Costs and Multiple Hypotheses

The utilities in my model of epistemic pursuit worthiness do not take into account the costs of pursuing the hypothesis. As mentioned in Section 2.5.3, one reason for this is that it is not clear whether the costs, in terms of the resources and time available to do scientific research, are commensurable with the epistemic value of learning more about the world. One attractive feature of the Simple Model (and the extensions discussed in Section 2.7.1) is that changes in utility—e.g. the added utility of accepting a true theory we had previously suspended judgment about—can be interpreted as representing how much we would have learned about the world. This is feature that allows it to represent the factor McKaughan (2008) calls informativeness, which is intuitively relevant to decisions about pursuit. This factor would be obscured if u(acc(h), h) instead represented some combination of the value of accepting the truth and the costs of finding this out.

One way to incorporate costs while avoiding this problem would be to assume that the total utility of a given outcome could be written as a linear combination of the epistemic value of the situation and its costs. If we let c(ph) be the costs of pursuing h we could, for example, replace each utility in equation (3) by one of the form: u[ejj(h), ti(h),

ph] = u(ejj(h), tj(h)) + c(ph). If we assume that the costs of pursuit are independent of the epistemic state reached, we can rewrite (3) as:

(9) EU(𝑝_ℎ) = ∑ (Pr(𝑡_{𝑖 𝑖}(ℎ)) × ∑ [u (𝑒𝑠_{𝑗 𝑗}, 𝑡_𝑖(ℎ)) × Pr (𝑒𝑠_𝑗 | 𝑡_𝑖(ℎ), 𝑝_ℎ)]) + ∑ Pr(𝑡𝑖 _𝑖(ℎ))× c(𝑝ℎ)

If we furthermore assume that the costs are independent of the truth value of h, this reduces to:

(10) EU(𝑝_ℎ) = ∑ (Pr(𝑡𝑖 𝑖(ℎ)) × ∑ [u (𝑒𝑠𝑗 𝑗, 𝑡𝑖(ℎ)) × Pr (𝑒𝑠𝑗 | 𝑡𝑖(ℎ), 𝑝ℎ)])

+𝑐(𝑝_ℎ)

Here, c(ph) will typically be a negative number although we could, in principle, imagine cases where a research project is expected to be so profitable that we want to say that pursuing h makes a net contribution to the available resources.

A different way to incorporate costs into a comparative notion of epistemic pursuit worthiness, which does not assume commensurability of epistemic value and costs, is to define the pursuit worthiness of a hypothesis in terms of the expected utility gain per unit of resources it would cost to pursue it. If c(ph) represents the total costs of pursuing h, we could say that h1 is more pursuit worthy than h2 if, and only if, EU(ph1)/c(ph1) > EU(ph2)/c(ph2). However, this has the undesirable consequence that we can no longer translate freely between expected utility, expected change in utility and pursuit worthiness. Since c(ph) is not the same for all hypotheses, it is not the case that EU(ph1)/c(ph1) > EU(ph2)/c(ph2) if and only if ΔEU(ph1)/c(ph1) > ΔEU(ph2)/c(ph2). For example, for hypotheses where ΔEU(ph) is negative, higher costs increase ΔEU(ph)/c(ph) and would thus increase pursuit worthiness if we define it in terms of this quantity. By contrast, higher costs always reduce EU(ph)/c(ph).53

A better approach may be to bring in costs as an external constraint on the decision problem. Here, the decision problem facing the agent is how to best spend a given amount of resources, including time and manpower. The agent then needs to consider the different

53 _{I here assume that both the costs and the utilities are expressed as positive quantities. For this reason, this}

quantity can also not be applied in any plausible way to cases where pursuing h would make a net contribution of resources.

sets of hypotheses that could be pursued for that amount of resources and choose the set

which has the highest expected (epistemic) utility.

This raises the question of how to represent the pursuit of multiple hypotheses in the framework. The natural way to do this is to include all possible combinations of truth- values of the relevant hypotheses in the background states, and to let the consequences include each possible assignment of epistemic states to the hypotheses. To illustrate this in terms of the Simple Model, this would give us the following expression for the expected utility of pursuing the set of two hypotheses H = {h1, h2}:

(11) EU(𝑝_𝐻) = Pr(ℎ₁&ℎ₂) × ∑ [u(𝑒𝑠_{𝑖 𝑖}(𝐻), ℎ₁&ℎ₂) × Pr(𝑒𝑠_𝑖(𝐻)| ℎ₁&ℎ₂, 𝑝_𝐻)] + Pr(ℎ₁&¬ℎ₂) × ∑ [u(𝑒𝑠𝑖 𝑖(𝐻), ℎ1&¬ℎ2) × Pr(𝑒𝑠𝑖(𝐻)| ℎ1&¬ℎ2, 𝑝𝐻)]

+ Pr(¬ℎ1&ℎ2) × ∑ [u(𝑒𝑠𝑖 𝑖(𝐻), ¬ℎ1&ℎ2) × Pr(𝑒𝑠𝑖(𝐻)| ¬ℎ1&ℎ2, 𝑝𝐻)]

+ Pr(¬ℎ₁&¬ℎ₂) × ∑ [u(𝑒𝑠_{𝑖 𝑖}(𝐻), ¬ℎ₁&¬ℎ₂) × Pr(𝑒𝑠_𝑖(𝐻)| ¬ℎ₁&¬ℎ₂, 𝑝_𝐻)]

Here, the esi(H) cover the six different ways to assign the three epistemic attitudes to the two hypotheses (accepting both, accepting h1 and rejecting h2, etc.). Thus, we take into account twenty-four different outcomes. Notice that the utilities here need not be equal to the sum of their constituents. For instance, we do not assume that u(acc(h1)&acc(h2),

h1&h2) = u(acc(h1), h1) + u(acc(h2), h2). There may be synergy effects between the two hypotheses, such that learning either would not be particularly interesting but where it would be very interesting if we knew that both were the case. The analogous point holds for the conditional probabilities: we allow for the possibility that pursuing the two hypotheses together may either increase or decrease the probability of obtaining (say) reliable evidence for either. For instance, we do not assume that Pr(acc(h1)&acc(h2) |

This way of incorporating costs into the model allows for the costs to be relevant to whether a hypothesis is pursuit worthy in the absolute sense. A hypothesis is absolutely pursuit worthy for a given agent if, and only if, it is part of the set of hypotheses H which maximises EU(pH), while staying within the constraints imposed by the resources available to the agent. It does not, however, provide a way to make costs of pursuit relevant to comparative pursuit worthiness of hypotheses.

How to best incorporate costs into a comparative definition of pursuit worthiness within this model will not be crucial in this thesis. For comparative pursuit worthiness, all I will require for my purposes is that hypotheses which are costlier to pursue are less pursuit worthy, all things being equal.