 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 Yoav 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 about 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 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 acknowledgement to you telling you that it received the message, and that acknowledgement 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 acknowledgement. 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 call 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. 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 a 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 will offer you some answers to. The TCP Backoff Game is just one example of a real-world situation that we can examine using Game Theory. Throughout the course, we'll describe many more real-world examples that Game Theory can be used to think about. Hi, folks. It's Matt again. Now we're ready to start solving games and making some predictions of how people will play in different settings. We're talking right now about Nash Equilibrium, which is probably the most basic and standard solution concept 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. It's a very basic and fundamental concept. In order to sort of motivate it, let's start by just talking through a particular game that was described and invented by another famous person. This is John Maynard Keynes' beauty contest game. What's the idea here? Let's think of a basic situation that you might be interested in. This is one that Keynes described in some detail. The idea was you have a stock, you're holding onto it, and the stock price is rising. That's great. You're an investor. You're trying to make profits off of your stock holdings. You begin to believe that maybe the stock is too high to be justified by the value of the company. You're thinking that it's possible that the stock is overvalued. Maybe there's a bubble in the market, and you're starting to think about selling. You'd like to sell it, but you'd like to wait until the price is at its peak. You'd want to wait until the price was just where it's going to hit its maximum before you sell. You wanted to get out of the market just before the other investors do. This is a game where now you have to predict what other people think about the stock price and what they're going to do and when they want to get out. How will they act? How should you respond to that? 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. These are going to be two key ingredients that we have. There's a very stylized version of this, which is known as the Keynes beauty contest game. Where did it come from? Well, actually Keynes described... There was a newspaper in England that had a contest where players had to guess which picture of several women other readers would think was the most attractive one. It wasn't to guess what you thought, but what you thought other people were thinking. Keynes likened investing to this. It's not only what you think of the stock, but what you think other people are thinking about the stock that's important in driving your decisions. This is now represented by a very simple game that is played by many people. What does this game look like? Each person gets to name an integer between 1 and 100. You get to pick a number between 1 and 100. It has to be an integer. 1, 2, 3, etc. Players are going to move simultaneously. The player who names the integer that's closest to two-thirds of the average integer wins a prize. The other players get nothing. To win this game, you have to guess the average and then two-thirds of it. You want to be right at two-thirds of whatever the average guess is. A little bit below the average guess. If there's two people that happen to hit the same integer that's 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, we'll roll a dice, a three-sided dice, 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 response to that. This video is going to introduce the idea of mixed strategies and explain why they're important to Nash equilibrium. So the example that I want to think about here 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 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 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, 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 the value of an attack is limited. So the Nash equilibrium of a game like this is going to involve the defender defending in a 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 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 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, we're doomed either to an unfair outcome where one member of the couple always gets their preferred activity or a miscoordination where sometimes 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 miscoordination. Sometimes time plays an important role in strategic situations. Things take step one after the other and not only do they do, but the actors know that they will and that influences how they behave. So the year was 1519 and Nan Cortez, Spaniard, was leading a flotilla of 11 boats and about 600 men about to invade a continent later to be known as America. They were vastly outnumbered and were well aware of the heavy odds they were facing and as is famously known, as they landed Cortez ordered that all the boats be burnt. There's some controversy whether this was done in complete coordination and agreement with these men or it was sprung on them, but either way it's clear what the logic behind it was. Cortez knew that as they faced these daunting odds the men would be tempted to turn back, board the boats 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 fact 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 Cortez and his men, the two actors or the sets of actors who somehow actions are intertwined, but even when there's a 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 in particular Ulysses to do things that are in his own, not in his own best interest. He would jump into the sea, he would crash the boat against the rocks and so what he does according to the tale, he orders all his men 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 his men to tie him to the mast and should he possibly get free from the ties to restrain him with his swords. So it comes to pass, they sail through the Straits, when he hears the Sirens song he goes temporarily insane and tries to escape from the bonds and fails and all is good. So here again is a 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 as game trees. Hello again folks, so this is Matt and we are talking about imperfect information in the extensive form now so we are 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 will start just to give you some ideas about this, let's talk a little bit about poker which is a game that has been becoming incredibly popular recently, both for people playing and on television and other kinds of things and is one of the oldest games which has very extensive experience for a lot of people and one of the critical aspects of poker is that there is actually sequential play in betting, calling, folding, so one player gets to make a decision in terms of how much they are going to make a bet at a certain point in time other players have to react to that, so 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 possible cards both from odds in the game and based on what they are 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 the cards. So when we think about these kinds of games, there are many possible hands that are going to make poker a fairly complicated game to keep track of. There are many betting strategies which means that the overall tree that we are going to have to work with is going to be quite complicated so it is actually going to be almost impossible to draw the tree in the sense of just drawing it out on the screen but there is nonetheless a lot that we can learn about analyzing such games and analyzing the types of strategies that they have how do we represent extensive form games within complete information how might we reason about these things and moreover there will be simpler settings. Poker is actually a fairly complicated game and there are other fairly complicated games but very high stakes games so for instance we could have one country thinking about invading another one a potential war or a conflict they are trying to decide what the other country is going to do in response so if you invaded what would they do? That is a game of incomplete information because you might not know exactly how strong they are or what is the willingness of the population to fight what might happen politically how strong are they if there was a war so these are situations where one party might have to move first anticipating reaction of the other the second one has to anticipate what the fact that they are being invaded means about the strength of the other do you surrender, do you fight so those are games that are going to have similar kinds of features to these and it is going to be very important to develop a set of a way of representing these things and some thoughts about analyzing those so that is where we are headed next and we will see a lot more of this very shortly Hi folks so it is time to start into a new topic and a very interesting one at that what we are going to talk about now is repeated games and so this will use some of the reasoning we have had in terms of sub game perfection and extensive forms but now think of situations where players are playing a game but they are playing it repeatedly over time so when we think of most interactions in the world there is a lot of them which occur more than once so when we think about different kinds of things firms in a marketplace they are interacting with their competitors they are doing it day after day after day when we think about political alliances countries deciding how they should negotiate with other ones whether they should have conflicts and so forth those are the things where this happens repeatedly over time so there is a long history and a long future ahead of them friends, do you exchange, do you help your friends out when they need help do they help you back when you need it so you are going through repeated interactions and how you have performed in the past if you have friends that have done very well for you you are more likely to reciprocate some of these kinds of things can involve repeated interactions workers, team production you have got to day after day you need to help some of your coworkers 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 be an important aspect understanding how the repetition affects the play involves a whole series of different kinds of things just to fix ideas let's talk a little bit about OPEC which was a cartel which was formed in the early 1970s and just in terms of background if you go back to 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 reflect say 2008 dollars so we adjust for inflation and other kinds of things but if you adjust everything to the same to adjust for inflation you get a price of roughly 20 dollars per barrel or less from the period of 1930 to about 1973 OPEC starts to form what happens they decide that they have just been acting as 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 price 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 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 dollars a barrel by the time you get to 1982 these are also in real terms so these are adjusted for inflation they've pumped the price up to 90 dollars a barrel and so in the early 1980s it does quite well in terms of OPEC lots of money coming in and then it collapses for a while so when you look at the period say between about 1986 to 2002 so between 82 and 86 it starts to erode it kind of falls apart between 86 and 2002 the price is basically 40 dollars a barrel or less and then they get back together again so actually during this period there was the Iran-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 things are getting back in terms of higher prices by late 2008 the price is pushed back up to over 100 dollars a barrel so what we need to begin to understand this understanding the repeated interaction 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 undesired behavior so if you form a cartel and say everybody here's our quotas don't pump more than this you need to be able to respond to somebody who ends up breaking their agreement and pumping too much you need to be able to observe what's going on you also need to care about the future so if you stop caring about the future and only care about today then it's very tempting for 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 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 by the cartel agreement you also need a stable set of players 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 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 stands in contrast to what's usually called non-corporative game theory and in fact coalitional game theory is often called cooperative game theory the names are a little misleading and I'll get back to that in a moment but let's first speak about what situations does coalitional game theory try to model you may recognize these to find looking gentlemen the person on the left is David Cameron Prime Minister of Britain and to his right as we look at it his left as they sit is Nicholas Clegg his coalitional 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 parliament and so that's a classical example where a coalition formed in fact we usually when we think about coalition we think about political parties but coalitions form not only in politics they certainly form in business so the wind coalition is a 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 do that they can't do alone for example lobby government established standards and things of that nature now coalitions aren't always among organizations or parties or firms 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 an electrician 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 presumably when they get paid for the house they build that should be reflected in how they divide the payment and so there's a competition of both cooperation and competition here and so for that reason calling these cooperative games as the common term is is a little misleading just as much as non-corporative game theory is misleading as well because for example if you look at a normal form game the canonical representation of a non-corporative game one can easily describe a completely harmonic situation of so called team games or common payoff games where the interests of the ages are completely aligned so both non-corporative 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 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 really are quite important for 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 spoil 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 Marshall service selling some horses that were seized by somebody who was convicted of embezzling money and so the US Marshalls got a bunch of guys in 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 a 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 7600 US dollars and this really was not a hoax this was an actual auction it was covered in the news so that's kind of an amazing thing it goes to show the power of auctions 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 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 look at the piece of paper you can see the moves of the players who have moved before you and then which is choosing to write down a 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 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 presumably less and you get the difference as your utility for having played the game 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 auctions 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 players 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 auctions 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 auction so Bayesian games are a formalism that allows us to model this kind of uncertainty about the utility functions and this week we'll go into Bayesian games in more detail