# 5-and-10

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One version of the 5-and-10 problem is "I have to decide between \$5 and \$10. Suppose I decide to choose \$5. I know that I'm a money-optimizer, so if I do this, \$5 must be more money than \$10, so this alternative is better. Therefore, I should choose \$5."

version, sometimes known as the heavy ghost problem, is a problem in certain types of UDT-like decision theories, when the fact that a counterfactual is known to be false makes the algorithm implement it.

Specifically, let there be a decision problem which involves the choice between \$5 and \$10, a utility function that values the \$10 more than the \$5, and an algorithm A that reasons something like:

"Look at all proposition of the type '(A decides to do X) implies (Utility=y)', and find the X that maximises y, then do X."

When faced with the above problem, certain types of algorithm can reason:

"The utility of \$10 is greater than the utility of \$5. Therefore I will never decide to choose \$5. Therefore (A decides to do 'choose \$5') is a false statement. Since a false statement implies anything, (A decides to do 'choose \$5') implies (Utility=y) for any, arbitrarily high, value of y. Therefore this is the utility maximising decision, and I should choose \$5."

That is the informal, natural language statement of the problem. Whether the algorithm is actually vulnerable to the 5-and-10 problem depends on the details of what the algorithm is allowed to deduce about itself.