Often the utility of a decision alternative is not a deterministic value but is subject to uncertainty. For example, in advance, it is unknown whether an insured event will occur. In order to identify the optimal decision alternative, the expected utility of all possible decision alternatives is evaluated. Formally, the expected utility is the expected value of the utility function; i.e., $\operatorname{E}\left[u(\mathbf{y})\right]$, where the set of attributes $\mathbf{y}$ is composed of random variables.
Illustrative example: You own an expensive bike that is worth 4000€. You consider the following options:
- (A) Buy a good bike lock for 100€. You estimate the probability of bike theft within the next year to 0.5%.
- (B) You buy insurance that covers theft for 110€ per year – additional to (A), as the insurance requires you to have a good bike lock.
- (C) You do not invest in a bike lock. For this scenario, you estimate the probability of bike theft to 5%. Note that the value refers to the probability of theft of the unprotected bike.
In order to identify the optimal decision, the expected utility of each decision alternative is evaluated. In this case, utility is assumed to be proportional to negative cost.
- For (A), the expected utility is $-100€ - 0.5\%\cdot 4000€ = -120€$.
- For (B), the expected utility is $-100€ - 110€ = -210€$.
- For (C), the expected utility is $5\%\cdot -4000€ = -200€$.
Thus, for a timeframe of one year, decision alternative (A) is optimal.
The one-year timeframe is not very realistic, as in reality, you will probably use the bike much longer. However, for longer timeframes the decrease in the bike's value over time needs to be considered – which complicates the example at hand. ■
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