When your gain is also my gain. A class of strategic models with other-regarding agents

Highlights

• Vector-valued games are applied in order to analyse other-regarding strategic models.

• The well-being of the opponent agents matters.

• Agents’ preferences are represented by a weighted Rawlsian function, as an alternative to traditional utilitarian interpretation.

• The behavioural attitudes of the agents and the social relationships of other-regarding characterization are defined and proven.

• The parameters in the Rawlsian representation of the agents’ preferences provide the characterization of equilibria.

Abstract

This paper explores the role of social preferences in a competitive framework. More precisely, we study other-regarding strategic models where agents show Rawlsian preferences and, therefore, they care about the best interest of the worst-off agent. The representation of preferences proposed is the most appropriate when the utilities of the agents are vector-valued and their components are not compensable but complementary. In these cases, the improvement of the result for each agent has to be reached by simultaneously improving all the components of the vector-valued utility. Depending on the attitude exhibited by the agents with respect to the results of the others, we distinguish different types of agents and relate them with the parameters of the Rawlsian preference function. An analysis of the sets of equilibria in terms of these parameters is presented. Particularly, in the case of two agents, the equilibria for all the values of the parameters are completely described.

Introduction

In many situations, the behaviour of individuals is inconsistent with the traditional assumption reflected in competitive models where the self-interest is the main motivation of decision-makers. In fact, people often care for the well-being of others and this behaviour may have economic consequences (see Cooper and Kagel, 2013, Nowack, 2006, Tabibnia et al., 2008). The literature in psychological and behavioural game theory shows that the assumption that in strategic situations agents are rational and self-interested does not exclude that an individual may take also into account the interests of others. Some contributions on this topic are Colman, Korner, Musy, and Tazdait (2011), where the other-regarding concerns of the agents are modelled as functions of their own and their opponents’ objective payoffs. They focused on Berge equilibria (Abalo and Kostreva, 2004, Abalo and Kostreva, 2005, Berge, 1957) for games in which utility-maximizing agents are motivated by social attitudes defined by payoff transformations. In Monroy, Caraballo, Mármol, and Zapata (2017) other-regarding preferences are also incorporated into the problem of the commons by assuming that the agents take into account the utilities of all of the group. They studied the situation under different degrees of concern of each agent with respect to the utilities of the others and presented results on the equilibria for different types of agents.

In order to analyse other-regarding strategic models, the methodological framework considered in the present paper is the theory of strategic vector-valued games. We deal with non-cooperative games where the preferences of the agents are incomplete and they can be represented by vector-valued functions. In the literature, the research regarding vector-valued utilities has mainly focused on the case in which the preferences of the agents are represented by weighted additive value functions, as in Keeney and Raiffa, 1976, Mármol et al., 2017 and Monroy et al. (2017). Recently, Rébillé (2019) has axiomatically characterized preferences which can be represented by pseudo-linear utility functions and also by additive separable pseudo-linear utility functions.

An important issue is to identify the contexts where other-regarding attitudes such as interpersonal altruism, fairness, reciprocity or inequity aversion describe the preferences of the agents and to what extent these other-regarding preferences have relevant economic and social effects. These attitudes are in fact significant to explain the behaviour of individuals when facing social dilemas as the sustaining of common resources (Monroy et al., 2017) or the provision of public goods (Kolstad, 2011). In many of these cases an utilitarian approach, which considers additive value functions, can capture the social nature of agents’ preferences. Nevertheless, in situations where inequity aversion has to be taken into account, maxmin value functions, in the spirit of the egalitarian approach proposed by Rawls (1971), seem to be more adequate (Charness and Rabin, 2002, Engelmann and Strobel, 2004). Egalitarianism wants to improve the worst case, thereby, in a general setting, it seeks to maximize the minimum of the weighted components of the vector-valued utility functions.

Zapata, Mármol, Monroy, and Caraballo (2019) provided the theoretical bases for the analysis of the equilibria under the assumption that the preferences can be represented by a weighted maxmin function, also known as Rawlsian function. This representation is the most appropriate when the components of the utilities are not compensable but complementary and, therefore, the improvement of the result for each agent has to be reached by simultaneously improving all the components of the vector-valued utility.

Particularly, Rawlsian preferences are appropriate to better analyse decisions related to the consumption of goods or services with positive externalities, such as the level of education achieved in an economy. In this case, the high level acquired by a part of the population could not compensate for the low or null level achieved by another part of it. Moreover, the egalitarian approach underlies relevant economic issues such as income distribution. When other-regarding attitudes, especially fairness and inequity aversion, are taken into account, agents are not only concerned about their own income but also about how the total income generated by the society is allocated among the population. However, agents could show a distaste for low relative income of others for reasons of fairness or altruism but also for the effects in their own wellbeing. Consider, for example, the case of neighbouring districts with very different level of income within the same city where the richer district can fear the social conflicts in the poorer district. The same could be applied for countries where the richer ones fear the arrival of immigrants of the poorer ones. Likewise, agents could seek improving the worst position when their own reputation or profits depends on that of others. This would also be the case for decisions in research groups that have to obtain financial support that depends on the reputation of both each individual and the group as a whole.

In this paper we will analyse the role of Rawlsian preferences for a wide class of strategic situations which includes Cournot oligopolies (Cournot, 1838) and the problem of the commons (Hardin, 1968), among others.

We first recall some results on the links between the equilibria of vector-valued games and the equilibria of scalar weighted maxmin games. We then address the interesting case in which the agents show social preferences that are represented as Rawlsian functions. We provide results which permit the description of different types of agents depending on their attitude with respect to the gains of the others, that is, in terms of the parameters of the maxmin representation. Given the vector-valued game that describes the other-regarding strategic model, and the corresponding weights assigned by the agents, we establish the associated scalar Rawlsian game. We also provide necessary conditions and sufficient conditions on the weights of the agents which enable the identification of equilibria relying upon agents social behaviour. The key result which allows us to perform a complete analysis of the equilibria for these strategic models is the description of the best response function of the agents in terms of best responses of the classical model and of the parameters that represent the importance that each agent attaches to the outcome of the others.

The remainder of the paper is organized as follows. Section 2 includes some basic concepts and previous results on the weighted maxmin approach for the equilibria of vector-valued games. Different types of agents are defined and characterized in Section 3. Section 4 is devoted to analyse a strategic model with Rawlsian preferences. The interesting case of two agents is specifically considered. Finally, Section 5 contains some concluding remarks and the proofs are included in Appendix.