Consider a game like the following linked one.
Neither player is aware of what the other player has played or will play.
Suppose then I model a slightly different game. In this game, player two moves first and can tell the future. She knows the move that player one will make and can, therefore, be represented to have moved after player one, thus:
While this conveys the strict rules of the game, it is misleading to the actual process of play because it is shown as if player two moves after player one. Is there a better way to show player two's extra knowledge while still having them play higher up in the tree?
I thought of using a baysian game where the one branch is the true game and the other is a contrived game to force player one to choose certain things with a probability of 1:0. But then I didn't know how to show that player one believes the probability to be 0:1.
Please ask if this last paragraph was not clear. Though it was not paramount to the question, it indicates what I have tried.
This was too long to be a comment:
There’s an issue here with talking about “knowledge”. Nash-style equilibrium concepts all revolve around players knowing other players’ strategies because they’re all flawless objective Bayesian rationalists. l think what you’re suggesting is that 2 is clairvoyant and knows 1’s actions in advance. Then 2 really has two information sets, one in which they know 1 plays U and one in which they know 1 plays D. 2 then chooses U’ or D’ and 1 is forced to choose what 2 foresaw, as 2 is correct. This isn’t a game because there’s nothing that determines which information set 2 winds up in. You could make nature decide, but that wouldn’t make it equivalent to the second game pictured. 2 can know that 1 will do something irrational forcing 1 to do it. You could insist that 1 can only play rational strategies, but then depicting the game in this way would go part of the way to solving it.
Addendum: You could abandon game trees altogether by thinking of 1 causing 2 to know 1’s action in the past. With retrocausality the linearity of a game trees falls apart. If moves represent causation and we want to keep the axioms of game trees except for perfect recall and the game tree having a root then there’s one way to do this. Let $\mathbb{Z}_z$ be the set of all integers less than or equal to $z$. Nodes are functions $f:\mathbb{Z}_z\to\{U,D,U’,D’\}$ such that even integers maps into $\{U’,D’\}$ and odds into $\{U,D\}$ (the choice of even vs odd is arbitrary). We have four information sets consisting of all nodes $f$ with domains $\mathbb{Z}_z$ (allowing $z$ to vary with $f$) with the same value of $f(z)$. We assign $f$ with $z$ even to $1$ and $f$ with $z$ odd to $2$. The information a player has at a given information set is represented by $f(z)$, which is what the other player has/will played/play. The player may then choose from their respective set of moves. After player $i$ players move $m$ at node $f:\mathbb{Z}_z\to\{U,D,U’,D’\}$, we move to node $f_m:\mathbb{Z}_{z+1}\to\{U,D,U’,D’\}$ such that $f_m(z+1)=m$ and for all $y\in\mathbb{Z}_z$, $f_m(y)=f(y)$. Note that most of the nodes only exists so you can keep the idea of both players making choices, all but eight of them could clearly never be reached since they’d involve different moves being played at the same information set. You’d then need to define solving this game “tree”. You could count some of the nodes as identical, for instance compressing both of 2’s information sets into one node and the members of particular information sets of 1 that follow members of particular information sets of 2. This is a lot more workable, but now there are loops of moves. Of course, there’s retrocausality, so perhaps that’s not so concerning. However, if you want to keep the idea of 1 freely taking an action and this causing something for $2$ in the past and have a normal game tree, well you can’t.