Thanks for posting, I had fun trying to solve it and I think I learned a few things.
My solution is below (I think this is correct but I’m no expert) but I’ve hidden it in a spoiler in case you’re still wanting to figure it out yourself!
M has preference order of 2a>1>2b. He wants to set r such that if S has b>tv then S will pass the test and then remain loyal. If S has b<tv then M wants S to fail the test and therefore not get the chance to defect in round 2. It is common knowledge that this is what M wants.
Starting by making S’s Payoff for 2b less than that for 1 gives a formula for r:
v−b−r<−b
v<r=v+ϵ for some small positive ϵ
With this value for r, S’s payoff matrix becomes:
1. −b
2a. −vt−ϵ
2b. −b−ϵ
We can see that if vt−ϵ<b then S’s best payoff is obtained by choosing 2a. Otherwise his best payoff is 1. This is exactly what M wants—he has changed S’s payoffs to make S’s preference order the same as his to the greatest extent possible.
Due to M’s preference being common knowledge, S knows that M will choose this value of r and therefore knows what v is before he chooses whether to pass the test (v=r−ϵ) and can choose between the three options simultaneously.
This is an interesting result as M’s decision on r does not depend on the tax rate—he must always set an obedience test to be slightly more aversive than the entire value that is at stake. The tax rate only affects whether S will choose to pass the test.
I wonder what would happen if one were to remove b and play the game iteratively. The game stops after 50 iterations or the first time S fails the test or defects.
b is then essentially replaced by S’s expected payoff over the remaining iterations if he remains loyal. However M would know this value so the game might need further modification.
I think we should still keep b even with the iterations, since I made the assumption that “degrees of loyalty” is a property of S, not entirely the outcome of a rational-game-playing.
(I still assume S rational outside of having b in his payoffs)
Otherwise those kind of tests probably makes little sense.
I also wonder what happens if M doesn’t know the repulsiveness of the test for certain, only a distribution of it (ie: CIA only knows that on average killing your spouse is pretty repulsive, except this lady here really hates her husband, oops), could that make a large impact.
I guess I was only trying to figure out whether this “repulsive loyalty test” story that seems to exist in history/mythology/real life in a few different cultures has any basis in logic.
Thanks for posting, I had fun trying to solve it and I think I learned a few things.
My solution is below (I think this is correct but I’m no expert) but I’ve hidden it in a spoiler in case you’re still wanting to figure it out yourself!
M has preference order of 2a>1>2b. He wants to set r such that if S has b>tv then S will pass the test and then remain loyal. If S has b<tv then M wants S to fail the test and therefore not get the chance to defect in round 2. It is common knowledge that this is what M wants.
Starting by making S’s Payoff for 2b less than that for 1 gives a formula for r:
v−b−r<−b
v<r=v+ϵ for some small positive ϵ
With this value for r, S’s payoff matrix becomes:
1. −b
2a. −vt−ϵ
2b. −b−ϵ
We can see that if vt−ϵ<b then S’s best payoff is obtained by choosing 2a. Otherwise his best payoff is 1. This is exactly what M wants—he has changed S’s payoffs to make S’s preference order the same as his to the greatest extent possible.
Due to M’s preference being common knowledge, S knows that M will choose this value of r and therefore knows what v is before he chooses whether to pass the test (v=r−ϵ) and can choose between the three options simultaneously.
This is an interesting result as M’s decision on r does not depend on the tax rate—he must always set an obedience test to be slightly more aversive than the entire value that is at stake. The tax rate only affects whether S will choose to pass the test.
Thanks, the final result is somewhat surprising, perhaps it’s a quirk of my construction.
Setting r to be higher than v does remove the “undercover agents” that have practically 0 obedience, but I didn’t know it’s the optimal choice for M.
I wonder what would happen if one were to remove b and play the game iteratively. The game stops after 50 iterations or the first time S fails the test or defects.
b is then essentially replaced by S’s expected payoff over the remaining iterations if he remains loyal. However M would know this value so the game might need further modification.
I think we should still keep b even with the iterations, since I made the assumption that “degrees of loyalty” is a property of S, not entirely the outcome of a rational-game-playing.
(I still assume S rational outside of having b in his payoffs)
Otherwise those kind of tests probably makes little sense.
I also wonder what happens if M doesn’t know the repulsiveness of the test for certain, only a distribution of it (ie: CIA only knows that on average killing your spouse is pretty repulsive, except this lady here really hates her husband, oops), could that make a large impact.
I guess I was only trying to figure out whether this “repulsive loyalty test” story that seems to exist in history/mythology/real life in a few different cultures has any basis in logic.