Political Science 30: Politics and Strategy, Lec 3, UCLA

Political Science 30: Politics and Strategy, Lec 3, UCLA

okay so as promised last week on on Thursday I set up and solve 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 and 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 okay 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 fundraising either okay 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 on Thursday the incumbents decision node has a branch for each possible action the incumbent can take and we were simplifying the situation a lot we said the incumbents choice was just raised 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 okay we set up the tree in the natural sequence 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 okay 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 okay 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 spend a lot of time on me explaining my assumptions to you about the outcomes but I 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 okay we can't if we 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 okay 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 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 okay 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 ok we're comparing the challenge is payoff for one outcome to the challengers payoff for another okay so this is a challenger payoff from what I'm going to abbreviate raise funds raise funds okay and come but raises funds challenger raises funds this is the Challenger payoff from the outcome raise funds not okay raise funds raise funds raised funds not we're comparing Challenger to challenger never in game theory do we need to compare one players payoff to another players 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 head 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 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 that 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 major philosopher 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 cliche ok and you may have heard this cliche 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 isn't orange I don't know you can't make that comparison okay so 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 there 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 is also and for this class more important a way for us to talk about what we think other people are doing ok 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 gain 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 the 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 dicey err 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 is 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 sad 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 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 okay 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 fund raising was worth three and losing raising funds that was the worst outcome we said that was worth one ok 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 ten minus three that's seven or eight minus one that's also seven okay raising funds we could say is worth what's raising funds worth to my utility with two um 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 cost you two and I'm emphasizing this what seems to be a minor point of 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 an economic applications of utility theory in political science good things happen in 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 sad I'm sure I will in a future one another way we could just say the same thing is that raising fun coughs 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 listen 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 run 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 okay 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 and this problem is three I think I sort of flip li said that three was the value of the 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 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 orally the best possible out can be zero and have all the other numbers be negative as long as the relay a shoe 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 one point eight point three point one that wouldn't change it I actually have a great deal of freedom and 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 slime 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 incumbents payoff is I won I raised funds it's a five what's the challengers payoff what is it negative two very good okay incumbents pay off here five right come but raise funds here okay what's the challengers payoff here zero is my baseline income its payoff here yeah challengers payoff five incumbents 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 street 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 ok 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 path 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 raised 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 challengers 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 note 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 challengers 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 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 inaction because the incumbent only controls one decision note 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 challengers 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 challengers 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 challengers for 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 arm for your homeworks okay another possible strategy that the fundraiser has is are F if RF RF if ant okay so I'm using shorthand here another strategy that the Challenger has it's not the challengers 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 okay 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 um 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 a ride 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 challengers strategy and let me decode it for you okay so this is action at naught right left left most 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 raids 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 before 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 left most node first in 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 okay challenger has one more arm strategy what's this challengers other strategy don't raise funds no matter what that's exactly right okay so number four is and if RF and if an for both of these abbreviations what you want to be thinking of is that for each decision know that the player has you're going to have that many strings that met many things strung together by commas okay strategies by themselves really is all your actions rather actions by themselves is really all you need to have the full strategy it's just a little helpful I think to have this level of background information in there okay alright so relationship one is strategies versus actions okay so actions are like our atoms they're the smallest thing one branch and a node strategies are combinations of actions sometimes a strategy as simple as in the case with the incumbent sometimes the strategies are a little more complicated if you get big game trees with lots of decision nodes just writing down the strategies can be a challenge okay thankfully we can do a lot of analysis I think we can actually get more insight from small games with few decision nodes than big hairy games with a million decision notes strategies more complicated than actions and then the next step up is equilibria are more complicated than strategies okay so let me which I think I want to leave this and um I've already started writing some stuff about equilibrium over here so let's get the the full definition here okay equilibrium is a strategy for each player such that neither player could do better given the other players ' could be here or could be here however many players there are one other many other strategies okay so we're building up action is composed of Stratton is composed of nothing action is just an action a strategy is composed of actions and when I wrote the definition of a strategy I had to specify that a strategy belonged to a player okay so when you say what's a strategy or you have to single out one player to pick a strategy equilibrium is made of strategies one for each player okay strategy action for each node equilibrium strategy for each player's components are simple but they build up it's not just any old strategies though if the strategy set that composes the equilibrium is really an equilibrium it means that neither player can change can never play can do better by changing unless the other player changes okay so if you keep on doing your thing you're the Challenger I'm the incumbent if you are playing this green strategy here okay if you're going to not raise funds when I do but raise funds if I don't then I can't do better by not raising funds okay similarly if I'm the Challenger and the incumbent is going to raise funds I can't do better by changing my strategy either okay there's nothing I can do if the incumbent is going to raise funds there's nothing I can do to get myself a higher payoff okay the Challenger is part of the equilibrium analysis is kind of obvious because it was part of solving the game we kind of have to step back to see the incumbents part of that that given the challengers 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 um the 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 okay in one sense the incumbent can do better in this game than the outcome we're predicting the incumbent could get a tenth but not when the income when the Challenger is playing this strategy okay that's why the incumbents choices equilibrium okay so let me write the punch line here equilibrium in this game is incumbent raise funds Challenger 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 are going to give the definition in a different slightly different language each strategy is a best response to the others okay my strategy is the best response to your strategy your strategy is the best response to my strategy our choices reinforce each other ok 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 incumbents 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 start part of your strategy given what you did and what you would do I don't regret what I did ok so the second mover 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 ok so let me just emphasize that this just a we stated definition of equilibrium okay same idea just in different words okay another thing another vocabulary thing I want to alert you to is the book calls this roll back equilibrium what we do when we solve the game with the roll back process is we find the equilibrium that's been at the key at the core of everything that we've done so far okay roll back later in the course we'll start to talk about what that means right now roll back is the only kind of equilibrium we know about I'll just sit tight with that when we do the roll back process what we get are the strategies in a sequential game that satisfy this idea of equilibrium okay 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 the probably the second two-thirds of the class are is the same idea of equilibrium but in cases where the players make their choices at the same time those are a little bit harder ok 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 action strategies or 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 an 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 income is going to do if you asked your the challenger what his strategy is going to be um he would say Snee 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 inaction okay 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 okay 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 one makes a choice player two makes a choice player two 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 to 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 sit 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 two's decision here player two's decision here so here's a strategy okay another one would be L comma R come out 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 two's action for each node there are three possible nodes the player to could find yourself at okay so you can either find yourself at the node where player one played L we're player one played a more where player one played R and her strategy has to tell her what to do it all three of them okay so player two'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 they 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 ok 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 two okay so again strategies whenever we think about strategies we have to think about which player kids because we have to know how many nodes will be included all right player two 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 two 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 v 2 has three nodes strategy needs three components yes there are more there are more yeah got a candidate for one our lr 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 switching 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 choose two can choose left or right if one chooses middle one can two can choose left middle or right and if one chooses right – 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 up 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 skis 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 ski 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

11 thoughts on “Political Science 30: Politics and Strategy, Lec 3, UCLA

  1. Just watched a brilliant UCLA professor guide me through the concept of game theory while doing the dishes. What a time to be alive!

  2. Seems like a great Professor intriguing class but i just don't think with a BA in political science you can make $$$

  3. I'm sorry but seems to me that you didn't understand the point of the game. The idea is to simplify the concept of politics in order to take reasonable decisions. To do that, she must take some assumptions of how world works and not argue about it. That's why the game is so simple, because, as a didatic method is much more efective. Of course the UCLA students have a general idea of what are the problems in USA politics and I believe they do talk about it in other classes.

  4. I'm really lost on something here. Where the fuck is she getting 8 possible realities at the end with player 1 player 2? It's L, L…END where is there 8 possible paths? She has L,L,L….So I don't see where this third L is coming from, there is no third node.

  5. oh great… it's camera guy from lec 1 is back. dude, you don't have to keep moving the camera to keep her in frame. it's totally fine if she's talking off screen. sometimes, she's referring to something on the board, but the camera man's attention shifts to her and we at home can't keep up with what she's explaining on the board.

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