> I'm not a mathematician, so I don't understand expressions like "counterfactual part of subgames emerging" or "there is an identity implied by being under imperfect information". I appreciate you writing back at length, but the majority of what you wrote went straight over my head.

# Defining Counterfactuals

Consider a fair coin flip. You have {Heads, Tails}. Lets assume you are going to get heads, take it as a given - that is what actually happens. It actually happens. It is factual. However, for the purposes of analysis, sometimes it doesn't really matter that we know what happened. We need to consider all the cases that didn't happen. Tails in our analysis would be the counterfactual. These two situations, the factual heads and the counterfactual tails, they're associated with each other. There is a set {HT} that contains both of them. E.g P(head) = |{H}|/|{H, T}|.

When I'm saying counterfactual I'm referring to the events that we didn't assume to be factual, but which we want to keep track of. Probability kind of drops these terms when it says "assume" because it says |{H}|/|{H, T}| becomes |{H}|/|{H}| which is equal to one. This is perfectly fine from a math perspective. The thing is that just like saying 5/x = y lets us move to 5=xy is valid, it has some assumptions built into it. Namely that x can't be zero. Our counterfactuals are a bit like the x, because they offer an easily hidden constraint on what it is valid to do. If you reintroduce uncertainty about whether or not you are in H, you have to do so in a way that takes you back to having the set {H, T}.

Let me show you a practical example of that to make the point very clear:

Let the value of heads be one and the value of tails be zero. P(Head) = 0.5 But assume heads on the same coin flip. P(Head|assumptions) = 1.0 But assume heads again on the same coin flip. P(Head|assumptions) = 1.0. Now since P(Head) is actually true with 1/2 probability: Ev(Head) = 1/2*P(Head|assumption) + 1/2P(Head|assumption)?

Well, it is certainty true that 1/2P(Head|assumption) is a correct term. So you can't say this is wrong on the basis of the assumptions alone. Being very precise, the problem is the neglected counterfactual wasn't handled. Every term leading up to the equation was technically true, but obviously we just did something really weird right? And to be very precise, we neglected the counterfactual associated with heads.

# Defining subgames

Most games can be written out as a game tree. I showed one in my previous post. When you move down the tree, you are in a subgame of that tree.

In a perfect information game like chess if you are in a subgame, a portion lower on the tree, the other parts of the tree don't matter anymore. Your results are independent of the rest of the game tree.

In imperfect information though, just because you are in a subtree doesn't mean you know which subtree you are in. Your actual view into the game is through the information you have. Just like in the {HT} case you have to consider the potential that you have both H or T, in a subgame you have to consider the potential you are in every subgame that is reachable given the information you've seen.

Check out this to get a better sense of what I'm talking about: https://www.youtube.com/watch?v=EbKmZLp5HvA