shooting while opponent blocks should yield u(0,0), right?
Well, I could make a table for the state where no one has any bullets, but it would just have one cell: both players reload and they go back to having one bullet each. In fact, the game actually starts with no one having any bullets, but I omitted this step.
Also, in both suggestions, you are telling me that the action that leads to state x should yield the expected utility of state x, which is correct, but my function u(x,y) yields the expected utility of the resulting state assuming that you’re coming from the original, neutral one. Otherwise, it would need an additional argument to say what state you’re currently in. Instead of writing the utility of each action as u(current state, next state), I wrote it as u(next state)- u(current state). Each state is an ordered pair of positive integers, the two player’s bullets. So, to write it the way you suggested, the function would need four arguments instead of two.
So the utility for S+B is 0 and the utility for R+R is 0.5. The equilibrium is where both players reload with probability = 2⁄3. The utility of the (1,0) state is +2/3.
In your last table, shooting while opponent blocks should yield u(0,0), right? And both reloading would be u(2,1).
Well, I could make a table for the state where no one has any bullets, but it would just have one cell: both players reload and they go back to having one bullet each. In fact, the game actually starts with no one having any bullets, but I omitted this step.
Also, in both suggestions, you are telling me that the action that leads to state x should yield the expected utility of state x, which is correct, but my function u(x,y) yields the expected utility of the resulting state assuming that you’re coming from the original, neutral one. Otherwise, it would need an additional argument to say what state you’re currently in. Instead of writing the utility of each action as u(current state, next state), I wrote it as u(next state)- u(current state). Each state is an ordered pair of positive integers, the two player’s bullets. So, to write it the way you suggested, the function would need four arguments instead of two.
This made me think the last table was just for the (1,0) state. Is this not the case?
I’m not sure why the previous state would matter.
By previous state, I meant current. I misspoke.
Yes, the last table is for the (1,0) table.
So the utility for S+B is 0 and the utility for R+R is 0.5. The equilibrium is where both players reload with probability = 2⁄3. The utility of the (1,0) state is +2/3.
Thanks. I now see my mistake. I shouldn’t have subtracted the expected utility of the current state from the expected utility of the next.