A has two ammunition stores, one of which is twice as valuable as the other. B is an attacker who can destroy an undefended store, but he can only attack one of them. A can successfully defend only one of them. A learns that B is about to attack one of the stores but does not know which. What should he do?

 The scenario given is a classic example of a decision problem where a decision maker has to choose between two possible options in an uncertain situation. In this case, A has to decide which ammunition store to defend when he knows that one of them is going to be attacked by B, but he does not know which one. To make a rational decision, A needs to weigh the pros and cons of each option and choose the one that maximizes his expected payoff.

In this situation, A has two options: defend the store with higher value ammunition or defend the store with lower value ammunition. Let's consider the expected payoffs of each option.

Option 1: Defend the store with higher value ammunition

If A defends the store with higher value ammunition, he has a 50-50 chance of defending the store that will be attacked. If he successfully defends the store, he will have saved ammunition that is twice as valuable as the ammunition in the other store. However, if B attacks and destroys the store, A will lose ammunition that is twice as valuable as the ammunition in the other store.

The expected payoff of this option can be calculated as follows:

Expected payoff of defending the higher value store = 0.5*(2x) + 0.5*(-2x) = 0

where x is the value of the lower value ammunition store.

As we can see, the expected payoff of this option is zero. This means that if A defends the store with higher value ammunition, he can neither gain nor lose anything on average.

Option 2: Defend the store with lower value ammunition

If A defends the store with lower value ammunition, he also has a 50-50 chance of defending the store that will be attacked. If he successfully defends the store, he will have saved ammunition that is half as valuable as the ammunition in the other store. However, if B attacks and destroys the store, A will lose ammunition that is half as valuable as the ammunition in the other store.

The expected payoff of this option can be calculated as follows:

Expected payoff of defending the lower value store = 0.5*(0.5x) + 0.5*(-1x) = -0.25x

As we can see, the expected payoff of this option is negative. This means that if A defends the store with lower value ammunition, he can expect to lose some value on average.

Based on the expected payoffs of each option, it is clear that A should defend the store with higher value ammunition. Although this option has an expected payoff of zero, it does not involve any expected loss, unlike the option of defending the store with lower value ammunition.

Another way to look at this is to consider the worst-case scenario for each option. If A defends the store with higher value ammunition and B destroys it, A will still have the store with lower value ammunition. However, if A defends the store with lower value ammunition and B destroys it, A will have lost ammunition that is half as valuable as the ammunition in the other store. Therefore, by defending the store with higher value ammunition, A can minimize his potential losses.

In conclusion, A should defend the store with higher value ammunition when faced with the decision of which ammunition store to defend. This decision is based on the expected payoffs of each option and the potential losses associated with each option. By making this decision, A can minimize his potential losses and maximize his expected value.

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