Maladaptation in an unequal world: an evolutionary model with heterogeneous agents

Annals of Operations Research (2024) 337:1089–1110 https://doi.org/10.1007/s10479-024-05863-3 ORIGINAL RESEARCH Maladaptation in an unequal world: an evolutionary model with heterogeneous agents Angelo Antoci1·Simone Borghesi2,3 ·Giulio Galdi4·Mauro Sodini5,6 ·Elisa Ticci3 Received: 31 January 2023 / Accepted: 24 January 2024 / Published online: 2 March 2024 © The Author(s) 2024 Abstract Maladaptation is steadily increasing its presence in agenda and debates about climate change and its impacts. The term denotes actions undertaken, at the individual or collective level, to defend against the adverse effects of climate change or environmental degradation, but that ultimately exacerbate the underlying risk factors. In this paper, we investigate the effects of maladaptation in terms of well-being and inequality in a two-population (North–South) evolutionary model. While agents in the South often face higher vulnerability to environmental degradation and limited defense mechanisms compared to their Northern counterparts, the latterstandtoenduregreater economic losses, in absolute terms. Our model demonstrates that the diffusion of maladaptive choices could result in a Pareto-dominated steady state, influencing inequality levels positively or negatively based on the scale of maladaptation impacts relative to the existing environmental degradation. These findings stress the imperative of integrating environmental risk studies with maladaptive effects and dynamics. Additionally, they advocate for international discourse not only on climate change mitigation but also on adaptive measures among countries. Keywords Maladaptation ·Inequality ·Negative externalities ·Economic growth · North–South interactions JEL Classification C70 ·D62 ·O13 ·O40 ·Q20 1 Introduction Environmentaldegradationisattractingincreasingattentioninmodernsocietiesforitsnumerous undesirable effects (in terms of health damages, productivity loss, reparation costs, etc.). Climate change accelerates and magnifies environmental degradation, and its effects are going to be larger and come sooner than estimated in previous assessments (IPCC, 2022). While each inhabited region is going to be affected in a different way, many ecosystems across the world are going to degrade irreversibly. In this context, the demand and scope for effective adaptation is growing and IPCC acknowledges benefits and progress in adaptaBSimone Borghesi [email protected] Extended author information available on the last page of the article 123 1090 Annals of Operations Research (2024) 337:1089–1110 tion implementation. However, there is an increasing awareness of both limits and potential negative effects of adaptation initiatives. Indeed, in some cases, environmental degradation stimulates behaviours perceived as rational at the individual level, i.e. capable of increasing personal well-being at least in the short run, which at the aggregate level or in the long run may generate a reduction in the well-being of the community or part of it. If so, such adaptive choices turn out to be “maladaptive” (Barnett and O’neill, 2010). The literature has highlighted several cases in which adaptive strategies can have relevant adverse impacts for the environment. Air conditioning is a paradigmatic example in this regard, as it is one of the most common adaptive strategies to reduce distress caused by rising temperatures. To the extent that the increase in cooling demand exceeds the reduction in energy demand due to warmer winters, air conditioning ultimately contributes to global warming. Deschênes and Greenstone (2011) and Auffhammer and Aroonruengsawat (2011), for instance, find a positive relationship between electricity consumption and heat shocks in the U.S. residential sector. Similar results are obtained by Davis and Gertler (2015)for Mexico. More recently, Van Ruijven et al. (2019) estimate that global energy demand is projected to increase between 11 and 58 percent by 2050 due to global warming. Other adaptation strategies that can generate increased energy consumption and growth in GHG emissions are snowmaking (Abegg et al., 2007), desalination and inter-basin water transfer projects (Barnett and O’neill, 2010), and pumping-based water efficiency schemes (Beilin et al., 2012). In other cases, adaptation processes do not cause a growth of GHG emissions, but still increase environmental pressures and the associated risks in other ways. The empirical literature provides a wide variety of case studies. In different contexts, from the United States and Australia (Hamin and Gurran, 2009), Great Britain (Fezzi et al., 2015) to Bangladesh (Pouliotte et al., 2009) negative consequences for emissions, freshwater and land fertility emerged as a result of changes in land use and plans due to adaptation against environmental risks. Increased use of fertiliser and pesticide as adaption strategies can generate new environmental risks (IPCC, 2018). The literature review by Müller et al. (2017) finds that some climate insurance schemes, by fostering the status quo or intensification in farming practices, can produce maladaptive outcomes. According to Magnan et al. (2016), the use of sandbags to reduce coastal erosion in Cape Town has released plastic into the sea with the consequent loss of recreational value of beaches. In other cases, adaptation choices do not need to produce additional environmental threats, but can exacerbate or generate new forms of exposure and vulnerability. In Mixteca Alta, Mexico, a mix of perverse economic and institutional factors is pushing smallholders to maladaptive cultivation practices, as they tend to substitute resilient with sensitive components in their cropping system and to reduced crop diversity making the system more rigid against the increasingly erratic weather conditions (Dobler-Morales et al., 2021). IPCC (2023) mentions the example of seawalls which can produce short-term gains but also long-term losses creating lock-ins of risk exposure. In short, the number of case studies on maladaptation is rapidly growing (for rich literature reviews showing several real-world examples see also Eriksen et al. (2021) and Thomas and Warner (2019)). In contrast, theoretical works on the subject are still rare, though in recent years important progress has been made in the conceptualisation of maladaptation (Holling et al., 2002; Barnett and O’Neill, 2013; D’Alisa and Kallis, 2016; Schipper, 2020). We contribute to this field of research with a theoretical model which analyses maladaptation through a distributional lens. For this purpose, this article focuses on the process of adopting maladaptive choices via an evolutionary model in which the interacting economic agentsbelong to twodistinct populations. Evolutionarymodelshavebeenadopted in different contexts to capture imitative behaviors of economic agents (e.g. Villani & Biancardi, 2019; Wang, Chen, & Szolnoki, 2019) also in presence of environmental dynamics (Xepapadeas, 123 Annals of Operations Research (2024) 337:1089–1110 1091 2005;Bischietal.,2013; Blanco and Lozano, 2015; Tilman et al., 2020; Antoci et al., 2021; Zhou et al., 2022; Ding et al., 2023). These eco-evolutionary games show the richness of possible dynamical outcomes which may arise when interactions among agents and between agents’ strategies and the environment are modeled. We integrate this area of research adding the ex-ante existence of different population groups in order to identify potential distributive implications of environmental feedback effects of adaptation choices in a context of bounded rationality,1 In the present model agents decide whether to self-protect from environmental degradation or not depending on what the others do. The agents are divided in the two populations according to (1) their ability to self-protect against environmental degradation; (2) the negative environmental effects generated by their adaptation choices; (3) the relative cost of adaptation compared to their gross output. The paper, in particular, investigates the dynamics underlying the diffusion of (mal)adaptive strategies, and the possible feedback effects that the propagation of such strategies in the two populations may have on the well-being of agents belonging to the whole community. The objective of the present analysis is to highlight the conditions under which maladaptive choices can make both populations worse-off and/or can generate an increase in inequality between the two populations. Earlier empirical research found that distributive effects of maladaptation can be the result of inequalities in agency and political power which affect the distribution of benefits from climate adaptation creating source of increased vulnerability and marginalization (Johnson et al., 2023; Sovacool, 2018). We show that an increase in inequality can emerge even without introducing a political perspective to maladaptation (as, for instance, in Glover and Granberg (2021)), but as the mere result of the interactions of agents who differ in terms of adaptation effectiveness and affordability. The paper is structured as follows. Section2discusses the related literature. Section3 describes the model, Sect. 4the dynamic regimes emerging from the analysis, Sect. 5examines the well-being of the two populations at the steady states and the factors driving the well-being differential between them. Section6summarises the main results derived from the model and discusses possible directions for future research. 2 Related literature The model builds upon and connects two strands of the literature: (i) the one on self-protective choices, and (ii) the one on the environment-inequality nexus. The concept of maladaptation can be traced back to the literature on self-protective choices, originated in the ‘90s from some seminal contributions Shogren and Crocker (1991) for instance distinguished between self-protection that reduces the chance and the severity of undesirable events and that which transfers them to others. The idea of counterproductive reactions to environmental threats was further developed by subsequent analytical studies (e.g. Antoci & Borghesi, 2012) which pointed out that the attempt to defend from environmental degradation may have perverse effects on the well-being of the economic agents. As such, these works can be considered as predecessors of the literature on maladaptation at a time in which the term “maladaptation” was still to be coined. The concept of maladaptation has successively made its way mainly in the literature on climate change. Introduced by Barnett and O’neill (2010) 1Some models in the literature analyse the interplay between mitigation and adaptation expenditures on environmental degradation (see e.g. Bahn 2010). However, in such works, negative externalities and inequality consequences of maladaptive choices in a context with heterogeneous economic agents are not explicitly taken into account. 123 1092 Annals of Operations Research (2024) 337:1089–1110 as “action taken ostensibly to avoid or reduce vulnerability to climate change that impacts adversely on, or increases the vulnerability of other systems, sectors or social groups”, the notion of maladaptation has been later disentangled into different meanings and manifestations and applied to different environmental threats. Eriksen et al. (2021) distinguish perverse adaptation measures according to the underlying reasons for the consequent increase in vulnerability: interventions of governments, NGOs and development agencies which reinforce vulnerability (for instance in case of elite capture), redistribute it (for instance infrastructural projects), or create new sources of vulnerability (as in the case of negative temporal rebound effects or resettlement policies). Analogously, Juhola et al. (2016) distinguish maladaptation types based on outcomes: rebounding vulnerability of targeted or implemented actors, shifting vulnerability to other actors, and creating common pool problems. Here, we use the term of maladaptation as in the third meaning proposed by Juhola et al. (2016), that is self-protection choices that dampen the effect of a global public bad (for instance climate change, global biodiversity loss, ocean acidification and so on) and increase the severity of the problem for the entire community. At the same time, the distribution of the outcomes is dynamically shaped by the combined effects of initial agents’ heterogeneity and the results of their interactions and environmental feedback. In this sense, the model conceptualises maladaptation in a general framework which encompasses also the other specifications and, in line with the maladaptation literature, captures the intrinsically distributive nature of this phenomenon. The present paper, therefore, extends the literature on the environment-inequality nexus in that the proposed modelling reflects some stylised facts identified by empirical literature in this research area. There is a consensus that poorer agents are more adversely affected by climate change, pollution, and other environmental hazards. At the local level, socially and economicallydisadvantagedpeoplelive in moremarginalareas that aremore exposedto environmental degradation (Barbier, 2010; Barbier and Hochard, 2018). At the global level, the impactof climate change on low-and middle-incomeregionsisstrongerandmore severe than on richer regions (see, for example IPCC, 2007, p. 13). The latter have greater possibilities to adapt and react more effectively to environmental damages due to better access to knowledge, resources, technologies, credit and insurance markets (Barbier, 2010,2015). Country rankings based on synthetic indicators mirrored this dichotomy. Countries with lower levels of Sustainable Development Goals Index tend to score higher in terms of Climate Change Vulnerability Index, based on vulnerability to weather-related natural disasters, sea level rise, and loss of agricultural productivity (Pigato, 2019). Similar patterns are shown by indicators elaborated within The Notre Dame Global Adaptation Initiative (ND-GAIN). Vulnerability scores, measuring a country’s exposure, sensitivity, and ability to adapt to the negative impact of climate change, are lower for higher levels of income. Along the same lines, the readiness scores, which measures a country’s ability to implement adaptation actions, declines moving from the group of upper income countries to that of low income countries.2It is worth noting that resource abundance may tend to exacerbate, rather than improve, the economic well-being of a population. This phenomenon, known as the “curse of natural resources”, accounts for variations in both average income and economic growth levels, operating both at cross-country (Sachs and Warner, 2001) and within-country levels (James and Aadland, 2011). Various factors contribute to this negative correlation between resource abundance and economic performance. The literature highlights price effects that undermine the profitability of local manufacturing, subsequently reducing the competitiveness of export sectors (Sachs and Warner, 2001; Amiri et al., 2019). Additionally, some scholars emphasise the signifi2World wide rankings are available at https://gain.nd.edu/our-work/country-index/rankings/. 123 Annals of Operations Research (2024) 337:1089–1110 1093 cance of low institutional quality as the primary factor in this relationship (Brunnschweiler and Bulte, 2008; Sala-i Martin and Subramanian, 2013; Amiri et al., 2019). Finally, the positive relationship between resource abundance or dependence and exposure to climate change impacts underscores the interaction between lower economic performance and vulnerability to climate change (Thomas and Twyman, 2005). In brief, poorer countries are disproportionately exposed to and impacted by the risks of climate change, and they have less capacity to adapt to these challenges. Both global and regional reports corroborate this dual burden (Hallegatte,2016; IPCC, 2022). Similarly, a recent UNEP report (UNEP, 2023) estimates that adaptation costs and the related financial requirements, as a percentage of GDP, are higher for low-income countries compared to lower-middle or upper-middle-income countries. 3Themodel Let us assume that there are two countries at different stages of development, which differ from each other in terms of three elements: per capita gross output, exposure to environmental hazards, efficacy in reducing such exposure.3This narrative of the problem allows us to refer to the asymmetric capability of less developed countries and more developed ones to cope with environmental degradation. We name the wealthier country Nand the poorer country S. Each agent in country k,k=N,S, has a gross output Yk, with YN>YS>0 The gross output Ykis reduced by the damages that environmental degradation generates. Agents can choose whether to defend against environmental degradation or not. Following Antoci and Borghesi (2012), this translates into two strategies available to agents from both countries: 1. Adapting to environmental degradation (strategy D); 2. Not adapting to environmental degradation (strategy ND). In order to adopt strategy D, agents have to incur into an additional cost equal to CD.The strategy chosen by each agent from country kdetermines the final level of the net output: YND k=Yk−ND k(P), if the agent in country kchooses ND (1) YD k=Yk−D k(P), if the agent in country kchooses D (2) where Pis an index of environmental degradation, and ND k(P)and D k(P)are damage functions measuring the economic damage suffered by each agent adopting strategies ND and D, respectively. The difference in net output yielded by the two strategies depends exclusively on the different form of the damage functions. In particular, we assume that for an agent from country kit holds that: ND k(P)=αkP,if the agent in the country kchooses ND (3) D k(P)=αkP 1+dk ,if the agent in the country kchooses D (4) where αkweighs the negative impact of Pon the economic activities of agents from country kand dkweighs the efficacy of protecting against environmental degradation. In line with 3Other characterisations of this context are plausible. For instance, the two populations of agents can also be interpreted as two social groups (e.g. rich and poor individuals) within a single country. 123 1094 Annals of Operations Research (2024) 337:1089–1110 Strömberg (2007)andBarbier(2010), we assume that dN>dS>0; that is, the economy of Sis less effective in contrasting the adverse effects of environmental degradation. Under the stated conditions on the parameters, we have that: D k(P)< ND k(P), k=N,S(5) Condition (5) states that the adverse effects of environmental degradation are stronger if an agent does not protect herself. More specifically, it holds that: dD k(P) dP <dND k(P) dP ,k=N,S(6) Condition (6) specifies that as the environment degrades, i.e. Pincreases, adaptive choices become increasingly convenient, as the damage function increases less rapidly for agents adopting D. We assume that the level of environmental degradation is only affected by the strategy distribution in the two countries. In this case, the dynamics of Pdepends on the shares of agents x(t)and z(t)adopting strategy D at time tin country Nand S, respectively: P=P+βN·x(t)+βS·z(t)(7) where P>0 is the environmental degradation when all agents in both countries adopt strategy ND, i.e. x=z=0, whereas βN,βS>0 measure the impact of adaptive choices of agents from Nand S, respectively. The mechanism described by Eq. (7) characterises strategy D as a maladaptive strategy. Given the positive sign of parameters βNand βS,ifthe share of agents adopting strategy D increases, then the environmental degradation increases. We remark that if all agents successfully coordinated on strategy ND, they could be better off, as they would enjoy a lower value of Pwhile not incurring into the increased cost of adopting D. Recalling that agents adopting strategy D pay a cost CD>0, the payoff of an agent from country kis: Uk=UND k(x,z)=Yk−ND k(P), for strategy ND UD k(x,z)=−CD+Yk−D k(P), for strategy D (8) whereas the time evolution of xand zis assumed to be described by replicator dynamics (see, for example Weibull, 1997): ˙x=x(1−x) N(9a) ˙z=z(1−z) S(9b) where ˙xand ˙zare the time derivatives of xand z, respectively, and kis the payoff differential in country k: k=UD k(x,z)−UND k(x,z)=−CD+αkdk 1+dk P,with k=N,S(10) Intuitively, the payoff differential is positive (k>0) if the cost CDof adopting strategy D is lower than its benefits, represented by the damage differential: ND k(P)−D k(P)=αkdk 1+dk P(11) 123 Annals of Operations Research (2024) 337:1089–1110 1095 We note that this damage differential is always positive according to condition (5). By substituting equation (7)into(10) we obtain: k=−CD+αkdk 1+dkP+βN·x+βS·z,k=N,S(12) According to dynamics (9a)–(9b), the share of agents in country kadopting strategy D increases when the payoff differential is positive (UD k(x,z)−UND k(x,z)>0), whereas it decreases in the opposite case (UD k(x,z)−UND k(x,z)<0). As noted by Levin et al. (2013), rationality depends on the context, and bounded rationality is a sensible assumption of human behaviour in social-ecological systems which are often complex adaptive systems. This may be the case in the context of global commons that typically result from a complex composition of choices of innumerable near and far away agents whose consequences accumulate gradually. For these reasons, we assume imitation dynamics based on the difference between the payoffs of strategies D and ND in each country. 4 Dynamic regimes 4.1 Steady states The dynamical system (9a)–(9b) is defined in the square Q: Q={(x,z):0≤x≤1,0≤z≤1} Each point (x,y)of the square Qrepresents a specific distribution of strategies D and ND, within the two populations considered. The vertices of the square (x,y)=(0,0),(1,1), (0,1),(1,0)are the states where each of the two populations “specialises” by adopting only one of the two possible strategies. In order to find the steady states of system (9a)–(9b), we study the points in which ˙x=0 and ˙z=0. We find that ˙x=0 holds when all agents in Nadopt the same strategy, i.e. for x=0, x=1. Furthermore, ˙x=0 holds if the two strategies D and ND yield the same payoff, which occurs on all couplets (x,z)belonging to the straight line: z=fN(x):= 1+dN αNβSdN CD−P βS −βN βS ·x(13) along which the two strategies yield the same payoff for agents in N, i.e. UD N(x,z)= UND N(x,z). It is easy to check that the share of agents in Nadopting strategy D increases (˙x>0) above the line (13), whereas it decreases (˙x<0) below it. Analogously, ˙z=0 holds when all agents in Sadopt the same strategy, i.e. for z=0, z=1, and in all couplets (x,z)belonging to the straight line: z=fS(x):= 1+dS αSβSdS CD−P βS −βN βS ·x(14) along which UD S(x,z)−UND S(x,z)=0. In analogy with country N, the share of agents adoptingDin Sincreasesabovetheline(14)and decreasesbelowit.Wenotethatthelines(13) and (14) have the same slope: f N(x)=f S(x)=− βN βS<0. Furthermore, fN(0)=fS(0) holds for: αSdS 1+dS =αNdN 1+dN (15) 123 1096 Annals of Operations Research (2024) 337:1089–1110 We shall see that (15) proves to be a necessary condition for the existence of steady states internal to Q. Indeed, we now summarise the steady states, i.e. the states (x,y)in which ˙x=˙z=0: 1. All vertices (0,0),(1,1),(0,1),(1,0)of Q; each of them represents a scenario in which a single strategy is adopted in both populations. 2. The intersection points (when they exist) between the line (13) and the sides of Qwith either z=0orz=1. 3. The intersection points (when they exist) between the line (14) and the sides of Qwith either x=0orx=1. 4. The points, when they exist, internal to the square region Qand belonging to both lines (13)and(14). According to the above analysis, no internal steady state exists if (15) is not satisfied. In the non robust case in which (15) holds, the lines (13)and(14) coincide and all the points belonging to the intersection between them and the interior of the square Qare steady states. For simplicity, in the following analysis we shall not consider such a non robust case. 4.2 Stability properties of the steady states The following proposition concerns the stability of the steady states (0, 0), (0, 1), (1, 0), (1, 1). The proof is straightforward and will be omitted. Proposition 1 The Jacobian matrix of the system (9a)–(9b)evaluated at the steady state (x,z)=(i,j),i=0,1and j =0,1,is: (1−2i)UD N(i,j)−UND N(i,j)0 0(1−2j)UD S(i,j)−UND S(i,j)(16) and has eigenvalues: (1−2i)UD N(i,j)−UND N(i,j) and (1−2j)UD S(i,j)−UND S(i,j) The analysis of the sign of the eigenvalues given in Proposition 1allows us to illustrate the stability properties of the steady states (0,0),(0,1),(1,0),(1,1). In what follows we will denote with Qx=0the side of Qwhere x=0, and with Qx=1the side where x=1. Similar interpretations apply to Qz=0and Qz=1. All sides of this square are invariant; namely, if the pair (x,z)initially lies on one of the sides, then the whole correspondent trajectory also lies on that side. Stability of the steady state (0,0) The eigenvalue in direction of Qz=0of the Jacobian matrix (16), evaluated at (0,0), is strictly negative if and only if (iff, hereafter): CD>αNdN 1+dN P(17) The eigenvalue in direction of Qx=0is strictly negative iff: CD>αSdS 1+dS P(18) 123 Annals of Operations Research (2024) 337:1089–1110 1097 Stability of the steady state (0,1) The eigenvalue in direction of Qz=1of the Jacobian matrix (16), evaluated at (0,1), is strictly negative iff: CD>αNdN 1+dNP+βS(19) The eigenvalue in direction of Qx=0is strictly negative iff: CD<αSdS 1+dSP+βS(20) Stability of the steady state (1,0) The eigenvalue in direction of Qz=0of the Jacobian matrix (16), evaluated at (1,0), is strictly negative iff: CD<αNdN 1+dNP+βN(21) The eigenvalue in direction of Qx=1is strictly negative iff: CD>αSdS 1+dSP+βN(22) Stability of the steady state (1,1) The eigenvalue in direction of Qz=1of the Jacobian matrix (16), evaluated at (1,1), is strictly negative iff: CD<αNdN 1+dNP+βN+βS(23) The eigenvalue in direction of Qx=1is strictly negative iff: CD<αSdS 1+dSP+βN+βS(24) Stability properties of the steady states in the interior of the edges of Q We note that the right sides of the inequalities from (17)to(24) define the threshold values for which changes in the stability properties of the steady states occur as the parameter CD varies (bifurcation values). The following propositions concern the stability properties of the steady states belonging to the interior of the edges of the square Q, i.e. those where both adoption choices coexist in Nwhile all agents in Splay the same strategy, or vice-versa. The proofs are straightforward and will be omitted. Proposition 2 The Jacobian matrix of the system (9a)–(9b)evaluated at the steady states (i,z)in the interior of the edges Qx=i(i =0,1)is: (1−2i)UD N(i,z)−UND N(i,z)0 z(1−z)∂UD S(i,z)−UND S(i,z) ∂xz(1−z)∂UD S(i,z)−UND S(i,z) ∂z(25) 123 1104 Annals of Operations Research (2024) 337:1089–1110 Fig. 4 Basins of attraction for different parameter values. Baseline: αN=0.23, αS=0.3, βN=0.4, βS=0.3, dN=0.9, dS=0.5, CD=0.16, P=1 Note that equation (46) always holds when the term on its right hand side is non-positive. We have: αS−αN αS 1+dS−αN 1+dN >1 (49) if and only if αNdN 1+dN <αSdS 1+dS that is, if condition (35) holds.5 We recall that the above inequality requires that the damage differential of Ndue to the adoption of strategy D must be lower than the one of S. If this condition holds, then condition (46) holds only for sufficiently high values of the ratio (βN+βS)/P. This occurs when negative externalities of agents adopting self-protective strategy D are much higher than autonomous degradation P. Under these circumstances, inequality is bound to increase moving from (0,0)to (1,1), as human action makes environmental degradation increase more sharply and thus causes the gap in well-being between agents from Nand Sto widen, by virtue of condition (46). 7 The general case In this work we employed a specification of the model whose simplicity allows for a more conciseillustrationofthe phenomenon.However,weremarkthat amoregeneralmodelisation does not alter our results, which are robust to other formal specifications. Although the complete study of the general case is beyond the scope of this work, we here show that the results hold also if only the following assumption is made on the payoff functions UD k(P) 5Note that, since dS<dN,ifαS>α Nthen condition (35) is certainly met. In other words, the latter condition is always satisfied if the South suffers a higher economic impact from environmental degradation and has a lower capacity to defend. 123 Annals of Operations Research (2024) 337:1089–1110 1105 and UND k(P): dUND k(P) dP <dUD k(P) dP <0,k=N,S(50) We recall that this condition requires that the payoff of agents adopting ND decreases more rapidly than that of agents adopting D, when the level of environmental degradation P increases. Assuming that Pis a function of xand z, with partial derivatives ∂P(x,z) ∂x>0and ∂P(x,z) ∂z>0 (that is, Pis increasing in xand z), we have that ˙x=0forx=0, x=1, and along the curve satisfying the equation: UD N(P)−UND N(P)=0 (51) Analogously, it holds that ˙z=0forz=0, z=1, and along the curve satisfying the equation: UD S(P)−UND S(P)=0(52) The two equations (51)and(52) implicitly define two functions: z=fN(x)and z= fS(x), with equal slope: f k(x)=− dUD k(P)−UND k(P) dP ·∂P(x,z) ∂x dUD k(P)−UND k(P) dP ·∂P(x,z) ∂z =− ∂P(x,z) ∂x ∂P(x,z) ∂z <0,j=D,ND Therefore, the graph of function z=fN(x)is a translation (either upward or downward) of the graph of z=fS(x). This implies that, just as in the model analysed in this work, there are generally no steady states internal to square Q, in which strategies D and ND coexist in both countries. As concerns the steady states internal to the sides of Q(where the two strategies coexist in only one of the countries), we have that the Jacobian matrix evaluated at (i,z), with i=0,1 and 1 >z>0, is given by: (1−2i)UD N[P(i,z)]−UND N[P(i,z)]0 z(1−z)∂UD S[P(i,z)]−UND S[P(i,z)] ∂xz(1−z)∂UD S[P(i,z)]−UND S[P(i,z)] ∂z and has a positive eigenvalue in the direction of the side of Qwith x=i: z(1−z)∂UD S[P(i,z)]−UND S[P(i,z)] ∂z>0 (by virtue of (50)) Analogously, the Jacobian matrix evaluated at the steady states (x,j), with j=0,1and 1>x>0, is given by: x(1−x)∂UD N[P(x,j)]−UND N[P(x,j)] ∂xx(1−x)∂UD N[P(x,j)]−UND N[P(x,j)] ∂z 0(1−2j)UD S[P(x,j)]−UND S[P(x,j)] and has a positive eigenvalue in direction of the side of Qon the side with z=j: x(1−x)∂UD N[P(x,j)]−UND N[P(x,j)] ∂x>0 (by virtue of (50)) Therefore, also in this more general context, we find that only the vertices (0,0),(0,1),(1,0), and (1,1)can be attractive and that all trajectories starting from the interior of Qconverge to one of such vertices, excluding the ones belonging to the stable manifolds of the steady states (i,z)and (x,j)above-mentioned, when they exist and are saddle points. The less 123 1106 Annals of Operations Research (2024) 337:1089–1110 general specification adopted in this work is thus meant to provide an easier interpretative framework. 8 Discussion and conclusions In this work we studied of the interconnection between inequality in the capacity to adopt adaptive strategies and inequality in well-being, highlighting the role that environmental degradation plays in this relation. In addition, we explicitly modelled the heterogeneity of agents, making the environmental damage they suffer, the efficacy, and affordability of their responses dependent on the groups they belong to. We recall that we chose a country-wise exemplification to illustrate our model only for the relevance of the issue at hand and for its descriptive efficacy. Other instances in which two groups are differentiated according to their capacity to cope with adversities would be equally valid, and so our results. As proved in the Sect.7, the model specification chosen does not affect the results, which are robust to a more general formulation of the problem. Two major conclusions can be drawn from our analysis. Firstly, the non-adoption case, in which no agent in either country adopts the maladaptive self-protective strategy, is Paretooptimal whenever it is attractive, i.e. it is individually beneficial. This represents the case in which the cost of adoption of the maladaptive strategy is too high to make it convenient for any agent, thus preventing agents from generating the negative externalities related to such strategy. Intuitively, no agent would thus be better off adopting the Pareto-dominated maladaptivestrategy.However, ouranalysis showsthatnon-adoption couldbePareto-optimal even if it is not attractive. This happens when the adoption cost of the maladaptive strategy is sufficiently low for at least the wealthier agents and its negative externalities are sufficiently high to make every agent worse off. In this case maladaptive strategies make the system reach a Pareto-dominated steady state. This undesirable outcome could be overcome if agents successfully coordinated on non-adoption or if an institution were established or a policy enforced to prevent agents from adopting maladaptive strategies. Secondly, we also found that the inequality between the different groups of agents may decrease or increase depending on the ratio between autonomous environmental degradation and that generated by agents adopting strategy D (see condition (46)). If maladaptation has severe environmental consequences (i.e. it causes environmental degradation to remarkably increase with respect to the case in which none maladapts), then inequality will increase as we move from full non-adaptation (0,0)to full adaptation (1,1). Indeed, even if agents from the relatively less developed country adopt the maladaptive strategy, the fact that they are also more vulnerable to environmental degradation makes inequality increase. In the full adoption scenario, the environment is further degraded by the negative externalities coming from all agents, which goes to disproportionally increase the burden on the most vulnerable. On the contrary, if the environmental consequences of maladaptation are relatively low with respect to existing pollution, then inequality may decrease as we move to full adaptation. In this case, although the richer agents have a higher capacity to defend from environmental degradation, they also have more to lose from environmental degradation which makes them suffer higher losses. Stated differently, the reduction of inequality observed in this case is not the result of a progressive redistribution policy but it simple mirrors the fact that everybody gets poorer thus reducing the initial gap between agents. These results therefore suggest the need to systematically integrate the study of indirect adaptationeffectsand theirdistributionalpotential into climaterisk assessments and interven123 Annals of Operations Research (2024) 337:1089–1110 1107 tions. Efforts in this direction can be important to build an awareness of possible maladaptive outcomes and to push policy makers and practioners to consider them. They also open the debate on how to distribute not only climate change mitigation, but also how to distribute adaptation actions. We remark that this paper describes the dynamics and well-being consequences of maladaptive strategies, but it does not model the introduction of maladaptive strategies in a group which does not have any already. An expansion of this model might indeed implement a cross-group imitation, discounting the payoff difference by a factor relating to the fact that imitators and imitated agents belong to different groups. Moreover, we did not include within-group heterogeneity since it did not seem to directly affect our results. However, relevant insights might be drawn from an analysis that studies both cross-group and stratified within-group externalities. Finally, in this work we were concerned with the degradation of an environmental variable which affected both groups, as it occurs, for instance, in the case of climate change. Maladaptive strategies could also increase environmental pressure at the local, rather than at the global level, e.g. a lake. Analogously, maladaptation might degrade and environmental indicator from which only a socially homogenous group benefits (e.g. in terms of income, education, or other socio-demographic variables). An analysis of the interaction of local degradation with respect to a global one would make for a compelling extension of this work, with a focus on the dynamics of maladaptive strategies affecting the former and/or the latter. Acknowledgements Angelo Antoci gratefully acknowledges financial support from MUR.(PRIN: PROGETTI DI RICERCA DI RILEVANTE INTERESSE NAZIONALE – Bando 2022, Prot. 20227N3BMK). Simone Borghesi and Elisa Ticci gratefully acknowledge financial support from the University of Siena under the PSR (Piano Sostegno alla Ricerca) support scheme. Mauro Sodini would like to thank the Czech Science Foundation (GACR) under the Project 23-06282S and SGS Research Project SP2024/003 of VSB Technical University of Ostrava for financial support of this work and the financial support of the European Union under the REFRESH – Research Excellence For REgion Sustainability and Hightech Industries project number CZ. 10.03.01/00/22_003/0000048 via the Operational Programme JustTransition. Mauro Sodini also acknowledges support within the project “The Impact of Crises on Complex Spatial Economic Systems (ICCSES)”, ProgrammaFRA 2022UniversitàdiNapoli‘FedericoII’(DR/2022/2055del17/05/2022).Finally,the authors are grateful to the anonymous reviewers and the participants in the 28th Annual Conference of the European Association of Environmental and Resource Economists (Cyprus, 27-30 June 2023) for their valuablefeedback on earlier drafts of this paper. Funding Openaccessfunding providedby Università degliStudi di Sienawithinthe CRUI-CARE Agreement. Declarations Conflict of interest No competing interests to declare. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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Authors and Affiliations Angelo Antoci1·Simone Borghesi2,3 ·Giulio Galdi4·Mauro Sodini5,6 ·Elisa Ticci3 Angelo Antoci [email protected] Giulio Galdi [email protected] Mauro Sodini [email protected] Elisa Ticci [email protected] 1Department of Economics and Business, University of Sassari, Via Francesco Muroni, 25, 07100 Sassari, Italy 2Florence School of Regulation, European University Institute, Via Giovanni Boccaccio, 121, 50133 Florence, Italy 3Department of Political and International Sciences, University of Siena, Via Pier Andrea Mattioli, 10, 53100 Siena, Italy 4Department of Economics and Management, University of Trento, Via Vigilio Inama, 5, 38122 Trento, Italy 5Department of Law, University of Naples “Federico II”, Via Porta di Massa, 32, 80133 Naples, Italy 6Department of Finance, Faculty of Economics, Technical University of Ostrava, Sokolská tˇr. 2416, 702 00 Ostrava, Czech Republic 123