Barros et al. (2009, 2011). This index is defined over a different set of advantages (which, confusingly, are sometimes referred to as ‘basic opportunities’), namely access to certain basic services, such as piped water, electricity or sanitation.

For a particular service _l_ the HOI, denoted by **_H_**<sup>**_l_**</sup> , is simply the average access rate in the population, penalized by the degree of dissimilarity in that coverage of that particular service across types. It is clearly analogous to the Sen welfare function, where mean outcomes are adjusted by one minus a measure of inequality. Sometimes an aggregate index is calculated as an average of **_H_**<sup>**_l_**</sup> across a number of different services.<sup>28</sup> Various versions of the HOI have now been computed for at least 39 countries, and the index is popular at the World Bank.

The motivation behind the HOI, as initially proposed by Barros et al. (2009), was to measure the extent to which children in various developing countries have access to “basic opportunities”. Although the authors do not motivate it this way, one could view the index as an example of the ex-ante approach applied to a multidimensional advantage space, with each dimension corresponding to access to a particular service – such as water or schooling – and the valuation of the opportunity set of each type being given by the coverage of the service in that type. The particular inequality index applied to that smoothed distribution of probabilities is the dissimilarity index, commonly used by sociologists. While the dissimilarity index might therefore be seen as a measure of inequality of opportunity, the HOI itself is closer in nature to a “development index” (like the Human Development Index – HDI - propounded by the United Nations Development Program), that incorporates an aversion to inequality of opportunity.<sup>29</sup> Indeed, Brunori et al. (2013) show that the HOI and HDI are very highly correlated over the current sample of countries for which they are available.

# **7. Conclusions**

What is, then, the state of the economic literature on individual responsibility and equality of opportunity? In one sense, it is possible to claim that much has been achieved in the twenty years since the pioneering works of Roemer, van de Gaer and Fleurbaey. After all, changing the space in which judgments of fairness are made across the social sciences is no small task, and incorporating respect for individual responsibility into an egalitarian framework no mean feat. The length of our reference list –as well as the fact that this paper is but one of three surveys of this literature being readied at the time of writing - are themselves indications of some impact, at least

> 28 However, see Ravallion (2011) on the potential pitfalls of such arbitrary aggregate indices or, as he calls them, “mashup indices” of development.

> 29 A possible caveat with viewing the dissimilarity index within the HOI as a measure of inequality of opportunity is that the index is typically calculated “for children”. This justifies the use of certain variables - like geographic location or education of the adults in the household - as circumstances, which are clearly in the realm of choices for adults. The argument is that the index applies to children, and these are circumstances from their perspective.  But this then raises the issue of the age of responsibility, and whether or not all inequalities in access to services for children below a certain age should not be considered inequality of opportunity. Under that view, unequal access to water or sanitation among five-year olds within the same type (i.e. sharing identical observed circumstances) should also be counted as inequality of opportunity.

**_30_**