summarizes the choice of measures for the simulation results, as well as the methodology and scenarios used. The following criteria are often identified as desirable in the literature on choosing a measure:<sup>48</sup>

- **Simple** . The measure must be understandable and easy to describe.

- **Coherent** . It must conform to a common-sense notion of what is being measured.

- **Motivated** . It must fit the purpose for which it is being developed.

- **Rigorous** . It must be technically solid.

- **Implementable** . It must be operationally viable.

- **Replicable** . It must be easily replicable.

- **Incentive compatible** . It must respect country incentives, meaning that a country will not be penalized when it improves the learning of its student population.

The choice among these characteristics is not always easy. Trade-offs and compromises might be necessary based on the main purpose of the measure. A headcount measure has many of the characteristics, including simplicity, but often at the cost of coherence and incentive compatibility. A measure might be incoherent if it does not conform to a common-sense notion of what is being measured. For example, it might seem plausible to argue that children or educational systems with lower scores among the poor are worse off, other things equal. Incentive compatibility might fail when an index is already too high or when the index misses critical progress at foundational levels of learning. When coherence and incentive compatibility are critical, measures that respect both the monotonicity axiom and the monotonicity sensitivity axiom will be in order. Alternative measures met those criteria at the expense of a very important one, simplicity. This is no trivial choice, and in some cases might not be warranted if only the focus axiom is needed.

For this simulation, initial conditions are radically different, as some regions have much higher levels of learning poverty than others. The simulation needs both simplicity and the ability to generate the correct incentives so different stakeholders will act on the results. Hence, simulation results will be presented based on three measures—the headcount rate, which is suitable for countries and regions with lower levels of learning poverty, and the learning poverty gap and learning poverty severity, which are the most relevant for Africa and the low-income countries aggregate.

However, one important empirical question remains. Are all these complementary different measures empirically relevant? The extend in which will depend if we find that countries with same learning poverty level have different learning poverty gaps (figure 3, panel A), or do countries with the same learning poverty gap have different learning poverty severity (figure 3, panel B). Figure 3 illustrates those points using the latest available data from 99 countries in the learning poverty database for which learning poverty gap and learning poverty severity indicators are available.<sup>49</sup> The figure shows a wide range of

> 48 See for example Alkire et al. (2015) and Cameron et al. (2019).

> 49 All learning assessments used are anchored in a standard deviation of 100 points, this should be sufficient to have the FGTclass of measures to be minimally comparable. Of course, within-country temporal comparisons, assuming temporal comparability of the assessments, are the ideal case. All gap measures are relative to the test-specific MPL. One interesting aspect is that once the gap conversion is made, the measure becomes test independent, and can be presented independently of any scale. One important assumption however, which is shared with the global poverty monitoring, when doing cross-country comparisons, is that the learning (income) marginal sensitivity of the cardinal variable is the same. Meaning, improving one learning point is equally hard (or equally well captured) across all assessments (or different measures of income and consumption).

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