...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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...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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...plays strategy given in Question 3 above, what is optimal strategy for wife? 5. For what possible reasons can Wife DEVIATE from playing her optimal strategy in Question 4? 6. Count total Number of strategies for Husband 7. Count total Number of strategies for Wife. 8. List all strategies for Husband. 9. List all strategies for Wife. 10. Make a Normal Form Game. 11. Identify Weak and Strong Nash Equilibrium. Also give reason. 12. From above Nash Equilibriums, identify subgame perfect equilibrium and explain why? 13. From above Nash Equilibriums, identify subgame imperfect Equilibrium and explain why it is subgame imperfect. Ultimatum Game: In an Ultimatum Game, the proposer must divide a pie worth $30, and shares must be multiples of $10. As usual, the responder gets to accept or reject. 1. Draw the extensive form game tree. 2. Use backwards induction and rollback to solve the game. 3. Is there a dominant strategy for the proposer in this game? 4. List strategies for both players, corresponding outcomes, and use to write the game in normal form. 5. Identify a Nash Equilibrium in strategies which is NOT subgame perfect. 6. Based on experiments, what is the most likely outcome of this game? 7. Four possible explanations for the conflict between observed outcomes and theoretical predictions are: (i) Fear of Rejection, (ii) Fairness...
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...Situations economists and mathematicians call games psychologists call social situations. While game theory has applications to "games" such as poker and chess, it is the social situations that are the core of modern research in game theory. Game theory has two main branches: Non-cooperative game theory models a social situation by specifying the options, incentives and information of the "players" and attempts to determine how they will play. Cooperative game theory focuses on the formation of coalitions and studies social situations axiomatically. This article will focus on non-cooperative game theory. Game theory starts from a description of the game. There are two distinct but related ways of describing a game mathematically. The extensive form is the most detailed way of describing a game. It describes play by means of a game tree that explicitly indicates when players move, which moves are available, and what they know about the moves of other players and nature when they move. Most important it specifies the payoffs that players receive at the end of the game. Strategies Fundamental to game theory is the notion of a strategy. A strategy is a set of instructions that a player could give to a friend or program on a computer so that the friend or computer could play the game on her behalf. Generally, strategies are contingent responses: in the game of chess, for example, a strategy should specify how to play for every possible arrangement of pieces on the board. An alternative...
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...Biography, a contract for a movie deal, and a nomination for The Pulitzer Prize for Biography. This book is based on a man by the name of John Nash who is a brilliant mathematician and also a paranoid schizophrenic. Even though Nash struggled with paranoid schizophrenia he was still able to revolutionize a concept called game theory. This work that he did with game theory greatly improved what was already established as well as created the Nash equilibrium. Game theory is the mathematical study of strategies used to win games. It began with games like tic-tac-toe and chess since they are easy to analyze because they are known as games of complete information. Complete information is when your opponent’s positions are in clear site. Game theory then went on to analyze card games where player’s cards aren’t seen, which is known as incomplete information. In card games there are elements such as bluffing that can become a variable in the analysis. Mathematicians continued to analyze more critical games such as economics, biology, philosophy, and even which girl to go after. This is around the time that John Nash developed a principle for mutual consistency of player’s strategies which is known as the Nash equilibrium. The Nash equilibrium can be applied to a wider variety of games than the original game theory. This equilibrium proposes that all players have a perfect strategy that becomes stable and if they stray it will make all positions worse off. In the games there...
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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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...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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... Game Theory and Strategic Behavior The main point of the paper is the paradox between theoretically predicted equilibrium and Nash equilibrium. Results of the research state that people often act differently and there is no model of behavior. Most of actions are spontaneous and it is provoke by kind of emotion conditions such as calmness or fearless which sometimes interferes to take the correct decisions. By supporting his ideas the author of the article, introduce ten examples driven by experiments. While I am reading the article, I understand that it covers the whole material that we study in class during the whole semester – for example, it introduces some of the main point of game theory – dynamic games, several exists of games or prediction how the game will end. As we know from the course that most of games are consisted by two payers which shows us at least two treatments. In each game, we have coincided point that is called Nash equilibrium. Without reaching the point of equilibrium, we go to the second treatment where the payoff changes and there is changing in the strategy of player 1. In fact, the article shows us a famous and significant game that also called one-shot traveler’s dilemma game. The main idea of the model is that the payoffs change constantly and there is not affected on the Nash equilibrium. By each round, the strategy and behavior of people are shifting from the lower limit to the upper limit. Another game is represented...
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...units, such as 0, 1, 2, 3, 4, 5, or 6 dollars. Suppose, furthermore, that costs of production are zero for both firms, and each firm aims to maximize its own profits. (a) Write down the strategic form of this game. (b) Is there a strictly dominant strategy equilibrium of this game? Explain. (c) Is there a weakly dominant strategy equilibrium of this game? Explain. (d) What are the action profiles that survive Iterated Elimination of Strictly Dominated actions? Explain. (e) What are the action profiles that survive Iterated Elimination of Weakly Dominated actions? Explain. (f) Is the game dominance solvable? 2. Two people are engaged in a joint project. If each person i puts in the effort xi , a non-negative number equal to at most 1, which costs her c(xi ), the outcome of the project is worth f (x1 , x2 ). The worth of the project is split equally between the two people, regardless of their effort levels. 1 ECO502/EE698A (a) Formulate this situation as a strategic game. (b) Find its Nash equilibria when i. f (x1 , x2 ) = 3x1 x2 , c(xi ) = x2 , for i = 1,2. i ii. f (x1 , x2 ) = 4x1 x2 , c(xi ) = xi , for i = 1,2. (c) In each case, is there a pair of effort levels that yields both players higher payoffs than the Nash equilibrium effort levels? 3. (Hotelling competition) Consumers are uniformly distributed along a boardwalk that is one mile long. Ice-cream prices are regulated, so consumers...
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...A game theory consists of a model that can help us understand how managers, investors and other related parties deal with economic consequences of financial reporting. There are many different types of games, such as cooperative and non-cooperative. The model that will be discussed in this report is on non-cooperative. A game theory consists of an interaction between two or more players. Each player is assumed to maximize their expected utility (profit). Most of the time, this interaction occurs when there is uncertainty and information asymmetry. On top of that, the players must also take into account the actions of the other players when making a decision. The fourth slide illustrates an example of two players involved in a competition. Suppose that there is player A and player B. The actions of player A can be difficult to predict, since his action depends on what he thinks player B will take, and vice versa. Therefore, players’ decisions greatly depend on assumptions. A non-cooperative game has no form of negotiation and no binding contracts. This means that no agreement has been made on purpose. Furthermore, both parties do not exactly know the other party’s action will be. A good example of a non-cooperative model will be the “chicken” game. It is basically about two drivers who drive their cars very fast towards each other. Both drivers have to make their decisions to either drive straight or to swerve at last minute and the one who swerves first will be the looser...
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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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...decide simultaneously whether to hunt for a stag or for hare. Each player has a choice between two strategies: (hunt stag, hunt hare). If both hunt stag, each gets half a stag If both hunt hare, each gets one hare If one hunts for hare while the other tries to take a stag, the former will catch a hare and the latter will catch nothing. If we suppose that the utility of catching a stag is 4 and for a hare is 1, then the payoff matrix for the players can be summarised as follows: Stag Hare (2,2) (0,1) (1,0) (1,1) Game theory is concerned with what outcomes arise in equilibrium. An equilibrium outcome is one which consists of a strategy for each player which is that player’s best response to the strategies of all players. An equilibrium is a stable state which holds because no rational player has an incentive to deviate from that state, where a rational player is one who always attempts to maximise their utility from any situation which requires a choice between alternatives. In the...
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... Explain. There is no dominant strategy from left or right. Both right and left could result in a higher payoff. b. Is there Nash equilibrium in this game? Explain. A Nash equilibrium exists when “players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy (Shor, 2006).” Since neither player has a dominant strategy, this game does not have a Nash equilibrium. c. Why this game is called a cooperative game? Both players have to cooperate with one another and choose the same side. It is in the best interest of both left and right to choose the same side. If not, they will have a big negative payoff. 2. a. What is the firm’s Total Revenue? MC equals MR to maximize the profits. The monopoly output level is at E. From the demand curve, the price should be at A. Total revenue equals Price times Quantity. This is represented at area of AJEO\ b. What is the Total Cost? The average output E on the graph is H. The total Cost equals ATC times Q. This equals area BHEO. c. What is the firm’s Total Profits? Total profits equal total revenue minus total cost. This is represented by area AJHB. d. If the above monopolist were to behave like a perfectly competitive firm (operating in the long run), determine its output. The long run equilibrium price would represent the minimum of the average total cost. This is where the level of MC crosses ATC. The output level would...
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...OLIGOPOLY AND MONOPOLISTIC COMPETITION Up to now, we have covered two extreme types of markets. We covered perfect competition with the highest degree of competition, then we covered monopoly with the lowest degree of competition. Now, we will cover oligopoly and monopolistic competition. These two market types are in between two extremes: they show some features of competition and some features of monopoly. Oligopoly Definition: Oligopoly is a market structure in which there are a few sellers and they sell almost identical products. There are barriers to entry in oligopoly. Oligopoly is characterized by the tension between cooperation and self- interest among these sellers. For example, if the oligopolist firms can cooperate, they can charge a high price and share profits. But if they cannot cooperate and instead they compete because of following their own self-interest, then price goes down and profits decline. We will give examples of this later. Oligopoly Examples: crude oil (Kuwait, Iraq, Saudi Arabia, Venezuela, Kazakhstan, Azerbeijan) , coke (Coca Cola, Pepsi, Cola Turka), GSM providers (Turkcell, Vodafone, Avea), inter-city bus transportation (between Istanbul & Denizli: Varan, Ulusoy, Pamukkale, Köseoğlu), airline travel (between Istanbul and Frankfurt: Turkish Airlines, Pegasus, …) etc. Monopolistic Competition Definition: Many firms sell products that are similar but not identical. There is free entry and exit like perfect competition. But at the same time, there is...
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