Your graph shows that expected utility tops when p=1. So does it mean that based on your analysis people should always take the bet?
What this is saying is that if everyone other than you always takes the bet, then you should as well. Which is true; if the other 19 people coordinated to always take the bet, and you get swapped in as the last person and your shirt is green, you should take the bet. Because you’re definitely pivotal and there’s a 9⁄10 chance there are 18 greens.
If 19 always take the bet and one never does, the team gets a worse expected utility than if they all always took the bet. Easy to check this.
Another way of thinking about this is that if green people in general take the bet 99% of the time, that’s worse than if they take the bet 100% of the time. So on the margin taking the bet more often is better at some point.
Globally, the best strategy is for no one to take the bet. That’s what 20 UDTs would coordinate on ahead of time.
What this is saying is that if everyone other than you always takes the bet, then you should as well. Which is true; if the other 19 people coordinated to always take the bet, and you get swapped in as the last person and your shirt is green, you should take the bet. Because you’re definitely pivotal and there’s a 9⁄10 chance there are 18 greens.
If 19 always take the bet and one never does, the team gets a worse expected utility than if they all always took the bet. Easy to check this.
Another way of thinking about this is that if green people in general take the bet 99% of the time, that’s worse than if they take the bet 100% of the time. So on the margin taking the bet more often is better at some point.
Globally, the best strategy is for no one to take the bet. That’s what 20 UDTs would coordinate on ahead of time.