Let us now revisit the solution concept for extensive-form games. And let's start by doing it let's start by looking at the following example. In this game there are a number of Nash equilibrium and here's one of them. (B,H), (C,E). What is (B,H)? Player 1 goes down here and down here, whereas player 2 goes down here and down here. Under this strategy profile, of course the outcome of the game is this one. And the path to both players is five. Let's first convince ourselves that this indeed a natural eqilibrium, so let's hold player one's stratagy fixed and see if player two can profitably deviat 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 wouldn't impact, the outcome at all, given that player 2 is going down b. And so, that's not a profitable deviation. It wouldn't change, their. payoff. The payoff to player 2. And the other thing they could do is say, I'm going to go down this, this way. But that would, in fact, worsen their payoff. Because they would end up here, with a payoff of zero rather than the 5 they're getting. So player 2 cannot profitably, deviate from their current strategy. What, what about player player 1? Can they profitably deviate? Well, what could they do? They could say, okay. Rather than go b, I'll go a. [SOUND] But then, they will get a payoff of 3 rather than the 5 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 would matter, though in any case end up in this this outcome of the path of five and so that's not profitable deviation either. So neither player has a profitalbe deviation then by definition it's a natural equilibrium. But there's something a little disturbing about this equilibrium. Let's clear the slide so it's a little less messy and let's again write down the strategy for player 1,going B H and let's focus on, in, on this node right there. Why would player 1 actually do H? Because a G dominates it. The G, they get a payoff of 2 rather than 1. And so even though It did lead to an actual equilibrium. There is something a little troubling about it. And the way to understand it is by claiming that they would go down H here, player 1 is threatening player 2 by telling him, listen do not consider going down here because I'm going to go down here. Therefore and you would get a 0, so you'd better go here and get a 5 is what player 1 is saying to player 2 but this strategy is not credible because after all player 2 says, player 1 says that but in fact, it would not be in their interest. I believe that player 1 actually would go down here. And so how do we capture this in a formal definition? That brings us to a, to the notion of subgame perfect equilibria or subgame perfection. So, let's first define the subgame. It's a very obvious notion a looking at some node in the game, node h, the subgame of G rooted at h is a restricted, a restriction of h to the descendants from, from that, from that node. And similarly what are the set of all subgames of G? Well look at all the nodes in G and the set of all subgrames. is simply all the subgame routed at sum node in g. And so, a Nash equilibrium is a subgame perfect, if its restriction to every subgame is also a Nash equilbrium for that, that subgame. Say for example we go to the previous slide and we consider again clearing the slide for a second. And if we look at the puh, the Nash equilibrium B H, c, e. And we just sort the Nash equilibrium. But among the subgames of this game. the subtrees of this of this tree, is this subtree. So here's a subgame. It's a game of a single player, player one. And the restriction of this [UNKNOWN] is simply the action of going H, but this is not an equalibrium in this very simple tree because theres a proffer of deviation of G to the player and so while there's a [UNKNOWN] of the whole tree is just friction to the sub-tree here. Is not a natural equilibrium and therefor this natural equilibrium is not a sub game perfect. And so, so we see that in fact that captures the intuition of non credible threat and notice also that one special case of the sub tree is the entire tree So subgame perfect equilibirium has got to also be Nash equilibrium. So let's test your understanding of this concept a little bit. Let's look at this tree and ask ourselves what are some of the subgame perfect equilibirium there. For example how about (A,G), (C,F)? Well, the claim is, this in fact is slightly imperfect.Now, why is that.What is a, g, c and f. So that gives you this outcome over here. And you can check that there is no possible deviation, but you can also ask in all the sub-games is there a possible deviation? Well look, let's look at some of the possible sub-games. Well, for example, here there is this this deviation over here. That would not be proper to compare to because they would go down from. eight to three. How about over here? Is there [INAUDIBLE] deviation, for example, to player 2? Not really, because, iIf they deviated over here, they would end up with a five rather than the ten they're getting. How about over here? Is there possible deviation at this node to the agent one? Well, no, because if they deviate they would get one rather than two. So, in all subgames, the restriction of the statuary profile, to that sub-game, is still a Nash equilibrium. And, a,g, c, f, is, in fact, a sub-game perfect Nash equilibrium. How about b, h, c, e? Well, the claim is, that it's not. Well let's first write down the strategy B, H, B, H and C, E and this is not something perfect for the reasons we saw before. We saw that in this subgame 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 subgame perfect. And, in fact, for the same reason, a, h, c, f. Will not be subgame perfect. Let's write down what (A,H), (C,F) is. (A,H), (C,F). You can check that it's a Nash equilibrium but it is not subgame perfect. Again, this subgame here is allows for a proper 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, that he never gets to. Visit this node by going down here. Even so, it's not subgame 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 subgame perfect Nash equilibrium is.
