In Our Time - Game Theory

Episode Date: May 10, 2012

Melvyn Bragg and his guests discuss game theory, the mathematical study of decision-making. First formulated in the 1940s, the discipline entails devising 'games' to simulate situations of conflict or cooperation. It allows researchers to unravel decision-making strategies, and even to establish why certain types of behaviour emerge. Some of the games studied in game theory have become well known outside academia - they include the Prisoner's Dilemma, an intriguing scenario popularised in novels and films, and which has inspired television game shows. Today game theory is seen as a vital tool in such diverse fields as evolutionary biology, economics, computing and philosophy. With:Ian StewartEmeritus Professor of Mathematics at the University of WarwickAndrew ColmanProfessor of Psychology at the University of LeicesterRichard BradleyProfessor of Philosophy at the London School of Economics and Political Science.Producer: Thomas Morris.

This BBC podcast is supported by ads outside the UK. Every Sunday, we cover the week's tech news on This Week in Tech. Hi, this is Leo Leport inviting you to join us for this Sunday's episode with Ian Thompson, Abraar Alhidi, and Patrick Bejja. We talk about Outlook crashing in space, open AI, spending hundreds of millions of dollars on a podcast, and the return, or maybe not, of Zombo.com. Join us every week for this week in tech. You'll find it at twit.com.

and wherever you get your podcasts. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the program. Hello, in 1928, a 25-year-old Hungarian student gave his first public lecture. Its title was On the Theory of Parlor Games,

and it explored features of basic board games that fascinated the young mathematician. The student's name was John von Neumann, and in the next 15 years he turned these observations into a new discipline, game theory. When Neumann realized that the apparently trivial decisions made while playing games could be used to study decision-making more broadly. Since the early 1950s, game theory has flourished. It provides a mathematical description of situations in which individuals have to make choices.

It's been used by economists to study how markets work, and also employed by biologists, political scientists and software designers. And thanks to film and television, Game theory scenarios such as the prisoner's dilemma have become familiar outside the realms of academia. With me to discuss game theory, Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick, Andrew Coleman, Professor of Psychology at the University of Leicester, and Richard Bradley, Professor of Philosophy at the London School of Economics and Political Science. Ian Stewart, perhaps you could begin by giving us some idea of why studying games

and which games you study is useful in the first place. We all play games all the time in this sense. It's about, as you said, decision-making. The idea is that you consider people who are faced with making choices between a small number, relatively small number, of different alternatives. Am I going to buy a car? Am I not going to buy a car? But then there's another player in the game, in this case the car manufacturer, who's deciding what models to produce and what price to sell them at.

And so this transaction and this decision process can be seen as two players in this case, is a very, very simplified version of it. Each has a certain number of different choices, strategies they can use. And the mathematics starts out with the idea that actually neither of them has any idea what the other one is going to do. Surely there are some idea. A manufacturer knows somebody is going to buy a car. That's right. If somebody's going to buy a card that's well made, and that's a bit of a start.

But in other examples and other models, you would add that sort of detail later and get a more complicated kind of game. But to understand the basic principles, you start with the simplest one, which is essentially each knows what the payoffs are, what the win and losses for particular combinations of choices. But they don't know what choice the other one is going to make. I'm intrigued as I think a lot of our listeners would be by the fact that this seems to have kicked off. with this young man talking about parlor games. Now, can you give us an example that he chose from that first lecture he made of which parlor game this was, or which two or three it was, which actually got him going? Well, I think the best example to choose, and I don't know whether he lectured about it,

but it's a game that children play, and it has all of the basic ingredients, and this is scissors paper stone, or rock paper scissors, it's sometimes called. Not only children is. That's right. It's a reflection of late-night holidays, I can tell you. don't need any apparatus except yourself. So we all know the rules. Scissors cuts paper, paper wraps stone, stone

blunts scissors, so that tells you what wins. So you put your hands behind your back and simultaneously you and the other player make the shape of the scissors or the paper or the stone. And if both of you make the same shape, that's a draw. But if I make scissors and you make paper, I've won.

If I make scissors and you make stone, you've won. So how do you play this game? And the more you think about this. Okay, I've watched children playing this and they look at each other and they see who's smiling and they're actually following, you know, trying to outwit the other player. But in the simplest version of this, if, for example, I play scissors rather too often, you will notice this and you'll start playing stone more because that's going to win. So I shouldn't play scissors

too often. I shouldn't play paper too often. I shouldn't play stone too often. And you rapidly realise that roughly speaking, I should play each of them about one third of the time. And I should also do this unpredictably. If I did them in order all the time, scissors, then paper, then stone, then scissors, then stone, you'll spot the pattern. So just thinking of the top of your head about this particular game, the best strategy would seem to be to choose at random from the three choices, with equal probabilities. And in fact, if you do this, what happens in the long run, nobody wins, nobody loses, the whole thing evens out.

So we have the whole thing encapsulated here. You have the structure of the game, this is a two-player game. You have the payoff, win, lose or draw for each combination of strategies, and each player has a certain number of strategies available. And we're going to explore how that gets turned to mathematics, not to do this in the future. Andra Coleman, so what are the origins of the game theory? Sorry, Andrew, Andrew, Andrew, Conman.

What are the origins of the game theory? And the theory itself got established, as you yourself mentioned in the mid-20th century. But the idea of strategic thinking, that's to say, taking into account in your decision-making, what decisions other people are making, is as old as the hills.

look at the records of ancient civilizations, there are documents, interesting documents, which show people not only being aware of this, but also aware that once you take into account other people's decisions in your decision-making, it's often quite tricky and paradoxical. And a good example comes from Pliny the Younger. This is the first century AD. He was a Roman senator, and some people were brought before the senator who were accused of committing a crime, and the Senate was split. In Pliny's faction, they wanted to acquit these people. There was a block that wanted to banish them to an island, and there was a block that wanted to condemn them to death, so you see, A, B, C, and Pliny was in the smallest group. And he and the other

senators in that group became aware that if they just voted by a show of hands, they'd get their worst option, because they wanted to acquit, and the biggest block was condemned to death, So it would be silly for them to, in fact, vote sincerely, and they'd be better of joining the banishers, then they'd get their second best option rather than their worst. And Pliny wrote about this at some length in a letter, and it shows that, and he did some what we would nowadays call game theoretic analysis. And then in late antiquity, the Babylonian Talmud has got a passage about how to distribute

the wealth of somebody who dies with debtors, but not enough money to pay all the debtors. And several examples are given that puzzled rabbinical scholars for centuries, and even it was seriously thought there must be transcription errors that they couldn't have meant what they said. And then recently, the Nobel Prize winning game theorist Robert Aumann showed that the distributions that were being proposed were actually corresponding to a game theoretic system of, it's called the nucleolus. It's quite a complicated game theory. And these old rabbis had somehow got to this intuitively. The forerunners of the modern movement were particularly in the 18th century, James Wardegrave, or Wargrave, I think he's pronounced, who's an ancestor of Lord Wargrave

in the current House of Lords of all people. He did an analysis of the game Leher, which is a card game, where people, simple card game, people select cards, and he worked out that the optimal strategy is a mixed strategy, what Ian was referring to, where you choose between your available strategies with a randomizing element. And that was a correct analysis that he gave in the 18th century. Did it mean that he won?

It meant that you wouldn't lose, that you'd maximize your chances. If both people adopt the optimal strategy, then nobody wins, as in scissors, paper, stone. But if you don't, you stand the chance of losing. In the 19th century, Kourneau, a French mathematician did an analysis of supply and demand of duopoly, two producers of the same goods. In his example, it was mineral water. trying to decide whether to restrict or increase production, optimal strategies of both end up in what we now call a Nash equilibrium,

which we might be coming back to later, but this was a kind of forerunner of the Nash equilibrium idea. And then Charles Latwich Dodgton, who we usually remember as Lewis Carroll, who was an Oxford Don, Mathematics Don. He was very puzzled by problems of voting in his college, and he worked out various voting systems to try to avoid strategic or tactical voting.

We need him now. We need him now. Sarah Malo, a German mathematician, proved the first major theorem in game theory in 1913, and then we get into Borel and the French Borel and von Neumann in 1928, who you mentioned in your introduction. I did, and he's regarded as the beginning at the start of the modern game theory notion.

Can you just tell you. us a little more, Andrew Coleman, about John Van Neumann. And not about him so much, is what he contributed. Hungarian 25 when he did his first lecture, and then he, not very long afterwards, he co-wrote a book about game theory, the first book about game theory. In 1928, he proved a very important theorem.

And I've actually looked up his, because I speak a little bit of journal, at least I read a bit of journal, tried to read his article where he proved this theorem, is very long, very complicated and impenetrable, but it was very important because it was, he proved that strictly competitive games, that is to say two player games in which one player's gains always equivalent to the other players' losses, like chess, for example, is a strictly competitive game. He proved that strictly competitive games always have a particular kind of

solution, and he showed what it is. And it was important partly because the mathematician Borrell, who I mentioned, the French mathematician, who'd been working a lot on games in the years preceding that, had struggled to do this, and it actually conjected that this wasn't the case, that you couldn't prove such a theorem. So von Neumann was the first to actually prove it. He was an immensely influential mathematician. He didn't just work on games.

He was one of the most talented mathematicians of the 20th century. And this was really, well, historical. of game theory often give 1928 as the birthday of modern game theory because he proved this first really important theorem. Thank you very much. Richard Bradley, I read

that game theory can be divided into two types, cooperative and non-corporative. Can you develop that please? Yeah, so it's a very basic distinction in the class of games. So the cooperative games are the games in which it's possible to make a binding agreement.

What binds the agreement that the players make between themselves is something that's enforced or ensured by some mechanism outside of the game. So two players can make a promise to each other to coordinate their strategies in a certain kind of way. And there's something, a police force or a conscience or something like that in the background, that makes sure that they stick to their agreements. And that's why these agreements are possible and can have consequences in the game. So that's why those are called the cooperative games, because it's possible to cooperate through agreements. non-cooperative games are just those games in which it's not possible to do.

There's no such mechanism. Cooperation is possible, but there's nothing. If there is cooperation, it has to be cooperation that's achieved in virtue of players playing their rational strategies, thinking about what's best for themselves, not in virtue of some outside mechanism ensuring that they cooperate. And many game theorists think that the fundamental class of games are the non-cooperative ones. I mean, ultimately, if there is some mechanism for ensuring agreement, it ought to be explained in terms of a bigger game outside, a bigger description, if you like,

or a richer description of the situation, which tells you why people would acknowledge their conscience or respect the authority of the enforcer. Can you give us some specific examples about what you're talking? Yeah, so there is very simple class. of games called coordination games in which players have common interests. And in these classes of games, so here's a very simple example, you and I are walking down the street towards one another and we don't want to run into one another, so we've got to either move to

the right or to the left. Both of us have a strategy of moving to the right or to the left. And that's a purely non-cooperative game because there's no, there's no, we can assume that it's a non-cooperative game. Let's say we can't talk to each other before. But if I can see you moving in one direction, I will coordinate my movement to match that so that we don't run into one another. And so the outcome there doesn't require any policing.

There doesn't have to be some third party that ensures that we both move in the same direction. We'll just do so because it's in our interest not to run into one another. In other classes, in other kinds of games. So let's take a littering, for instance. We don't even need to talk again, just the social phenomenon of littering.

It would be in all of our interests, as it were, if none of us littered. But left around devices, we know what people do. They litter, essentially they litter because if everybody else is littering, why should I not litter? If everybody else isn't littering, well, why shouldn't I littered? Because my little bit of littering makes almost no difference. And in that kind of non-cooperative game, the outcome will be a suboptimal one. If you look at that situation as a cooperative game, on the other hand, you can see that clearly the thing that we should agree on doing

is that all of us should not litter. So in the cooperative game, the outcome would be we all agree not to litter because we can make such an agreement. In the real world, if you like, where it's impossible to police littering all the time, the outcome will be that everybody litters. And that's because we can promise each other not to litter, but as soon as Ian's gone around the corner, I'll just chuck my...

cease. No agreement can be sustained there unless it's in my interest to respect the agreement. It's not in my interest to respect the agreement, so the outcome will be that we all litter. And you see this all over the place, actually. Some places you don't, which is interesting as well, isn't it? In some places, people make an attempt

effort not to litter, but we'll leave that for the moment. Richard Bradd, in 1951, a 21-year-old mathematician, John Nash came up with an idea in this area that's, I'm told it's very, very important to game theory. The Nash equilibrium. Now, where did that take the theory? Well, so the Nash equilibrium. So Nash lent his, John Nash was an American mathematician.

I mean, famously wrote a dissertation to Princeton of only 28 pages, out of which came four seminal papers in games theory. And in a Nobel Prize. And Nobel Prize, so the best 28 pages. So the concept of Nash, equilibrium predates him, but he lent his name to it because he was the first to prove some very general results about it. And the idea of a Nash equilibrium is very simple. It's a Nash equilibrium in a game, so this is a situation in which, let's say, two people are independently making

choices and the outcome of the game is the pair of choices of the two of them. The pair of choices that the two of them make is called a Nash equilibrium just in case both players are making a choice that is the best response to what the other player's choice is. So let me give the, let's go back to the example of us walking down the street towards one another. If you step out of the way to your right, then my best response is to step out of the way to my right. So that we pass each other. And if you move to the left, then my best response to what you're doing is to move to my left, so that we pass one another. So those two moves where you go right and I go right, and you go left and I go left, are both Nash equilibrium,

in that terribly simple game that we face called walking down the street towards a name I just gave to that. So the idea is here is that it's best response on the assumption that the other player is playing as it were their role in the Nash equilibrium that you're looking at.

And as you say, it's very simple and it gets more and more complicated. Let's start to get a bit more complicated. Ian Stewart. But let's begin again by talking to, I think, what a lot of the listeners will know about, the prisoner's dilemma.

Now, I want two things here. I haven't got all that much time, but I really, really, briskly tell us what it is. And then I'd like to know where mathematics comes into it. I mean, why we can't talk about it, it's like Faraday being a bit angry

with Clark Maxwell saying, I understood you when you told me about your theories. Why did you have to turn it into algebra? And so, prisoners dilemma, please, and mathematics. It's the iconic game where exactly this business of cooperation and non-cooperation is very clear. So, okay, Melvin, you and I have been both hauled in by the police.

We have committed a crime together, but the only evidence that pins it on either of us is basically if one of us confesses. The police haven't got a lot. They know we've done it, but they don't really have the evidence. Okay? So if we both keep quiet, we'll each be in the slumber for about a month until it's all sorted out,

then they'll have to let us go. Now, if I decide to confess, then they'll let me free and you will go to jail for 12 months. If you confess, then you get let free and I go to jail for 12 months. If we both confess, then we'll both get six months. Okay, so what do we do? And I sit here thinking, you know, it's actually rather obvious. It doesn't matter what Melvin does. I do better if I confess.

Because if you don't, then when I confess, you're going to go in the slam mode, I get set free. And on the other hand, if you do confess, it's still better for me to confess because if I don't, I get a longer sentence. So this is what's called a dominant strategy. One of these, confession is always better than staying silent, for me, given whatever you do.

but by exactly the same argument confession is the best thing for you as well therefore we both confess but in fact if we'd both stayed silent we would have both got off with a shorter sentence so the game theoretic pursuing the Nash equilibrium in this game

leads to a rather paradoxical result we don't end up with what's best for both of us we think we're working rationally to work out what's best for both of us for me and I don't know what the other person's doing. And there's a very clear argument that says the best thing for me is to confess. It always is better than not confessing.

But if we both decide to confess, it turns out that we actually do better than the whole of that argument seems not to apply. People have written over 2,000 research papers on this particular game because this whole issue of

cooperation, non-cooperation, what is rational strategy, what is not, all shows up in this game. We'll come back to another in a minute. Can I just take that game with Andrew Coleman? Why is it attracted so much interest? Ian explained it so simply. A lot of people would be saying, well, I confess,

and it's quite simple. But why is it intrigued with its complication? I think there are two main reasons. One is that it's inherently paradoxical. It's very, very strange that there should be, a game in which it's clearly rational for both players to act in one particular way when we define rationality as doing what's best for themselves. And yet, if they both did something different, they'd have both been better off.

I mean, ever since David Hume and the Scottish Enlightenment, we define instrumental rationality as choosing the best option to pursue your own interests. And if both players do that in the Prisoner's Dilemma game, then they end up worse off than two irrational players who haven't done that, who cooperate and don't confess to the police. So it's deeply and fundamentally paradoxical. I think that's one reason it's intrigued people. There are three reasons I said that were two.

The second reason is that it provides a wonderful way of studying cooperation and competition in the laboratory. So experimental games, which are initiated by social psychologists, but nowadays conducted also by behavioural economists have often used prisons dilemma game because it provides you with a beautiful laboratory example of a situation in which you can study cooperation, competition and very numerous factors to see what increases or decreases cooperation and so on. And the third reason is that it's ubiquitous in everyday life.

Ever since it was discovered 1950 at the Round Corporation in California, California, as a kind of anomaly, people have begun to realize that everywhere you turn their prisoners' dilemmas, I mean, an arms race between two countries like a prisoner's dilemma, because it's in both countries' interests to increase their arms production, irrespective of what the other does, you either get a strategic advantage or you avoid falling back, but they'd both be better off if they didn't. In the Cold War, the two blocks spent billions on just maintaining the balance of terror. They could have maintained the same balance of terror by not doing it,

but it's a dominant strategy because it had a prisoner's dilemma type structure. And it crops up all over the place in everyday life. And that's another reason why economists particularly are absolutely intrigued by it. Richard Bradley, can you take us, can you give us something from the idea of coordination? What's the game element in that? Can you give us a specific example and then tell us the theory, please? Sure. So, I mean, we've already seen a very simple example, which is the example of... Walking down the street, right. It was said first on this program.

Right. And of course, the sort of which side of the road do you drive on is a very, very similar thing. There are more interesting coordination in games when there's still an advantage to both parties to coordinate on some kind of, some pair of actions. but there's somewhat more advantage to one rather than the other. So there's a very famous game in game theory called Battle of the Sexes, which perhaps really should have been called Battle of the Spouses or Battle of the Partners, which is a game in which there's a coordination problem, but also it's an element of competition in it. So it extends what Andrew was saying about, bringing out some sort of fundamental features of social interaction.

So in the Battle of the Sexes game, the idea is the husband and wife. for trying to decide where to go out that evening and their husband would like to go out to dinner and the wife would like to go to the opera. And they would like to go out with each other. They're still at that phase of their marriage. And so what they really want to avoid is going off to separate activities. But of course, you know, the husband would prefer

that they both agree to go out to dinner and the wife would prefer that they both go out to the opera. Well, so they're in a coordination game and there are two Nash equilibrium. So, I mean, it's clearly the husband's best response to follow his wife to the opera if that's what she's going to do. It's clearly the wife's best strategy to go to the dinner if that's what her husband's going to do. But in fact, I mean, this is really a case in which neither have any idea what the other is going to do

because it depends on each other. So there's this sort of what I do depends on you, what you do, and what you do depends on why I do, and so there's no. So that's a very sort of clear example of, a coordination problem in which there's more than one match equivalent, so it's not very clear what people should do in this situation. Before we move on, can you briskly, and I asked to do this before, where does mathematics come into it? I mean, you've been talking about a very eloquent thing,

clearly. Why do we need mathematics? Okay, the mathematics comes in, firstly because actually calculating these mixed strategies, what probability do you play which strategy is, especially if you have the significant number of options more than just two, you actually need to do some sums to get the numbers right. And so, for example, with Prisoner's Dilemma, we have a zero-month sentence, one-month sentence, a six-month sentence, and a 12-month sentence

sitting there in the description of the game. Now, in that particular case, because of these dominant strategies, you just get this rather paradoxical result. But in very similar games, The strategy that would be adopted would be to play various of these possibilities with certain probabilities, and you need to know what the numbers are. I'm sorry to be a sick way. How does having numbers help? Can you just tell us that? Is that possible, sir? How does it help? It helps because it actually gives you. This is how you play the game.

If, for example, the calculation says you should use a particular strategy 10% of the time and the other strategy 90% of the time, you can actually take your decision by choosing a random number between 1 and 10. And if that number is 1, you play the first strategy. And if it's anything from 2 up to 10, you play the other strategy. so you can actually you mathematically roll the dice and use the probabilities in the way that

this mathematical calculation tells you and presumably they get hideously complicated as the games include more and more people if you have more and more people or more and more strategies more and more strategies and two people is not too bad, it's just a bigger table of numbers more and more people and it starts to get really complex so there's another advantage I think of using

mathematics in these things is that it allows you to separate the particular story that you're telling about a game. So this prison's dilemma starts off with this nice story in which there's sort of number of years that you spend in the prison. And then once you turn it into a bit of mathematics where you just write down some numbers to indicate payoffs, that's when you realize actually this game is all over the place

because there's lots and lots of social situations in which those numbers represent payoffs are of a different kind. Sometimes they're monetary payoffs, sometimes their lengths of prison sentence, sometimes their honours that are rewarded to. The numbers can stand in for so many different things, so it makes the analysis very, very general. So I think the generality is really one of the most important things

that you get out of it, allows you to study lots of situations in sort of one go, as it were. One of the attractions about this programme is the new games that have come up. There's a hawk and dove game, Andrew Coleman. Can you tell us about that? Can you tell us how the British biologist John Maynard Smith, so you thought it used to be a big picture of this programme, John Maynard Smith, until his death.

How he seized on that for his contribution to evolutionary biology. So what's the Hulk and the Dove about and what did John Maynard Smith do with it in the 1970s? So he was the one who brought it to prominence, but it was actually invented rather interestingly by a strange American, unemployed American scholar who was wandering around London submitted an article to the journal Nature, and it was sent to John Maynard Smith to referee, blind referee, and it was completely unpublishable, but Maynard Smith recognised that it was a work of genius nonetheless, so very unusually he asked the editor to put him in touch with this author who he didn't know,

and he was given an address, and he went there, and it was like a Doss House. This man was living like a champ, and he collaborated. Price. He was called George Price. died shortly afterwards by cutting his own throat. He was a very disturbed man, and a biography of him recently has come out. But they collaborated together, and they produced an article,

and that's where the hawk-dove game was first introduced. And what it was intended to do was to solve a problem about how cooperation could evolve. It's very striking that Darwin's theory of natural selections based on the idea of a competitive process in which only the

strongest survive or the most successful the fittest survive. And yet cooperation abounds in nature. Birds give alarm calls to warn their conspecifics if there's a cat in the garden, although it can only

increase their own chances of being predated on. There's animals which have fights with cons specifics usually use or very frequently use ritualized forms of fighting rather than all-out aggression where they could kill each other very easily and so on. So there's a lot of cooperation in nature and this was always a puzzle and always a great embarrassment for the theory of evolution. Even Darwin recognized that he was very puzzled by social insects and so on.

And the hawk-tab game provides a partial solution to this because what it shows is that if there are two ways of engaging in a combat or a competition with a cons specific, another member of your own species, one being all-out aggression, and the other being displaying or ritualized, some form of ritualized interaction. And then with reasonable assumptions, again, we need mathematics here. There's quite a lot of mathematics in the Horktuff game. If you make reasonable assumptions, it turns out quite surprisingly that the optimal solution is a mixed strategy

in which each individual will play hawk with a certain probability, depending on the actual payoffs in the game, the cost of being wounded and the value of the prize. And so we'll play hawk with a certain probability and dove with a certain probability. So the evolutionarily stable strategy as, as John Maynard Smith called it, arises as a mixed

strategy outcome from this game. So it had a huge influence on the development of behavioral biology and it solved a important problem in the theory of evolution. And it went all over the intellectual waterfront, didn't it, this theory, and moving on rather briskly,

and that was a wonderful explanation. It was important economics. More important in recent years in economics, where one would expect it to be. So pitifully and pointedly, and precisely. I'll give you one example, because one of the places where it really is used in economics,

in practical circumstances, is in the design of various systems for economic transactions. For example, auctions. One of the big auctions not so long ago in this country was selling off bits of the radio spectrum for third generation mobile phones. The mobile phone companies had to bid for a particular range of frequencies they could use for their transmissions. And the government set the

auction up in a way that actually brought in tens of billions from these companies. Now, not so long ago in Australia there was an auction. This kind of kicked the whole thing up. They were auctioning something very similar, radio spectrum. And the rule was high-speed win, but you can withdraw your bid after that if you wish. And the company that won it put in the first, second, third and fourth highest bids and withdrew the first three. They were just hedging their bets. And everyone looked at that and said, well, that's not actually got the taxpayer the best amount of money for this. So in 1993, when the American government was auctioning off the frequency spectrum parts of it,

they got some game theorists and some economists in and said, design this auction for us. And using game theoretic principles, they came up with something, I won't go into details, but it involved, everyone has to pay a deposit just to be in the game, and if they drop out, they lose it.

And if you have the highest bid and then withdraw, there is a penalty. Basically, you don't gain by doing that. And that led to a much better outcome in terms of the revenue to the taxpayer. And there's lots of these games, they're terrific. I could play them all, morning with this. I'm going to just concentrate

it too now. First of all, Richard Bradley, the social contract. The game theory is being applied to the social contract, that which keeps a civilised society driving on the right side of the road, etc. Can you just explain how it figures there? Yes, so, I mean, one of the reasons why

people got so excited about the Prisoner's dilemma game is that quite early on political theorists recognised that this was a wonderful way of understanding why we need a social contract. the situation described in there looks very much like Hobbes's vision of the state of nature in which everybody is aggressively attacking one another

partly to gain the goods that they have but also to protect themselves and to gain a reputation for being aggressive and this has political scientists recognised something of the structure of the prisoner's dilemma because it's in everybody's interest to behave aggressively for whatever the other person is going to do. If the other person is going to be aggressive, well, you better be aggressive so that you're ready for the fight.

If they're going to be passive by being aggressive, you can take their goods and acquire a reputation as a bully, which can be quite a good thing in these circumstances and so on. And again, paradoxical outcome, life in the state of nature is in the sense suboptimal for everybody because everybody is running around and attacking each other in life. Hobbes is fernous, raves is nasty, brutish and short. Well, so the conditions there are ripe for a contract

in which people can agree to take themselves out of it, out of this situation and put themselves in a situation where they are in the better pair of outcomes where there's some cooperating. We all agree not to carry guns. We all agree not to carry guns. Well, in Hobbes thought we would all agree to just obey one person, and that was the only way of guaranteeing it. Other political theorists, of course, said,

well, we must look at the state of nature more carefully and we might agree to different things. the problem that political theorists want to address, namely what contract should we agree to, can be thought about by thinking about the problem that we're trying to avoid by making such a contract. And that's what game theory allows us to examine. I was intrigued by the ultimate game, Andrew Coleman, which I knew nothing about it. Actually, of all the games, I find it most intriguing and rather touching in a way. Could you tell people about the ultimate game?

Ultimatum game, is it? Ultimatum game, is it? Right. I can't read my own writing. It's almost childishly simple, and that's part of its attraction. It's just, all right, it's a two-player game. One player has given a sum of money, let's say, for sake of argument, it's £10. And that player, the proposer, then makes a proposal to how to divide this money between herself and the other player. And the other player can either accept this proposal or reject it,

but there's no other alternative. Can I just come in to make it clear? Sorry, if I'm making it muddier, you'll tell me. Isn't it when two people walk along and there's a £10 note on the pavement, one will then pick it up, and they have to do something about it. That just sort of sharpens it a bit.

Okay, yeah, exactly. So there's two of them now from the start, there's one £10 note, and what happens? So the proposal proposes a division of this money, and the responder either accepts the proposal or rejects it, But if the responder rejects it, then neither player gets anything. So you have to assume that although they've picked up this money, it's going to be, if they can't agree, then neither of them will get anything.

Now, from a purely game theoretic point of view, it's clear that the proposer should offer the responder the minimal amount that makes it worthwhile for the responder to accept it, namely one penny. And the responder should accept this, because one penny, is better than nothing, and we're assuming that players are rational in the sense they always do what's best for themselves. The alternative's nothing, except a penny. And that's trivial. However, of course, that's not what happens.

The interest in this game is not inherent in any paradox in the game itself. It's in the fact that it doesn't correspond to human psychology. And the fact that in practice there have been hundreds of experiments done in this game. all over the world in different locations, often with huge endowments, as they called, because they take American dollars to very, very poor parts of the world, and then offer an amount of money that's equivalent to more than a month salary. And the results are basically the same, namely, that proposes very seldom offer less than a quarter of the prize to the responder.

And if they do, even with these very large prizes, it's liable to be rejected. So if you find 100,000 and you offer 25,000, or now 18%, 18,000 people say now I've been insulted. Yes, exactly. And this is interesting from the psychological point of view. Well, one way to think of that mathematically is that there's two different kinds of payoff involved. There's the amount of money, but there's also honour or prestige or sense of fairness. and if one of these players offers one penny

the other one looks at it and says I would rather deprive you of the rest of the money that's worth more to me than getting the penny and up to about £2.50 let's say you still feel that way I'm not getting... Yeah, so with the £10 note

up to some level people are thinking yeah sure I get a bit more than I would otherwise have got, but the other person is getting away with murder here. I'm not going to allow that. And one of the ways this is shown up in experiments is if you start playing the game in sequence many, many times, observing what happened in the previous one, because now you get punishment strategies. If you do something nasty to me this time, then the next time we play

the game, I'm going to do something nasty to you. And both players can then learn, don't do that. and in a lot of experiments this is what happens. What's extraordinary to me is that this happens all over the world in tribes in the Amazon and on the stock market and that people won't be insulted in that way. It throws a whole new spanner into the works which is just the thing to say when we've come to the end of the program. So thank you very much Richard Bradley Ian Stewart and Andrew Coleman.

And next week we were talking about Carl von Klauswitz and his book On War. Thank you for listening.