"Why should we be skeptical of my simplified puppies/kittens example?"

Great post. So the utility function assumes there is no value in voting your preference. There is value assigned to the payout and the outcome, but what about the act of voting your conscious. I'd propose that in addition to your vi (value if your preference wins) and p (payout), there is additional value to your voter or "agent" in simply having voted their preference. This is why people vote for third party candidates in US presidential races, right? If we introduce another variable, say "ci" for how much the voter values voting there conscious, I think you potentially end up with multiple Nash Equilibrium, particularly where ci is greater than p.