By Erik Angner

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Would anybody fail to act in the same way at nodes #2 and #3? There is a wide class of cases where people do. 5 Multi-stage investment problem beyond recovery at the time when the decision is made. Consider the R&D scenario outlined earlier in this section. 5. Faced with these two problems, many people say they would invest at node #2 but not invest at node #3. Yet, at node #2, the $9M is a sunk cost: it cannot be recovered. Whether or not making the further $1M investment is worth it should not depend on whether you find yourself at node #2 or #3.

Utility is nothing but an index or measure of preference. Given a rational preference relation, you may ask whether it is always possible to find a utility function that represents it. When the set of alternatives is finite, the answer is yes. The question is answered by means of a so-called representation theorem. If the set of alternatives is finite, then >,:: is a ratio11a/ preference relation just in case there exists a utility f11nctio11 representing >,:=. 30 Representation theorem D Proof.

Y>-x (fora/Ix, y) (iii) 11ot x >- x (for nil x) Proof (i) Suppose that x >- y & y >- :. 13 tells us that we need to show that x >,,, : and that it is not the case that:>- x. 15. The second part goes ns follows: suppose for a proof by contradiction that:>,:, x. From the first assumption and the definition of strict preference, it follows that x >,:, y. 5, it follows that: ;;: y. : >,:, y. : >,:, x. (ii) Begin by assuming x >- y. Then, for a proof by contradiction, assume that y >- x. 13 implies that x >,:, y.