It’s more subtle than that. Utility functions, by design, encode preferences that are consistent over lotteries (immune to Allais paradox), not just pure outcomes.
Or equivalently, they make you say not only that you prefer pure outcome A to pure outcome B, but also by how much. That “by how much” must obey some constraints motivated by probability theory, and the simplest way to summarize them is to say each outcome has a numeric utility.
It’s more subtle than that. Utility functions, by design, encode preferences that are consistent over lotteries (immune to Allais paradox), not just pure outcomes.
Or equivalently, they make you say not only that you prefer pure outcome A to pure outcome B, but also by how much. That “by how much” must obey some constraints motivated by probability theory, and the simplest way to summarize them is to say each outcome has a numeric utility.