During a family dinner today, we played a game, more like a question really. It went like this:
Imagine you (and everyone else on Earth) have to pick either a red or a blue button.
If more than half the people pick the blue button, everyone survives.
If not, only those who picked the red button survive.
I believe the people who brought up this game remembered it incorrectly. Clearly, the game was meant to present a tricky choice, but this version has an obvious dominant strategy - choosing the red button.
For this game to be a good dinner-table topic, we need a more challenging scenario. So, I quickly came up with a modification, with a weak Nash equilibrium:
Imagine you (and everyone else on Earth) have to pick either a red or a blue button.
If more than half the people pick the blue button, those who picked the red button survive.
If not, everyone dies.
Here’s a slightly more complex version if you want to ruin the family dinner:
Imagine you (and everyone else on Earth) have to pick either a red or a blue button.
If more than half the people pick the blue button, all those who picked the red button survive, along with $\alpha = 0.6$ of the blue button pickers.
If not, everyone dies.
Pretty tricky, isn’t it? If there’s a finite number of people, having exactly half of the population press the blue button while the rest press the red button would indeed create a unique correlated pure strict Nash equilibrium. But what happens when we expand the strategy space to include mixed strategies? I believe it will no longer be unique; there will likely be asymmetric results, and we might end up with numerous potential Nash equilibriums.
Let me know what discussions it sparks if you really are going to use this in your next family dinner, lol.