Could someone explain how does the defending general in the Metropolis—Podunk example calculates his strategy?
I understand why the expected loss stays the same only if he defends Metropolis 10 /11 times and Podunk 1⁄11 times, but when I tried to calculate his decisions with the suggested procedure I ended up with the (wrong) result he should defend Podunk 10⁄11 times and Metropolis 1⁄11, and I’m not understanding what I’m doing wrong, so I’d likely botch the numbers when trying to calculate what to do for a more difficult decision.
Think of the attacker as deriving the same utility whether he/she attacks Podunk 10 times, or Metropolis once. That’s why the attacker is ready to attack Metropolis 10 times before occupying it, or Podunk once.
Likewise, the defender derives a utility of 10 for defending Metropolis and 1 from defending Podunk. What this means is that the defender is able to bear 9 blows from the enemy and it would still be worth it to defend Metropolis. After the 10th attack, the defender would derive the same utility from defending Metropolis as if he/she were to defend Podunk.
I hope that sorta explains it. Someone correct me if I got it wrong.
I think I should calculate the loss of utility multiplying or the missed defences, then, since the utility is different only when the defence is unsuccessful.
This way I get −10 x 1⁄11 and −1 x 10⁄11, so the utility stays the same and I’m defending Metropolis 10 times out of eleven as I’m supposed to.
Could someone explain how does the defending general in the Metropolis—Podunk example calculates his strategy?
I understand why the expected loss stays the same only if he defends Metropolis 10 /11 times and Podunk 1⁄11 times, but when I tried to calculate his decisions with the suggested procedure I ended up with the (wrong) result he should defend Podunk 10⁄11 times and Metropolis 1⁄11, and I’m not understanding what I’m doing wrong, so I’d likely botch the numbers when trying to calculate what to do for a more difficult decision.
Think of the attacker as deriving the same utility whether he/she attacks Podunk 10 times, or Metropolis once. That’s why the attacker is ready to attack Metropolis 10 times before occupying it, or Podunk once.
Likewise, the defender derives a utility of 10 for defending Metropolis and 1 from defending Podunk. What this means is that the defender is able to bear 9 blows from the enemy and it would still be worth it to defend Metropolis. After the 10th attack, the defender would derive the same utility from defending Metropolis as if he/she were to defend Podunk.
I hope that sorta explains it. Someone correct me if I got it wrong.
I see, expressed like this it’s obvious…
I think I should calculate the loss of utility multiplying or the missed defences, then, since the utility is different only when the defence is unsuccessful.
This way I get −10 x 1⁄11 and −1 x 10⁄11, so the utility stays the same and I’m defending Metropolis 10 times out of eleven as I’m supposed to.