Game theory by john nash pdf

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. What has enabled information economics to burgeon in the last twenty years is the parallel development of concepts and techniques in game theory. 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.

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.

In conclusion, I offer some suggestions for further reading to those whose appetites are whetted by my sampler of examples. General interest: Dixit, Avinash, and Barry Nalebuff. New York: W. Norton, Schelling, Thomas. The Strategy of Conflict. The dissertation is provided for research use only. Nash Equilibrium is a game theory Game Theory Game theory is a mathematical framework developed to address problems with conflicting or cooperating parties who are able to make rational decisions.

When it was rst introduced, Science A Bane Essay Game Theory focused soley on two-person zero-sum games, but has since evolved to encompass strategies and game play between more players. Preview this book » What people are saying - ….

John Nash: Playing Nice is Good for Everyone February 9, by Kate Vitasek Next in my mini-series about the great economic thought leaders who were seminal in the development and success of modern outsourcing is one of my favorites, the mathematician John F.

Dur-ing the s Game Theory was largely advanced by many scholars researching this area of mathematics. Cooperative or non-cooperative Symmetric and asymmetric Zero-sum and non-zero-sum Simultaneous and sequential Perfect information and imperfect information Combinatorial games Infinitely long games Discrete and continuous games Many-player and population games Metagames.

The player is specified by a number listed by the vertex. Player1 chooses Player 1 0,0 3,4 Rows choosesDown Player2 chooses Normal form or payoff matrix Columns of a 2-player, 2-strategy game. John Nash was a mathematician and an economist. He developed several theories in economics. He was a Princeton and CMU graduate. His most important contribution was the theory of Nash equilibrium He is the person portrayed in the movie A beautiful mind.

What is Nash Equilibrium? For any two groups that do not co-operate there will be a point at which neither group can benefit from unilateral action , and that the groups will hold their strategies constant at this point. The Nash equilibrium is not usually the most effective strategy; it is only the best one without co-operation.

Prisoners Dilemma Cooperation is usually analysed in game theory by means of a non-zero-sum game called the "Prisoner's Dilemma. The prisoner's dilemma is meant to study short term decision-making. Each player gains when both stay silent. Friend or Foe? It is an example of the prisoner's dilemma game tested by real people.

Chicken Game Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will Driver B Driver B both crash. Analysis of Chicken Game Both lose when both swerve.

Game theory is exciting because although the principles are simple, the applications are far-reaching. Game theory is the study of cooperative and non cooperative approaches to games and social situations in which participants must choose between individual benefits and collective benefits.

Game theory can be used to design credible commitments, threats, or promises, or to assess propositions and statements offered by others. Open navigation menu. Close suggestions Search Search. User Settings. On the contrary, if a solution can be obtained from a con- vincing set of axioms, this indicates that the solution is appropriate for a wider variety of situations than those captured by the specific non-cooperati- ve model.

Nash breaks radically with this tradition. He assumes that bargaining between rational players leads to a unique outcome and he seeks to deter- mine it. He solves the problem in the 2-person case and he derives his solu- tion both by means of the axiomatic approach and as the outcome of a non- cooperative model. Three axioms specify the relation w hich should hold betw een the solution and the set B: i Pareto efficiency, ii symmetry and iii inde- pendence of irrelevant alternatives.

Axion iii states that, if the set of feasible utility pairs shrinks but the solu- tion remains available, then this should remain the solution.

It is more dif- ficult to defend than the others and there has been considerable discussion of it in the literature. Recent developments in non-cooperative bargaining theory w hich build o n the seminal p ap er [42] hav e co nfirmed this interp retatio n.

Namely, assume players alternate in pro po sing po ints fro m B until agree- ment is reached. Assume that if an offer is rejected there is a small but posi- tive probability that negotiations break down irrevocably. The equilibrium co nditio n states that each time each responder is indifferent betw een accepting the current proposal and rejecting it.

Indeed the equilibrium conditions imply that both equilibrium proposals have the same Nash product, hence, since they have the same limit, they converge to the Nash solution. However, even more important is that this application of the Nash program may clarify some ambiguities concerning the choice of the threat point in applications of the Nash bargaining model.

See [5] for further discussion. Nash In the variable threat case, each party has a choice how much pressure to put o n the o ther. Tw o additional axioms allow reduction of the problem to the case with fixed threats and, hence, determine the theory. The first is equi- valent to assuming that each player has an optimal threat strategy, i. The second says that a play- er cannot improve his payoff by eliminating some of his strategies.

In the non-cooperative approach, Nash assumes that players first commit themsel- ves to their threats. Players will be forced to execute their threats if they can- not come to an agreement in the second stage. Each pair of threats induces a fixed-threat bargaining subgame in which the distinguished equilibrium that maximizes the product of the utility gains is selected. Applying back- w ards induction and using this selection i.

Consequently, the overall game has a value and optimal threat strategies. Needless to say, the solution obtai- ned by the non-cooperative approach coincides with that obtained by means of the axioms. It imposes no conditions concerning behavior after a deviation from the theory has occurred.

It turns out that not all Nash equilibria have this desirable property: After a deviation from the equilibrium has occurred, a believer in this equilibrium may prefer to deviate from it as well. The equilibrium relies on a non-credible threat of- player 2. Otherwise it will have little meaning.

For, in general, to execute the threat will not be something A would want to do, just of itself. To eliminate equilibria that rely on non-credible threats, various refine- ments of the Nash equilibrium concept have been proposed, which will not be surveyed here see [50]. Let us just note that tw o papers of Reinhard Selten were fundamental. Selten [45] argues that a theory of rational beha- vior has to prescribe an equilibrium in every subgame since otherwise at least one player would have an incentive to deviate once the subgame is reached.

He calls equilibria having this property subgame perfect. They can be found by a backwards induction procedure. Selten [46] suggests a further refinement that takes the possibility of irrational behavior explicitly into account, i. Formally, he considers slightly perturbed versions of the original game in which players with small probabilities make mistakes and he defines a trembling hand perfect equilibrium as one that is a limit of equilibrium points of perturbed games.

It is interesting to note that Nash already discussed a game with an imperfect equilibrium see Ex. Consequently, a game typically allows multiple perfect equilibria. At present, the debate is still going on of whether these strong stability requirements indeed capture necessary requi- rements of rational behavior.

N ash alread y enco untered this p ro blem and in his study of the fixed-threat bargaining problem.

Clearly, any pair of demands that is Pareto efficient constitutes a pure equilibrium of the game. But if we discriminate between them by studying their relative stabilities we can escape from this troublesome non-uniqueness.

Any maximizer of the function d,d,h d is an equilibrium of this perturbed game and all these maximizers converge to the unique maximizer of the function uluZ on B as the noise vanishes. Furthermore, if h varies regularly, the per- turbed game will have the unique maximizer of d,d,h d as its unique equili- brium.

It follows that the Nash bargaining solution is the unique necessary limit of the equilibrium points of the smoothed games. The Nash-property is not an unintended by- product of our theory. For example, in the game of Figure 1, the equilibrium a, a has a Nash product of 30, while the N ash-p ro d uct o f b, J?

This p ap er w ould be incomplete if it w ould not also mention the pioneering w ork of Nash, to gether w ith Kalisch, Milno r and Nering [25] in experimental eco - nomics. That paper reports on a series of experiments concerning n -person games and aims to compare various theoretical solution concepts with the results of actual play, i. The authors find mixed support for various theoretical solu- tion concepts and they discuss several reasons for the discrepancy between theoretical and empirical results.

Among others, the role of personality dif- ferences, the fact that utility need not be linear in money and the importan- ce o f apparent fairness co nsideratio ns are mentio ned. In additio n, several regularities are documented, such as framing effects, the influence of the number of players on the competitiveness of play, the fact that repetition of the game may lead to more cooperative play, and the possibility of inducing a more competitive environment by using stooges.

A s documented by the importance of the above mentioned concepts in current experimental eco- nomics, the paper is an important milestone in the development of descrip- tive game theory. See [41]. They did no t co n- stantly play the one-shot equilibrium, but they also did not succeed in rea- ching an efficient outcome either. The experimenters view their experiment as a test of the predictive relevance of the one-shot equilibrium and they interpret the evidence as refuting this hypothesis.

Nash, however, argues that the experimental design is flawed, that the repeated game cannot be thought o f as a sequence o f independent games and he suggests that the results would have been very different if the interaction between the trials had been removed.

Nash in [12] Hence, he not only specifies an appropriate design for testing static equilibrium predictions; he also describes the essential insight in the theory of repeated games and he points to a specific form of bounded rationality as an explanation for obser- ved discrepancies between theoretical predictions and empirical outcomes.

Indeed, much of the modern literature in economics and related disciplines takes the follo- w ing form: A social situation is modeled as a non-cooperative game, the Nash equilibria of the game are computed and its properties are translated into insights into the original problem. The Nash bargaining solution can also be considered a very successful solution concept since it has also been applied frequently.

Furthermore, because of its more abstract nature, it is associated with ambi- guities, which might inhibit successful applications. Such ambiguities may be resolved by application of the Nash program, i. The problems associated with multiplicity of equilibria and with the fact that not all equilibria need correspond to rational behavior, have hampered successful application of the Nash program.

Nash resolved these difficulties for the special case of 2-person bargaining games. Inspired by his ideas and building on his techniques an important literature dealing with these issues has been developed, which enables the analysis and solution of more com- plicated, more realistic games.

Hence, the domain of applicability of non- cooperative game theory has been extended considerably. It is to be expec- ted that, as a result, our insights into the workings of society will be enhan- ced as well. Why did you not publish the interpretations which are in your thesis w hen it w as mad e the basis o f the article [38], in the A nnals o f Mathematics?

So that became a separate publication in Econometrica, differentiated from the rest of it, while that which could be presented more as a mathematical paper went into the Annals of Mathematics. The meeting is now open to questions from anyone.

I found the assumptions that there are only finitely many strategies very intuitive, very natural, but of course to prove it you have to assume that you can vary the mixed strategies continuously. A nd if I now think that having only finitely many actions available is very natural.

I also have to assu- me that only finitely many options in randomizing are available. Would you agree that this should be viewed as an assumption for the definition of ratio- nal players to justify that a player can continuously vary probabilities in choo- sing pure actions?

How do you justify it? Otherwise I would have the con- ceptual philosophical problem. I think I can live w ith this finitely many actions, but the Nash theorem somehow has to rely on continuous variation of probabilities.

Would you also see it as an assumption of rational players, so it is more philosophical. Thank you. Mathematically of course it is clear that you must have the continuity. You can get quite odd numbers in fact. Nash numbers. There is something I just wanted to say. When I heard about the Nobel awards, and I heard that the persons were who they were, I wondered how they were connected. And talking to some persons I found that impression is sort of con- firmed, that there may be specific aspects of it that are not immediately accepted.

But something can be more interesting if it is not immediately accepted. So there is the problem; the possibility that all cooperative games could really be given a solution. This could be analogous to the Shapley value. If it were really valid you would be able to say, here is a game, here are the possibilities of cooperation, binding agreements, threats, everything, this is what it is worth to all the players. You have a management-labor situation, you have international politics, the question of membership in the common market; this is what it is worth.

That is, if everything could be measured or given in terms of utilities. So the possibility that there could be something like that is very basic, but Shapley would have had that very early if the Shapley value were really the true value.

But one example in this book I stu- d ied sho w s ho w the so lutio n co nsid ered there in fact d iffers fro m the Shapley value, and so it is a very interesting comparison. I think there will be further work.

N O TES 1 Actually, Nash also assumed that in a non-cooperative game, the players will be unable to communicate with each other. Yet, in my own view, this would be a needlesssly restrictive assumption. For if the players cannot enter into enforceable agreements, then their ability to communicate will be of no real help toward a cooperative outcome. But this is the term used by many other game theorists to describe this class of Nash equilibria. Existence of an equilibrium for a competitive eco- nomy.

Econometrica 3: - What is game theo ry trying to acco mplish?