Transcription
Okay, so as promised last week, on Thursday, I set up and solved the game for you. We talked about it a little bit, but as I said on Thursday, I didn't say all I wanted to say about that game. Indeed, you might not have had the opportunity to say all you want to say about the game. So I'm going to just put it up very quickly, go through it more quickly than we did on Thursday, and then I'll be jumping into, first, start off with some observations about payoffs.
But just to refresh your memories, this game has two players: an incumbent member of Congress and a challenger. They're both deciding whether to raise funds or not. They both want to win the election; only one of them can. Um, they both, although what they both don't like, is fundraising either.
The game had a sequence to it, so we represented it with a game tree, with the first mover's decision depicted at the top decision node. I'm trying my best to use all those vocabulary terms I was throwing at you on Thursday. The incumbent's decision node has a branch for each possible action the incumbent can take, and we were simplifying the situation a lot. We said the incumbent's choice was just raise funds or not (I'm abbreviating), and then whatever the incumbent did, the challenger got to react. Okay, so the challenger has two decision nodes, but again, I'll emphasize that only one of these is going to happen. Okay, I'm ever going to go down this path of the tree; the incumbent raises funds, and the challenger is then going to decide whether she wants to raise funds or not, or we're going to go down this branch of the tree. Okay, so the challenger has two possible decisions that she will have to make, but only one of them is actually going to appear in reality; only one, only one branch is going to be taken. We set up the tree in the natural sequence: the thing that happens first at the top and later decisions coming lower in the tree. At the very bottom of the tree, we put the terminal nodes, and what's in the terminal nodes are the payoffs. We put numbers, one for each player, that indicates how well they like the outcome associated with this particular set of decisions by all players.
Thursday, we spent a lot of time talking about the outcomes, talking about what would happen in the case that both candidates raised funds; what would happen if just the incumbent did; if just the challenger did; neither did. We spent a lot of time on me explaining my assumptions to you about the outcomes, but, as I said, the outcomes don't appear directly in the tree; we need to think about them in order to put the payoffs in the tree. And I'm going to, as I'm talking, just put in those payoff numbers that we had last time, but the outcomes don't actually occur. We can't—if somebody has walked into the room right now for the first time and looked at the tree on the board—it's a new game theory; they could solve the tree, but they wouldn't necessarily know what it's about. They would miss that part of it. All right, so the payoffs—first mover's payoff first—that's just a convention; you could do it the other way. That people don't. So when in Rome, do as the Romans do; when in game theory, do as the game theorists do: put the first mover's payoff first. And these were the numbers we were using. The idea of payoffs is very intuitive; it's easy to think of them as points in a game that you might play for fun. And I think you know, buddy was hung up by the idea that players want higher payoff, and in solving the game, the way we solved it, working from the bottom up, was for each decision node to ask which branch would give the player who controlled the decision node the higher payoff.
So using that idea, if we get to this node, the challenger compares her payoff from raising funds to her payoff from not raising funds. And I really want to emphasize that we're always comparing payoffs that belong to the same player. Okay, we're comparing the challenger's payoff for one outcome to the challenger's payoff for another. Okay, so this is a challenger payoff from what I'm going to abbreviate: raise funds, raise funds. Okay, incumbent raises funds, challenger raises funds; this is the challenger payoff from the outcome: raise funds, not. Raise funds, raise funds; raise funds, not. We're comparing challenger to challenger. Never in game theory do we need to compare one player's payoff to another player's payoff; we never do that. If you find yourself doing that, pinch yourself; you're doing something wrong. Okay, we only compare payoffs for the same player. Okay, I'm getting a little ahead in my outline, but I think this is something I'm probably going to emphasize more than once: we don't make interpersonal comparisons of payoffs. Okay, very taboo to do that. And the reason why is this idea of payoffs: you may or may not find it reasonable—a reasonable way to represent people's payoff. So I'll talk about how reasonable it is, when it might not be reasonable, later on. Okay, but it's much more reasonable to think that I, as one person, can give a higher or lower number to different things—different things that I could experience—in a way that would reflect my preferences than to think that I can compare how much it means to me to avoid fundraising versus how much it means to you to avoid fundraising. Okay, we can't compare my happiness, my utility, to somebody else's utility.
One way to see this is to sort of back off from this idea of payoffs and link them to a broader concept: the concept that, if you've taken any economics classes, I'm sure you've heard the word utility. What we're doing with payoffs is exactly the same thought experiment—actually, the exact same useful fiction—that we do in microeconomics when we use utility. It's the same idea: a number that represents how good you feel about something. In economics, utility numbers are often assigned to how good you feel about something you can buy. Do you get a higher utility from spending all your money going out to dinner and having a small apartment, or would you rather have a big apartment, stay home and cook your dinner? Is what that—that's the kind of scenario that we have encountered in a microeconomics class. Here, we're more interested in assigning utility to political situations, but it's the same idea; it's a very old idea. The term utility is usually associated with Jeremy Bentham, a major philosopher who wrote in the, I guess, late 18th century. I don't actually know that he coined the term utility, but he certainly popularized it, and the use that we make of it today owes much to Bentham. Um, one way that people often understand the concept of utility is that it gives us a way to compare apples and oranges; that's the cliché. Okay, and you may have heard this cliché expressed the other way: people tend to say that if they don't know how to make a choice, or they are uncomfortable with a choice that they have to make, they'll say, "Well, is this a better apple than that orange? I don't know; you can't make that comparison." Okay, so, in one sense, you can't make that comparison in a global sense of what is the perfect apple, what is the perfect orange; we don't need to do that. What you can't say, though—what we do all the time; we could not get through the day without making comparisons of: Do I want an apple now? Do I want an orange now? Do I want to speed up and go through the yellow, or do I want to stop? And we're just constantly comparing situations that are different. And the idea of a utility function, the idea of assigning utility numbers to different things that can happen, it's just a way for us to organize what we think ourselves are doing when we're making comparisons and making choices based on those comparisons all the time. It's a way to think about what we are doing ourselves; it's also, and for this class more important, a way for us to talk about what we think other people are doing. Okay, so when we're doing social science, we're constantly talking about what other people are doing, and as I try to emphasize last week, we need to be thinking about what our assumptions are about the people we're trying to understand preferences. Okay, utility numbers are the conventional way in economics, in game theory, in decision theory, and management sciences—you sort of throughout the social sciences—the conventional way to understand people's preferences, to understand how people make choices—choices that involve comparing apples and oranges, comparing not alike things.
Okay, so I hope you can kind of see how the utility idea gives us a way to compare apples and oranges. If the comparison is apples to apples, and say it's quantity: Do I want three apples or one apple? Let's say I like apples; I'll choose three; that's easy; I don't need utility for that; I can just count apples. Do I choose one apple or one orange? No, I don't know what I'm going to do, because they're different things. If I map both of those things—apple and oranges—into the same thing, into a number, I can compare numbers. Okay, so by assigning numbers to every possible thing, we can compare numbers; that's the idea. Let me emphasize again: I don't—I can't feel like I can't emphasize enough—nobody thinks utility is real. Okay, we don't even think it's something that, you know, these days they could do an MRI and see what's—what part of your brains are lighting up and measure how much dopamine you have floating around in your brain; that may—and some of those things may indeed give us some insight to how good the person is feeling at a particular point in time; that's not what we think utility is. Okay, utility is a useful fiction; we don't think it's real; it just is a way of helping us talk sensibly and coherently about preferences. And we do—and game theory is based on the idea that preferences are real, not that they're always stable, but there is enough reality to preferences that we need to think about them to understand how people behave and to understand how people interact. So the fiction is useful; it's tolerable to think that an individual person can compare apples and oranges. I think it's reasonable to assume that it's unreasonable to say the enjoyment I get from an apple is more than the enjoyment you get from an apple; we just can't know that. Okay, now if you were taking an analytic philosophy course—there are courses in the philosophy department that deal with issues involving utility that actually deal with some issues in game theory as well—that question of interpersonal comparisons would be on the table. Okay, so maybe I shouldn't come on too strong and say that we absolutely can't do it. What is a more reasonable thing for me to say, a more judicious thing for me to say, is that in game theory we don't have to do it; it's much, much dicier to think that, ah, we can compare one person's preferences to another. And the nice thing is that in game theory we just don't have to; so that's—that's very good. Okay, that's what I wanted to say about interpersonal comparisons. I think some of why you might care about interpersonal comparisons, why you might be tempted to make them, will be clearer once you start working on your homework problem. Okay, so take-home message here is: never, never compare one person's payoff to another person's payoff; it doesn't make sense; it's not part of the standard utility thought experiment.
The other thing I wanted to say about utility is about these—these payoff numbers; I was gliding over them a little bit quickly on Tuesday. And something that I said is that the numbers represent the order of preference. I think I very glibly said what you get from the utility number is: high number good, low number bad; and that's true; you do get that, but you get a little bit more. Okay, utility numbers for a person represent not only the order of preference, okay, it's not just the higher number is better for the person, but it also represents the person's intensity of preference. So over here, let me just summarize the assumptions I made about the preferences for both candidates; I had written it separately on Thursday, but there was a clear parallel: both candidates, we said, got a payoff of 10 from the outcome of where they win the election, no fundraising. Implicit in this little box of payoffs that I'm going to put up here is that they care about whether they win, and two, they care about their own fundraising. Okay, so if I'm the incumbent, I'd prefer that I don't raise funds, but I don't care if you do, challenger. I care about what the challenger raises funds; don't care about whether the incumbent does. The other possible outcomes—now I've got both candidates together, so actually all of these is going to be possible for at least one player—on winning with raising funds was worth eight; losing, no fundraising was worth three; and losing, raising funds—that was the worst outcome—we said that was worth one. Okay, so just as I wrote it up there, it's from best outcome to worst outcome, but in passing what I mentioned was you could think about these numbers as capturing two independent aspects of the situation. The way I set it up, there are two independent aspects. Okay, so that winning is worth seven points. Okay, how do I know? If I hold the fundraising constant, the difference between winning and losing is 10 minus 3; that's seven, or 8 minus 1; that's also seven. Okay, raising funds we could say is worth—what's raising funds worth to my utility?—with two—negative two. All right, raise it—the way I wrote it here, people—if I said not raising funds, that would be two, but raising funds costs you two. And I'm emphasizing this—what seems to be a minor point of language—because you're going to have to deal with these negative numbers in this problem set and all the problem sets you get. Okay, more so than in economic applications of utility theory, in political science, good things happen, bad things happen. Okay, it's not just giving up money to get stuff that is good; it's making choices that bring about good consequences, that bring about bad consequences. So when we are putting together the composite numbers that represent our net payoff from whatever outcome we're studying, we have to remember that good things increase our utility; bad things decrease them. Okay, another way to think about this is: when you're reading your problem set, I'm pretty sure I use the word cost at some point, and that if I didn't in this problem set, I'm sure I will in a future one. Another way we could just say the same thing is that raising funds costs two units of utility. Okay, it might seem obvious, but every year people get confused with dealing with aspects of choices that cost the players some utility. It cost the players some utility; it means you subtract that amount of utility. So if this was a problem set and I was giving you the list, Arial, analogous to the one that I have given you today, I might actually write something like what I just said: each player values winning at rough—at seven points; each player dislikes fundraising, and that costs the player two units of utility. All right, that's the kind of—that's the way to take that kind of ordinary language and translate it into a game.
Again, I said this on Thursday, but I think I said it really quickly: if we think about winning is worth seven points and fundraising worth negative two points—utility units, whatever you want to call it—there's sort of an implied baseline here. What's your payoff when you don't win and you don't raise funds? It's up there: three. Okay, baseline in this problem is three. I think I sort of flip—ly said that three was the value of a day job outside of politics, not being in office but not raising funds either. Most of the time we will use a baseline of zero. Okay, we get to pick what our baseline is. Okay, when I set up this example, I deliberately didn't pick a baseline of zero because what I want to show you—and I think I will show you explicitly in just a second—is that it doesn't matter. Okay, but usually having some kind of neutral outcome associated with payoffs—with a baseline payoffs—make it easy for yourself; make it zero. Yes, the baseline—the question is, what's your name? Neve asked whether the baseline is when you don't have any cost, you don't have any benefits; that is a natural one to choose as a baseline. Okay, and you will not go wrong by choosing that. In some scenarios, it's hard to figure out what that should be. What I want to let you know is it doesn't matter. Okay, you could pick any one of these outcomes as the baseline, and you would get different numbers to represent the utility. Okay, so, for example, if we said, "I'm going to let—um—the best possible outcome be a hundred," and that's pretty close to ten; you just multiply everything. I'm going to let the best possible outcome be five and make everything else relative to that, or—ly, the best possible outcome can be zero and have all the other numbers be negative; as long as the relative ship between the numbers is the same, I'll get the same answer in the game. And if you think about what I did to solve the game, of course that makes sense; I'm just comparing the numbers here, so you can: A, pick any—um—outcome as your baseline; B, you can pick any numbers you want; I could pick these numbers; it wouldn't change it; I could pick 1, 8, 3, 1; that wouldn't change it. I actually have a great deal of freedom in picking my numbers. What we are going to see, probably by the end of the week, is that most of the time in social science when we do game theory, we don't actually use numbers; we use variables, and the variables will allow us to do a little bit more. Okay, and those variables are sensitive to the fact that the numbers themselves don't matter as much as the relationship between them. You delay the baseline is—so let me step back and see the baseline is not part of the game; you could think of it as a stepping stone between the story and ordinary language and the game. And something that you'll be doing—your homework—something I'm going to be doing for you throughout this class is explaining the logic of how you go from the story to the game; that's an important part of knowing how to do game theory, and it's actually—let me step back and say—it's something you want to do in your homeworks. Okay, the homeworks will say, "Write down the game tree," and yes, you should write down the game tree, but it's actually a good idea to write a little paragraph explaining how you're setting up the game tree; that—that kind of support statement is part of using game theory to say something about politics. Picking a baseline outcome is a good way to explain why you're setting up the game the way you do, and in many contexts the baseline will sort of be natural—like most people would think that that would be a natural one. It's actually not clear to me that this has a natural baseline; losing without raising funds, you can sort of see that, but I could also see a logic of maybe picking this as a baseline and having everything be relative to it; needs suggesting. I think it's a good idea that if the game has kind of a status quo, okay, it's about—this is what happens if nobody does anything, and here's different ways that people can change it and then react to what another player does; that status quo is often a natural thing to use as a baseline. Okay, let me actually—I'm gonna leave these numbers here, and let's see—how am I—helping—not such a good color—let's use blue here. Another possibility would be just to take the advice that I gave you a minute ago: don't make the baseline 3, make it zero. Remember, I'm making these numbers up; they're just designed to convey my idea, so I can pick a number, and 0 is actually a nicer number. I'm going to pick—I'm going to leave these other two numbers alone—that winning is worth 7 points and raising funds gives me negative 2. Okay, so now I've got a different set of payoffs. Okay, my baseline here is zero now. Okay, if going from that baseline, I win and I don't raise funds, what's that going to give me for a payoff? Seven. Okay, if going from this baseline of zero, I win but I have to raise funds, what's my payoff? Five. Okay, if going from this baseline, I raise funds and I lose, what do I get? Negative two. Okay, so all fine—this is actually a good example because one thing it illustrates is that nothing special about negative numbers; just treat them the same. Let's put those alternate payoffs here in the game—that doesn't look alternate; that looks the same—my alternate payoffs here would be: in this situation, the incumbent's payoff is—I won, I raised funds—it's a five. What's the challenger's payoff? What is it? Negative two. Very good. Okay, incumbent's payoff here: five. Right, incumbent raise funds here. Okay, what's the challenger's payoff here? Zero is my baseline. Incumbent's payoff here? Yeah, challenger's payoff five; incumbent's payoff here—there's that seven—is that lovely payoff end right here; baseline again. Okay, so it's a different game now; instead of the game with the black payoffs, it's the game with the blue payoffs, but if we solve it the same way, we do the same algorithm. Let's just do it now. I'm the challenger here; if I get to this node, would I rather have a payoff of negative two or zero? I'll take the zero. Thank you; 0 is nothing, but at least it's not negative, so I will not raise funds. If we get to this—pat—point in the game, we go down this path in the tree; I'm the challenger; I'm comparing five to zero; five is good; I'll take that. I'm not—everyone say halfway—two-thirds of the way through solving the game right now; um, I've decided what I'm going to do at each of these nodes. Now I replace these decision nodes with the strategic equivalent; the strategic teaching equivalent of this decision node is the payoffs associated with the optimal choice of this node; the optimal choice at this node for the challenger is no—the payoffs associated with that choice are eight, three. Okay, it's a little bit tricky here is that the strategic equivalent has a payoff for both players, but the reason why this is the strategic equivalent and not that is only based on the challenger's—which Weis. Okay, so I've just pruned this whole part of the tree; I don't have to think about it anymore; all I need to know about this node is that the strategic equivalent is 8, 3. Over here, all I need to know is that the strategic equivalent is 0, 5. Yes, um, the reason is—okay, I'm even early today; I'm starting to make those mistakes. What's your heart? Okay, um, I did it right here. Okay, it's—um—I was pointing to the right place, but right now—green—we emphasize this green is supposed to be solving the blue payoff game, okay, and the blue payoff game doesn't have this payoffs in it; the strategic equivalent is five, zero. Okay, yeah. Oh, that's a particularly pernicious mistake because I think I would have gotten the right answer anyway, and those are the worst—the ones where I get the wrong answer; I can usually correct myself, so thank you. Okay, so strategic equivalent now of both of those nodes. Final step in solving the game: back up a level to this decision node; the incumbent looks at the two choices and if the strategic equivalent associated with it. Okay, this is the key ingredient in thinking strategically; the incumbent does not look at raise funds or not and say, "Oh, raising funds cost me two units of utility." Doing that would be wrong; it would be not strategic. When we use game theory to understand politics, when you assume that people are too smart to do that. Okay, we assume that the incumbent will say, "Raising funds is a pain in the short run, but it is the equivalent of a payoff of five; not raising funds is nice in the show run, but if I don't do it, if I anticipate what my challenger's going to do, I'm going to get a payoff of zero." Okay, so this—by replacing this row of decision nodes with the strategic equivalents—I can solve the higher node; if it was a game with even more nodes, then I would replace this decision node with its strategic equivalent; this five, zero would bump up even higher, and I would just do that till I got to the very top of the tree. Okay, now the point of this example was to emphasize that we get the same result when we use different numbers. Okay, so I hope that—ah—that part came through clearly here. Okay, all right, I think I'm just going to leave the game with kind of a double set of payoffs here—um—and let me kind of catch my breath a second and emphasize two basic points. Okay, one basic point is that by solving the game we get a prediction about what's going to happen. Okay, we predict that the incumbent is going to raise funds; the challenger is not going to raise funds. Okay, so one thing we get from a game is a prediction about what the players are going to do, and sometimes that is indeed what we use game theory for. Okay, that use of game theory is—it's used practically in what sometimes called questions of institutional design. Okay, if you don't like what's happening in a situation—if you're the professor and you don't like the way the students are performing—you might redesign the one institution you can control with it, which is the syllabus. Okay, and if you're a game theory professor, you think strategically when you do that and you know—try to put you guys in a position where your optimal choice will be the one that's going to make you learn the most. In more serious situations, if you are the United and you are sending blue helmets to a country whose people are killing each other and you're trying to figure out some way to rearrange how people move about, how people trade with each other, how people make their livings so that there's not such an incentive for violence and crime—this actually—this happens in a wide variety of situations—you might actually set—use game trees—try to figure out how—what people's preferences are; that's always a key component on that, but then figure out how they would respond to different types of choices. And sometimes you're in a position where you can change the choices that are available to people, or you're part of a conversation—more likely—where a group of interested parties are trying to change the choices people can make and how those choices add up to outcomes in a way that would make everybody better off, and in that case using a game to figure out what the outcome would be would be helpful. So I don't want to denigrate that fact that the game gives us a prediction; it's not worth a whole lot in this particular context because we started with the observation that this is what happens. Okay, I didn't need any game theory when I first told you this little puzzle on Thursday—that incumbents raise funds so insatiable even though their challengers don't seem to do much and they always win anyway. So we didn't need the game to tell us that incumbents would raise a lot of funds and that often challengers would not. What we were looking for was an explanation for why. Okay, that's something else we get from game theory; we get an understanding of why the predicted outcome occurs. More important, that understanding of why the predicted outcome occurs always depends on the outcome that didn't occur. Okay, I've been emphasizing this all along, and I will continue to because I think it's one of the most important lessons from game theory: if we want to understand what is going on in the world, if we want understand why it's going on, we have to ask ourselves what else could have happened and why not. Okay, what that means: whenever we're analyzing a game, we don't—we're not just satisfied with predicting what's going to happen; we also want to make sure and say something about what we think would happen in the counterfactual. Okay, that's—that's key to the logic of game theory. Okay, may not get to the full explanation of this today, but we're going to be developing this idea of an equilibrium, okay, a situation where neither player by themselves can improve by making a different choice. Given that the incumbent has raised funds, the challenger is not going to do better by making different choices than she did; and given that the challenger's decision is to raise funds—of the raised funds—that the incumbent doesn't—and not raise funds—if she does, there's nothing the incumbent can do that would bring about a better outcome. Okay, the idea of equilibrium brings with it an idea of what we'll call the equilibrium path; the equilibrium path—all right, it over here—what we expect at each decision node. Okay, another phrase for this is—well—the predicted outcome. Okay, what do we think the guys are going to do? What do we think we're going to see in the situation? This, but just as important is what's going on off the equilibrium path. What's going on off the equilibrium path over here is what helps us understand why. All right, so I want to say some things about—I may add some more vocabulary terms here that are going to help us manage this counterfactual part of the solution of a game. Okay, this idea that when we understand the strategic situation, we not only understand what's happening on the equilibrium path but what's happening off the equilibrium path, and the reason why we need to know what's going on—what would be going on off the equilibrium path—is that that's the reason why we do observe what we do. Okay, all right, so on that—you use—I'm not gonna use green; that's not a good—this—all right, so I think I will just put some—a couple of definitions here. First of all, I'm going to make a distinction between strategies and actions. In ordinary language, um, those two things could mean the same thing; in this context, what's your strategy? What are you going to choose on it? They almost seem synonymous. In game theory, a strategy is a more complicated thing than an action. Okay, so let's say—Y—okay, so they like—mine—this tray with these fake markers that don't work. An action is a choice at a node. Okay, so action is a really simple thing; an action corresponds to one branch. Okay, so this is a decision node with two possible actions; here's another one with two possible actions. At some point we'll probably do an example where the decision-maker has three possible actions at one node; guess what they choose—the highest payoff of the three or the four or however many—nothing—nothing too strange about that, but action is just this one simple choice, one branch in a tree. A strategy is composed of actions. Okay, so I'm going to do this definition and more of a complete sentence: a player's strategy is an action for each node she controls in the game. All right, so let's do—let's go back to the black payoffs now; I'm going to get rid of the blue ones, and let's use this board space here to talk about strategies and what is an example of a strategy for the challenger. It would be a strategy for the challenger—no—people don't know how to put it in words; I think you know—what would be an example of a strategy for the incumbent? Raise funds. Okay, the incumbent only has two possible strategies in this game: raise funds or not. For the incumbent, there is no difference between a strategy and an action because the incumbent only controls one decision node. Okay, let me put it—the question to you in a little more precise term: the challenger has some strategies in this game. What's the challenger's best strategy? What's the strategy we think the challenger is going to play? Say it louder. Raise funds. Is that a full strategy? When incumbent does not, and not when incumbent does. The challenger's strategy is more of a mouthful; that's why it was kind of unfair for me to ask you guys to say—I've been usually asking you things that you can answer with one word. Okay, this is one of the challenger's four possible strategies. Okay, I'm going to write down another one, and I'm going to use a kind of shorthand that you're going to find very useful for your—for your homeworks. Okay, another possible strategy that the fundraiser has is: RF if RF, RF if not. Okay, so I'm using shorthand here. Another strategy that the challenger has—it's not the challenger's best strategy, but it's a possible strategy—is to raise funds if the incumbent raises funds and to raise funds if he doesn't. Challenger could just be a fundraising Energizer Bunny; that's—it's a strategy. Good. Say not the best one; not as good as this, but it's a possibility. Another even shorter way to abbreviate. Okay, so one way to abbreviate—and this is actually the way I like to do it, and I would encourage you to do it this way on your homeworks and even more so on your exam—because this way, to me, is still staying in touch with the story here. Okay, I raise funds if the incumbent raises funds; I raise funds if the incumbent doesn't. I'm saying especially on the test because on the test—especially on the midterm—you know, we only have an hour and 15 minutes here; you guys are going to be pressed for time; you're going to think, "I just got to write as fast as possible," and so you're going to be tempted to use the form that they use in the book, and let me just show you what it is. So another possibility would be: on—to raise funds if the incumbent does not raise funds. Okay, this is the most terse abbreviation of the challenger's strategy, and let me decode it for you. Okay, so this is action at node—right—left—leftmost node; action at next node. Okay, in coming up with a strategy for a player that's the second move or a third mover, what you're going to do is you're going to go across this row of decision nodes that is controlled by the challenger, and you're going to list all the combinations of actions at those nodes. And one way to do it would just be to say, "A strategy is: raise funds here, not here; raise funds here, raise funds here; not here, not here; not raise funds here, raise funds here." I think those were the four—yeah—I don't want to say this is wrong; people definitely do it, but what I will say is that if you're doing it too fast on an exam, even though it seems like you're saving time, if you get confused, you get to an answer that doesn't make sense, you want to go back and check your work; you're more likely to understand what's going on in the game if you have it set up like that; this is just going to look like symbols. Okay, so this is my preferred way to denote a strategy. When you see people denoting strategies just by a string of actions, the way to interpret it is that it's the action associated with the leftmost node first and the rightmost node second. In this case, where there's just two—if there were—there was a third decision node over here, we'd have to have three actions. Okay. Challenger has one more—arm—strategy. What's this challenger's other strategy? Don't raise funds no matter what. That's exactly right. Okay, so number four is: and if RF, and if not. For both of these abbreviations,
A higher payoff? Okay. The Challenger is part of the equilibrium analysis; this is kind of obvious because it was part of solving the game. We kind of have to step back to see the incumbent's part of that. That given the Challenger's full strategy, the incumbent cannot do better by changing. Okay. And again, that's how we're getting this explanation of why things are the way they are from the game. Okay.
After class on Thursday, I got a question that sort of pointed to this ten payoff over here. Okay, great payoff, right? Why can't the incumbent get to that payoff? Because of the strategy that the Challenger is playing. In one sense, the incumbent can do better in this game than the outcome we're predicting; the incumbent could get a ten, but not when the incumbent—when the Challenger is playing this strategy. Okay. That's why the incumbent's choices are equilibrium.
So let me write the punch line here: Equilibrium in this game is incumbent raises funds, Challenger does not; if incumbent raises funds, but raise funds if not. On your homeworks, on the exams, in any application of game theory, anytime somebody asks you what the equilibrium in a sequential game is, this is the kind of thing they're looking for. They're looking for a pair of strategies for each player, and each strategy—I'm going to give the definition in a slightly different language—each strategy is a best response to the others. My strategy is the best response to your strategy; your strategy is the best response to my strategy. Our choices reinforce each other. You're the incumbent; given what you did, I'm the Challenger; I'm glad I didn't raise funds. It's a little bit harder to see from the incumbent's side, though, but now we're able to see it. If I'm the incumbent, given that this is your whole strategy, including the off-the-equilibrium-path part of your strategy, given what you did and what you would do, I don't regret what I did. So the second mover's strategy has to be both what they do in equilibrium and what they would do otherwise, because what they would do otherwise is often the reason why the first mover makes the choice that she does.
So let me just emphasize that this is just a stated definition of equilibrium. Same idea, just in different words. Okay. Another thing, another vocabulary thing I want to alert you to is the book calls this rollback equilibrium. What we do when we solve the game with the rollback process is we find the equilibrium that's been at the key, at the core of everything that we've done so far. Okay. Rollback. Later in the course, we'll start to talk about what that means. Right now, rollback is the only kind of equilibrium we know about. I'll just sit tight with that. When we do the rollback process, what we get are the strategies in a sequential game that satisfy this idea of equilibrium. What makes rollback special is that it only applies to games where first one player moves and then another player moves. What we're going to be getting to in probably the second two-thirds of the class are the same idea of equilibrium, but in cases where the players make their choices at the same time. Those are a little bit harder.
Okay, see what else I—all right, yeah. I want to do my—I want to do one last thing. I think we've just got the right amount of time to do it: this difference between strategies and actions. Then action is a simple thing; strategies are composed of actions. Strategies are more complicated things. Um, you guys are all nodding right now. I know you get it right now. I know some of you are going to lose it. It's—it's—I don't understand why it's slippery, but it is slippery. It's—it's not the natural way we think about strategy. So just to maybe juxtapose that in ordinary language: this is a Challenger; you're working for this Challenger, or your journalist interviewing them, and you say, "What's your fundraising strategy?" In ordinary language, the Challenger could either give a game-theoretic definition of the strategy. In ordinary language, the Challenger might say, "Well, my strategy is: if the incumbent raised funds, I will alone; if the incumbent didn't, I will." But more likely, especially if we already know what the incumbent is going to do, if you asked the Challenger what his strategy is going to be, um, he would say—he wouldn't necessarily just say, "I'm not going to fundraise." He'd say, "You know, I'm above money politics, so I'm going to rely on my—the strength of my ideas and my grassroots support." But you know what his strategy was, and but more importantly, that ordinary language response that he would give would just be an action. So an ordinary language—in the newspaper, when we talk about strategy and politics—I think it's also true when we talk about strategy and games and sports as well—but a full strategy in a football game or something like that would have branches in the trees: What are we going to do if they do this? What are we going to do if they do that? But in the middle of the game, what's the strategy going to be on this play? The ordinary language answer to that would be more like an action.
So because keeping the distinction between strategy and action distinct is a little tricky—it's unusual for us; it's doing something that we don't otherwise do—um, I think on your homework I've asked you to do some strategy counting, and if I haven't, I will. Let's count some strategies in this game. Okay. This game is going to be kind of quick and dirty here; it's not going to have a story. We're just going to have Player 1 here. Player 1 has three possible actions. Okay, they are left, middle, and right. How interesting is this story? Once Player 1 makes a choice, Player 2 makes a choice. Player 2, regardless of what Player 1 does, Player 2 can just choose left or right, right? Okay. So those are the—and we're not going to solve this game, so I'm not even going to put payoffs here—but something that you sometimes might want to ask yourself about a game, if you're trying to figure out whether you set it up right, is to ask yourself how many strategies each player has, okay, or to enumerate all the strategies just to make sure they're all there. Okay. So Player 1, how many strategies—excuse me, question in the back? Three. What are they? Left, middle, and right. Okay, Player 1, no problem. Player 2, what is one strategy the player 2 could play? And as much as I don't like this format for writing down—it's okay to call it out as an answer. Okay, L comma L. What—actually, see the whole thing—see the whole thing. Let me not quite—so you were thinking of L comma R. Okay. There are games that would have that, but in this game we've got Player 2's decision here, Player 2's decision here, so here's a strategy. Okay. Another one would be L comma R. Come on. Okay. Um, how about this one? Another one. All right. Do you guys see what I'm—what I need to do here? The strategy needs to tell Player 2's action for each node. There are three possible nodes the Player 2 could find themselves at. Okay. So you can either find yourself at the node where Player 1 played L, where Player 1 played M, where Player 1 played R, and her strategy has to tell her what to do at all three of them. Okay. So Player 2's strategies have three components. Okay. One way that I find helpful, um, and to think about strategies is: a strategy is something you could program into a computer. Okay. You know, if you've ever done any programming, the frustrating thing about computers is they have no common sense; they won't see a pattern; you have to tell the computer what—you have to tell the program what to do for every possibility that can arise; they have no ability to think for themselves. Okay. That's what a strategy does. Okay. Strategy really is something that you could program into a computer. Link. Okay. So what I need here—we go back to the definition of the strategy—is an action at each decision node. Okay. Each decision node controlled by Player 2. Okay. So again, strategies—whenever we think about strategies—we have to think about which player it is because we have to know how many nodes will be included. All right. Player 2 has—I'm going to use letters here—it could be at node A, node B, or node C. If I wasn't already using numbers to represent the players, I'd probably number these one, two, and three. Not necessarily part of a game, but it helps me remember that all three of these nodes belong to Player 2. So what I'm doing here is I'm thinking, what is a set of actions for each node? And the key here is Player 2—V2—has three nodes; strategy needs three components. Yes, there are more; there are more. Yeah, got a candidate for one: R, L, R. That's a good one. It's not just those three. Anybody want to think about how many strategies there are total? Yeah, eight. Eight is right. How'd you get eight? It's two to the third power. That's exactly right. The total number of strategies is the number of choices here times the number of choices here times the number of choices here: two times two times two. If we switch to black here—added medium here. Okay. So now I'm making kind of a very funky game. One can choose left, middle—I guess middle, not medium—sort of the same thing. Well, I can choose left, middle, or right. If one chooses left, two can choose left or right. If one chooses middle, one—two can choose left, middle, or right. And if one chooses right, two can choose right or left. Like maybe one choosing middle is actually paving a road that two can only walk on—ephah one has already started it. Now what do we have in terms of how many strategies in that game? Twelve. Okay, because we have two times three times two. Okay. The total number of strategies is the product across all nodes of the number of actions at each node. Okay. This looks weird, doesn't it? Okay. It—you'll do a couple of examples. If you feel like you're frustrated with homework—example one of the first two to three problems in Dixit and Skees—those early problems in the back of the chapter have some strategy counting problems, and I would recommend them. I'm going to recommend some problems in Dixit and Skees for you before the midterm anyway, but certainly doesn't hurt to—um—to start doing some now. Okay. So stay tuned Thursday; we're not done with this game yet. We're almost done with it. Bell.