How could a cat even give me the money in the first place? I mean physically holding it? Can a cat write cheques?
If I don't lose anything for saying a simple yes I would definitely let them win. And I believe most people would do that too. So yeah, I'd wager. But there's the chance of this. Not anyone's fault. So I'd wager only an amount I'm ok with losing and would be useful if I do win, like I don't mind losing a month's salary, but an extra month's salary can be really really useful for me.
Wage more than 1000x of my income if only the cat can tell that random person that "my name is Liz and I'll give you half of my money if you say yes if you can contact me at NUF.".
...if I were the random person, I would say no. Anyway, I don't like gambling, so I wouldn't wager anything.
Are you picking all these polls out from some documentary?! There was literally an old Brit gameshow on this topic, loosely based on Game Theory. Golden Balls (2007) ... dumb name I know. Two contestants would choose the ball for "Split" or "Steal" for a pot of money. If they both chose Split they divvy the money. If one person chooses Steal he gets all of it. If they both chose Steal they get nothing. Even though Game Theory tells you that the most logical answer is to split/pay it forward/do good (i.e. random person should rightfully agree LET YOU WIN) .... the fear of being screwed over FAR outweighed the desire to do good (i.e. the random person will NOT LET YOU WIN). In Golden Balls, it was statistically more likely that someone would screw you over when the pot gets big enough. So the correct answer is that you should bet small or nothing. Here's the podcast on RadioLab. Won't spoil the ending on what one genius contestant did. Would be great if you can achieve that. https://www.wnycstudios.org/podcasts/radiolab/segments/golden-rule