Welcome to Week 4's review session on decision trees. I'm Noah Gantz your instructor for the week. Before we get started, I wanna point out that this review session is completely optional. If the decision tree concepts we introduced in Session 1 are clear to you, then you can skip this review and move onto the practice problems. After that, go ahead and tackle week 4's homework. However, if before trying the practice problems, you'd like an additional look at how to set up and analyze a decision tree, then join me for a review. Our review is going to use an example from patent litigation. The two companies involved are Cellectis and Precision BioSciences. Let's start with some background on Cellectis and Precision BioSciences. Both companies developed genomic editing technologies, that they sell for use in genetic engineering. In 2008, Cellectis sued Precision BioSciences for patent infringement, and the litigation ended in 2015. Imagine now that it's 2008. Cellectis's legal advisors note that the litigation process has three well established stages. In stage 1, the court decides whether or not Cellectis's patent is valid. In stage 2, if the patent has been considered valid, then the court will determine whether or not Precision BioSciences has infringed on select business patent. In stage 3, if there had been a determination of infringement by the court, then the court will determine the magnitude of the damages payable to Cellectis. Note that if the court rules against Cellectis, in either the first or the second phase, then the litigation ends. Cellectis estimates the probabilities and the cash flows together with it's attorneys. First of all, they estimate the probabilities of litigation advancing from one stage to the next. They estimate that the chance of the patent being found valid in stage 1 or 2 and 3. So there's a 1 in 3 chance that it won't be found valid. Then, given a valid patent, they estimate the chances that Precision BioSciences will be found to have infringed as only 1 in 4, so there's a 3 in 4 chance that Precision BioSciences will be found not to have infringed. Finally, they estimate the possible damage awards in phase 3 by multiplying royalty percentages by the value of Precision BioSciences contracts. And when they do the multiplication, they come out with four equally likely damage estimates. $20 million, $30 million, $40 million, and $50 million, each occurs with a probability of 1/4th. The legal team also estimates its fees for each of the three trial stages. It estimates $3 million dollar charge for the first stage, patent validity. It estimates its expenses as $5 million for the second stage, the determination of infringement. And it estimates a $1 million dollar charge for the third stage, damages. Cellectis has the following decision problem, when moving from one stage of litigation to the next, if Cellectis loses at that stage, it has to stop and pay its attorney's fees up through that stage, and that's it. Even if Cellectis wins at that stage, it can decide to stop litigation, and pay only its attorney's fees up through that stage. More generally, Cellectis must decide the following, whether or not to litigate in the first place. That is, whether it should sue Precision BioSciences for patent infringement, or not. And if it does litigate, then after each stage it wins, should it continue litigation, or stop? So, we want to do two things. First, we'd like to construct a decision tree for Cellectis. We wanna construct the tree with decision nodes, event nodes, and payouts. We also wanna write down the name and cash flows associated with each decision and event. Next, for each event node, we wanna write down the probabilities that they occur. We wanna make sure that the probabilities at each event node add up to one. Then, for each payout, we wanna use the revenues and costs on the branches that lead up to the payout, to calculate its value. Having constructed the decision tree, we then want to analyze it. We'd like to determine Cellectis's maxi-min set of decisions. We'd then like to determine Cellectis's maxi-max set of decisions. Finally, we'd like to determine Cellectis's expected value maximizing set of decisions. And then we can compare the three. So let's get started. First we're going to construct the tree with decision nodes, event nodes, and payouts. Remember that decision nodes are squares, event nodes are circles, and outcomes or payouts are triangles. Cellectis's first decision is whether to enter into phase one of the litigation, whether to litigate it at all. All right, and if it doesn't, then it pays nothing. On the other hand, if it does, it's going to pay its legal team $3 million dollars. Having decided to go into litigation, we encountered the first event node; that's whether the patent is found valid or not. If its not found valid, then litigation is over. If it is found valid, then Cellectis can decide whether to proceed to phase 2 of the litigation. At that point, it can decide to stop, or it can decide to proceed with phase 2, in which case it's going to pay its attorneys an estimated fee of $5 million dollars. Having entered into phase two, the court will then decided whether the patent is infringed or not. If it's not infringed, then the litigation ends. If it is infringed, then Cellectis can decide whether to proceed into phase 3 of the litigation. That's the damages phase. At that point it can decide not to, or it can decide to proceed into the damages phase, and pay it's attorneys an estimated fee of $1 million dollars. In the damages phase, there are four equally likely outcomes. Cellectis might be awarded damages of $50 million, $40 million, $30 million or $20 million, so that's the full tree. Next, we want to decide what the probabilities are of the different event nodes, so let's start at the beginning, and work towards the end. In the first event, whether the patent's found valid or not, we know that there's a two-third chance that the patent will be found valid. And a one-third chance that the patent will not be found valid. So notice that those two probabilities add up to one. In the second litigation phase, that's patent infringement, we know that there's only a 1 in 4 chance that the patent will be found to have been infringed upon. And that leaves a three fourths chance that the patent will not be found to have been infringed upon. Again, the two probabilities add up to one. In the final damages stage, there are four equally likely outcomes of $50, $40, $30 and $20 million dollars in damages. And each of those is going to accrue with an equal probability, or 1 in 4. So those are all the probabilities in the event nodes. At each event node they add up to one. Finally, at each outcome we want to determine what the payout is for Cellectis. To do that we're going to calculate the payouts associated with the various outcomes. We're going to start at the initial root of the tree, and work through all the cash flows on the branches that lead up to the outcome. We start at the lower left, in the litigation phase. If Cellectis decides not to litigate, then it pays nothing to it's attorneys. On the other hand, if Cellectis decides to litigate, it's going to pay $3 million dollars to its attorneys. And then one of two things can happen. It could lose phase one of the litigation in which case it would have paid three million and got nothing back and so the outcome will be negative three million. On the other hand, it may decide to litigate and pay the three million and then The patents found to be valid, at that point Cellectis has a second decision, it can decide not to proceed with these to the patent infringement. In which case it's paid the three million and got nothing back, so again the outcome is negative three million for Cellectis. Alternatively having initiated litigation and found it's patent to be valid, Cellectis may decided to continue with litigation in phase two and then pay an additional $5 million to it's attorneys. At that point, there are two potential outcomes. The patent may be found to not have been infringed upon. In that case, Cellectis will have paid the $3 million plus another $5 million or it will have lost $8 million. On the other hand, it could be that the patent is found to have been infringed upon. At which point Cellectis has yet one more decision. And that is whether to engage in phase 3 of the litigation, the damages phase. If Cellectis having won the patent validity and won the patent infringement phases decides at that point to stop litigation, it will have paid $3 million and $5 million, got nothing back, and it will still have loss $8 million. On the other hand, if at that point Cellectis continues with the damages phase, it will pay an additional $1 million to its attorneys and then face one of four equally likely outcomes. For example, if the outcome is $20 million in damages, then Cellectis will have paid three million and five million and one million. And received 20 million. Or it would have a net payout of $11 million. That's 20 minus one, minus five, minus three. If the damage is at 30 million, you can see that the payout would just be ten million more or 21 million. If the damages are 40 million, it would be a 31 million dollar payout. And, if the damages are 50 million, it would be a 41 million dollar payout. So now, we've constructed our decision tree. We've included decision nodes, event modes, and outcomes. Along the branches we've written the associated payout, the associated cash flows along the way. At each the event nodes we've included probabilities that sum to one and we've calculated the net payout to select this at each of the outcomes. We can now move on to analyzing the decision tree. So now that we have constructed a decision tree, we want to identify three different sets of decision strategies. The first set are called maxi-min decisions. Remember that these are decisions that maximize the minimum outcomes. So they're a risk averse strategies and what they're doing is they're trying to take the worst outcomes and make sure the worst outcome is as good as possible. To do that we start at the end of the tree and we work backwards towards it's root. At each event node we choose the minimum outcome, the worst outcome. And then at each decision node, we choose the decision that maximizes the outcomes of those decisions. So let's take a look. Here's our tree. We're gonna start up the first node at the end is the event node so we're going to select the minimum [COUGH] outcome. And you can see that, that's 11 million, we'll substitute the event node with the 11 million. And now there's a decision for Cellectis to make. And that is does it proceed to phase three in litigation. And it's gonna choose the decision that maximizes the outcome. In this case it's the 11 million. So yes, Cellectis is going to choose to enter interface re-litigation if it makes it that far. And so it will choose the litigation and will replace the decision node with the 11 million. We're back to an event node, and again we're going to select the minimum event outcome. Here that would be if Cellectis loses phase two of the litigation. That is, if the patent's determined not to have infringed, in which case, it will have lost $8 million and we'll substitute that for the event node and now, Cellectis has the decision of whether to enter phase 2 or not, given that it made it through phase one. And here you can see that the maximum value for Cellectis would be not to enter into phase 2 litigation and just take the $3 million loss, rather than the chance at an $8 million loss. We'll substitute the $3 million loss for the decision node. And we're finally at the first outcome, which is whether the patent is valid or not. Here, in either case, you can see the outcome is negative $3 million, so we'll substitute that value. For the event node, and now Cellectis has it's first decision, that is whether to enter into litigation at all or not. And if it does, it will lose $3 million, and if it doesn't, it will lose nothing. So to maximize the outcome, it will choose not to go into litigation. And we see that Cellectis's maxi-min strategy is to not to enter into litigation at all. The next set of strategies that we're going to look at are maxi-max strategies. Those are the opposite. These are risk-seeking, or reward seeking strategies that are intended to maximize the maximum possible outcome without regard to how bad things can get. Begin, we're gonna start at the end of the tree and work backwards. At each event node we're now going to choose the maximum outcome rather than the minimum cuz we wanna see how good things might be. Then at each decision node, selectors will choose the outcome maximizing decision. And again we'll work backwards towards the root. So here's the decision tree. We'll start at the end, and we'll select the maximum event outcome. So that in this case would be 41 million and we'll substitute that for the event node, and now Cellectis must decide whether to enter into phase 3, if it makes it that far. If it does, would have a maximum outcome of 41 million. If it doesn't, it would have a maximum outcome of negative eight million. So it's going to choose to enter into phase 3. We're going to substitute the 41 million for the decision node, and now we're back to an event node. Whether the patent was infringed or not in phase 2. And here, you can see the maximum outcome, again would be $41 million. So we'll substitute that for the event node. That's as good as things could get for Cellectis. And now Cellectis has a decision of whether to enter into phase 2 if it makes it that far. And you can see again that if it makes it that far, if it enters into phase 2 it can earn up to 41 million. And if it decides to stop at that point, it would lose 3 millions. So, again, the value maximizing decision would be to enter into litigation. And so we'll take that $41 million and we'll substitute it for the decision node. And, here, we're at the outcome of the event of the Phase one of the trail, determining whether the patent's valid or not. Again, we're going to select the maximum event outcome, which is $41 million, substitute it for the event node, and finally, Cellectis has a decision of whether to enter into litigation at all. If it does enter into litigation, it would have the chance of earning up to $41 million. If it doesn't, its gonna have the chance of earning zero. So Cellectis in this case would choose to enter into litigation. And we see that to maximize its maximum payout, Cellectis should proceed through all stages of litigation. That's the only way it can earn a positive payout. And it has the chance of running up to $41 million, if things work out for it. [COUGH] So we've seen the maxi-min set of decisions, maximizing the minimum outcome is to not enter into litigation at all. And really, at any stage of the litigation, to stop. The maxi-max set of decisions to maximize the maximum possible outcome, would be to proceed through all stages of litigation to have a chance at that $41 million. Finally, we're going to look at the expected value maximizing decision for Cellectus. Those are the risk neutral decisions that place equal weight on good and bad outcomes. As before, we're going to start with the tree's outcomes and work backwards to its root. At each event node, we're gonna calculate the expected value of the outcome. So that's the weighted average of the outcomes. We'll use the probabilities as the weights. At each decision node, then, we'll choose the expected value maximizing decision. So here we go again. We're gonna start at the end, and we'll calculate the expected value of phase three of the trial where we'll equally weight the $41, $31, $21, and $11 million outcomes. And when we calculate the expected value, it's $26 million. And we'll substitute that for the event node. So now, Cellectis must decide whether to enter into phase three litigation, given that it made it that far. If it does, the expected value is $26 million, if it doesn't it's negative eight. So Cellectis is going to choose to continue litigation if it makes it that far. And we'll substitute the $26 million for the decision node. Again, we have an event node and we're gonna calculate the expected value of the two events which would be if it wins phase 2, ten the expected value from then on out would be $26 million. If it loses phase 2, then the expected value is negative $8 million. So we'll calculate according to those probabilities of one-fourth and three-fourths, the expected value of the two outcomes, and it's just a half a million. We'll substitute the half a million in for the event node and now Cellectis needs to decide if it makes it to phase 2, should it enter into phase 2 litigation. If it does, the expected value's a half a million, if it doesn't, the expected value is negative $3 million. So we'll decide, should it make it to phase 2 to enter into litigation? And we'll substitute the half million for the decision node. The last event is whether Cellectis's patent will have been decided to be valid or not. And here, Cellectis needs to calculate the expected value of the two outcomes. If the patent's valid, then the expected value for Cellectis is a half million, if it is invalid, it is negative $3 million. And when we take the expected value across those two outcomes, with two thirds and one third probabilities, the expected value is negative $0.67 million. Will substitute the negative $0.67 million for the event node, and select this as first decision as whether to enter into litigation at all. If Cellectis does enter into litigation, its expected value is negative $0.67 million and if it doesn't the expected value is $0. In choosing to maximize the expected value Cellectis would choose to not enter into litigation at all. And so, to maximize its expected value Cellectis again would not enter into litigation. That's the same initial strategy as the risk minimizing, maxi/min strategy, not to enter into litigation. So what have we seen? We built a decision tree. We used decision nodes, event nodes and outcomes. We made sure that the probabilities that each of the event nodes sum to one. We added up the cash flows from the roots of the tree to each leaf to calculate the appropriate pad at each leaf. We then identified three strategies. The maxi-min, which is the risk minimizing strategy. The maxi-max, which is the reward maximizing strategy. And the expected value maxing strategy, which is risk neutral. We saw that the initial maxi-min and expected-value maximizing decisions coincided. In both cases, it was optimal not to litigate and to collect no money. The maxi-max strategy differed however. To have a chance at collecting damages, it had to continue through all stages of litigation. So, that's it for our review problem. Now that you've got it under your belt, it's time for you to go and work a couple of practice problems in preparation for the quiz. Thanks.
