welcome to game theory online I’m Kevin Leighton Brown I’m one of the three instructors for this course the other two are Matt Jackson and yohave shoham both from Stanford University you’ll see them both in subsequent videos this video is going to give you a high-level sense of what game theory is all about and the kinds of concepts that we’re going to think about in the rest of the course first of all before I go on let me tell you a little bit more about what game theory isn’t game theory doesn’t use the word game in the way that most of us are used to in common life and it certainly doesn’t think particularly about computer games instead game theory is a way of thinking about strategic interactions between self-interested people for this reason it’s very important for economics and also for computer science political science psychology and a variety of other disciplines what ties all of these disciplines together is a concern for thinking about how self-interested participants would behave in strategic interactions and also thinking about how those interactions should be structured for example by a government or by the designer of a computer system in order to lead to good outcomes I’m going to begin by thinking about one such example from computer science this is an example that involves networking but don’t be scared off by the computer science content it isn’t representative of what will come in the rest of the course and in any case I’m not going to assume that you have any particular knowledge about how computers work in this example I’m going to begin by thinking about this pop up which you might have seen in your browser before and if you’re like most people you realize that a pop-up in your browser that promises slow connection detected click Next to correct maybe shouldn’t be trusted it might install a virus or otherwise harm your computer so you probably wouldn’t click on this but the interesting thing is if you did this particular pop-up might actually help you I’d like to think about how it works and we can use this example to illustrate something interesting game theory before I do that I need to tell you a little bit about how the TCP protocol works which is one of the backbones of the internet so as you probably know if you’re over here on the internet and you want to communicate with some other computer which is over here what happens is that your communication gets broken up into a bunch of different packets which conceptually are kind of like envelopes with a message inside them that get delivered across the network to your recipient and when I say delivered across the internet I mean you don’t actually have a direct connection between your computer and your desired recipient instead there’s a whole sequence of different computers along the way who pass the message one to the next to get it from you to your recipient so you pass the network the message along the network to some computer you’re connected to it passes it to another computer and so on down the chain until it reaches your recipient at that point your recipient sees that the message is addressed to it and it sends back an acknowledgment to you telling you that it received the message and that acknowledgment likewise passes through a whole sequence of computers until it gets back to you so far so good here’s the catch sometimes a computer in the Internet is overwhelmed with messages let’s say this one right here and when that happens it handles this congestion in a pretty surprising way it takes some of the messages that it receives and it just throws them away and it doesn’t tell anyone it just deletes the messages until it gets down to a level that it can handle again and then with the stuff that it can handle again it continues behaving as it should passing messages on appropriately well you might wonder then how it is that you end up with reliable communication over the Internet given that every now and then some computer on the internet throws away your messages well the way that this works is that your computer waits a certain amount of time after sending a message to see if it gets an acknowledge and if it doesn’t it assumes that the message was never received and sends it again here’s the part that’s important for our discussion of game theory your computer also does something else in this situation it slows down the speed at which it continues to send messages in the future on the assumption that there’s some congestion somewhere in the network and that this congestion can be reduced by bombarding the network with fewer messages per unit time and likewise other computers on the Internet are doing the same thing that’s why we don’t have the network completely saturated that’s why most of the time we get pretty reasonable throughput on the internet because everybody is balancing the speed that they send messages out using what’s called this back off mechanism in the TCP protocol okay that’s all you need to know about the back off mechanism I’d like to think about the strategic problem that you face in deciding whether to install this somewhat suspicious-looking piece of software that is I’d like to ask should you send your packets on your network connection using a correctly implemented version of the TCP protocol which does have the back off mechanism inside it or should you run this program and instead use a defective implementation which disables the back off mechanism and just blasts the network all the time without any concern for the congestion that it will cause other people or you well this is a bit of a surprising use of language but problems like this one are what game theorists called games a game in general is any interaction between two or more people where the outcomes of the interaction depend on what everybody does and everybody has different levels of happiness for the different outcomes so let’s think about a two-player version of this interaction which a game theorist would call a two-player game you might incidentally worry that the Internet has a lot more than two people using it and so that this two-player restriction is going to be a problem you’ll have to trust me but this example scales very naturally to larger numbers of players and everything interesting about it would remain true so in the two-player case we have a question of whether each of the players should use a correct implementation whether one of them will use a correct implementation and the other one a defective implementation or whether both of them will use defective implementations in the case so we so we need to say what happens in order to analyze this let’s say that when both players use correct implementations they both experience a delay of one millisecond let’s say that if one person uses a correct implementation and the other person uses a defective one then the person with the defective implementation manages to flood the network with packets in a way that causes the other person to back off pretty heavily causing the person who backed off to experience a much longer delay and the person with the defective implementation to get their packets through virtually immediately lastly let’s say that if both people use defective implementations of TCP then we’re again in a symmetric situation where they both experience the same delay and they both experience a bigger delay than they would have before because there’s now a greater chance that their packets will be lost at every stage in the chain and so it will take them longer to send a message well I’d like to encourage you to play this game with a friend or to play it even just in your head or best of all to play it on the online system that we’ve provided where you can interact with other students in the class what do I mean to play a game well this game might not seem very exciting to play as compared to other things that you would call games like soccer or chess but in principle all of these games are the same there are sets of actions that players can take and after everybody has chosen what they’re going to do in the game there’s some result where everybody feels at different levels of happiness this very simple game has each player choose either to use a correct implementation or to use a defective implementation and once we know what both players will do we can look at these rules that I’ve given here and decide how happy both players would be of course nobody likes delay so the players are trying to minimize the amount of delay that they experience in the network so if you wanted to minimize the amount of delay that you experienced how would you play this game that’s kind of the most natural question to think about when you’re thinking about a game theoretic setting but I’d like to invite you to think about a bunch of other more abstract and philosophical questions which we’ll also address throughout this course first of all do you think it’s the case that all users should be expected to behave the same in a situation like this relatedly if you’re not one of the players of the game but rather you’re someone who cares about how the whole system works from the outside for example the designer of the network what kinds of global patterns of behavior would you expect to see emerge you’ll notice that these numbers that I came up with here are a little bit arbitrary and they’re not very precise it’s reasonable to wonder how much these predictions that we can make about how the game should be played and what behavior would occur depend on those numbers is it the case that for slightly different numbers we would expect to see very different behavior what effect would there be if players could communicate with each other before they played the game in a non binding way what effect would there be if players could repeatedly play the game against each other either for a finite number of repetitions or infinitely finally how important is it how I model my opponent is it different if I think my opponent is rational and does something that is in his or her own best interest or would I play this game in the same way regardless of how I believe my opponent is thinking about the game these are examples of the kinds of questions that this course will help you to think about and we’ll offer you some answers to and the tcp back off game is just one example of a real-world situation that we can examine using game theory throughout the course will describe many more real-world examples that game theory can be used to think about hi folks its Matt again and now we’re ready to start solving games and making some predictions of how people will play in different settings and so we’re talking right now about Nash equilibrium which is one probably the most basic and standard solution concept of all in all of game theory it’s named after John Nash who was a mathematician at Princeton and actually some years back won the Nobel Prize for his work on this subject and it’s a very basic and fundamental concept and in order to sort of motivate it let’s let’s start by just talking through some a particular game that was described and invented by another famous person so this is John Kane John Maynard Keynes beauty contest game so what’s the idea here so let’s let’s think of a basic situation that you might be interested in and this was one that Keynes described in some detail so the idea was you have a stock you’re holding on to it and the stock price is rising that’s great you’re an investor you’re trying to make profits off of your your stock holdings and you begin to believe that maybe the stock is too high to be justified by the value of the company so you’re thinking that it’s possible that this stock is overvalued maybe there’s a bubble in the market and you’re starting to think about selling okay well you’d like to sell it but you’d like to wait until the price is at its peak right so you’d want to wait until the the price was just where it’s going to hit its maximum before you sell so you wanted to get out of the market just before the other investors do so this is a game where now you have to predict what other people think about the stock price and and what they’re going to do and when they want to get out so how will they act how should you respond to that so this is the basic ingredients of Nash equilibrium are going to be having some prediction of what other players are doing and then choosing the optimal strategy in response to that so these are they’re going to be too Keating means that we have so there’s a very stylized version of this which is known as the the Cannes beauty contest game where did it come from well actually Cannes described the there was a newspaper in the in England that had a contest where players had to guess which picture of several women other readers would think was the the the most attractive one so it wasn’t to guess what you thought but what you thought other people were thinking so Keynes likened investing to this it’s not only what you think of this the stock but what you think other people are thinking about the stock that’s important and driving your decisions okay so so this now is represented by a very simple game that is is played by many people so what’s this game look like each person gets the name on integer between 1 and 100 ok so you get to pick a number between 1 and 100 s to be an integer so 1 2 3 etc players are going to move simultaneously and the player who names the integer that’s closest to 2/3 of the average integer wins a prize and the other players get nothing so to win this game you have to get you have to guess the average and then 2/3 of it right so you want to be right at 2/3 of whatever the average guesses so a little bit below the average guess if there’s two people that happen to hit the same integer that it’s the the right one then ties are going to be broken uniformly at random so we’ll just flip a coin or if there’s three people will we roll a dice a three-sided die etc okay so how would you play this game you have to think about what other players are going to do and then forecast what you think the best integer is in a response to that this video is going to introduce the idea of mixed strategies and explain why they’re important to Nash equilibria so the example that I want to think about here there is the United Nations setting up checkpoints to defend against terrorist attacks at a port in Somalia and you can see that what happens is they they place a checkpoint on the road they stop all cars like this taxi here and they go through the contents of the car to make sure that it doesn’t contain any explosives or other dangerous materials now let’s think about this situation as a game so there are a variety of different roads that the UN could choose to set up its checkpoint on every hour and for each one of those roads the the potential attacker could decide to attack that road and if it’s the case that the defender defended that road and the attacker attacks it then the attacker gets a large negative payoff because they’re captured and their attack is not successful if on the other hand the attacker attacks any other road then the attack is successful and then the utility of the attacker depends on the value of the target that was attacked and wasn’t defended now it’s pretty clear that if the UN were to commit to any deterministic strategy if they were to choose their action in any deterministic way things are going to go pretty badly for them because the defender sorry the attacker would be able to look at what they’re doing then they’d be able to watch for a while see what strategy the UN is following and then attack something different and attacks would always be successful so it must be that this is not really how checkpoints get set up and indeed what really happens is that the checkpoints are set up in a randomized way so that even if the attackers watch for a while and figure out what the randomized strategy is they’re not able to know on a given hour where the checkpoints are going to be and this means that their value of an attack is limited so the Nash equilibrium of a game like this is going to involve the defender randomized way like this and this kind of a randomized strategy is called a mixed strategy and that’s going to be the topic of the next sequence of videos in this video we’re going to look at some additional solution concepts other than the Nash equilibrium so these are different ways of talking about which outcomes of a game make sense from a game theoretic perspective first of all I want to talk about a solution concept called iterated removal of dominated strategies and I want to illustrate this by the example of grace shown in this picture here who decided to jump out of a plane to celebrate her 91st birthday so I want to think about a game between Grace and the guy that she chose to strap herself to who you can also see in the picture and in particular I want to think about his decision of whether to pack the parachute safely or not and her decision about whether to I jump out of the plane or not now in principle she might worry that he would choose not to pack the parachute safely and she would choose to jump out of the plane and if that were to happen then she would never get to celebrate her 92nd birthday but you can see in fact she did choose and indeed she landed safely and her choice was a good one so how was she able to reason that this was sensible well if she looked at the payoffs of the game she would see that that this guy let’s call him Bruce Bruce’s action of not packing the parachute safely was very bad not only for grace but also for himself in fact it was a dominated strategy and knowing that he’s rational grace reasoned that he would never play a dominated strategy and so she was able to change the game by removing this dominated strategy and instead to reason that she only had to care about the remainder of the game in which his dominated strategies didn’t exist this is the idea of iterated removal of dominated strategies which you’ll hear about more formally later secondly I’d like to revisit our question of soccer goal kicking and I’d like to ask is it really the case that when a player prepares to take a penalty kick he’s really solving for the Nash equilibrium now we did see experimental evidence that shows that the Nash equilibrium is a pretty good description of what actually happens in these situations but is it the case that the players are really thinking about the idea of Nash equilibrium that doesn’t seem right it seems like the players are thinking about how best to kick the ball into the goal in order to hurt the other guy as much as possible or in order to do as well for themselves as possible it turns out that this isn’t an accident in the case of zero-sum games these three ideas doing as well for yourself as possible hurting the other player as much as possible and being in Nash equilibrium all turn out to coincide finally I want to revisit the battle of the sexes and ask is it really the case that as we saw before with the Nash equilibria of this game were doomed either to an unfair outcome where one member of the couple always gets their preferred activity or a miss coordination where sometimes the the two members of the couple end up doing different activities it doesn’t seem like this is a good model of how people really do solve disputes like this between themselves so I want to think about a new solution concept called correlated equilibrium in which we don’t have this problem and we’re able to achieve fairness without miss coordination sometimes time plays an important role in in in strategic situations things take step one after the other and not only they do they do but the actors know that they will and that influences how they behave so the year was 1519 and m-man Cortez Spaniard was leading a flotilla of eleven votes and about 600 men about to invade a continent later to be known as America they were vastly outnumbered and well aware of the the heavy odds they were facing and as is famously known as they landed Cortes ordered that all the both be burnt this not controversy whether this was done in complete coordination and agreement with his men or it was sprung on them but either way it’s clear what the logic behind it was the quartet Cortes knew that as they faced these daunting odds the men would be tempted to turn back board the boat and and flee and by removing that option that increased their fighting resolve going forward and so again there’s not only the fact that time passes between actions but that one reasons about that facts and impacts how the strategic situation unfolds we see it not only when there are multiple agents involved but even in a single decision-maker here it’s not only Cortes and his men the two actors or the two sets of actors who are somehow actions are intertwined but even when the single actor the fact that time unfolds can impact the situation here’s another famous historical tale in this case of Ulysses and the sirens Ulysses is captaining his boat and about to pass through the Straits of the sirens and as is commonly known the sirens song are so seductive that they would cause any person imperialist to do things there are in his own noth-nothing his old best interest he would jump into the sea you would crash the boat against the rocks and so what he does according to the tale he order is always meant to first put wax in the ears so they will not be seduced by the song he himself who in fact wants to hear the song knowing that he would not be able to withstand the seduction orders he’s meant to time to the mast and should he possibly get free from the from the tithe to restrain him with his swords and and and and so it comes to pass they sailed through the Strait when they when he hears a siren song he goes temporarily insane and tries to escape from the bond and fails and always good so here again is single actor in this case Ulysses reasoning about the future thinking about what will the situation be and taking action now to impact what the strategic situation will look like in the future to model such situation we turn to games in extensive form as the term is called sometimes simply known out of game truth hello again folks so this is Matt and we’re talking about imperfect information in the extensive form now so we’re going to be talking now about games where we have some sequential moves and there can be some uncertainty in players minds about both the possible payoffs of others and the strategies that others might be following so we’ll start just by you know to give you some ideas about this let’s talk a little bit about poker which is a game has been becoming incredibly popular and recently both for people playing in on television and other kinds of things and is one of the oldest games which has very extensive experience and for a lot of people and the the you know one of the critical aspects of poker is that there’s actually sequential play in betting calling folding so one player gets you know two to make a decision in terms of how much they’re going to make a bet at a certain point in time other players have to react to that so the sequential play you see some cards in many of these games but not all so you might see some of the cards that the other players are holding but you don’t know how strong their hand is and you have to be inferring things about their their possible cards both from odds in the game and based on what they’re doing in terms of their strategy so you see the bets and you react to them and you have to make inferences based on that so that involves having beliefs about the motivations the rationality of other players what their payoffs are what their potential payoffs are which in poker might come from from the cards so when we think about these kinds of games you know there’s many possible hands that’s going to make poker fairly complicated game to keep track of there’s many betting strategies which means that the overall tree that we could have to work with is going to be quite complicated so it’s actually going to be almost impossible to draw the tree in the sense of just drawing it out on the screen but there’s nonetheless a lot that we can learn about analyzing such games and analyzing the types of strategies that they have how how do we represent extensive form games with incomplete information how might we but reason about these things and moreover there’ll be simpler settings you know poker is actually a fairly complicated game and there’s other fairly complicated games but very high stakes games so for instance you know we could have one country thinking about invading another one a potential war or a conflict there trying to decide what the other country is going to do in response so if you invaded what would they do that’s a game of incomplete information because you might not know exactly how strong they are was the willingness of the population to fight what might happen politically how how strong are they and if there was a war so there’s these are situations where one party might have to move first anticipating reaction of the other the second one has to anticipate what the the the fact they’re being invaded means about the strength of the other do you surrender do you fight so those are games that they’re going to have similar kinds of features to these and it’s going to be very important to develop a set of a way of representing these things and some thoughts about analyzing those so that’s where we’re headed next and we’ll see more a lot more of this very shortly folks so it’s time to start into a new topic and a very interesting one at that what we can talk about now is repeated games and so this will use some of the reasoning we’ve had in terms of sub game perfect and extensive forms but now think of situations where players are playing a game but they’re playing it repeatedly over time so you know when we think of most interactions in the world there’s a lot of them which occur more than once so you know when we think about different kinds of things you know firms in a market place they’re interacting with their competitors they’re doing it day after day after day we think about political alliances countries deciding how they should negotiate with other ones whether they should should have conflicts and so forth those are the things where you know this happens repeatedly over time so there’s a long history and a long future ahead of them friends you know do you exchange you to help your friends out when they need help do they help you back when you when you need it so you know you’re going through repeated interactions and how you’ve performed in the past of your friends that have done very well for you you’re more likely to reciprocate some of these kinds of things can involve repeated interactions workers team production you’ve got a you know day after day you need to help some of your co-workers sometimes you have to do tasks that don’t go noticed and sometimes they do help you out and so forth so these are situations where repetition can make a difference and understanding it will an important aspect of understanding how the repetition affects the play involves a whole series of different kinds of things I’m just to fix ideas let’s talk a little bit about OPEC which was a cartel which is formed in the early 1970s and and just in terms of background if you go back to you know the the period between 1930 and up to 1973 when OPEC started being put together the price of oil so these are all adjusted numbers to reflects a 2008 dollar so we adjust for inflation and other kinds of things but if you just everything to the same to adjust for inflation you get a price of roughly $20 per barrel or less from the period of 1930 through about 1973 OPEC starts to form what happens they decide that you know they’ve just been acting it’s a bunch of players a bunch of different producers they’ve been acting on their own for many years and just pumping lots of oil and they pump so much oil that the prices is fairly low it’s very easy to buy oil so they’re going to start by restricting production cut production back down and that’ll drive the price up ok what’s the difficulty is if all the other ones are producing very little and driving the price up I’d like to cheat on the agreement and pump more oil so you can think of this as looking somewhat like a giant prisoner’s dilemma game where we’d all like to pump less oil if we’re OPEC or oil-producing countries drive the price up but we all have an incentive to cheat a little bit and pump more oil and the repetition can help us police that if we’re OPEC so what happens by 1976 the price goes up to about $50 a barrel by the time you get to 1982 these are also in real terms so that these are adjusted for inflation they’ve pumped up the price up to $90 a barrel and you know so it’s in the early 1980s it does quite well in terms of Oh packet lots of money coming in and then it collapses for a while so when you look but the period per say between about 1986 to 2002 so 19 to 82 and 86 it starts to a road it kind of falls apart between 86 and 2002 the price is basically $40 a barrel or less and then they get back together again so actually during this period there is a ran Iraq war a whole series of things go on in the Middle East that make it much more difficult to sustain cooperation things deteriorate and then after a while you know things are getting back in terms of higher prices by late 2008 the price is pushed back up to over $100 a barrel okay so what we need to begin to understand this understanding the repeated interaction and understanding the motivations of the players and so forth can help us quite a bit so when we think about this a cartel is much like a repeated prisoner’s dilemma and in order to make sure that other people behave if you want to get people abiding by this you need to be able to observe other people’s play and react quickly to that and punish them desired behavior so if you form a cartel and everybody hears our quotas don’t pump more than this you be able you need to be able to respond to somebody who ends up breaking their agreement and pumping too much so you need to be able to observe what’s going on you also need to care about the future right so if you stop caring about the future and only care about today then it’s very tempting from one of these countries just to say okay forget the agreement I’m just pumping as much as I can today the price is very high I’m going to grab what I can and war certainly didn’t help this right so if you have a war you need cash up front you’re going to want to pump much more oil and that makes it much more difficult to abide by the cartel agreement you also need a stable set of players and and stationarity so the more complicated the setting becomes and the more sources of oil there are in the world and so forth so constantly changing sources of production can hurt growing demand can help actually right so if the world is getting more and more oil dependent and countries come online that have more and more demand for oil that can help drive the price up so in understanding repeated games we’ll have to account for a lot of things over time but we can begin to analyze this using the the tools of game theory and looking at situations where people are playing games but they’re playing them over and over and over again and trying to understand the consequences of that how does threats of future play impact play today so that’s the basic idea and let’s get into that in a little bit more detail coalitional game theory is an approach to modeling strategic situations that stand in contrast was usually called non cooperative game theory and in fact coalitional game theory is often called cooperative game theory the names are Loomis leading and I’ll get back to that in a moment but let’s first speak about what situations does coalition game theory try to model you may recognize these two fine-looking gentlemen the person on the left is David Cameron Prime Minister of Britain and to his to his right as we look at it his left as they sit is Nicholas Clegg he is cultural partner now here are two political rivals who nonetheless come together and presumably there’s a reason there’s something where they can accomplish together that they cannot accomplish alone in particular in this case command a majority in the in the parliament and so that’s a classical example where a coalition formed in fact we usually when you think about coalition we think about political parties but coalition’s form not only in politics they certainly form in business so the wind coalition is a the coming together of a number of firms in the United States to promote the joint agenda of wind energy turbines and such again these are companies who are competitors and nonetheless they feel that together there’s things that can exist do that they can’t do alone for example Lobby government established standards and things of that nature now coalition’s aren’t always among organizations or parties of ferns or heavyweight things we as individuals routinely come together to accomplish things together whether it’s to enter in marriage or for example build a house when you have a carpenter Allen efficient and a painter that come together they together can accomplish something that they can’t on their own now the mere fact that people come together doesn’t mean that their interests are aligned or that they bring the same amount of value to the coalition they’ve formed it could be for example that the framer of a construction crew is irreplaceable but electricians are easy to find and what could easily replace and presumably when they get paid for the house they build that should be reflected in how they divide the the payment and so there’s a competition of both cooperation and competition here and and so for that reason calling these cooperative games as the common term is it is a little misleading just as much as non cooperative game theories misleading as well because for example if you look at the normal form game the canonical representation of a non cooperative game one can easily describe a completely harmonious situation of so-called team gains or common payoff games where the interests of the ages are completely aligned so both non cooperative game theory and coalitional or cooperative game theory model both competition and coordination the essential difference is the basic modeling unit in coalitional game theory the basic modeling unit is the group the team and what they kind of can accomplish and the analysis is based on this basic modeling this video will help to introduce the idea of Bayesian games which is a new game representation so here I want to think about auctions now canonically when we think about auctions we think about something like this wood cutting from 1885 showing the auction of tea in Melbourne Australia we’ve got a guy in a top hat standing at the front of the room and he’s got a gavel in his hand and he’s probably talking in a funny voice and at some point he bangs the gavel and somebody just won a bunch of tea but auctions are a really practical thing they they really are quite important in the world and they’re used for a lot of different things here’s an example of a much more modern auction this is an auction of bluefin tuna at a fish market in Tokyo and you can see this is from 2008 and auctions are used to sell fish because fish spoiled quickly and their value changes quickly and so it’s important to find a way of determining what the prices should be on a day-by-day basis here’s an auction by the US Marshal Service selling some horses that were seized by somebody who was convicted of embezzling money and so the US marshals got a bunch of guys and cowboy hats to round up this person’s horses and sell them off to recover some of the embezzled money again you can see why an auction might be used here because the value of something like a horse is pretty unclear it depends on supply and demand it depends on intangible things like the value of the horse the number of other good horses on the market at a given time and so on here’s yet another auction this is an auction of rugby players presumably only for a limited time and again you can see why an auction might need to be used because it’s pretty unclear what this guy would be worth of course not all auctions happen in person here’s a famous auction that occurred on eBay a few years ago this was a situation where a woman in Florida was eating a grilled cheese sandwich that she had made for herself you can see picture of it right here and she took a bite out of it and then she was amazed to discover what she considered to be a perfect likeness of the Virgin Mary in the shape of the burn mark on the grilled cheese sandwich right there and so she was so amazed she decided that this was a religious relic she stopped eating the sandwich and like anyone would do if they discovered a sign from God she posted it on eBay to sell to the highest bidder this is kind of interesting both because it’s ridiculous but also because it goes to show how the Internet allows auctions to create markets in places where they really wouldn’t have existed before so if this had happened before the internet it seems pretty clear that this woman would have had a hard time finding a buyer for her grilled cheese sandwich but you can see that when I took this screenshot the auction still had almost four days to go and already the top bid was seven thousand six hundred US dollars and this this really was not a hoax this was an actual auction it was covered in the news so that’s that’s kind of an amazing thing it goes to show the power of auctions to to match up buyers and sellers here’s the last auction picture I want to show you this is what’s called a silent auction in a charity auction so what’s going on here is there there are actually two different bundles for sale so there’s a gift basket here for sale and all of the people who are interested in buying it are able to go up and inspect it decide what they think it’s worth to them and then they write a bid they write their name and an amount that they’re willing to pay on this sheet of paper here and the reason that I think the silent auction is particularly interesting is we can really see how this might look like a game we here have a well-defined action space you come you you look at the piece of paper you can see the moves of the players who have moved before you and then you you take an action which is choosing to write down number and at the end presumably you don’t write down a number that is worth more than the gift basket is worth to you at the end if you have the highest number you win the gift basket you realize an amount of utility which is which depends on how good the gift basket is from your point of view and you pay whatever amount it is that you wrote down on the piece of paper which is a presumably less and you get the difference as your utility for having played the game so all of this looks like something we can really model in the context of this course and if we realize we can model the silent auction this way we can probably see that we can model the other options that I’ve just described in this sort of way as well however there’s something really critical about the case of the silent auction which is different from the games that we’ve talked about in this course already and that is that when I want to reason about what other people will do in this game I need to think about what they think the gift basket is worth to them that’s going to be something very important to the way they choose to act in the game and critically this is something that will affect their utility and it’s not something that I know and so this is a case where I’m not quite sure what other player’s utility functions are even in cases where I can imagine what all of the actions taken in the game are so this is a different sort of setting than we’ve ever looked at before and hopefully you can see that this is really necessary to model the case of options it’s really fundamental to an auction that I’m not quite sure what the good is worth to all of the other participants in the auction and that fact is really critical to my strategic reasoning in the option so Bayesian games are a formalism that allows us to model this kind of uncertainty uncertainty about the utility functions and this week we’ll go into Beijing games in more detail you

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