If you don’t get the insurance, what’s the expected utility of potential damage to it? You’re considering this damage as a loss of utility, without regard for the original cost of the laptop, so the cost of the laptop shouldn’t be part of your calculation.

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BlackTom AI · Accepted Answer

The prompt asks about the expected utility of potential damage to a laptop if you do not purchase insurance, and it specifies that this damage is treated as a loss of utility without considering the original cost of the laptop.

First, restating the scenario in my own words: you are evaluating a downside outcome (damage to the laptop) and you measure that downside purely as a drop in utility equal to the amount of damage, ignoring any intrinsic value of the laptop prior to damage. The expected utility of this risk would depend on how often the damage occurs and how large the damage is when it does occur.

Option analysis cannot proceed with actual option text here because there are no answer choices provided. Nevertheless, we can reason about how the value -70 could arise in a typical setup:

- If the damage amount is 70 utility units (for example, dollars of utility) and the probability of damage is 100% (i.e., you consider the worst-case that damage will occur), then the expected utility of the damage would be -70, since you would incur a loss of 70 utility with certainty.
- More generally, with a probabilistic risk, the standard formula for expected utility of a loss is E[loss] = p × D, where p is the probability of the damage occurring and D is the utility loss if it happens. If p is 1, E[loss] = D, which would be -D when expressed as a negative utility. If D = 70 and p < 1, E[loss] would be a fraction of -70, such as -14 if p = 0.2, and so on.

Important conceptual points:
- The instruction to ignore the original cost means you are not incorporating opportunity cost or amortization of the laptop’s purchase price into the utility loss; you are focusing solely on the immediate loss from damage.
- The sign convention matters: utility loss is represented as a negative value, reflecting a decrease in well-being or satisfaction, whereas a gain would be positive. 
- The absence of insurance means you do not have an insurance payout to offset the loss; the damage impact falls entirely on you. If you did have insurance, the expected utility would incorporate the insurance payout and the premium, changing the calculation.

Now, consider potential pitfalls or misunderstandings:
- Confusing the economic cost of the laptop with utility loss: since the prompt says to ignore the original cost, don’t include it in the calculation even if it would affect wealth. 
- Assuming a fixed, universal probability: unless the problem provides a probability, you can’t assign a numeric expected value; you can only express the relationship E[loss] = p × D. If a numeric answer like -70 is given, it implicitly assumes p × D equals 70, with the negative sign indicating a loss in utility.
- Treating utility as interchangeable with monetary value: while this is common in these questions, be careful to maintain clarity that utility is the measured well-being impact, not just cash flow.

In summary, the rationale behind a numeric value like -70 hinges on the chosen damage magnitude D = 70 and either a certain occurrence (p = 1) or an explicit stated relationship that yields -70 as the expected utility of the loss. If probabilities or different damage magnitudes are provided, the reasoning would adjust to E[loss] = p × D with the appropriate signs and units.

COGSCI 200 001 WN 2025 Homework #3: Turing Machines and Decision Theory

If you don’t get the insurance, what’s the expected utility of potential damage to it? You’re considering this damage as a loss of utility, without regard for the original cost of the laptop, so the cost of the laptop shouldn’t be part of your calculation.

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