilibrium stage-II play (see Theorem B.19 in Appendix B.2), we show that although the MCG has infinitely many SPEs, the unique pair of matching factors $$({m}_{1}^{{\rm{c}}},\,{m}_{2}^{{\rm{c}}})$$ is played at all SPEs. Moreover, the equilibrium abatements Ai (and, therefore the payoffs ∏i) for the MCG are uniquely defined in the sense that they do not depend on which SPE is chosen. Specifically, with m and a denoting the unique positive solutions of

$${B{\prime} }_{1}({\bar{e}}_{1}-a)=(1+m){D{\prime} }_{1}(\bar{e}-(1+m)a),$$
(5)
$$m{B{\prime} }_{2}({\bar{e}}_{2}-ma)=(1+m){D{\prime} }_{2}(\bar{e}-(1+m)a),$$
(6)

the equilibrium matching factors are $$({m}_{i}^{{\rm{c}}},\,{m}_{j}^{{\rm{c}}})=(1/m,\,m)$$, and the emissions profile at any SPE is

$$({e}_{1},\,{e}_{2})=({\bar{e}}_{1}-a,\,{\bar{e}}_{2}-ma).$$
(6)

Both countries are better off playing the matching climate game than the basic climate game

Comparing the subgame-perfect emissions profile of the matching climate game (MCG) given by Eq. (7) with the baseline emissions profile ($$\bar{e}$$, the Nash equilibrium of the basic climate game, abbreviated BCG), we find:

• At the equilibrium emissions profile of the MCG, both countries’ emissions are lower, but their payoffs are higher, than at the baseline equilibrium (i.e., the subgame-perfect emissions profile is a Pareto improvement on the Nash equilibrium of the BCG).

• The equilibrium emissions profile of the MCG is locally Pareto efficient, i.e., a small deviation from the MCG’s equilibrium abatement levels cannot increase both countries’ payoffs.

• The MCG’s equilibrium emissions profile is socially optimal iff the two countries’ marginal benefits at the equilibrium emissions profile are equal, which need not generally be the case; it then follows that from the perspective of a social planner, the country having lower marginal benefits at the subgame-perfect emissions profile emits too much, while the one with higher marginal benefits emits too little (Lemma A.8).

Discussion

Matching-commitment agreements have been garnering increasing interest as a way to construct international agreements to lower global greenhouse gas (GHG) emissions without requiring extensive international monitoring and enforcement. Our main contribution in this paper is to show the theoretical viability of such an approach assuming that countries behave strategically in the absence of an agreement. We show that matching-commitment agreements can indeed result in increased abatement and welfare for the countries involved, relative to a baseline of strategic behavior in the absence of agreements. Our results suggest that matching-commitment agreements can be a promising option for addressing other public goods problems when enforcement is difficult.

To conclude, in the next few subsections we position our results in a broader context. First, we highlight some appealing features of the matching climate game (MCG). Next, we touch on related models from the literature on matching-commitment agreements. We then discuss the main simplifying assumptions made in our model, and mention how our assumptions regarding the damage and benefit functions could be relaxed. Lastly, we discuss how our methods can be applied to the provision of public goods in general.

In addition to improving on the baseline scenario, the matching climate game (MCG) has a number of attractive features in comparison with other mechanisms (including other matching mechanisms) suggested to address international public goods problems such as reducing global GHG emissions, which we describe below.

The MCG asks countries to relinquish very little of their sovereignty and entails no negotiation. First, a country that agrees to participate in the MCG may still choose not to match, and hence does not commit to any policy change, other than refraining from increasing its emissions in response to other countries’ emissions reductions, a phenomenon known as carbon leakage. (Similar to Barrett’s9 approach, including a clause stipulating that the matching agreement does not come into force unless all countries are signatories can help assuage countries’ fears that their environmental policies would be undone by defecting countries, and hence stabilize participation in the treaty and avoid leakage.) Second, unlike many other climate-agreement schemes (e.g., refs. 8,13), signatories are not asked to expose themselves to the possibility of economic sanctions. Third, countries decide on their own policies for matching factors and unconditional abatements, rather than submitting to the authority of an institution or coalition of countries (see, e.g., ref. 34 for such a model without matching; see refs. 35,36, respectively, for matching mechanisms in which a coalition of signatories decides on a common matching factor at which all members will match the total unconditional abatement of all countries, or of other coalition members). Fourth, countries are only asked to commit to conditional actions (as opposed to the Paris Agreement’s pledges), and to refrain from emitting more than they would in the absence of an agreement. In other words, the only encroachments on national sovereignty that the MCG makes is requiring countries to keep their word and fulfill a conditional commitment that they choose to make.

The fact that the MCG does not employ costly punishments is also appealing, because it sidesteps the second-order free-riding problem: when punishment is costly, players are tempted to let others do the punishing37,38. While in the MCG countries can implicitly punish other countries for insufficiently high matching commitments by reducing their unconditional abatements, at the subgame-perfect emissions profile doing so is not costly in the sense that the countries choose Nash equilibrium unconditional abatements (given the matching factors chosen in stage I).

In an analysis of a similar matching-commitment agreement between n identical countries, Rübbelke39 points out that, because countries commit only to matching factors, they can still adjust their unconditional abatements if they discover (after having made their matching commitments) that they underestimated the benefits of abatement. This flexibility allows countries to at least partially (and sometimes fully) correct their emissions abatements based on updated estimates of benefits, avoiding the need for costly renegotiations. Although we do not repeat Rübbelke’s analysis for our model, the flexibility in the choice of unconditional abatements in the MCG is likely to give rise to a similar ability to compensate for updated estimates of the benefits of abatement.

Lastly, many suggestions for climate agreements involve international monetary (or in-kind) transfers through which countries pay to subsidize the abatement efforts of other countries13,26,34,40,41,42. The MCG similarly subsidizes abatements in that positive matching factors effectively reduce the marginal costs of abatements. However, in the MCG, countries’ subsidies are implemented not by transferring resources, but through additional abatements provided conditionally. This type of subsidy is essentially unilateral and seems more likely to be implemented than one that involves monetary or in-kind transfers, especially in the case of countries between which diplomatic relations are tense or nonexistent.

Related literature

To our knowledge, Rübbelke’s39 was the first to analyze a matching-commitment agreement to reduce GHG emissions, in which each country declares a single matching factor at which it will match any other country’s abatement. As is common in the literature on matching-commitment agreements, his analysis is restricted to identical countries and symmetric equilibria, whereas we allow for heterogeneous countries and asymmetric equilibria. The style of our model is also different from Rübbelke’s39 in that—similarly to Boadway et al.21—we use Finus’s22 well-established emissions game to specify payoffs from emissions profiles (ref. 39 instead builds on the standard model of public goods, see, e.g., ref. 43). Our analysis is closest in spirit to Boadway et al.’s21, with the most important difference being this: Boadway et al.’s main result concerns the efficacy of the matching climate game (MCG) in emissions reductions relative to an exogenously-determined baseline scenario in which at least one country would benefit from unilaterally reducing its emissions—in which case emissions would be reduced even without an agreement, if this country simply followed its incentives. By contrast, our main contribution here is to show that the MCG incentivizes countries to lower emissions relative to an endogenously-determined baseline scenario from which they have no incentive to deviate.

Matching mechanisms are most commonly studied in the general context of public goods or externality problems, and in that context our analysis is closest to Guttman and Schnytzer44. However, those authors (a) show that a subgame-perfect equilibrium (SPE) exists for positive externalities, and show that their approach may not work for negative externalities (such as GHG emissions, analyzed here), and (b) do not establish that matching commitments yield unique equilibrium actions (i.e., an emissions profile in a climate agreement), whereas we are able to do so. Note also that in the context of public goods provision, Buchholz et al.45,46 have shown that when players are heterogeneous, matching often leads to inefficient equilibria at which not all players contribute. Our analysis shows that this problem is avoided in our approach, regardless of any possible inequalities between the countries.

Lastly, we mention that another growing branch of the research into social dilemmas in general, and climate governance in particular, involves the application of methods from statistical physics (see ref. 47 for a review, or ref. 48 for a notable example). Broadly, this literature takes a dynamic, evolutionary game-theoretic perspective, in which the independent agents (e.g., people, but also possibly countries) interact many times, and over time alter their strategies based on the payoffs they and others receive (but do not strategically analyze the game they are facing). To our knowledge, these methods have not yet been applied to matching-commitment agreements, and are a promising avenue for circumventing the technical difficulty of analyzing matching-commitment agreements among many countries in the “classical” game-theoretic setting.

Main simplifying assumptions

Our analysis makes use of a number of simplifying assumptions about climate agreements, the most important of which are discussed below.

One-shot game

Similar to, e.g., ref. 49, our simple formulation of the choice of GHG emissions levels in a one-shot game forces countries to make all environmental policy decisions at one point in time. It thus removes the possibility of countries responding to either observed environmental changes or one another’s behaviour. In Barrett’s49 words, it “can be interpreted as compressing perhaps a century of decision-making into a single period,” and the damage and benefit in question “should be interpreted as capturing the full consequences, including into the distant future, of decisions taken this century”. Some of the unforeseen environmental consequences of these emissions will only be experienced in the far future, so no observations of such consequences are available to affect policy in the near future. However, the assumption that countries cannot observe and respond to each other’s behaviour is more restrictive. Even if the full consequences of emissions are not immediately apparent, countries can still assess other countries’ emissions, and would likely vary their own emissions in response. We therefore regard this model as a first step and proof-of-concept for the use of matching-commitment agreements to lower GHG emissions.

There is, however, some experimental evidence that matching-commitment agreements can increase the provision of public goods. In particular, Guttman50 and Bracht et al.51 observe that matching-commitment mechanisms similar to that employed in our matching climate game (MCG) yield significant improvements in public good provision in repeated public goods games played in groups, and between pairs, respectively (see refs. 52,53 for reviews of the experimental literature on matching).

Commitment

Because our model is a one-shot game, it only addresses the issue of commitment in a limited fashion: even if country 2 could renege on the matching factor it committed to in stage I, it has no incentive to do so, and in particular, it does better playing a cooperative subgame-perfect equilibrium (SPE) than it does by reverting to its baseline emissions, i.e., choosing m2 = 0 and a2 = 0. This is because, even if country 1 continues to play $${m}_{1}^{{\rm{c}}}$$, it can still respond to country 2’s defection by decreasing its unconditional abatement, $${a}_{1}({m}_{1}^{{\rm{c}}},{m}_{2})$$. But, if both countries are free to renege on their matching commitments, stage I is merely “cheap talk” and the original basic climate game (BCG; Definition A.1)—in which the baseline emissions profile is the only equilibrium—is recovered.

In the language of Barrett14, the ability of countries to commit assumes the “enforcement problem” is solved. As Barrett49 notes, this is justified by the international-law custom of treaties being binding, pacta sunt servanda. This custom is codified in the Vienna Convention on the Law of Treaties: “Every treaty in force is binding upon the parties to it and must be performed by them in good faith” (ref. 54, p. 339). Thus, if countries can agree that whatever matching commitments they themselves select be legally binding, they are more likely to keep their word. Since countries are also free to refrain from any matching (i.e., matching factors may be zero), and since the matching commitments are conditional on other countries’ actions, such commitments are weaker—and hence safer from being exploited, and thus more likely to be kept—than commitments to specific emissions abatements.

Two recent examples suggest that legally binding commitments to match emissions reductions would likely be an improvement on the status quo, even without an explicit enforcement mechanism in place. First, some parties to both the Kyoto Protocol and the Paris Agreement rejected legally binding targets55,56; this suggests that countries ascribe more weight to commitments that are explicitly legally binding, possibly due to the risk of reputation loss57. Second, although the Paris Agreement’s emissions targets are not binding, other aspects of the agreement are, and even the withdrawal of the US from the Paris Agreement will conform to the agreement’s rules for withdrawal58.

Two vs. many countries

While it is widely acknowledged that an effective agreement must include as parties the two largest emitters, the US and China, their combined emissions amount to about 45% of global emissions, so a climate agreement between only these countries will likely be insufficient3. We leave the analysis of matching-commitment agreements among many countries for future work, but note that a number of studies suggest that such agreements among many countries can also be effective in reducing emissions. For instance, encouraging analytical results include those of Fujita35 and Rübbelke39 (described in the sections “Advantages over other climate-agreement schemes” and “Related literature”). Using the STACO model40 to estimate countries’ damages and benefits from emissions, Kawamata and Horita59 simulate a matching scheme similar to the matching climate game (MCG) for six countries and regions and show that emissions are substantially reduced. Lastly, in Guttman’s50 experimental study, matching commitments also increased public good provision in repeated games in groups of 3–6 players.

In the context of multilateral matching-commitment agreements, it will likely be imperative that all countries be signatories. This is because although matching-commitment agreements do not strictly require signatories to perform any abatement, they do require that countries refrain from emitting more than they would in the absence of an agreement. If some countries opt out of the agreement, and signatories collectively abate their total emissions, then non-signatories will have an incentive to increase their emissions from their no-agreement baseline emissions (this is the carbon leakage mentioned in the section “Advantages over other climate-agreement schemes”).

Increasing damages from climate change

Ricke et al.60 recently estimated the country-level social cost of carbon (CSCC)—a measure of the long-run economic cost of a tonne of CO2 emitted in 2020. The CSCC is analogous to the marginal damage suffered by each country, $${D{\prime} }_{i}$$: it quantifies the marginal cost of emissions by comparing a reference emissions pathway (describing how emissions are distributed over a length of time) to a pathway that begins with an additional pulse of one tonne of CO2 emissions, but is otherwise identical to the reference pathway. Ricke et al.60 repeat these calculations for a variety of different socioeconomic scenarios (that result in different emissions pathways) and for different empirical estimates of the economic impacts of warming. (Note that warming, and consequently damage due to climate change, does not depend strongly on the emissions pathway and is well described by the cumulative emissions61,62. Thus, our formulation assumes that damages result from global cumulative emissions over a long time-horizon, ignoring the possible impact of the emissions pathway).

While one of Ricke et al.’s60 model specifications results in a negative CSCC (i.e., a climate benefit) for some countries, they note that “[t]hese results are among the most sensitive in the analysis, as under the BHM long-run and DJO damage model specifications all countries have positive CSCC”, where BHM and DJO refer to the damage-function estimates of Burke et al.63 and Dell et al.64, respectively. Moreover, they find that “under the long-run impact model specifications [...] countries like Russia, Canada, Germany and France that have negative CSCC under the reference case switch to having among the highest positive CSCCs”. Another analysis by Kompas et al.65 applies a “computable general equilibrium” approach and finds that all of the 139 countries included in their analysis benefit from complying with the Paris Agreement, providing additional support for our assumption that the marginal damages of emissions are positive.

Relaxation of assumptions on the damages and benefits of emissions

The assumption that global emissions are bounded (Definition A.2) can be relaxed: its function is to ensure that the basic climate game (BCG) has a unique Nash equilibrium, but all that is required for our results to hold is that the BCG has a Nash equilibrium, and that it is chosen as the baseline emissions profile. In contrast, the assumption of bounded abatement with matching (BAM) is both necessary and sufficient for the matching climate game’s (MCG) stage-II best-response functions to be well-defined, and hence cannot be relaxed.

Application to the provision of general public goods

While the matching climate game (MCG) and our analysis are framed in terms of climate agreements, they can be applied equally well to other public goods problems. The public good provided in our model is the abatement of GHG emissions, and so reduced emissions correspond to increased abatement. This in turn implies that our assumption that the benefits and damages of emissions are decelerating and accelerating, respectively, translate into requiring that the costs and benefits of providing the public good being accelerating and decelerating, respectively. Our analysis then shows that the MCG may also be useful in promoting the provision of public goods satisfying these assumptions, in situations in which enforcing cooperation is difficult46,53,66.

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Acknowledgements

The preparation of this paper was supported by grants from the Warren Center for Network and Data Sciences at the University of Pennsylvania to C.M., from the US Army Research Office (W911NF-12-R-0012-03) to E.A., from the Defense Advanced Research Projects Agency NGS2 program (grant no. D17AC00005) to E.A. and S.A.L. and from the National Science Foundation (grant no. GEO-1211972), the Nordforsk-funded project Green Growth Based on Marine Resources: Ecological and Socio-Economic Constraints (GreenMAR) and the US Army Research Office (grant no. W911NF1810325) to S.A.L. We thank Robin Boadway for his valuable comments on an earlier draft.

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C.M. devised the project, analyzed the model and drafted the manuscript in collaboration with E.A., U.D., S.A.L. and E.A.R. All authors interacted in model development, provided input and edited the manuscript.

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Correspondence to Chai Molina.

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Molina, C., Akçay, E., Dieckmann, U. et al. Combating climate change with matching-commitment agreements. Sci Rep 10, 10251 (2020). https://doi.org/10.1038/s41598-020-63446-1

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• Dynamics of informal risk sharing in collective index insurance

• Fernando P. Santos
• Jorge M. Pacheco
• Simon A. Levin

Nature Sustainability (2021)


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# Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are isomorphic.

I know three infinite families of such groups: (1) Free abelian groups $\mathbb Z^n$ (2) Free groups $F_n$ (3) Fundamental groups of closed orientable surfaces $\Gamma_n = \pi_1 (\Sigma_n)$.

The way to see that these are indeed examples is via topology: the classifying spaces of these groups are tori, graphs and surfaces and we understand their covering theory.

[Note: some geometric group theorist claim that these are the easiest torsion-free groups that exist. Their outer automorphism groups are given by $GL_n(\mathbb Z)$, $Out(F_n)$ and $Mod^{\pm}_n$, groups that generated a vast body of research.]

Are there more (than the above mentioned) examples of such groups?

similar, but in fact a different question

• I presume that constructing new examples from old, by taking the direct product with a group that has no proper finite index subgroups, is cheating? – Jeremy Rickard May 10 '16 at 13:19
• In a similar vein to Jeremy's comment, Baumslag gave an example of a one-relator group with every finite quotient cyclic. This group has a unique subgroup of each finite index. – HJRW May 10 '16 at 18:53
• The second two examples are mediated by the following phenomenon: you have a class of groups closed under taking finite index subgroups which have the property that they are determined up to isomorphism within the class by their Euler characteristic. This seems pretty rare, e.g. I wouldn't expect it for fundamental groups in any other dimension. – Qiaochu Yuan May 10 '16 at 20:02
• If one removes the somewhat rigid assumption that the group is torsion-free and finitely presented (and instead just assume finitely generated), we can reduce to assuming that the group is residually finite, which discards all the trivial counterexamples given so far. – YCor May 10 '16 at 20:19
• Thanks for the idea to change the conditions, it really seems more natural to ask for residual finiteness. I have edited the question as suggested. – Jens Reinhold May 10 '16 at 22:07

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# Calculating relative angular velocity for missile lock on...

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SO, I had this function to calculate whether or not a target can be "tracked" by a spaceship's missile launcher. (This is a pseudocode version)
function tgtIsTrackable (tgt as Object, me as object, coverage As Single, range As Single, trackingrotspeed As Single) As Integer

vector RelativeVelocity = vecSubtract(Tgt.Velocity, me.Velocity)
bool WithinCone = PointIsWithinCone (Tgt.Coords, me.Coords, Me.ForwardVector, Coverage, Range)

bool SlowEnough = ((magnitude(RelativeVelocity) / distance (Tgt.Coords, me.coords)) <= (TrackingRotSpeed))

If WithinCone And SlowEnough Then Return TRUE
end function
PointIsWithinCone is a function that returns TRUE if a point is within an area defined by a vector (Forward), a distance (range), and an angle (coverage). SlowEnough is set to TRUE if the angular velocity of the target (relative to Coords) is less than TrackingRotSpeed -- the speed (in rad/sec) at which the launcher can track a target. (It's based on the fact that LinearVelocity = AngularVelocity*Radius. In this case, LinearVelocity is RelativeVelocity, Radius is the distance between the two objects, and I'm solving for AngularVelocity). This works perfectly if my ship is not rotating. However, I've come to realize that I have to take into account MY rotational velocity, since my current rate of rotation will have an effect on whether or not I can track the target. In 2D, I'd simply add my rotation rate to my TrackingRotSpeed to come up with a a value that represents how fast the launcher is tracking the target. However, this is a 3D game, and my rotation rate is stored as three euler angles. I'm not sure how to make this work. Thanks...

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Quote:
 my rotation rate is stored as three euler angles. I'm not sure how to make this work.

Me neither. I'm going to use a vector for my part of the discussion. The direction of the vector is the axis upon which "me" is rotating. The magnitude of the vector gives its rotational velocity in radians/sec.
Also, velocity/distance = 1/time. Oh, wait, that's actually almost the correct unit for comparison; never mind. Do keep in mind that one rotation equals 2*PI radians.
You're doing the same thing in 3d that you would do in 2d but there's more to it.
Use the relative linear velocity(in meters) of the target to determine a fake rotational velocity(in rad/sec). To do this, "relative" means relative to the tracking ship, find the relative linear velocity(vector, m/s) of the target and the relative distance(vector, in meters). Perform a cross product. This gives you the fake rotational velocity of the target(vector). Take this result and do a dot product with the tracking ship's rotational velocity vector. Take that result(magnitude, rad/sec) and subtract it from the target's fake rotational velocity magnitude. Compare that result with the maximum rotational velocity magnitude that the ship's tracker can perform(rad/sec). If the tracker's max is higher, the target can be tracked.

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Interesting. I think my terminology was wrong when I said I was storing the rotational velocity as "Euler angles." What I am using is this:

AngularVelocity = Vector ( AngularVelocityAroundX, AngularVelocityAroundY, AngularVelocityAroundZ )

It looks like I wasn't all that far off. I'll try this as soon as I get back to a coding machine.

Thanks a lot

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v = w x r
v = linear velocity of point
w = angular velocity of point (supposed to be lowercase omega, not "w")
x = cross product operator
r = position of point

so I think it's safe to change it to
w = r x v
I have nothing to base this on though except
i × j = k j × k = i k × i = j.
So r might be from the target to the tracker.

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Ok, this is going to involve a bit of math so stick with me here. The overall strategy is this: 1) first we will define all the terms. 2) Next we are going to write the velocity vector of the target in the coordinate system of the ship. In other words, how would someone on the ship describe the way the target is moving. 3) Finally, we will use this to calculate the angular velocity of the target in the ship's coordinate system.

1)
Here are the initial definitions (these are all vectors unless otherwise specified):
Known:
* s - position of the ship in ambient space coordinates
* t - position of the target in space coordinates
* vs - velocity of the ship in space coordinates
* vt - velocity of the target in space coordinates
* the euler angles
* the first derivatives of the euler angles

Unknown:
* r(space) - displacement from the ship to the target in space coordinates
* v(space) - relative velocity of the ship and target, written in space coordinates
* ws - the angular velocity of the ship (in space coordinates of course)
* r(rotating) - displacement from the ship to the target, written in the rotating coordinates of the ship
* v(rotating) - relative velocity of the ship and target, written in the rotating coordinates of the ship

Goal:
* wtr - the angular velocity of the target relative to the ship, written in the ship's rotating coordinates
* wtr - magnitude of wtr
--

2)
The relative positions and velocities in space coordinates are found by subtracting that of the ship from the target:
r(space) = t - s
v(space) = vt - vs

ws, the angular velocity of the ship, can be found from the euler angles and the first derivatives of the euler angles by using equation 21 on the following page:
http://mathworld.wolfram.com/EulerAngles.html

To convert the relative velocity in the space coordinates into the rotating ship coordinates, use the main equation of rotating coordinate systems, which says that
v(rotating) = v(space) - ws x r

This is the velocity vector of the target in the coordinate system of the ship.
--

3)
wtr, the angular velocity of the target in the ship's coordinate system, may be found with the following equation:
wtr=( r(rotating) x v(rotating) )/length( r(rotating) )^2

The magnitude of the angular velocity of the target from the perspective of a person on the ship is then just wtr, then length of wtr.

Whew. Hopefully I didnt make any mistakes there.

[Edited by - Maze Master on July 8, 2008 10:30:07 PM]


text:
Timezone: »

Workshop
Human Computation for Science and Computational Sustainability
Theodoros Damoulas · Thomas Dietterich · Edith Law · Serge Belongie

Fri Dec 07 07:30 AM -- 06:30 PM (PST) @ Emerald Bay 4, Harveys Convention Center Floor (CC)

http://www.cs.cornell.edu/~damoulas/Site/HCSCS.html

Researchers in several scientific and sustainability fields have recently achieved exciting results by involving the general public in the acquisition of scientific data and the solution of challenging computational problems. One example is the eBird project (www.ebird.org) of the Cornell Lab of Ornithology, where field observations uploaded by bird enthusiasts are providing continent-scale data on bird distributions that support the development and testing of hypotheses about bird migration. Another example is the FoldIt project (www.fold.it), where volunteers interacting with the FoldIt software have been able to solve the 3D structures of several biologically important proteins.

Despite these early successes, the involvement of the general public in these efforts poses many challenges for machine learning. Human observers can vary hugely in their degree of expertise. They conduct observations when and where they see fit, rather than following carefully designed experimental protocols. Paid participants (e.g., from Amazon Mechanical Turk) may not follow the rules or may even deliberately mislead the investigators.

A related challenge is that problem instances presented to human participants can vary in difficulty. Some instances (e.g., of visual tasks) may be impossible for most people to solve. This leads to a bias toward easy instances, which can confuse learning algorithms.

A third issue with crowdsourcing is that in many of these problems, there is no available ground truth because the true quantities of interest are only indirectly observed. For example, the BirdCast project seeks to model the migration of birds. However, the eBird reports only provide observations of birds on or near the ground, rather than in migratory flight (which occurs predominantly at night). In such situations, it is hard to evaluate the accuracy of the learned models, because predictive accuracy does not guarantee that the values of latent variables are correct or that the model is identifiable.

This workshop will bring together researchers at the interface of machine learning, citizen science, and human computation. The goals of the workshop are i) to identify common problems, ii) to propose benchmark datasets, common practices and improved methodologies for dealing with such phenomena, iii) to identify methods for evaluating such models in the absence of ground truth, iv) to share approaches for implementing and deploying citizen science and human computation projects in scientific and sustainability domains, and v) to foster new connections between the scientific, sustainability, and human computation research communities.

There will be two awards (250 book vouchers) for Best Contribution for the oral and/or poster presentations sponsored by the Institute for Computational Sustainability (www.cis.cornell.edu/ics)

We welcome submissions* related to (but not limited to) the following topics:

• Biases, probabilistic observation processes, noise processes, and other imperfections in citizen science and human computation
• Novel citizen science and human computation projects in science and sustainability
• Human-machine interactions and human computation in science and sustainability
• Novel modeling paradigms for citizen science and human computation projects in science and sustainability
• Methods for recruitment, retention, and modeling of human participants
• Dataset shift and domain adaptation methods for citizen science and human computation projects
• Spatio-temporal active learning and general inference techniques on citizen science and human computation projects

*Every submission should include both a paper and a poster. The paper should be in standard nips format (final format, not a blind-author submission), up to 4 pages maximum and the poster in portrait format and up to 33” x 47“ (A0) dimensions.

Expected outcomes of this workshop include: a list of open problems, acquaintance with relevant work between scientific domains, a 5-year research roadmap, emerging collaborations between participants, and attracting more people to work in Computational Sustainability and Human Computation. In addition, we will explore avenues for organizing a future machine learning competition, on a large-scale HC problem, to foster and strengthen the community around a prototypical HC effort.

#### Author Information

##### Thomas Dietterich (Oregon State University)

Tom Dietterich (AB Oberlin College 1977; MS University of Illinois 1979; PhD Stanford University 1984) is Professor and Director of Intelligent Systems Research at Oregon State University. Among his contributions to machine learning research are (a) the formalization of the multiple-instance problem, (b) the development of the error-correcting output coding method for multi-class prediction, (c) methods for ensemble learning, (d) the development of the MAXQ framework for hierarchical reinforcement learning, and (e) the application of gradient tree boosting to problems of structured prediction and latent variable models. Dietterich has pursued application-driven fundamental research in many areas including drug discovery, computer vision, computational sustainability, and intelligent user interfaces. Dietterich has served the machine learning community in a variety of roles including Executive Editor of the Machine Learning journal, co-founder of the Journal of Machine Learning Research, editor of the MIT Press Book Series on Adaptive Computation and Machine Learning, and editor of the Morgan-Claypool Synthesis series on Artificial Intelligence and Machine Learning. He was Program Co-Chair of AAAI-1990, Program Chair of NIPS-2000, and General Chair of NIPS-2001. He was first President of the International Machine Learning Society (the parent organization of ICML) and served a term on the NIPS Board of Trustees and the Council of AAAI.


text:
# Given a list of strings, find every pair $(x,y)$ where $x$ is a subsequence of $y$. Possible to do better than $O(n^2)$?

Consider the following algorithmic problem: Given a list of strings $$L = [s_1, s_2, \dots, s_n]$$, we want to know all pairs $$(x,y)$$ where $$x$$ is a subsequence of $$y$$. We can assume all strings are of length at maximum $$m$$, where $$m << n$$ and are all over a finite alphabet $$\Sigma$$ with $$|\Sigma| << n$$. We may also assume that the number of pairs $$(x,y)$$ where $$x$$ is a subsequence of $$y$$ is much smaller than $$n$$.

A trivial algorithm would be this:

1. foreach x in L:
2.   foreach y in L:
3.      if x is subsequence of y:
4.         OUTPUT x,y

However, this has complexity $$O(n^2 \cdot m)$$ - I am curious to know whether there is a faster algorithm (Faster given that the number of pairs $$(x,y)$$ is much smaller than $$n$$, so for example an algorithm with complexity depending on the number of output pairs).

Note that this question is a follow up to this question, which is about the same problem but for substrings (not subsequences). There, the Aho-Corasick algorithm solved my problem perfectly - is there maybe somethign like this but for subsequences?

No, it is not possible to do better unless the Strong Exponential Time Hypothesis (SETH) fails. If we could solve this problem substantially faster than $$O(n^2)$$ we would immediately obtain a much faster algorithm for solving the NP-complete problem Satisfiability. This is true even for $$m$$ slightly more than $$\log(n)$$ and the case in which we want to decide whether such a pair $$(x,y)$$ exists at all.

See, e.g., these lecture notes under section 3 "Tight Lower Bounds for Orthogonal Vectors". The proof is analogous to the proof of Theorem 2 in these lecture notes.

First, we consider the more general problem of given two sets of strings $$X,Y$$, finding whether some string in $$X$$ is a subsequence of a string in $$Y$$.

Given a SAT formula, we split its $$n$$ variables into two equal sets of $$n/2$$ variables. In $$\Sigma$$ we take a character corresponding to every clause. In $$X$$ we add a string for every possible assignment to the first half of the variables, with a character corresponding to every clause not satisfied by those variables. Meanwhile, in $$Y$$, we add a string for every assignment to the second half of the variables, with a character for every clause that is satisfied by those variables. Clearly, the formula is satisfiable if and only if some string in $$X$$ is a subsequence of some string in $$Y$$.

If this problem can be solved substantially faster than $$O(n^2)$$, then this gives a substantially faster algorithm for Satisfiability than $$2^n$$. Suppose the problem could be solved in $$O(n^{1.99})$$ time, then Satisfiability could be solved in $$(2^{n/2})^{1.99}=O(2^{0.996n})$$ which contradicts SETH.

In your problem, there is only a single set of strings, all of which may be a subsequence of each other. This is however not a problem, as we can simply modify the strings in our instance such that no string $$Y$$ is a subsequence of any other string (for instance by padding all strings in $$Y$$ to have the same length), and similarly padding every string in $$X$$ to the same length as other strings in $$X$$ (but substantially shorter than strings in $$Y$$).

This can probably also be done with a constant-size (likely even binary) alphabet but this requires more clever encoding.


text:
# Homogeneous polynomial explained

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.[1] For example,

x5+2x3y2+9xy4

is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial

x3+3x2y+z7

is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function.

An algebraic form, or simply form, is a function defined by a homogeneous polynomial.[2] A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.

A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.[3] A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics.[4] They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.

## Properties

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then

P(λx1,\ldots,λ

 dP(x x 1,\ldots,x

n),

for every

λ

in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many

λ

then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then

P(x1,\ldots,xn)=0   ⇒   P(λx1,\ldots,λxn)=0,

for every

λ.

This property is fundamental in the definition of a projective variety.

Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial.

R=K[x1,\ldots,xn]

over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d forma vector space (or a module), commonly denoted

Rd.

The above unique decomposition means that

R

is the direct sum of the

Rd

(sum over all nonnegative integers).

The dimension of the vector space (or free module)

Rd

is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree d in n variables). It is equal to the binomial coefficient
 \binom{d+n-1}{n-1}=\binom{d+n-1}{d}= (d+n-1)! d!(n-1)!

.

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if is a homogeneous polynomial of degree in the indeterminates

x1,\ldots,xn,

one has, whichever is the commutative ring of the coefficients,
 n dP=\sum i=1
x
 i \partialP \partialxi

,

where
 style \partialP \partialxi
denotes the formal partial derivative of with respect to

xi.

## Homogenization

A non-homogeneous polynomial P(x1,...,xn) can be homogenized by introducing an additional variable x0 and defining the homogeneous polynomial sometimes denoted hP:

 hP}(x { 0,x

1,...,xn)=

 d x 0

P\left(

 x1 x0

,...,

 xn x0

\right),

where d is the degree of P. For example, if
 3 P=x 3

+x1x2+7,

then
 3 3

+x0x1x2+7

 3. x 0

A homogenized polynomial can be dehomogenized by setting the additional variable x0 = 1. That is

P(x1,...,

 hP}(1,x x 1,...,

xn).


text:
# Fullerenes of the Δ n=10 series

Авторы:
Аннотация:

Fusion reactions of cupola half-fullerenes C10, C12, C16, C18, C20, C24, C30 and C36 with each other are considered on the basis of Arrhenius.s postulate. It means that at first there forms an intermediate compound and only afterwards a usual chemical reaction is going on. The final structure of fullerenes is obtained with the help of geometric modeling and is optimized through the use of Avogadro package. In general, the fullerenes which tend to take the appearance of a perfect spheroid have lesser formation energy. However, in some case selforganization of unshared electron pairs prevails. The energy of fullerenes C26, C30, C40, C50 and C60 obtained is calculated and is explained in the context of strain-related stability as well as in the framework of interaction of electron valence pairs.


text:
# Condition number for a symmetric positive definite matrix

I am trying to get an estimate for the induced 2-norm condition number $$\kappa_{2}(M)$$ of this matrix $$M$$: $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{x^{n-j}}{(n-j)!}\:{\rm{d}}x,$$ which is symmetric positive definite of size $$n\times n$$. The above link gives the elements of the inverse. We also know that its determinant is the reciprocal of this OEIS sequence.

I want to obtain something like $$\kappa_{2}(M) = \Omega(f(n))$$ for some to-be-determined $$f(n)$$.

From numerics, it seems that $$\kappa_2$$ grows very fast with respect to $$n$$. But I am not sure what might be an analytic way to obtain the asymptotic for $$\lambda_{\max}/\lambda_{\min}$$.

• Numerics seem to suggest that $\lambda_{\text{max}}$ converges to approximately $1.30384272207$ as $n \rightarrow \infty$, so we "just" need to figure out what's happening with $\lambda_{\text{min}}$. – Nathaniel Johnston Oct 12 '20 at 1:00

(Too long for a comment.)

I managed to numerically extract the condition numbers up to $$n = 112$$. I have plotted this on a log scale:

It looks a bit faster than linear on a log scale; maybe a small quadratic term? But it looks no faster than $$e^{n^2}$$. The best fit line is dashed. The best fit parabola of the natural logarithm of the condition number is $$-33.5838 + 8.09351n + 0.0217842n^2$$. I also tried fitting a cubic polynomial as a sanity check, obtaining $$-19.6223 + 6.71268n + 0.0516761n^2 - 0.000174806n^3$$; that the coefficient of $$n^3$$ is negative and small might be indicative that it isn't "real", but of course we can't rule that out numerically.

Code:

#include <Eigen/Dense>
#include <Eigen/Eigenvalues>
#include <boost/math/special_functions/factorials.hpp>
#include <boost/multiprecision/cpp_bin_float.hpp>
#include <iostream>
#include <vector>
#include <cmath>
#include <fstream>

using Real = boost::multiprecision::number<boost::multiprecision::cpp_bin_float<1000>>;
using Mat = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
using boost::math::factorial;
using std::log;
using std::sqrt;

int main() {
std::ofstream of{"data.csv"};
std::cout << std::setprecision(std::numeric_limits<Real>::digits10);
of << "n, cond(M)\n";
for (int n = 2; n < 200; ++n) {
std::cout << "n = " << n << "\n";
Mat X = Mat::Zero(n,n);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n; ++j) {
Real d3 = Real(2*n - i - j - 1);
X(i,j) = 1/(factorial<Real>(n - i - 1)*factorial<Real>(n - j - 1)*d3);
}
}
es.compute(X);

std::vector<Real> v(es.eigenvalues().data(), es.eigenvalues().data() + n);
if (!std::is_sorted(v.begin(), v.end())) {
std::cerr << "Expected invariant is broken.\n";
return 1;
}
assert(v.front() > 0);

Real cond = v.back()/v.front();
if (cond > 0.01/sqrt(std::numeric_limits<Real>::epsilon())) {
std::cerr << "Precision must be increased to get more samples; fails at n = " << n << "\n";
break;
}
of << n << ", " << cond << std::endl;
}
of.close();
}

Generated data:

n, cond(M)
2, 19.2815
3, 1181.56
4, 165823
5, 4.18166e+07
6, 1.65669e+10
7, 9.47936e+12
8, 7.39574e+15
9, 7.54511e+18
10, 9.7498e+21
11, 1.55626e+25
12, 3.00702e+28
13, 6.91676e+31
14, 1.86767e+35
15, 5.84992e+38
16, 2.10375e+42
17, 8.60899e+45
18, 3.97753e+49
19, 2.06044e+53
20, 1.18933e+57
21, 7.60721e+60
22, 5.36477e+64
23, 4.15244e+68
24, 3.51305e+72
25, 3.2363e+76
26, 3.23507e+80
27, 3.49783e+84
28, 4.07854e+88
29, 5.11458e+92
30, 6.88024e+96
31, 9.90493e+100
32, 1.52261e+105
33, 2.49406e+109
34, 4.34471e+113
35, 8.03433e+117
36, 1.57442e+122
37, 3.26413e+126
38, 7.14851e+130
39, 1.65132e+135
40, 4.01803e+139
41, 1.02846e+144
42, 2.76577e+148
43, 7.80505e+152
44, 2.30875e+157
45, 7.15071e+161
46, 2.31656e+166
47, 7.84207e+170
48, 2.7714e+175
49, 1.02155e+180
50, 3.92398e+184
51, 1.56944e+189
52, 6.5308e+193
53, 2.82524e+198
54, 1.26967e+203
55, 5.92328e+207
56, 2.86664e+212
57, 1.43826e+217
58, 7.4761e+221
59, 4.02365e+226
60, 2.24086e+231
61, 1.29065e+236
62, 7.68353e+240
63, 4.72539e+245
64, 3.00063e+250
65, 1.96638e+255
66, 1.3292e+260
67, 9.26348e+264
68, 6.65305e+269
69, 4.92194e+274
70, 3.74915e+279
71, 2.93919e+284
72, 2.37052e+289
73, 1.96612e+294
74, 1.67632e+299
75, 1.46866e+304
76, 1.32174e+309
77, 1.22145e+314
78, 1.15867e+319
79, 1.12785e+324
80, 1.12618e+329
81, 1.15317e+334
82, 1.21051e+339
83, 1.30226e+344
84, 1.43536e+349
85, 1.62041e+354
86, 1.87313e+359
87, 2.21652e+364
88, 2.68421e+369
89, 3.32574e+374
90, 4.21478e+379
91, 5.46219e+384
92, 7.23698e+389
93, 9.80034e+394
94, 1.35617e+400
95, 1.91725e+405
96, 2.76844e+410
97, 4.08212e+415
98, 6.14524e+420
99, 9.44279e+425
100, 1.48074e+431
101, 2.36913e+436
102, 3.86672e+441
103, 6.43655e+446
104, 1.09254e+452
105, 1.89067e+457
106, 3.33506e+462
107, 5.99552e+467
108, 1.09826e+473
109, 2.04958e+478
110, 3.89608e+483
111, 7.54267e+488
112, 1.4869e+494

(These number break every plotting program I've tried past $$n = 75$$ as that's where they start overflowing double.)


text:
# Members & Guests

## Prof. Dr. Guofang Wang

Albert-Ludwigs-Universität Freiburg im Breisgau

E-mail: guofang.wang(at)math.uni-freiburg.de
Telephone: +49 761 203-5584
Homepage: http://home.mathematik.uni-freiburg.de/w…

## Project

22Willmore functional and Lagrangian surfaces

## Publications within SPP2026

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds in Rn+m. We first provide, following Cabré \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritoré \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in Rn+m, which is optimal when the codimension m≤2. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension m≤2, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.

Related project(s):
22Willmore functional and Lagrangian surfaces

Abstract: For an immersed Lagrangian submanifold, let \check{A} be the Lagrangian trace-free second fundamental form. In this note we consider the equation \nabla^*T=0 on Lagrangian surfaces immersed in \mathbb{C}^2, where T=-2\nabla^*(\check{A}\lrcornerω), and we prove a gap theorem for the Whitney sphere as a solution to this equation.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

 Journal Math. Ann. Volume 374 Pages 1845--1882 Link to preprint version Link to published version

Related project(s):
22Willmore functional and Lagrangian surfaces

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

 Journal Annales de l'Institut Henri Poincare (C) Non Linear Analysis Link to preprint version Link to published version

Related project(s):
22Willmore functional and Lagrangian surfaces

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