## Game Theory (lectures, articles, etc.)

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Game Theory (lectures, articles, etc.)

Game Theory

Some students, after watching the movie A Beautiful Mind , got interested to study game theory. Perhaps you've seen that movie, too!?

Game theory is interesting because you can use its results to make better strategic decisions.

History

Game theory is a branch of economics, with roots in applied mathematics, that studies how players of a mathematically formalized game can best maximize their returns. The field formally began with the 1944 publication Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. The book addressed types of games known as “zero-sum” games in which the gains of one player must be the losses of another. A game like poker falls in this category because money you earn is money that some one else loses.

In the 1950s, the field of game theory grew substantially, and new topics were introduced like cooperative games, non-cooperative games, and repeated games. It was during this time that the famous “prisoner’s dilemma” (1) game was created and John Nash wrote his dissertation at Princeton about a solution concept, now known as the “Nash Equilibrium.”

(1) “prisoner’s dilemma” ... http://en.wikipedia.org/wiki/Prisoner's_dilemma

Game theory has even affected the fields political science, business, evolutionary biology, computer science, and philosophy. The field has gathered such fame that the Nobel prizes in 1994 and 2005 were awarded to economists primarily for their contributions to game theory.

Avinash Dixit, University Professor of Economics at Princeton University, and is John Nash's colleague and friend, stated :

Game theory studies interactive decision-making, where the outcome for each participant or “player” depends on the actions of all. If you are a player in such a game, when choosing your course of action or “strategy” you must take into account the choices of others. But in thinking about their choices, you must recognize that they are thinking about yours, and in turn trying to take into account your thinking about their thinking, and so on.

It would seem that such thinking about thinking must be so complex and subtle that its successful practice must remain an arcane art. Indeed, some aspects such as figuring out the true motives of rivals and recognizing complex patterns do often resist logical analysis. But many aspects of strategy can be studied and systematized into a science — game theory.

Giacomo
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Formal Definition of a Game

Game theory is about making actions based on what you think others might do, taking into account that other people are acting based on what you might do. The circularity is what makes game theory interesting.

For completeness, here is a formal definition of a game. A game involves three components:

1. A list of players
2. Moves each player can make (strategies)
3. A listing of payoffs, or benefits, for each player on each eventual outcome (a utility function for outcomes)

Game theory assumes that each player only cares about maximizing personal payoff. Sounds like a reasonable enough assumption for a business environment.
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Prisoner's dilemma

http://en.wikipedia.org/wiki/Prisoner%27s_dilemma

Nice guys don't have to finish last!

Robert Axelrod, Ph.D., Professor of Political Science and Public Policy at the University of Michigan, calls the game the "E. coli of social psychology". It has been used to study everything from the effects of Westernization in Central Africa to the levels of aggression in career women.

Played as originally conceived in 1950 as a one-shot affair, the Prisoner's Dilemma's optimal strategy seems cynically simple: rat your buddy out, if you want to get ahead, stab your friend in the back. Defecting, that is, appears to provide the highest percentage in oneshot play, regardless of what the other player does.

The only problem is, the other player's no fool and will logically come up with the same strategy: to defect as well. The result: a three year sentence for both, instead of the one year you'd both receive if you'd cooperated.

But what happens if you had to play the game over again?

Professor Anatol Rapoport, Ph.D., of the University of Toronto. It was one of the simplest and most famous: tit for tat. A player cooperates on his first move and, on all subsequent moves, simply mirrors his partner's previous move.

Cooperation by the partner ends up being rewarded and defection punished, while redemption (a change of heart on the next move) is always possible. In addition to tit for tat's victory, nice strategies in general- never being the first to defect- took the top eight spots in Axelrod's tournament.

Whoever declared that "nice guys finish last" was wrong after all.

Nice Guys Finish First

http://en.wikipedia.org/wiki/Nice_Guys_Finish_First

Can cooperation every occur without the state?

http://journal.ilovephilosophy.com/Arti ... tate-/1130
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Nash equilibrium

http://en.wikipedia.org/wiki/Nash_equilibrium

==================================================

The Nash Equilibrium

The theory constructs a notion of "equilibrium," to which the complex chain of thinking about thinking could converge. Then the strategies of all players would be mutually consistent in the sense that each would be choosing his or her best response to the choices of the others. For such a theory to be useful, the equilibrium it posits should exist. Nash used novel mathematical techniques to prove the existence of equilibrium in a very general class of games. This paved the way for applications. Biologists have even used the notion of Nash equilibrium to formulate the idea of evolutionary stability. Here are a few examples to convey some ideas of game theory and the breadth of its scope.

(1) The Prisoner's Dilemma

In Joseph Heller's novel Catch-22, allied victory in World War II is a foregone conclusion, and Yossarian does not want to be among the last ones to die. His commanding officer points out, "But suppose everyone on our side felt that way?" Yossarian replies, "Then I'd certainly be a damned fool to feel any other way, wouldn't I?"

When cops interrogate two suspects separately, and suggest to each that he or she should rat on the other and turn state's evidence. "If the other does not rat, then you can cut a good deal for yourself by giving evidence against the other; if the other rats and you hold out, the court will treat you especially harshly. Thus no matter what the other does, it is better for you to rat than not to rat -- rating is your uniformly best or 'dominant' strategy."

Of course, when both rat, they both fare worse than they would have if both had held out; but that outcome, though jointly desirable for them, collapses in the face of their separate temptations to rat.

Yossarian's dilemma is just a multi-person version of this. His death is not going to make any significant difference to the prospects of victory, and he is personally better off alive than dead. So avoiding death is his dominant strategy.

(2) Real-World Dilemmas

Once you recognize the general idea, you will see such dilemmas everywhere.

Competing stores who undercut each other's prices when both would have done better if both had kept their prices high are victims of the dilemma. (But in this instance, consumers benefit from the lower prices when the sellers rat on each other.) The same concept explains why it is difficult to raise voluntary contributions, or to get people to volunteer enough time, for worthwhile public causes.

How might such dilemmas be resolved?

If the relationship of the players is repeated over a long time horizon, then the prospect of future cooperation may keep them from rating; this is the well-known tit-for-tat strategy. A "large" player who suffers disproportionately more from complete rating may act cooperatively even when the small fry are rating.

Thus Saudi Arabia acts as a swing producer in OPEC, cutting its output to keep prices high when others produce more; and the United States bears a disproportionate share of the costs of its military alliances. Finally, if the group as a whole will do better in its external relations if it enjoys internal cooperation, then the process of biological or social selection may generate instincts or social norms that support cooperation and punish cheating. The innate sense of fairness and justice that is observed among human subjects in many laboratory experiments on game theory may have such an origin.

(3) Mixing Moves

In football, when an offense faces a third down with a yard to go, a run up the middle is the usual or "percentage" play. But an occasional long pass in such a situation is important to keep the defense honest. Similarly, a penalty kicker in soccer who kicks exclusively to the goalie's right, or a server in tennis who goes exclusively to the receiver's forehand, will fare poorly because the opponent will anticipate and counter the action. In such situations it is essential to mix one's moves randomly, so that on any one occasion the action is unpredictable.

Mixing is most important in games where the players' interests are strictly opposed, and this happens most frequently in sports. Indeed, recent empirical studies of serving in tennis grand slam finals, and penalty kicks in European soccer leagues, have found the behavior consistent with the theory.

(4) Commitments

Greater freedom of action seems obviously desirable. But in games of bargaining that need not be true, because freedom to act can simply become freedom to concede to the other's demands. Committing yourself to a firm final offer leaves the other party the last chance to avoid a mutually disastrous breakdown, and this can get you a better deal. But a mere verbal declaration of firmness may not be credible. Devising actions to make one's commitments credible is one of the finer arts in the realm of strategic games. Members of a labor union send their leaders into wage bargaining with firm instructions or mandates that tie their hands, thereby making it credible that they will not accept a lower offer. The executive branch of the U.S. government engaged in international negotiations on trade or related matters can credibly take a firm stance by pointing out that the Congress would not ratify anything less. And a child is more likely to get the sweet or toy it wants if it is crying too loudly to hear your reasoned explanations of why it should not have it.

Thomas Schelling pioneered the study of credible commitments, and other more complex "strategic moves" like threats and promises. This has found many applications in diplomacy and war, which, as military strategist Karl von Clausewitz told us long ago, are two sides of the same strategic coin.

(5) Information and Incentives

Suppose you have just graduated with a major in computer science, and have an idea for a totally new "killer app" that will integrate PCs, cell phones, and TV sets to create a new medium. The profit potential is immense. You go to venture capitalists for finance to develop and market your idea. How do they know that the potential is as high as you claim it to be? The idea is too new for them to judge it independently. You have no track record, and might be a complete charlatan who will use the money to live high for a few years and then disappear. One way for them to test your own belief in your idea is to see how much of your own money you are willing to risk in the project. Anyone can talk a good game; if you are willing to put enough of your money where your mouth is, that is a credible signal of your own true valuation of your idea.

This is a game where the players have different information; you know the true potential of your idea much better than does your prospective financier. In such games, actions that reveal or conceal information play crucial roles. The field of "information economics" has clarified many previously puzzling features of corporate governance and industrial organization, and has proved equally useful in political science, studies of contract and tort law, and even biology. The award of the Nobel Memorial Prize in 2001 to its pioneers, George Akerlof, Michael Spence, and Joseph Stiglitz, testifies to its importance. What has enabled information economics to burgeon in the last twenty years is the parallel development of concepts and techniques in game theory.

(6) Aligning Interests, Avoiding Enrons

A related application in business economics is the design of incentive schemes. Modern corporations are owned by numerous shareholders, who do not personally supervise the operations of the companies. How can they make sure that the workers and managers will make the appropriate efforts to maximize shareholder value? They can hire supervisors to watch over workers, and managers to watch over supervisors. But all such monitoring is imperfect: the time on the job is easily monitored, but the quality of effort is very difficult to observe and judge. And there remains the problem of who will watch over the upper-level management. Hence the importance of compensation schemes that align the interests of the workers and managers with those of the shareholders. Game theory and information economics have given us valuable insights into these issues. Of course we do not have perfect solutions; for example, we are just discovering how top management can manipulate and distort the performance measures to increase their own compensation while hurting shareholders and workers alike. This is a game where shareholders and the government need to find and use better counterstrategies.

(7) From Intuition to Prediction

While reading these examples, you probably thought that many of the lessons of game theory are obvious. If you have had some experience of playing similar games, you have probably intuited good strategies for them. What game theory does is to unify and systematize such intuitions. Then the general principles extend the intuitions across many related situations, and the calculation of good strategies for new games is simplified. It is no bad thing if an idea seems obvious when it is properly formulated and explained; on the contrary, a science or theory that takes simple ideas and brings out their full power and scope is all the more valuable for that.
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A conversation with
authors of CO-OPETITION

CO-OPETITION is a new approach to business strategy, one based on game theory and tested in the field. Competition isn’t dead and it isn’t dying. But that concept is only half the picture. CO-OPETITION explains how to compete without destroying the pie and how to cooperate without getting your lunch eaten.

What is CO-OPETITION?

CO-OPETITION is a new way of thinking about business. Some people see business entirely as competition. They think doing business is waging war and assume they can't win unless somebody else loses. Other people see business entirely as co-operation-teams and partnerships. But business is both co-operation and competition. It's CO-OPETITION. That's why we've chosen CO-OPETITION as our title, a word coined by Ray Noorda, founder of the networking software company Novell: "You have to compete and cooperate at the same time."

How does someone put this idea of CO-OPETITION into practice?

The key is looking at business as a "game." Once you do, you can much more easily spot both the competitive and cooperative aspects of what's going on. Then, after you have a more complete picture of what's going on, there are some simple concepts from game theory that allow you to change the game so you'll do better.

What is game theory?

A branch of applied mathematics that provides a systematic way to develop strategies when one person's fortune depends on what other people do.

How did game theory begin?

Game theory began fifty years ago, with the publication of the book Theory of Games and Economic Behavior by mathematical genius John von Neumann and economist Oskar Morgenstern. The book was immediately heralded as a major scientific achievement and led to numerous technical papers in the fields of economics, politics, military strategy, law, computer science and, even, evolutionary biology.

In 1994, three pioneers in game theory were awarded a Nobel Prize. The Federal Communications Commission has used game theory to design a \$7-billion auction of radio spectrum for personal communication services. (Of course, the bidders used game theory too.) Now, we're working with many businesses and consulting firms applying game theory to help them change the game.

So the key to all this is "changing the game?"

That's exactly right. The greatest rewards in business don't come from accepting passively the game you find. Rewards come from choosing the game you want and adapting it to serve your own purposes. A great strategist said "Philosophers have only interpreted the world. The point, however, is to change it." That strategist was Karl Marx.

You can do your job extremely well, work hard, and still find your efforts aren't rewarded. When that happens, your problem isn't that you're playing the game poorly. The problem is that you're playing the wrong game. The answer in such cases is to change the game. That doesn't mean giving up what you're doing. But it probably means going about it differently. Real success in any business comes from making the game you want, not taking the game you find. In CO-OPETITION, we lay out the principles for doing this.

What are the principles?

The first step is drawing a map of the game. After all, you can't change what you don't see. Our map, the Value Net, identifies all the players in the game of business and the interdependencies among them. The second step is changing the game. We have a method called P.A.R.T.S.-which stands for Players, Added values, Rules, Tactics, and Scope. The key is to find a systematic way to change the game. People say "think out of the box." We provide a method for thinking out of the box.

Who has applied these principles?

Many businesses have done it and profited greatly as a result. Our book is full of examples and our clients range from the big companies, such as Bell Atlantic, Citibank, Merck, Proctor & Gamble, and Xerox to smaller companies, such as United Aluminum and Flexmag. Management consulting firms are bringing our approach into their practices. CO-OPETITION is also changing the way people are taught strategy at business schools.

What businesses would benefit most from applying these principles?

We think almost any business would benefit, from a one-person start-up, to a giant multinational, to a non-profit. As word of our work spreads, we are particularly gratified by the wide range of people who find it useful: people at the top of the largest corporations, people who are described as "middle management," even engineers and artists, people who don't naturally think of themselves as strategists.

Can you give me some specific examples of businesses that have changed the game?

The book is full of examples-dozens. Here's a small sample. Many products don't take off because complementary products or services are either too expensive or don't exist at all. Take online services: a key complementary service is local phone service. Online services have yet to take off in Japan. One reason: NTT dominates the local phone market and charges a lot. That makes using online services expensive. By contrast, local calls in the U.S. are free, and this has helped fuel the explosive growth of America Online, CompuServe, and all the Internet-access providers.

Our strategic recommendation: Don't accept the game as it is; change it. Enter the complementary business yourself or, at the least, negotiate more favorable prices for complementary products and services on behalf of your customers. You're typically a bigger, more expert negotiator than your customers. Use your clout and expertise on their behalf.

Many, many businesses can benefit by changing the game this way. The car makers might consider negotiating cheaper auto insurance for car buyers. That would help boost the demand for new cars. Sallie Mae, the largest student loan provider, is helping its students get cheaper air travel, phone calls, and textbooks. That makes Sallie Mae's loans look better than the competition's. As for small business, there's a used-car magazine in Paris that negotiates favorable rates for its readers on various ancillary services, such as financing, mechanical warranties and the like. That business is so successful that it's expanding to Canada, Hungary, Poland, Sweden, Thailand and elsewhere.

Andy Grove, CEO of Intel, is a big devotee of changing the game this way. He's actually getting into complementary businesses himself. One of Intel's big challenges is that existing software applications don't push the limits of existing microprocessors. That's a problem because then Intel won't be able to sell its next-generation chip. Grove's answer: Develop a complementary product that will boost demand for more powerful chips. That's ProShare, Intel's desktop videoconferencing system, which is being heavily subsidized in an attempt to get the system to take off, and the demand for more powerful chips along with it.

How would you sum up your book?

CO-OPETITION is a book that develops a new approach to business strategy, one based on game theory and tested in the field. Competition isn't dead and it isn't dying. But that concept is only half the picture. CO-OPETITION explains how to compete without destroying the pie and how to cooperate without getting your lunch eaten.
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All is fair in love, war and poker

I'm posting the article to make it more convenient:

What do love, war and poker have in common?

High stakes, perhaps.

Certainly, in all three you spend a lot of effort trying to work out what the other side is really thinking.

There is another similarity: economists think they understand all three of them, using a method called "game theory".

Threats and counter-threats

Game theory has been used by world champion poker players and by military strategists during the cold war.

Real enthusiasts think it can be used to understand dating, too.

The theory was developed during the second world war by John von Neumann, a mathematician, and Oskar Morgenstern, an economist.

Mr von Neumann was renowned as the smartest man on the planet - no small feat, given that he shared a campus with Albert Einstein - and he believed that the theory could be used to understand cold war problems such as deterrence.

His followers tried to understand how a nuclear war would work without having to fight one, and what sort of threats and counter-threats would prevent the US and the Soviets bombing us all into oblivion.

Since the cold war ended without a nuclear exchange, they can claim some success.

Understand the world

Another success for game theory came in 2000, when a keen game theorist called Chris "Jesus" Ferguson combined modern computing power with Mr von Neumann's ideas on how to play poker.

Mr Ferguson worked out strategies for every occasion on the table.

He beat the best players in the world and walked away with the title of world champion, and has since become one of the most successful players in the game's history.

Game theory is a versatile tool.

It can be used to analyse any situation where more than one person is involved, and where each side's actions influence and are influenced by the other side's actions.

Politics, finding a job, negotiating rent or deciding to go on strike are all situations that economists try to understand using game theory.

So, too, are corporate takeovers, auctions and pricing strategies on the high street.

Financial commitment

But of all human interactions, what could be more important than love?

The economist using game theory cannot pretend to hand out advice on snappy dressing or how to satisfy your lover in the bedroom, but he can fill some important gaps in many people's love lives: how to signal confidence on a date, or how to persuade someone that you are serious about them, and just as importantly, how to work out whether someone is serious about you.

The custom of giving engagement rings, for instance, arose in the US in the 1930s when men were having trouble proving they could be trusted.

It was not uncommon even then for couples to sleep together after they became engaged but before marriage, but that was a big risk for the woman.

If her fiance broke off the engagement she could be left without prospects of another marriage.

For a long time, the courts used to allow women to sue for "breach of promise" and that gave them some security, but when the courts stopped doing so both men and women had a problem.

They did not want to wait until they got married, but unless the man could reassure his future wife then sleeping together was a no-no.

The solution was the engagement ring, which the girl kept if the engagement was broken off.

An expensive engagement ring was a strong incentive for the man to stick around - and financial compensation if he did not.

Not committed

Modern lovers might think the idea of engagement ring as guarantee is a thing of the past, but they can still use game theory to size up their partners.

When a couple with separate homes move in together, selling the second home is an important signal of commitment.

That second home is an escape route - valuable only if the relationship is shaky.

If your partner wants to hang on to his bachelor pad, do not let him tell you it is merely a financial investment.

Game theory tells you that he is up to something.
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Game theorists share Nobel prize

US citizen Thomas Schelling and Israeli Robert Aumann have won the 2005 Nobel prize in economics for their work in an area known as game theory.

...................

Professor Schelling has specialised in explaining strategies of international conflict, such as nuclear war.

Professor Aumann has developed the theoretical underpinnings of bargaining, co-operation and conflict.

'Totally overwhelmed'

Professor Schelling, 84, a US citizen, is distinguished university professor at the Department of Economics and the School of Public Policy at the University of Maryland, and emeritus professor of political economy at Harvard University, where he had taught for 20 years.

Professor Aumann, 75, who holds both US and Israeli citizenship but was born in Germany, teaches at Centre for Rationality at the Hebrew University of Jerusalem. He and his family fled Germany for the United States in 1938.

He told reporters that "this was a total surprise. I'm totally overwhelmed."

He said that he hoped that his work could be applied to conflict resolution even in situations like the Middle East.

"I think game theory creates ideas that are important in solving and approaching conflict in general," he added.

Thomas Schelling told reporters that "they (the Nobel committee) linked us together because he is a producer of game theory and I am a user of game theory."

He said his work focused on "using game theory to help myself understand conflict situations and opportunities."

Game theory was first developed by Hungarian mathematician John von Neumann in the 1940s and 1950s as a way to understand decision-making in the real world where several parties were bargaining, perhaps with unequal resources and information.

It developed into a mathematical theory of strategy, which helped explain which decision, to cooperate or not with rivals, had the best pay-off..

John Nash, John Harsanyi and Reinhard Selten won the Nobel economics prize in 1994 for their pioneering work in this area.

Mutually assured destruction

The Royal Swedish Academy of Sciences, which awards the Nobel economics prize, said in its citation that the two current winners had enhanced understanding of co-operation and conflict through game theory analysis which they applied to real-world problems.

Professor Schelling was among the first to apply the insights of game theory to international relations, looking at the nuclear arms race in his classic book The Strategy of Conflict.

He argued that the capability to retaliate was more useful than the ability to resist an attack, and that uncertain retaliation was more credible than certain retaliation.

These insights formed the theoretical underpinnings for the strategy of nuclear deterrence "and proved of great relevance for conflict resolution and efforts to avoid war", the Nobel Prize committee said.

Indeed, both the USA and the former Soviet Union adopted such a strategy - known as mutually assured destruction - during the Cold War, when they developed long-range nuclear weapons but agreed not to develop defensive weapons such as ABMs.

Professor Schelling also used game theory to develop an explanation of why segregation occurs.

Recently, his work has focused on building coalitions for climate change.

Co-operation or conflict

Professor Aumann's work has centred on a different element of game theory, the question of whether co-operation increases if games are continually repeated.

He showed that co-operation is less likely when there are many participants, when interactions are infrequent, when the time horizon is short or when others' actions cannot be clearly observed.

"Insights into these actions help explain economic conflicts such as price wars and trade wars, as well as why some communities are more successful than others in managing common resources," his citation said.

Professor Aumann received a PhD in mathematics from MIT, and his work, in contrast to that of Professor Schelling, is highly mathematical.
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Hi,

In A Beautiful Mind, Nash talks about all his friends wanting to pick up the blonde girl but for the best outcome they should all go after her friends. What did this inspire him to write? Was it the Nash Equilibrium?

Thanks.
straightjacket
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straightjacket wrote:Hi,

In A Beautiful Mind, Nash talks about all his friends wanting to pick up the blonde girl but for the best outcome they should all go after her friends. What did this inspire him to write? Was it the Nash Equilibrium?

Thanks.

Hello and welcome to the math forum. I saw the movie once and read the book several times. I may say things that were not in the movie.

Do you recall that John Nash and his buddies, all of them Princeton graduate students, were sitting in a bar. They saw a beautiful blonde sitting with a group of pretty girls.

http://magazine.richmond.edu/winter2003 ... ture3.html

In the movie A Beautiful Mind, mathematical genius John Nash boils down competitive behavior to the simple terms of sexual conquest. Trying to pick up women at a bar, Nash and several other Princeton graduate students are captivated by a beautiful blonde surrounded by other pretty girls.

“Recall the lessons of Adam Smith, father of modern economics,” says one of Nash’s cohorts. “Individual ambition serves the common good. Every man for himself!”

In a flash of revelation, the brilliant Nash disagrees.

“Adam Smith needs revision,” he says. “If we all go for the blonde, we block each other. So then we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes for the blonde? We don’t get in each other’s way, and we don’t insult the other girls. That’s the only way we win.”

Suddenly more excited about economic theory than sexual opportunity, Nash lapses into an intellectual reverie that summarizes his Nobel Prize-winning dissertation. “Adam Smith said the best result comes from everyone in the group doing what’s best for himself. Incomplete! Incomplete! Because the best result will come from everyone in the group doing what’s best for himself and the group. Adam Smith was wrong!”

Did it really happen that way? The real John Nash could not be reached for comment on this story, but University of Richmond economist Jonathan B. Wight sees at least one flaw in the Hollywood version — not in the movie’s portrayal of Nash, but in Nash’s understanding of Adam Smith.

Wight argues that Smith would be appalled by the suggestion that he advocated selfishness. In his recent novel, Saving Adam Smith, Wight brings the father of modern economics back to life to set the record straight by making the critical distinction between selfishness and self-interest tempered by self-restraint.

Adam Smith would have readily agreed with Nash’s “original” idea. In fact, centuries before Nash devised his equilibrium for non-cooperative games, Smith came up with his own equilibrium for the game of life: “Superior prudence,” he said, “is the best head joined to the best heart.” But over the years, economics instructors have edited out Smith’s “moral sentiments” — leaving only the impression that the “invisible hand” of free markets can magically convert individual greed into mutual benefit.

Giacomo
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### Evolutionary Game Theory

One of the exciting branches of modern game theory is known as evolutionary game theory, which applies traditional game theory to biological systems.

Here is a lecture on the subject by Prof. Martin Nowak of Harvard:
http://athome.harvard.edu/programs/evd/

Here is a website his associate, Christoph Hauert, has designed for exploring evolutionary dynamics using Java applets:
http://www.univie.ac.at/virtuallabs/

There is also an entry in the Stanford Encyclopedia of Philosophy:
http://plato.stanford.edu/entries/game-evolutionary/

xcthulhu
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At the chance of sounding parasitic, that's what I like about this website a free education. I hope you guys don't mind getting nothing in return but a thank you. What makes it better than just reading the material is the chance to see if you understood what is being presented to you by actually communicating with the authors.

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Can the lecture continue with some basic math behind the theory?

wolfhnd
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wolfhnd wrote:Can the lecture continue with some basic math behind the theory?

Yes, it does.

I'm so busy working on the Riemann Hypothesis, so I'll post a link to MIT Open course ware :

Search results

For example, the first item on the list: Game Theory: Lecture Notes

http://ocw.mit.edu/OcwWeb/Economics/14- ... ourseHome/
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### Re: Game Theory (lectures, articles, etc.)

Here's an interesting article that illustrates the distinction between ordinary decision theory and game theory:

http://www.scientificamerican.com/artic ... y-decision

For convenience, I'm posting the article

I heard this tale in India. A hat seller, on waking from a nap under a tree, found that a group of monkeys had taken all his hats to the top of the tree. In exasperation he took off his own hat and flung it to the ground. The monkeys, known for their imitative urge, hurled down the hats, which the hat seller promptly collected.

Half a century later his grandson, also a hat seller, set down his wares under the same tree for a nap. On waking, he was dismayed to discover that monkeys had taken all his hats to the treetop. Then he remembered his grandfather's story, so he threw his own hat to the ground. But, mysteriously, none of the monkeys threw any hats, and only one monkey came down. It took the hat on the ground firmly in hand, walked up to the hat seller, gave him a slap and said, "You think only you have a grandfather?"

This story illustrates an important distinction between ordinary decision theory and game theory. In the latter, what is rational for one player may depend on what is rational for the other player. For Lucy to get her decision right, she must put herself in Pete's shoes and think about what he must be thinking. But he will be thinking about what she is thinking, leading to an infinite regression. Game theorists describe this situation by saying that "rationality is common knowledge among the players." In other words, Lucy and Pete are rational, they each know that the other is rational, they each know that the other knows, and so on.

The assumption that rationality is common knowledge is so pervasive in game theory that it is rarely stated explicitly. Yet it can run us into problems. In some games that are played over time, such as repeated rounds of Prisoner's Dilemma, players can make moves that are incompatible with this assumption.

I believe that the assumption that rationality is common knowledge is the source of the conflict between logic and intuition and that, in the case of Traveler's Dilemma, the intuition is right and awaiting validation by a better logic. The problem is akin to what happened in early set theory. At that time, mathematicians took for granted the existence of a universal set-a set that contained everything. The universal set seemed extremely natural and obvious, yet ultimately several paradoxes of set theory were traced to the assumption that it existed, which mathematicians now know is flawed. In my opinion, the common knowledge of rationality assumed by game theorists faces a similar demise. -K.B.

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Giacomo
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### Re: Game Theory (lectures, articles, etc.)

Speaking of chess, here's Conway's Angel problem. It is played on an infinite chessboard.

Angel problem

Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move?

Giacomo
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### Re: Game Theory (lectures, articles, etc.)

CHESS

Chess Masters Are Quick On The Trigger

http://www.sciencedaily.com/releases/20 ... 060812.htm

Chess is typically envisioned as a game of concentration and deliberation, a game not to be taken lightly and a game definitely not to be rushed. But some recent research suggests that it's actually a player's split-second intuitions that make the master.

Bruce D. Burns of Michigan State University, in an article to be published in the July issue of Psychological Science, a journal of the American Psychological Society, compared chess players' rankings at normal tournament chess to their rankings at fast-paced blitz chess. In blitz chess, players have 5 minutes to complete all of their moves, which gives them an average of 7.5 seconds for each move. Because of that limitation, they don't have the time to mull over their moves and are forced to rely on their immediate intuition.

What Burns found was that players' rankings at normal chess were remarkably accurate predictors of their rankings at blitz chess, especially among higher-ranked players. Among lower-ranked players, performance at normal chess didn't seem to relate quite as strongly to their performance at blitz chess. This suggests that the skills chess masters use in normal chess are the same as those they use in blitz chess: lightning-fast intuition. Less-skilled players' instincts, on the other hand, aren't as developed as those of the experts, and the time constraints of blitz chess demonstrate the differences between their intuitive and ruminated game play

So if it's the quick thinkers that always win at chess, why do all the chess experts still spend hours on a game? Even though the pros can use their instincts to think of a good move in a matter of seconds, it takes a while to consider all the other possible moves and decide on the best one.
Giacomo
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### Re: Game Theory (lectures, articles, etc.)

jimmy123
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### Re: Game Theory (lectures, articles, etc.)

jimmy123
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### Re: Game Theory (lectures, articles, etc.)

Giacomo wrote:Speaking of chess, here's Conway's Angel problem. It is played on an infinite chessboard.

Angel problem

Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move?

Everytime I read one of your posts I wish I could go back in time and study mathematics in university instead of Economics. Although reading game theory note again was quite the pleasure. Thanks for the free education Giacomo.

asocialnorm
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### Re: Game Theory (lectures, articles, etc.)

Everytime I read one of your posts I wish I could go back in time and study mathematics in university instead of Economics. Although reading game theory note again was quite the pleasure. Thanks for the free education Giacomo.

Alot of math student wish they could start over and study accounting.
jimmy123
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### Re: Game Theory (lectures, articles, etc.)

I've recently learned about an applied game theoretician named Bruce Bueno de Mesquita who uses his bargaining model to (surprisingly reliably) predict the outcome of political events, foreign policy, as well as litigations, mergers and acquisitions. He has a really interesting talk on TED, and other talks/publications, some of which can be accessed by following the author tab of the website for this recently published book, "The Predictioneer's Game" (http://www.predictioneersgame.com/author).

I finished reading this book a couple of days ago, and found it very interesting and informative, and I'd recommend it to anyone provided that they are at least mildly interested in politics or law. It's certainly made me more likely to study more game theory in the future. I'm taking a Game Theory in economics course next semester, and am really looking forward to it :)

vzamfir
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### Re: Game Theory (lectures, articles, etc.)

Here's a link to the book mentioned:
http://www.amazon.com/Predictioneers-Ga ... 1400067871

Here's a link to the TED talk:
http://www.ted.com/talks/bruce_bueno_de ... uture.html

The book looks very neat! I'll probably try to read it next!

~XCT

xcthulhu
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### Re: Game Theory (lectures, articles, etc.)

Wouldn't Finishing first as the nice guy be a perception value only.

Each person would view the guy who finishes first with their intrepretaion of how the events unfolded and who the nice guy used along his way and how he used them to finish.

Ah a game of symentics.

Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move?

The angel may be able to jump 1000 squares per move but is only moving around to avoid the inevitability of the devil removing enough squares that will lead the angel into not being able to move thus trapping the angel and making the devil the victor.
Bucephelus
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### Re: Game Theory (lectures, articles, etc.)

Giacomo » May 24th, 2009, 7:22 pm wrote:Speaking of chess, here's Conway's Angel problem. It is played on an infinite chessboard.

Angel problem

Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move?

Forgive the neophyte approach ...

If the board size and the maximum allowable number of moves are both infinite, and the Devil has infinite range with regards to the squares it can choose to eliminate, but the Angel is limited to distances of up to a thousand squares per turn, then that would seem to offer an important limiting factor in the direction of a solution, since it entails the Devil having greater power than the Angel.

Question: Can the Angel see beyond the 1000 squares it is limited to moving per turn ?

If the answer is no, then the Devil can simply spend a large chunk of eternity slowly erecting a 1000 square wide barrier at fantastic range while the Angel moves unopposed however it wishes, uncertain where the Devil actually is. Once the Devil's outer barrier is complete the Angel's ultimate fate is sealed because the Devil can slowly work a checkerboard pattern inwards until the Angel comes within view, and eventually trap him at some finite point prior to the end of time, by virtue of his range power exceeding the Angels.

However, if the answer (on visibility) is yes, the game gets quite a bit more complicated. I'm guessing the answer is yes, otherwise it wouldn't be much of a problem. ;)

Correct me if I'm wrong, but I suspect the problem has only thus far been solved for fairly modestl values of range ?

Darby
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