 Let us now revisit the solution concept for extensive forum games. And let's start by doing it. Let's start by looking at the following examples. In this game, there are a number of Nashiculurbia, and here is one of them. BHCE. What is BH? One goes down here and down here, whereas player two goes down here and down here. Under this strategy profile, of course, the outcome of the game is this one. And the payoff to both players is five. Let's first convince ourselves that this is indeed a Nashiculurbian. So let's hold players one strategy fixed and see if player two can profitably deviate from their current response. Well, what can they do? They can say, here, I will, instead of going C, I would go D. They could say that. But that would impact the outcome at all, given that player two is going down B. And so that's not a profitable deviation. It wouldn't change their payoff to player two. And the other thing they could do is say I'm going to go down this way. But that would, in fact, worsen their payoff because they would end up here with a payoff of zero rather than the five they're getting. So player two cannot profitably deviate from their current strategy. What about player one? Can they profitably deviate? Well, what could they do? They could say, OK, rather than go B, I'll go A. But then they will get a payoff of three rather than the five they're getting. That's not profitable. And they could also say, I'm going to go down G over here. But given that player two is going down E, that wouldn't matter. They would, in any case, end up in this outcome with a payoff of five. And so that's not profitable deviation either. So neither player has a profitable deviation. Then by definition, it's a natural equilibrium. But there's something a little disturbing about this equilibrium. Let's clear the slides. It's a little less messy. And let's, again, write down the strategy for player one going B, H. And let's focus on this node right there. Why would player one actually do H? Because G dominates it. In G, they get a payoff of two rather than one. And so even though it did lead to a natural equilibrium, there's something a little troubling about it. And the way to understand it is by claiming that they would go down H here, player one is threatening player two and telling him, listen, do not consider going down here because I'm going to go down here. Therefore, and you would get a zero. So you'd better go here and get a five. Is what player one is saying to player two. But this stretch is not credible because after all, player two says, player one says that. But in fact, it would not be in their interest. So I believe that player one actually would go down here. And so how do we capture this in a formal definition? That brings us to the notion of sub-game perfect equilibria or sub-game perfection. So let's first define the sub-game into a very obvious notion. Looking at some node in the game node H, the sub-game of G rooted at H is a restriction of H to the descendants from that node. And similarly, what are the set of all sub-games of G? Well, look at all the nodes in G. And the set of all sub-games is simply all the sub-game rooted at some node in G. And so a natural equilibrium is a sub-game perfect. If it's restriction to every sub-game is also a natural equilibrium for that sub-game. So if, for example, we go to the previous slide and we consider, again, clearing the slide for a second, and if we look at the natural equilibrium B, H, C, E, and we just saw it's a natural equilibrium, but among the sub-games of this game, the sub-trees of this tree is this sub-tree. So here's a sub-game. It's a game of a single player, player one. And the restriction of this natural equilibrium is simply that action of going H. But this is not an equilibrium in this very simple tree because there's a profit deviation to G for the player. And so while it's a natural equilibrium of the whole tree, its restriction to the sub-tree here is not a natural equilibrium and therefore this natural equilibrium is not a sub-game perfect. And so we see that, in fact, that captures the intuition of non-credible threat and noses also that one special case of a sub-tree is the entire tree. So a sub-game perfect equilibrium has got to also be a natural equilibrium. So let's test our understanding of this concept a little bit. Let's look at this tree and ask ourselves what are some of the sub-game perfect equilibria there. For example, how about AGCF? Well, the claim is this, in fact, is a sub-game perfect. Now why is that? What is AGC and F? So that gives you this outcome over here. And you can check there's no profit deviation. But you can also ask all the sub-games, is there a profit deviation? Well, let's look at some of the sub-games. Well, for example, here there is this deviation over here, but that would not be profitable for player 2 because it would go down from 8 to 3. How about over here? Is there profit deviation, for example, to player 2? Not really because if they deviated over here, they would end up with a 5 rather than the 10 they're getting. How about over here? Is there profit deviation at this node to the agent 1? Well, no, because if they deviate, they would get 1 rather than 2. So in all sub-games, the restriction of the strategy profile to that sub-game is still a Nash equilibrium, and AGCF is in fact a sub-game-perfect Nash equilibrium. How about BHCE? Well, the claim is that it's not. Well, let's first write down the strategy BH, BH, and CE. And this is not sub-game-perfect for the reasons we saw before. We saw that in this sub-game right here, there is a profitable deviation for player 1, namely to deviate over here and get 2 rather than 1. And so it's not sub-game-perfect. And in fact for the same reason, AHCF will not be sub-game-perfect. Let's write down what AHCF is. AHCF. You can check that it's a Nash equilibrium, but it is not sub-game-perfect. Again, this sub-game here allows for a profitable deviation on the part of the player 1. So even though it's what's called off-path, even though player 1 makes sure that he never gets to visit this node by going down here, even so it's not sub-game-perfect because had he gotten here, he would not have done what he claims he would have done. And that gives us a good sense for what a sub-game-perfect Nash equilibrium is.