...Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright © 1995–2002 by Martin J. Osborne. All rights reserved. No part of this book may be reproduced by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from Oxford University Press, except that one copy of up to six chapters may be made by any individual for private study. 2 Nash Equilibrium: Theory 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Strategic games 11 Example: the Prisoner’s Dilemma 12 Example: Bach or Stravinsky? 16 Example: Matching Pennies 17 Example: the Stag Hunt 18 Nash equilibrium 19 Examples of Nash equilibrium 24 Best response functions 33 Dominated actions 43 Equilibrium in a single population: symmetric games and symmetric equilibria 49 Prerequisite: Chapter 1. 2.1 Strategic games is a model of interacting decision-makers. In recognition of the interaction, we refer to the decision-makers as players. Each player has a set of possible actions. The model captures interaction between the players by allowing each player to be affected by the actions of all players, not only her own action. Specifically, each player has preferences about the action profile—the list of all the players’ actions. (See Section 17.4, in the mathematical appendix, for a discussion of profiles.) More precisely, a strategic game...
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...In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.[1] If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision, and Will is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others in the game. Contents [hide] * 1 Applications * 2 History * 3 Definitions * 3.1 Informal definition * 3.2 Formal definition * 3.3 Nash's Existence Theorem * 4 Examples * 4.1 Coordination game * 4.2 Prisoner's dilemma * 4.3 Network traffic * 4.4 Competition game * 4.5 Nash equilibria in a payoff matrix * 5 Stability * 6 Occurrence * 6.1 Where the conditions are not met * 6.2 Where the conditions are met * 7 NE and non-credible threats * 8 Proof of existence * 8.1 Proof using the Kakutani fixed...
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...A Nash equilibrium is a pair of strategies, one for each player, in which each strategy is a best response to other. These represent the likely outcome of the game. According to Roger B. Myerson, “If we can predict the behavior of all the players in such a game, then our prediction must be a Nash equilibrium, or else it would violate this assumption of intelligent rational individual behavior. That is, if our predicted behavior does not satisfy the conditions for Nash equilibrium, then there must be at least one individual whose expected welfare could be improved simply by re-educating him to more effectively pursue his own best interests, without any other social change.” The above argument does not prove that Nash equilibrium should be the only methodological basis for analysis of social institutions. But it does explain why studying Nash equilibria should be a fruitful part of the critical analysis of almost any kind of social institution. The prisoners dilemma is one of the best examples of Nash Equilibrium. | | Jack | | | | C | NC | Tom | C | -10,-10 | 0,-20 | | NC | -20,0 | -5,-5 | | | | | *Numbers represent years in prison If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the outcome will be a Nash equilibrium.According to the above game both players know that 10 years is better than 20 and 0 years is better than 5; therefore, C is their dominant strategy and they will both...
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...Lecturas Matem´ticas a Volumen 24 (2003), p´ginas 137–149 a John Nash y la teor´ de juegos ıa Sergio Monsalve Universidad Nacional de Colombia, Bogot´ a Al profesor y acad´mico Don Jairo Charris Casta˜ eda e n In memoriam Abstract. In the last twenty years, game theory has become the dominant model in economic theory and has made significant contributions to political science, biology, and international security studies. The central role of game theory in economic theory was recognized by the awarding of the Nobel Price in Economic Science in 1994 to John C. Harsanyi, John F. Nash, & Reinhard Selten. The fundamental works in game theory of John F. Nash together with a brief exposition of them are included in this article. Key words and phrases. John Nash, History of Mathematics, Game Theory 1991 Mathematics Subject Classification. Primary 01A70. Secondary 91A12. Resumen. En los ultimos veinte a˜os, la teor´ de juegos se ha ´ n ıa convertido en el modelo dominante en la teor´ econ´mica y ha ıa o contribuido significativamente a la ciencia pol´ ıtica, a la biolog´ ıa y a estudios de seguridad nacional. El papel central de la teor´ ıa de juegos en teor´ econ´mica fue reconocido con el premio Nobel ıa o en Econom´ otorgado a John C. Harsanyi, John F. Nash & ıa Reinhard Selten en 1994. Se presentan los aportes de John Nash a la teor´ de juegos conjuntamente con una exposici´n ıa o elemental de ellos. 138 SERGIO MONSALVE 1. Introduction La Real Academia Sueca para...
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...this rebate into account, your marginal cost of a midsized automobile is $11,000. What price should you charge for a midsized automobile if you expect to maintain your record sales? 2) In a two-player, one-shot simultaneous-move game each player can choose strategy A or strategy B. If both players choose strategy A, each earns a payoff of $500. If both players choose strategy B, each earns a payoff of $100. If player 1 chooses strategy A and player 2 chooses strategy B, then player 1 earns $0 and player 2 earns $650. If player 1 chooses strategy B and player 2 chooses strategy A, then player 1 earns $650 and player 2 earns $0. a. Write the above game in normal form. b. Find each player's dominant strategy, if it exists. c. Find the Nash equilibrium (or equilibria) of this game. d. Rank strategy pairs by aggregate payoff (highest to lowest). e. Can the outcome with the highest aggregate payoff be sustained in equilibrium? Why or why not? a. Write the above game in normal form. | |Player 1 chooses A |Player 1 chooses B...
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...Information k = units of Capital l = units of Labour rK = Cost of Capital in dollars = $20 wL = Labour cost in dollars = $10 Output: Q=K^(1/4) √L Solution Min: C_((K,L))=r_K .k+ w_L .l=20k+10l Constraint: Q=k^(1/4) √l Using Lagrangian method L_((k,l,λ))=20k+10l- λ(k^(1/4) .l^(1/2)-Q) dL/dk=20- 1/4 λk^(-3/4) .l^(1/2)=0 dL/dl=10- 1/2 λk^(1/4) .l^(-1/2)=0 dL/dλ=- 1/4 λk^(1/4) .l^(1/2)-Q=0 Equating 20- 1/4 λk^(-3/4) .l^(1/2)= 0 20= 1/4 λk^(-3/4) .l^(1/2) λ= (80k^(3/4))/l^(1/2) 10- 1/2 λk^(1/4) .l^(-1/2)=0 10= 1/2 λk^(1/4) .l^(-1/2) λ= (20l^(1/2))/k^(1/4) (80k^(3/4))/l^(1/2) = (20l^(1/2))/k^(1/4) 80k=20l 4k=l Substituting Lagrangian relation into Output equation Q=K^(1/4) √L Q=k^(1/4) 〖(4k)〗^(1/2) Q=k^(1/4) 〖2k〗^(1/2) Q/2=k^(3/4) k= Q^(4/3)/2^(4/3) = Q^(4/3)/∛16= Q^(4/3)/(2∛2) Solving for l 4k=l 2^2 (Q^(4/3)/2^(4/3) )=l l= Q^(4/3)/2^(2/3) = Q^(4/3)/∛4 Minimum total expenditure on capital and labour in terms of Q C_((K,L))=20k+10l C_((K,L) )=2^2.5.(Q^(4/3)/2^(4/3) )+10(Q^(4/3)/2^(2/3) ) C_((K,L) )= (5Q^(4/3)+10Q^(4/3))/2^(2/3) = 〖15Q〗^(4/3)/2^(2/3) = 〖2^2.5.Q〗^(4/3)/2^(2/3) = (5Q^(4/3))/∛2 Information x denotes units of good X y denotes units of good Y Cx denotes the unit cost of good X in dollars = $1 Cy denotes the unit cost of good Y in dollars = $1 0 < β < 1 M denotes the maximum...
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...Ball Like Steve Nash Standing at 6’3 and weighing in at 178 lbs most people know the infamous Steve Nash from the past fifteen years that he has dedicated his life to professional basketball. Before pursuing the American dream of becoming a professional athlete, Nash was born in South Africa but grew up in Victoria, Canada where he played soccer till his late teens. He later discovered a passion for basketball where he hooped at St. Michaels University School with his younger brother Martin. He comes from a family of athletes including his father John Nash who played minor professional soccer in South Africa. His sister Joann Nash held the title as captain of the soccer team for The University of Victoria for three years. It was only right for him to keep on the family legacy. Nash took his talent to Santa Clara University in California where he led his team to the NCAA tournament as a freshman. He established a name for himself by his sophomore year with his quick feet, fundamentals, and amazing skills. Nba scouts were all over him and at no surprise he was drafted in the first round by the Phoenix Suns in 1996. He played two seasons with the Suns before being traded to the Dallas Mavericks. There Nash made a name for himself in the league with Dirk Nowitzki and Michael Finley by his side. They formed an unstoppable trio that carried the team to the Western Conference Finals. Although they didn’t get much farther Nash participated in the NBA All-Star Game and was recognized...
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...MGF1106 “John Nash Biography” “I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia”. – John F. Nash Jr. John Nash Jr. was born on June 13, 1928 in Bluefield, West Virginia. Mr. Nash was the older of the two children that his father, John Nash Sr. and mother Martha Martin had. John and his sister grew up in the Great Depression yet they were fortunate that their father was able to keep his job as an electrical engineer with Appalachian Power Company and they never suffered through that time. They actually lived in a beautiful house not very far from a country club, so the family never suffered in any way,...
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...Running head: CASE STUDY OF JOHN FORBES NASH, JR. 1 Case Study of John Forbes Nash, Jr. Lauren Shipp PSY410 May 26, 2014 Kidd Colt, Ed. D., LMHC CASE STUDY OF JOHN FORBES NASH, JR. 2 Case Study of John Forbes Nash, Jr. John Forbes Nash Jr. is a renowned and accomplished mathematician. He received his Ph.D. from Princeton University and taught at MIT and Princeton. He wrote The Equilibrium Point, later becoming known as the Nash Equilibrium, which revolutionized economics. In 1994, he received the Nobel Peace Prize in Economic Science for his pioneering work in game theory. He is one of the most brilliant mathematicians of modern time, but most of his life he suffered from schizophrenia (Meyer, et al., 2009). The following is a brief account of a case study depicting his struggle with schizophrenia. Overview Early in Nash’s life he showed signs of abnormal behavior. He was extremely intelligent and could read by age 4, but was unsociable and had problems with concentrating and following simple directions. As he grew older, his behavior became more bizarre. He would do such things as eat grass, torture animals, and use explosives in chemical experiments. He still showed sign of unsocial behavior (Meyer, et al., 2009). When he entered Carnegie Institute of Technology to study chemical engineering, his abnormal behavior continued. He acted childish, and would do such things as repeatedly hit a single key on a piano for hours. After receiving his Ph.D. from Princeton...
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...Definitions and illustrations of the concepts of Pareo and Nash equilibria. PARETO OPTIMUM/EQUILIBIRUM/EFFICIENT The idea of pareto optimum runs through all aspects of economics, for example, comparing different tax rules, and not just in game theory. It gained currency as a test to be applied in selecting economic policy as a result of a contradiction in the theory of the market deriving from the philosophy of utilitarianism. Utilitarianism could be construed as a justification for the oppression of the individual in the name of the greater good of society. This approach sits uncomfortably with the libertarian ideals of market economics, which put the individual above church and the (feudal) state, and by implication above the monarch in direct challenge to feudal rule. (see, eg, http://www.google.co.uk/#hl=en&source=hp&biw=1126&bih=425&q=Shanti+Chakravarty+neoliberal&btnG=Google+Search&aq=f&aqi=&aql=&oq=Shanti+Chakravarty+neoliberal&fp=45f26fda8f9185dd). To get around the difficulty posed by classical utilitarianism, market efficiency theorems came to rely on the idea of ordinal utility which does not allow for inter-personal comparison of utility. The paretian criterion explicitly rejects inter-personal comparison in arriving at economic policy. The paretian criterion is focused entirely on the individual. A pareto optimum is a state of affairs whereby NO ONE, no individual, can be made better off without making someone worse off. Since no one is above anyone in the...
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...team 1 and team 2 would invest money until they reach Nash equilibrium and their profits are maximized. Since this model is a symmetric model, both teams have the same incentive to win and therefore at equilibrium it can be assumed that win% for team 1 will equal the win% for team 2. Knowing these givens and the equation given, profits for team 1 can be found as followed: π1 = Vw1 – t1 → V(2-g)/4 π1 = (500)(2-.5)/4 π1 = 187.5 The profit can then be plugged back into the initial equation to determine team 1’s investment: π1 = Vw1 – t1 The win percentages for both teams will be .5 because .5 + .5 = 100% in a symmetric model 187.5 = 500(.5) – t1 t1= 62.5 Since this is a symmetric model, it is assumed that the investments and profits for team 1 will equal team 2’s investments and profits. π1 = π2 187.5 = π2 t1 = t2 62.5 = t2 2. For model 1, profits are larger when g=1/2 because the investments made by both teams are lower and therefore π1 = Vw1 – t1 will result in higher profits. The reason team 1 and team 2’s investments will be lower is because there is less incentive to invest when the advantage for investing goes down, as it does with a lower sensitivity parameter. A lower g (sensitivity parameter) means that teams need to investment less to yield a higher win%. The following graph shows the difference in profits for g=1 and g= ½: 3. To calculate the Nash equilibrium for an asymmetric model win % and team...
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...1. David Nash was born in Esher, Surrey in 1945. He studied at Kingston College of Art (1963-64), Brighton College of Art (1964-67) and Chelsea School of Art (1969-70). After finishing school at Brighton, Nash moved to North Wales before returning to Chelsea for one year in 1969. In Wales he purchased a chapel which has remained both his studio and home ever since. Wales was ideal for Nash, because he was surrounded by abundant resources and plenty of time to further develop his wood works. 2. The most interesting thing I discovered about David Nash was that he uses a chainsaw and axe as his primary tools as well as using a blowtorch in many of his works.David Nash uses wood as a medium for all his works. His interest in working with wood began as a child, when Nash helped clear and replant a forest his father owned. He also worked for the Commercial Forestry Group, where he learned about many kinds of wood. He is known for doing land art involving wood that remains in nature, as well as displaying his wood sculpture in studios. He carves wood from fallen trees as well as creates sculptures from growing plants. 3. Nash first decided to move to Wales after graduating from Brighton College due to the extremely low cost of living. In these years he experimented with making tower like sculptures and some very abstract works. He used paint to give more detail to these sculptures as well, which was unique to this period for Nash. He continued experimenting with this tower theme...
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...It is no secret that during the post-Cold War era there was a wave of skepticism that arose and questions about moving forward. After the Cold War there was a period of stillness where people wondered if there was going to be another global up rise. With WWI, WWII, then the Cold War, many people were left uncertain about the future. One of those people was active scholar Kate Nash, a professor of Sociology at Goldsmiths College, University of London and faculty of the Center for Cultural Sociology at Yale University. In The Cultural Politics of Human Rights Nash talks about why she is skeptical of the ability of global solidarity movements to address rights issues including: poverty, gender inequity, and structural violence. Nash also gives...
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...JOHN FORBES NASH JR. John Forbes Nash Jr. was born June 13, 1928 in Bluefield, West Virginia. Mr. Nash Jr. is an American mathematician who won the 1994 Nobel Prize for his works in the late 1980’s on game theory. Game theory is the study of strategic decision making or more formally known as the mathematical models of conflict and cooperation between intelligent and rational decision makers. Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. Mr. Nash Jr. has also contributed numerous publications involving differential geometry, and partial differential equation (PDE). Differential geometry is a mathematical discipline that uses differential calculus and integral calculus, linear algebra and multi linear algebra to study geometry problems. Partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. These are used to formulate problems involving functions of several variables. Mr. Nash Jr. used all of these skills and is known for developing the Nash embedding theorem. The Nash embedding theorem stated that every Riemannian manifold ( a real smooth manifold equipped with an inner product on each tangent space that varies smoothly from point to point) can be isometrically embedded into some Euclidean space ( a three dimensional space of Euclidean geometry, distinguishes these spaces from the curved spaces...
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...Essentials of game theory 1. Introduction Game theory is the study of strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."[1] An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.[2] Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of class relations, and has developed into an umbrella term for the logical side of science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better his results, given the other approach. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided...
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