The Logic Problem
AI's (at least, the ones we've developed) and robots sometimes do act irrationally. Sure, people can justify their actions and claim those are logical, but psychotic individuals can act logically according with what they believe.
It all has to do with whatever values / axioms / mandates / programming the AI has. For your case study, this makes sense ("is logical") if:
- The police-bot is mandated/programmed to save lives and aid those who are attempting to save lives. Thus, it saves lives of ants by stopping traffic. Note that this mandate never said which lives to save.
- The police-bot is also programmed to "spin" any public events it may be involved in. Thus the selfie being taken. It'll likely be posted to facebook with the caption "look at this police bot saving lives!" or "I'm here serving the public good at ROADNAME1 and ROADNAME2."
That's rational, right? No? Perhaps we can define irrationality (of the non-math sort) as:
Attempting actions which are inconsistent with your beliefs, fundamental axioms, or purpose.
You could develop a statistic on how often such actions occur, both for a group at large and individuals. This could develop into some metric (perhaps comparing a person's logic number to the "average" logic number), so we know how logical/illogical someone is.
Does That Help Us Get Along?
There is also a lot of police who can think irrationally like us and people don't like them, either. Why should an irrational robot be any better? If anything, an irrational robot is worse because it has the potential to do so many more things wrong in the same amount of time.
It certainly wouldn't help me get along with robots.