Right, welcome back. So, in the last video, we used a game tree to systematically develop and approach a competitive situation in which firms take decisions. But they don't do it simultaneously, but they do it one after the other. In this video, we're going to use the game tree and a technique we call backward induction to find the optimal strategy for the firm that has to decide first. So, let's have a look. So, we know that Mars have the first choice. They have the choice of signing the product placement contract or not signing it. And if they sign it, the game is over. Hershey's does not have a choice. Whereas if they don't sign it, then Hershey's gets to decide if they want to accept or not accept the deal. As the game played out, we found this result. So, Mars declined and Hershey's accepted. This turned out to be a loss maker for Mars. They lost half a million and Hershey's won by $200,000. Okay. So, what we're going to do is we're going to apply that concept of backward induction and try to go back and see if this is an outcome that is optimal in any sense or, and or, if it's something that we should have expected. But let's first see what backward induction is in the first place. Backward induction is the process of simplifying a sequential game. You remember looking for dominant strategies. You remember looking for dominated strategies and eliminating them. It's a bit like that, right. So, you're trying to simplify a particular type of game. We're eliminating actions at the final node and we work our way forward. So, anything that isn't going to be played is something we can ignore from the game, just like a dominated strategy. We will choose the actions that would not maximize an individual profit at that point and we eliminate them. And why do we do that, because a rational player will always try to maximize their own profit. So therefore, we will always rely on a rational rival never to choose these actions. Okay, anything that doesn't maximize my own profit is not something that I'm going to choose. Okay? So, let's take that example again. Let's take the chocolate wars again, and see if this is the right outcome, if this is an outcome we expected. Okay, so if Mars chooses to sign, then there's no contest, no choice for Hershey's, status quo, nothing to decide, okay. What if Mars declines the offer? If Mars declines the offer then, Hershey's has the choice of signing the deal, which is going to gain them 200,000, or not signing the deal, which is going to mean business as usual, and zero change in profits. So, for Hershey's the situation is, do I want to make 200,000 dollars or do I want to make nothing? Yes, so of course, the best outcome, the best solution for Hershey's is to accept the deal if offered the deal, okay. Therefore, we can eliminate this part of the game, it's never going to happen. And Mars should not expect this to happen because, of course, Hershey's wants to maximize their own profits. Now, what does this mean? It means for Mars that if they do not sign the deal, Hershey's will accept the deal. And this is going to mean a loss of half a million for Mars. If they accept the deal, so they sign the contract, Hershey's does not have a choice, and it's going to mean a loss of 200,000 for Mars. Now, comparing a loss of 200 and a loss of 500, of course, it's clear that Mars should have chosen this strategy, so they should have signed the deal. But I think we're probably losing out some sort of realism here, because if you go to your manager, if you go to your boss, and say, well, here's a product placement offer I got, and I believe it's going to lose us $200,000. Can we please do that? It's going to be a difficult sell, right? Typically, you would not engage in any project, you would not engage in any contract, you would not sign any contract that's going to lose you money. Okay? But if you think strategically, if you think ahead and actually make that calculation, and make that anticipation that Hershey's, if we don't accept it, Hershey's is going to accept it, and this would be an even worse outcome, then it all of a sudden makes a great deal more of sense to sign that deal, okay. So, looking forward, looking ahead and trying to think strategically what the other firm is going to do can help make better decisions in this case, right? So, the right outcome, the outcome we would've expected is for Mars to sign straight away. Let's use backward induction again for another game. Okay, it's a game of price setting. Firm A can choose a high price or a low price. Firm B can make their decisions dependent on what firm A has done. So, they can choose a high price if firm A has chosen a high, or, they can choose a low price and so on and so forth. Payoffs are 8 million for both, if both charge a high price, they are zero for firm A and 10 million for firm B. If firm B charges a low price and firm A charges a high price, they are 10 million for A and zero for B. If A charges a low price and B charges a high price, and if both charge low prices, they both made 5 million. What is the likely outcome of that game? Well, if we look at it just like that, it would seem reasonable that both A and B should choose high prices, because that gives them the highest profit jointly. Okay? But let's use backwards induction to try and see what's most likely to happen. If Firm A has chosen the high price, Firm B will have the choice of either getting 8 million, if they choose a high price or getting 10 million if they choose a low price. So, the best response, the best strategy for firm B is to charge a low price here. Okay, because 10 million is bigger than eight. If firm A chooses a low price, then firm B can choose between charging a high price and getting zero, and charging a low price and getting 5 million. So, the best choice for them again, is to charge a low price. Okay. Now, firm A is going to take a step back. They're going to try to anticipate what's going to happen. And charging a high price means profits of zero for them. Charging a low price means profits of 5 million for them. So, what we expect in this game via the technique of backward induction is that firm A charges a low price because they anticipate that firm B is going to charge a low price as well. So, the outcome of the game would be that both firms charge a low price. We're basically in a Prisoner's Dilemma type situation as you've seen in previous videos where both first do something that is individually rational, but not collectively profit maximizing. So, in this video, we've used the concept of the previous one, game tree. We've used the technique called backward induction to find what the optimal strategy for a firm is, okay. We've found, the optimal strategy for firm B, for the second mover, and from that, we could derive what the best strategy for the first mover was. So, we're going to have an exercise on this just after this, but for now, thanks very much and I will see you very soon.