A brave new world? Kantian–Nashian interaction and the dynamics of global climate change mitigation

Abstract

We model game-theoretic situations in which agents motivated by Kantian ethics interact with conventional Nashian agents. We define the concept of generalized Kant–Nash equilibrium which encompasses the Nash equilibrium, the Kantian equilibrium and the Kant–Nash equilibrium as special cases. We show the existence and uniqueness of a Kant–Nash equilibrium in a static game of climate change mitigation, and derive its properties. In a dynamic setting, we analyze a differential game of climate change mitigation in the presence of both types of agents, and compare their Markov emission strategies. For both the static model and the dynamic model, as the population share of Kantians increases, the quality of the environment improves, and both types of agents are better off.

Introduction

The standard non-cooperative game approach to climate change mitigation yields a dismal prediction: the incentive to free ride is so strong that there is little hope for any significant improvement over the business-as-usual scenario. Theoretical models of international environmental agreements (IEAs), using the internal and external stability concept based on Nash behavior (d’Aspremont et al., 1983), predict that the size of a stable coalition of countries forming an IEA is very small. Indeed, Barrett (1994), Carraro and Siniscalco (1993) and others have showed that in a world with identical countries and linear quadratic costs, only two or three countries would join a coalition (see Benchekroun and Long, 2012 for a survey). Moreover, when the model is modified such that larger coalitions are possible outcomes, the welfare gain to the world is very small (see e.g. Eichner and Pethig, 2013).¹

The reality is perhaps not so alarming: member countries of the EU have agreed to an agenda to curb emissions, and outside the EU, several (non-EU) European countries, several states in USA, and a couple of provinces of Canada have taken action to introduce cap-and-trade programs or carbon-tax legislations.² More recently, China and the USA have come to some understanding on the need to curb CO₂ emissions, which provided momentum for the COP 21 Paris Agreement.³ What factors can account for these developments? If we take the view that the political elite has some incentive to satisfy the wishes of the citizens, we might explain these changes as reflecting the fact that in some countries citizens have appreciated their responsibilities toward the environment.

We argue that analysis based on Nash behavior may not be entirely appropriate in contexts where some subset of agents are aware of their responsibility toward the provision of a public good. In fact, Laffont (1975) has questioned the validity of the Nashian assumption in predicting agents’ behavior. He asked, “Why is it that (at least in some countries) people do not leave their beer cans on the beaches?”

That question is difficult to answer using standard economic models. The impact on an individual’s own ‘welfare’ from leaving his beer cans on the beach is certainly negligible while the time and effort to properly dispose of them may not. Yet many people would make the required effort. Laffont’s explanation is very simple, yet compelling: “Every economic action takes place in the framework of a moral or ethics.” According to Laffont, Kant’s rule can explain the pro-social behavior observed in the beach example and in many other instances of human interactions. Laffont was, of course, referring to the categorical imperative formalized by Kant (1785).⁴ While Kant’s classic book offers several formulations of this concept, for our purposes it seems adequate to present it as follows: ‘Act as if the maxim of your action were to become through your will a general natural law.’⁵

Laffont’s explanation is concordant with Adam Smith’s view on social norms. Adam Smith (2002) finds that co-operation and mutual help are incorporated in established rules of behavior, and that “upon the tolerable observance of these duties, depends the very existence of human society, which would crumble into nothing if mankind were not generally impressed with a reverence for those important rules of conduct.”⁶

It is important to formalize how individuals motivated by Kantian ethics choose their actions, and to characterize the equilibrium reached. Recall that a Nash equilibrium is a situation in which if a player deviates from her intended action, she will get a worse payoff, assuming that others do not deviate. A player’s Nash counterfactual describes the payoffs that she would receive if she would deviate while all other players do not. A profile of actions is a Nash equilibrium if for each player all Nash counterfactuals give her a lower payoff. Laffont’s point is that Kantian agents do not make decisions on the basis of Nash counterfactuals. Instead, they consider only Kantian counterfactuals: If I were to deviate, what payoff would I get, assuming that all other agents would also deviate likewise?

Laffont’s model assumes identical individuals. Recently Roemer (2010, 2015) proposes a formulation of the concept of ‘deviating likewise’ in a more general model where Kantians are heterogeneous. In Roemer (2010), an agent is at a Kantian equilibrium if and only if she would receive a lower payoff upon a deviation (i.e., increasing or decreasing her equilibrium activity level by a factor λ ≥ 0) assuming all other players would deviate likewise.⁷

Laffont (1975) and Roemer (2010, 2015) only consider games in which all players act in the Kantian way. However, as argued in Long (2016), in most real world situations there are interactions between Kantian and Nashian agents. Thus, it is important to model the behavior of Kantians when both types of agents exist. What kind of counterfactuals would Kantians contemplate when they want to determine whether a proposed course of action should be adopted? Would each Kantian suppose that when she increases or decreases her activity level, only other Kantians would do likewise? Or would she contemplate a hypothetical (and obviously unrealistic) scenario where Nashians would also deviate? These issues were discussed in Long (2016), where the notion of a Kant–Nash equilibrium was proposed.

In this paper, we propose a more general formulation that encompasses the concepts of Nash equilibrium, Kantian equilibrium, and Kant–Nash equilibrium as special cases. For this purpose, we introduce the notion of a player’s set of co-movers. This is defined as the set of agents that a player supposes would do likewise if she were to scale up or down her equilibrium activity level. The formal definition is worded such that any player has at least one co-mover, namely herself. In our model, when deciding whether she should scale up or down her activity level, the agent considers the counterfactual scenario where all her co-movers would deviate in a similar way, and all others would stay put. We also allow for the possibility that a player may suppose that others would deviate like her, but not to same extent.

Assuming common knowledge, we define a generalized Kant–Nash equilibrium (GKNE) as a profile of activity levels such that for each individual, staying put offers a better payoff than what she would obtain under her counterfactual scenarios. After imposing some additional consistency requirements, we focus on three main cases. First, if every player’s set of co-movers is a singleton, a GKNE is a Nash equilibrium. Second, at the other extreme, if every player’s set of co-movers is the universal set $\mathcal{M}$ (the set of all players) then a GKNE is a Kantian equilibrium. The third case we consider is arguably the most relevant one: it is an intermediate situation in which there is a proper subset $\mathcal{K}$ of players (called the set of Kantian players) such that all individuals in $\mathcal{K}$ share $\mathcal{K}$ as their common set of co-movers, while each player outside $\mathcal{K}$ has only one co-mover, namely herself. In this situation, a GKNE is called a Kant–Nash equilibrium. We apply our new equilibrium concept to games of emissions, first in a static setting, then in a dynamic one.

The plan of this paper is as follows. In Section 2, we introduce the concept of GKNE, discuss its meaning and significance, and contrast the Kant–Nash explanation of pro-social outcomes with explanations provided by alternative theories. This is followed by an analysis of a static model of emissions when Kantians and Nashians co-exist. We provide a proof of existence and uniqueness of Kant–Nash equilibrium for a static emissions game, and show that under some weak assumptions a Kant–Nash equilibrium is Pareto efficient for the set of Kantian players. In other words, it is as if the Kantian players were co-operating to maximize a weighted sum of their individual payoffs. In Section 3, we introduce the concept of dynamic Kant–Nash equilibrium in Markov strategies and apply it to a dynamic game of climate change mitigation. Section 4 concludes with our perspective as to what the equilibria imply for social changes and the provision of public goods. Technical materials are relegated to the Appendix.