In the world of perfect decision makers, there is no risk to doing so, because the Countess will hand over the money, so the Baron will not take the hit from the revelation.
Since the payoffs are symmetrical (in sense that both baron and countess lose the same amount of utility if the secret is revealed), in the world of perfect decision markers the winner is clearly the person who decides to blackmail. But the countess can retaliate easily: after she pays, she can blackmail the baron, and get the money back with certainty, even with some bonus. So, say that the baron has Xb money, and prefers having at least Lb money to keeping the secret (this is the lower bound; it can be 0 if the baron is willing to pay anything he has). For the countess, the analogous quantities are Xc and Lc.
1st step: The baron blackmails, and rationally demands Xc-Lc money. Countess pays. After that, the baron has Xb+Xc-Lc, the countess has Lc. Now, the baron knows that the countess will reject any further threat.
2nd step: The countes blackmails the baron, and demands Xb+Xc-Lc-Lb money. Baron, obviously, pays. After that, the baron is at Lb, while the countess is at Xb+Xc-Lb.
3rd step is obvious.
Now, both the countess and the baron are reasonable enough that they can predict the outcome being an infinite oscillation of the bank account, and that the result is only loss of time. (Assume that it is impossible for the countess to trick the baron by spending the money somehow to get some utility while lowering her accessible possessions to Lc, or vice versa). So, what is the solution in the timeless decision theory? How would the ideally rational baron and countess behave?
Since the payoffs are symmetrical (in sense that both baron and countess lose the same amount of utility if the secret is revealed), in the world of perfect decision markers the winner is clearly the person who decides to blackmail. But the countess can retaliate easily: after she pays, she can blackmail the baron, and get the money back with certainty, even with some bonus. So, say that the baron has Xb money, and prefers having at least Lb money to keeping the secret (this is the lower bound; it can be 0 if the baron is willing to pay anything he has). For the countess, the analogous quantities are Xc and Lc.
1st step: The baron blackmails, and rationally demands Xc-Lc money. Countess pays. After that, the baron has Xb+Xc-Lc, the countess has Lc. Now, the baron knows that the countess will reject any further threat. 2nd step: The countes blackmails the baron, and demands Xb+Xc-Lc-Lb money. Baron, obviously, pays. After that, the baron is at Lb, while the countess is at Xb+Xc-Lb. 3rd step is obvious.
Now, both the countess and the baron are reasonable enough that they can predict the outcome being an infinite oscillation of the bank account, and that the result is only loss of time. (Assume that it is impossible for the countess to trick the baron by spending the money somehow to get some utility while lowering her accessible possessions to Lc, or vice versa). So, what is the solution in the timeless decision theory? How would the ideally rational baron and countess behave?