"Exterminate the brutes!"
-- Kurtz, Heart of Darkness
one, really:
First, pull out machine gun and yell "say hello to my little friend" and point at them.
Second, ask them, quietly, to let go of all this pretense bulls**t, and tell you who's who. If they don't they get a bullet in the head.
Watch them tell you who's who.
Nazaire:
one, really:
First, pull out machine gun and yell "say hello to my little friend" and point at them.
Second, ask them, quietly, to let go of all this pretense bulls**t, and tell you who's who. If they don't they get a bullet in the head.
Watch them tell you who's who.
How would you know who was lying and who was telling the truth? Who would you shoot? Seems like there's only one equilibrium in your scenario in which they all die and you never find out the truth.
Matthew:
^^ I don't think you understand what "equilibrium" means.
Let there be 4 players in the game: the Knight, the Knave, the Crazy, and you. Now let each of the three villagers derive at least some positive utility from playing their type (over telling the truth), except for the Knight, who always tells the truth. Otherwise, they would all probably just tell the truth.
Now suppose your optimal strategy is don't kill if you know they are telling the truth, kill otherwise. This assumes that you derive at least some utility from killing when you don't know they are telling the truth. Let's call this the frustration bonus.
Now we know that the knight only plays one strategy, which is to tell the truth all the time.
The Knave should tell the truth if the Crazy tells the truth, but lie if Crazy plays his random answer strategy.
Similarly, the Crazy should tell the truth if the Knave tells the truth, but choose his random answer strategy if the Knave lies.
Now suppose that you ask the Knight who everybody is first. The Knight will tell the truth.
Next, you will ask either the Knave or the Crazy. If you ask the Knave, the Knave will probably lie given that he has no credible commitment from the Crazy that the Crazy will tell the truth. If you ask the Crazy, the Crazy will probably choose the random answer strategy for the same reason. So both Knave and Crazy will lie and you will blow their brains out. That is the Nash equilibrium.
Now I suppose there could be other equilibria if
1) We define the utility functions in such a way that death is by far the worst outcome. However, we also have to make an assumption that you can credibly commit to killing everybody, or that your utility functions are known.
2) You ask the Crazy second and the crazy happens to answer correctly on your 1 (or 3 depending on how you ask) questions and you can credibly commit to killing and the death penalty loss is sufficiently high.
Zeus:
Harry:
Now let each of the three villagers derive at least some positive utility from playing their type
You should've failed intro game theory.
You should've failed the GREs.
See I can also make blanket statements without actually engaging the substance of the post.
I addressed all these issues, albeit not succinctly.
1. Killing liars is an assumption for sure, but all models are built on assumptions. If you don't kill liars, then basically it's much harder for you to credibly commit to do anything, so there's no point in you having a gun at all.
2. I didn't define the utility for staying alive, but I did mention it in my caveats about the tradeoffs between staying alive and playing their type. The key is really about whether you can credibly commit to doing anything. If you cannot credibly commit to killing them if they lie, then everybody will play their type. If you cannot credibly commit to not killing them if they tell the truth, then everybody will still play their type. If you can credibly commit to both, then it will depend on how much they value life to playing their type. Also, since there are three villagers, there can be varying utility functions with varying weights, and you can also choose to kill a subset of the villagers. Clearly, my model is a simplification that doesn't deal with these moving parts.
3. I used "type" here but it's not really "type" in the traditional game theory sense. Playing your type is just an action here. So replace the word "type" with "kind".
"If you can credibly commit to both, then it will depend on how much they value life to playing their type. "
So in equilibrium, then, they will tell the truth as long as the weight attached to staying alive is large enough relative to the weight of playing their type. Thank you.