there was a cool chapter in my decision theory textbook about how if you have N states of nature with some probability distribution, M possible actions you can take, and an NxM table of utilities for each combination of nature+action, you can quantify the utility of information about the state of nature.
for example. suppose you have a banana on your counter that looks kinda ripe. you have the choice of eating it today or tomorrow. it's either fully ripe and tomorrow it will be overripe, or today it is underripe and tomorrow it will be just right, but you don't know. Though you give it a 50:50 chance.
Let's say you HATE underripe bananas, and give them a utility of -1. If it's just right, that has a utility of 5, if it is overripe, that has a utility of 3. our utility table is:
today 5 -1
tomorrow 3 5
right now, based on expected utility, you should eat the banana tomorrow. eating it today gives you an expected utility of 2, and tomorrow the expected utility is 4.
now suppose an oracle appears in your kitchen, and tells you they know with 100% certainty if the banana is ripe today or tomorrow. However, they tells you that you must perform an unspeakable act to get the information (a little worse than eating an underripe banana, and you calculate it has a utility of -2)
if you know for certain when the banana will go ripe, you are guaranteed to get 5 units of utility. however, you must experience the unspeakable act of -2, so the true utility is 3, which is less than if you just wait till tomorrow! so even though you will always get to experience the bliss of a ripe banana, you're no better off.
you tell the oracle you think that's a bad deal. they roll their eyes and mutter something derogatory about decision theorists. then, they ask you what price you think is fair.
From our earlier calculation, if we know when the banana will be ripe, we are guaranteed 5 units of utility. if we don't, we have to rely on expected utility to give us about 4 units of utility. therefore, the *knowledge* of when the banana will be ripe has 1 unit of utility!
you hatch a scheme in your head. obviously you don't want to say that a fair price is -1 utility, since then you break even and nothing is gained. instead you say to the oracle:
"I'll eat half of a different, underripe banana"
that should be -0.5 units of utility for you. and to the oracle, that sounds like ironic justice. they agree, and you've just gained 0.5 units of utility from the deal!
@SuricrasiaOnline ...I've never thought about it like this but I'm actual none surprised that information theory links up to game theory through expected utility
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