Independence of irrelevant alternatives (IIA) says that adding another option shouldn't change a voter's (or the election result's) relative preferences among all other options.
"The table's waiter comes by and says their desert options are blueberry pie and vanilla ice cream. You order the pie. Then the waiter comes back and says, we also have chocolate cake. 'In that case, I'll have the ice cream!'"
I don't see what this has to do with intensity?
Arrow's theorem does assume that intensity doesn't matter, but this isn't IIA but is rather the unnamed predicate that voters express their preferences by ranking candidates with no ties allowed, rather than an approval voting scheme, or a "on a scale of 1-10" scheme.
Your example only looks irrational when one looks only at the overall preference list and furthermore thinks that the only important result is the winner. i.e. when looking at elections as an extended opinion poll and applying a purely mathematical logic ignoring politics, just like the Arrow Theorem does.
If a voter went for chocolate cake despite polls showing that this may cause blueberry pie to lose the election, maybe the voter does not care that much for blueberry pie over vanilla ice cream, or the voter really really cares for cake far more than the other choice - both of these are intensity issues and the election method de facto taking them into account is not irrational.
Maybe the cake was the only nut free option, and some voters deciding for the cake despite not having nut allergy themselves and despite losing convince the restaurant to offer more nut free foods later.
99% of the 'irrelevant choice' cases actually have some political logic like the above behind them which Arrow theorem just blithely ignores. The 1% are usually some attempt to mislead the voters (say by running a hopeless candidate with a similar name to another in order to siphon off votes), and should be handled by the electoral commission.
Independence of irrelevant alternatives (IIA) says that adding another option shouldn't change a voter's (or the election result's) relative preferences among all other options.
"The table's waiter comes by and says their desert options are blueberry pie and vanilla ice cream. You order the pie. Then the waiter comes back and says, we also have chocolate cake. 'In that case, I'll have the ice cream!'"
I don't see what this has to do with intensity?
Arrow's theorem does assume that intensity doesn't matter, but this isn't IIA but is rather the unnamed predicate that voters express their preferences by ranking candidates with no ties allowed, rather than an approval voting scheme, or a "on a scale of 1-10" scheme.