Project 3: Inequality Sensitive Optimal Treatment Assignment

Below I very briefly summarize at a high level what these different decision theorists (or evaluators) do and how they differ.

• The Bayesian evaluator selects the treatment that leads to the highest expected welfare, where the expectation is taken with respect to a probability measure (called the prior) over the set of states.

• The maximin evaluator avoids the specification of priors and defines the security welfare of a treatment rule to be given by the worst welfare across states that the rule could produce. The evaluator then selects the rule with the greatest security welfare.

• The minimax regret evaluator also avoids the specification of priors and defines the regret of a treatment rule at a state to be the difference between the welfare achieved by the best treatment given knowledge of the state and the welfare achieved by the treatment rule at the state. The worst regret of a treatment rule is the largest regret the treatment rule achieves across states. The evaluator then selects the rule with the smallest worst regret.

Let's illustrate how these decision theories work with a variant on the main example in Social Preferences Under Uncertainty and Ambiguity.

Assume that the evaluator represents the uncertainty about what outcome arises with a given treatment through a set of two states of the word. Treatment \(a\) is associated with outcome \(y^{s}(a)\) in state \(s\) and a prospect, \(y(a)\), collects the possible outcomes induced by treatment \(a\) across the two states. Similarly for treatment \(b\) and associated prospect \(y(b)\).

Our example is depicted in the table below:

The interpretation is as follows: treatment \(a\) yields outcome \(4.93\) in state 1 and \(2\) in state 2, whereas treatment \(b\) yields outcome \(6.21\) in state 1 and \(1\) in state 2. One can think about thhose numbers as providing a specific summary of how different individuals fare in each state, and where they come from need not concern us here, although I will have more to say about that question below. Let's for now take these numbers as given.

A treatment rule assigns individuals to treatments and in the example I focus on treatment rules that assign all individuals to the same treatment.

Let's first investigate how a Bayesian evaluator would choose among treatments. In addition to the state space described above, the Bayesian evaluator would assign probabilities to the states. Assume that our evaluator assigns probability \(\frac{2}{3}\) to state \(1\) and probability \(\frac{1}{3}\) to state \(2\). Then the expected outcome would be \(\frac{2}{3} 4.93 + \frac{1}{3} 2 = 3.95\) for treatment \(a\) and \(\frac{2}{3} 6.2 + \frac{1}{3} 1 = 4.48\) for treatment \(b\). On the basis of these expected outcomes, the Bayesian evaluator would use a treatment rule that assign everyone to treatment \(b\).

Now let's investigate how a maximin evaluator would choose among treatments. The security welfare of the treatment rule that assigns everyone to treatment \(a\) is \(2\), whereas the security welfare of the treatment rule that assigns everyone to treatment \(b\) is \(1\). Therefore, the maximin evaluator would use a treatment rule that assign everyone to treatment \(a\).

Finally, consider how a minimax regret evaluator would choose among treatments. The treatment rule that assigns everyone to treatment \(a\) exhibits a regret of \(6.21 - 4.93 = 1.28\) in state \(1\) and a regret of \(0\) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(a\) is then \(1.28\) (the maximum between \(1.28\) and \(0\)). On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0\) in state \(1\) and a regret of \(2 - 1 = 1 \) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(b\) is then \(1\) (the maximum between \(1\) and \(0\)). This means that the treatment rule with the smallest worst regret is the one that assigns everyone to treatment \(b\) (since \(1 < 1.28\)), and that is the rule that a minimax regret evaluator would choose.

First, notice that treatment \(b\) is ranked above treatment \(a\) in state 1. One can see this by looking at the \(\mathrm{E}[\mathrm{log} \, y]\) row in the table above, since \(1.83 > 1.60\). Next, notice that treatment \(a\) is ranked above treatment \(b\) in state 2: looking at the \(\mathrm{E}[\mathrm{log} \, y]\) row in the table above, we obtain that \(0.69 > 0\). Therefore, when a sufficiently large amount of data reveals the true state:

A Bayesian evaluator would put probability 1 on the true state and select a treatment rule that assigns everyone to treatment \(b\) in state 1 (since \(1.83 > 1.60\)) and assigns everyone to treatment \(a\) in state 2 (since \(0.69 > 0\)).

When the known state is state 1, a maximin evaluator would trivially assing a security welfare of \(1.60\) and \(1.83\) to assigning everyone to treatments \(a\) and \(b\), respectively. In turn, when the known state is state 2, the maximin evaluator would trivially assing a security welfare of \(0.69\) and \(0\) to assigning everyone to treatments \(a\) and \(b\), respectively. Consequently, the maximin evaluator would select a treatment rule that assigns everyone to treatment \(b\) in state 1 (since \(1.83 > 1.60\)) and assigns everyone to treatment \(a\) in state 2 (since \(0.69 > 0\)).

A minimax regret evaluator would exhibit, in state 1, a regret of \(1.83 - 1.60 = 0.23\) to the treatment rule that assigns everyone to treatment \(a\), and that would trivially be the worst regret, since the only state considered possible at state 1 is precisely state 1. On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0\) in state \(1\), and that would trivially be the worst regret for the same reason given above. A minimax regret evaluator would exhibit, in state 2, a regret of \(0\) to the treatment rule that assigns everyone to treatment \(a\), and that would trivially be the worst regret, since the only state considered possible at state 2 is precisely state 2. On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0.69 - 0 = 0.69 \) in state \(2\), and that would trivially be the worst regret for the same reason given above. Therefore, if the state is known, and happens to be state 1, the treatment rule that minimizes worst regret is the treatment rule that assigns everybody to treatment \(b\). Similarly, if the state is known, and happens to be state 2, the treatment rule that minimizes worst regret is the treatment rule that assigns everybody to treatment \(a\).

In other words, when the state is known, all three decision theories execute the same comparison of known distributions of the outcome variable across treatment rules, and therefore arrive at the same optimal treatment rule.

As we discussed in Social Preferences Under Uncertainty and Ambiguity, the egalitarian equivalent representation of the social preferences discussed above is given by the geometric mean of outcomes, shown in the row labeled \(\mathrm{GM}[y]\) in Table 2 below:

Now let's consider how the information in Tables 1 and 2 help us see how the choice of representation of the social preferences under certainty is irrelevant under point identification.

First, notice that the treatment rule that assigns everyone to treatment \(b\) is ranked above the treatment rule that assigns everyone to treatment \(a\) in state 1 according to both the \(\mathrm{E}[\mathrm{log} \, y]\) and the \(\mathrm{GM}[y]\) measures: looking at the corresponding rows from Tables 1 and 2 above, in the state 1 columns, we obtain that \(1.83 > 1.60\) and \(6.21 > 4.93\). Notice also that the treatment rule that assigns everyone to treatment \(a\) is ranked above the treatment rule that assigns everyone to treatment \(b\) in state 2 according to both the \(\mathrm{E}[\mathrm{log} \, y]\) and the \(\mathrm{GM}[y]\) measures: again, looking at the corresponding rows from Tables 1 and 2 above, in the state 2 columns, we obtain that \(0.69 > 0\) and \(2 > 1\). Therefore, both of these measures rank the prospects in the same way, on a state by state basis. This is simply due to the fact that these representations can be obtained from one another via a monotone transformation. In other words, when we have reached certainty or near certainty about the true state of the world, it does not matter which representation we use. They all give the same, correct answer.

Assume that, even after gathering very large amounts of data, the state space still consists of states 1 and 2. Our evaluation is summarized in our expanded version of Table 2, as seen below:

Let's first investigate how a Bayesian evaluator would choose among treatments. If the evaluator assigns probability \(\frac{2}{3}\) to state \(1\) and probability \(\frac{1}{3}\) to state \(2\), the expected outcome would be \(3.95\) for treatment \(a\) and \(4.48\) for treatment \(b\). On the basis of these expected outcomes, the Bayesian evaluator would use a treatment rule that assign everyone to treatment \(b\).

Now let's investigate how a maximin evaluator would choose among treatments. The security welfare of the treatment rule that assigns everyone to treatment \(a\) is \(2\), whereas the security welfare of the treatment rule that assigns everyone to treatment \(b\) is \(1\). Therefore, the maximin evaluator would use a treatment rule that assign everyone to treatment \(a\).

Finally, consider how a minimax regret evaluator would choose among treatments. The treatment rule that assigns everyone to treatment \(a\) exhibits a regret of \(6.21 - 4.93 = 1.28\) in state \(1\) and a regret of \(0\) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(a\) is then \(1.28\). On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0\) in state \(1\) and a regret of \(2 - 1 = 1 \) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(b\) is then \(1\). This means that the treatment rule with the smallest worst regret is the one that assigns everyone to treatment \(b\), and that is the rule that a minimax regret evaluator would choose.

In this example, the Bayesian and the minimax regret evaluators both assign everyone to treatment \(b\), whereas the maximin evaluator would assign everyone to treatment \(a\).

To provide further clarity on extent of their agreement or disagreement of different kinds of evaluators among themselves, assume now that the Bayesian evaluator assigns probability \(\frac{1}{3}\) to state \(1\) and probability \(\frac{2}{3}\) to state \(2\), the expected outcome would be \(2.98\) for treatment \(a\) and \(2.74\) for treatment \(b\). Were this to be the case, the Bayesian and the maximin evaluators would both assign everyone to treatment \(a\), whereas the minimax regret evaluator would assign everyone to treatment \(b\).

The example therefore shows how all three of these decision theories arrive at potentially different optimal treatment rules under partial identification.

The first example in Social Preferences under Uncertainty and Ambiguity made the case for a Bayesian evaluator that assigns probability \(\frac{2}{3}\) to state \(1\) and probability \(\frac{1}{3}\) to state \(2\). We had two representations of the evaluator's social preference over known outcome distributions. These representations are equivalent in the sense that they both correctly represent the social preference over known income distributions. However, once we add uncertainty about what outcome distribution arises with a given treatment, their recommendations as to which is the preferred treatment differ.

Let's review the example again by examining an expanded version of Table 1:

From the examination of the expanded Table 1 we learn that, if the evaluator computes, for each state \(s\), the magnitude \({\frac{1}{3}}(\ln y_1^s + \ln y_2^s + \ln y_3^s)\), the Bayesian evaluator would assign everyone to treatment \(a\), since the expected value of these (average) log measures across states is \(1.29\) for treatment \(a\) and \(1.22\) for treatment \(b\).

On the other hand, consider the expanded Table 2 again. This table summarizes the performance of the two treatments for each state using egalitarian equivalent measures.

From the examination of the expanded Table 2 we learn that, if the evaluator computes, for each state \(s\), the magnitude \((y_1^s y_2^s y_3^s)^{\frac{1}{3}}\), the Bayesian evaluator would assign everyone to treatment \(b\), since the expected value of these geometric mean measures across states is then \(3.95\) for treatment \(a\) and \(4.48\) for treatment \(b\).

This shows that the choice of representation of the social preferences under certainty matters under partial identification when solving a Bayesian optimal treatment assignment problem.

Let's now see whether the optimal treatment assignment for the maximin evaluator depends on the choice of representation of the evaluator's social preferences under certainty.

Table 1 reveals that the security welfare of the treatment rule that assigns everyone to treatment \(a\) is \(0.69\), whereas the security welfare of the treatment rule that assigns everyone to treatment \(b\) is \(0\). Therefore, the maximin evaluator would use a treatment rule that assign everyone to treatment \(a\).

Similarly, Table 2 reveals that the security welfare of the treatment rule that assigns everyone to treatment \(a\) is \(2\), whereas the security welfare of the treatment rule that assigns everyone to treatment \(b\) is \(1\). Therefore, the maximin evaluator would use a treatment rule that assign everyone to treatment \(a\).

We learn from this example that, under both representations of the social preferences of the evaluator under consideration (the \({\frac{1}{3}}(\ln y_1^s + \ln y_2^s + \ln y_3^s)\) representation and the \((y_1^s y_2^s y_3^s)^{\frac{1}{3}}\) representation), the maximin evaluator assigns everyone to treatment \(a\).

Last, let's consider the minimax regret evaluator. Using the information in Table 1, we see that the treatment rule that assigns everyone to treatment \(a\) exhibits a regret of \(1.83 - 1.60 = 0.23\) in state \(1\) and a regret of \(0\) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(a\) is then \(0.23\). On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0\) in state \(1\) and a regret of \(0.69 - 0 = 0.69 \) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(b\) is then \(0.69\). This means that the treatment rule with the smallest worst regret is the one that assigns everyone to treatment \(a\), and that is the rule that a minimax regret evaluator would choose under the \({\frac{1}{3}}(\ln y_1^s + \ln y_2^s + \ln y_3^s)\) representation.

We also saw earlier, using the information in Table 2, that the treatment rule that assigns everyone to treatment \(a\) exhibits a regret of \(6.21 - 4.93 = 1.28\) in state \(1\) and a regret of \(0\) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(a\) is then \(1.28\). On the other hand, the treatment rule that assigns everyone to treatment \(b\) exhibits a regret of \(0\) in state \(1\) and a regret of \(2 - 1 = 1 \) in state \(2\). The worst regret of the treatment rule that assigns everyone to treatment \(b\) is then \(1\). This means that the treatment rule with the smallest worst regret is the one that assigns everyone to treatment \(b\), and that is the rule that a minimax regret evaluator would choose under the \((y_1^s y_2^s y_3^s)^{\frac{1}{3}}\) representation.

This shows that the choice of representation of the social preferences under certainty matters under partial identification when solving a minimax regret optimal treatment assignment problem.

To conclude, the choice of representation of the social preferences under certainty matters under partial identification in this example for the Bayesian and the minimax regret evaluator. It has no effect on the choice of the maximin evaluator.