...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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...the public domain, and to Ed Sznyter for providing critical help with the macros we use to execute our numbering scheme. Version 1.1, 97/4/25 Contents Preface xi 2 Nash Equilibrium 1 Exercise 18.2 (First price auction ) 1 Exercise 18.3 (Second price auction ) 2 Exercise 18.5 (War of attrition ) 2 Exercise 19.1 (Location game ) 2 Exercise 20.2 (Necessity of conditions in Kakutani's theorem ) 4 Exercise 20.4 (Symmetric games ) 4 Exercise 24.1 (Increasing payo s in strictly competitive game ) 4 Exercise 27.2 (BoS with imperfect information ) 5 Exercise 28.1 (Exchange game ) 5 Exercise 28.2 (More information may hurt ) 6 Exercise 35.1 (Guess the average ) 7 Exercise 35.2 (Investment race ) 7 Exercise 36.1 (Guessing right ) 9 Exercise 36.2 (Air strike ) 9 Exercise 36.3 (Technical result on convex sets ) 10 Exercise 42.1 (Examples of Harsanyi's puri cation ) 10 Exercise 48.1 (Example of correlated equilibrium ) 11 Exercise 51.1 (Existence of ESS in 2 2 game ) 12 Exercise 56.3 (Example of rationalizable actions ) 13 Exercise 56.4 (Cournot duopoly ) 13 3 Mixed, Correlated, and Evolutionary Equilibrium 7 4 Rationalizability and Iterated Elimination of Dominated Actions 13 vi Contents Exercise 56.5 (Guess the average ) 13 Exercise 57.1 (Modi ed rationalizability in location game ) 14 Exercise 63.1 (Iterated elimination in location game ) 14...
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...Christian Schenk (MiKTEX), Ed Sznyter (ppctr), Timothy van Zandt (PSTricks), and others, for generously making superlative software freely available. The main font is 10pt Palatino. Version 6: 2012-4-7 Contents Preface 1 xi Introduction 1 Exercise 5.3 (Altruistic preferences) 1 Exercise 6.1 (Alternative representations of preferences) 1 2 Nash Equilibrium 3 Exercise 16.1 (Working on a joint project) 3 Exercise 17.1 (Games equivalent to the Prisoner’s Dilemma) 3 Exercise 20.1 (Games without conflict) 3 Exercise 31.1 (Extension of the Stag Hunt) 4 Exercise 34.1 (Guessing two-thirds of the average) 4 Exercise 34.3 (Choosing a route) 5 Exercise 37.1 (Finding Nash equilibria using best response functions) 6 Exercise 38.1 (Constructing best response functions) 6 Exercise 38.2 (Dividing money) 7 Exercise 41.1 (Strict and nonstrict Nash equilibria) 7 Exercise 47.1 (Strict equilibria and dominated actions) 8 Exercise 47.2 (Nash equilibrium and weakly dominated actions) 8 Exercise 50.1 (Other Nash equilibria of the game modeling collective decision-making) 8 Exercise 51.2 (Symmetric strategic games) 9 Exercise 52.2 (Equilibrium for pairwise interactions in a single population) 9 3 Nash Equilibrium:...
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...CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY EXERCISES 3. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast, high-quality system (High), or a slower, low-quality system (Low). Market research indicates that the resulting profits to each firm for the alternative strategies are given by the following payoff matrix: | | |Firm B | | | |High |Low | | |High |50, 40 |60, 45 | | | | | | |Firm A | | | | | |Low |55, 55 |15, 20 | a. If both firms make their decisions at the same time and follow maximin (low-risk) strategies, what will the outcome be? With a maximin strategy, a firm determines the worst outcome for each action, then chooses the action that maximizes the payoff among the worst outcomes. If Firm A chooses High, the worst payoff would occur if Firm B chooses High: A’s payoff would be 50. If Firm A chooses Low, the worst payoff would occur if Firm B chooses Low: A’s payoff would be 15. With a maximin strategy, A therefore chooses High. If Firm B chooses Low, the worst payoff would...
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...Maßberg Simon J¨ger a Institut f¨r Optimierung und Operations Research u summer semester 2015 Mathematics of Games Exercise Session 1 Exercise Session 1 due on 23.04.2015, by 8:15am, N24-226. Total : 20 Points Hand-in in groups of at most 3 persons! 1. Apply the iterated strict dominance elimination to the following 2-person game. Write down for each iteration, which strategies are strictly dominated and thus get deleted. Hint: In one step a strategy is only strictly dominated by a mixed strategy, but not by a pure one! A1 A2 A3 A4 B1 B2 B3 B4 5,7 11,2 4,7 4,8 6,6 10,1 3,2 5,0 3,0 8,4 5,6 2,2 4,7 5,7 6,8 3,10 [8 Points] 2. Each of n players gets a unique marker color. Now each player i (i ∈ {1, . . . , n}) simultaneously chooses his position: a real number xi ∈ [0, 1]. All points in [0, 1] are colored: Each point y gets player i’s marker color for that i with xi closest to y. If a point y ∈ [0, 1] has the same distance to more than one player’s position xi , its color is determined at random (uniformly). If one position xi is chosen by more than one player, the points with minimum distance to xi are colored randomly (uniformly) with one of those players’ marker colors. Each player wants to color a largest possible part of the interval with his marker color. (i) If there are two players, does a pure-strategy Nash Equilibrium exist for? If so, is it unique? Either give a pure-strategy NE (with justification) or explain why one does not exist...
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...a player chooses each action with some probability 2. “The game where players undertake actions with uncertain outcomes” is a definition of the game with asymmetric information. Falls 3. In a separating equilibrium with two types of informed player and two possible actions each type of informed player chooses different action.. True 4. Auction is a public sale in which property or merchandise are sold to the highest bidder. True 5. Private Value Auction is one where True * Each bidder knows his or her value for the object * Bidders differ in their values for the object 6. Common Value Auction is one where the item has a single though unknown value. True 7. English auction is one where: Falls * Bidders call out prices (outcry) * Auctioneer calls out prices (silent) * Bidders hold down button (Japanese) * Second-Highest bidder gets the object 8. Second-price auction is one where: True * All buyers submit bids * Buyer submitting the highest bid wins and pays the second highest bid 9. In a second-price auction bidding your true valuation is a dominated strategy. Falls 10. More bidders usually lead to higher prices. True Part 2 (50%) Explanations and/or calculations are required in all exercises!!! Without it you can get maximum only...
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...Vol. 27, No. 5, September–October 2008, pp. 811–828 issn 0732-2399 eissn 1526-548X 08 2705 0811 informs ® doi 10.1287/mksc.1080.0398 © 2008 INFORMS Supermarket Pricing Strategies Department of Economics, Duke University, Durham, North Carolina 27708, paul.ellickson@duke.edu William E. Simon School of Business Administration, University of Rochester, Rochester, New York 14627, misra@simon.rochester.edu Paul B. Ellickson Sanjog Misra M ost supermarket firms choose to position themselves by offering either everyday low prices (EDLP) across several items or offering temporary price reductions (promotions) on a limited range of items. While this choice has been addressed from a theoretical perspective in both the marketing and economic literature, relatively little is known about how these decisions are made in practice, especially within a competitive environment. This paper exploits a unique store level data set consisting of every supermarket operating in the United States in 1998. For each of these stores, we observe the pricing strategy the firm has chosen to follow, as reported by the firm itself. Using a system of simultaneous discrete choice models, we estimate each store’s choice of pricing strategy as a static discrete game of incomplete information. In contrast to the predictions of the theoretical literature, we find strong evidence that firms cluster by strategy by choosing actions that agree with those of its rivals. We also find a significant impact of...
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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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...Game Theory is undeniably new to me. Its concept is just so brilliant that it made me rethink how I ought to see a business’ road to success. In the past, my key idea of winning in the industry was by toppling down competitors, and rising as the sole survivor in the war. Plainly, it’s a winner-take-all perspective. The real target was to capture the entire market then. After reading the theory and the cases suitably alluded to, realizations came to me that I’m way too far from the wisdom good strategists possess. Way too far from making it to the corporate executives’ seat. Way too far from a business’ lifelong success. For Filipinos, it’s always been a “here-and-now” match. Typically overlooked are the impacts of strategies in the long run, and how competitors and other players in the game would tend to respond. Game theory offers the notion of coopetition — cooperative and competitive ways to change the game. The primary insight of game theory is focusing on others -- namely allocentrism. It further states that the game of business is all about value: creating it and capturing it. Many are the so-called mental traps that must be killed in order that one be set for the game or set to change it. We often think that it’s hard and it’s beyond our competencies to dare change the game, and that we should do just what others do — only in a differently-tailored fashion. We go with the flow and no new routes are shaped to arrive at a better position and standing for us and the...
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...K.B B.U 24.02.2014 1) Briefly discuss the similarities between real world situations and the concepts of prisoners’ dilemma, battle of sexes, matching pennies and hawk-dove games. Prisoners' dilemma Example 1 Barbers shop and Hairdressing are located in the same area. Each barber wants to have more clients. Each Decided to discount. Barbers shop made a discount 20% because he thought that Hairdressing will make 10% discount, and Hairdressing also made a discount because he also thought that Barbers shop will make 10% discount. Each barber's dominant strategy is 20%, which gives an inferior outcome if both use their dominant strategy. Barber's situation is similar to prisoner's dilemma because in this game if each player uses its dominant strategy it gives an inferior outcome. Of course they can cooperate and choose the best option for both of them. Example 2 Warsaw University and Politechnick University are two famous universities in Poland. Both universities want to have more international students. Each decided to discount for international students. Warsaw University made a discount 30% because he thought that Politechnick University will make a 20% discount, and here Politechnik University also made a discount because he also thought that Warsaw...
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...In the movie The Princess Bride, the outlaw Vizzini is challenged to a game of wits by Wesley. Vizzini is offered two goblets of wine and must determine which one holds the poison. This situation includes all the four elements of a game as described by Professor Stevens. The first element of a game is players. A game must have at least two players. In the movie The Princess Bride, Vizzini and Wesley are the players in the game. The second element of a game is common knowledge. Common knowledge is where every player has the same information and is aware that the other players have the same information. It was common knowledge to both Wesley and Vizzini that only Wesley knew which goblet had poison in it. It was also common knowledge that only one goblet could be chosen by Vizzini in that they would both drink their wine at the same time. The third element of the game is payoffs. The payoff of the game is the degree to which each player is satisfied with an outcome. The payoff for Vizzini and Wesley was very similar in that the opponent would be dead from drinking the poison and the winner would get the Princess. The fourth element of the game is strategy. Strategy is what decision would be made by the player in every situation the player finds himself in. When Vizzini found himself in this situation of choosing a cup and whether to be a clever man or a great fool, he decided to use a strategy that would create a diversion and switch the cups ensuring he alone knew which cup he...
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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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...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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...mathematician John Forbes Nash. His contributions to mathematics are outstanding. When he was an undergraduate, he proved Brouwer's fixed point theorem. He then broke one of Riemann's most perplexing mathematical problems and became famous for the Nash Solution. From then on, Nash provided breakthrough after breakthrough in mathematics. In 1958 John Forbes Nash was described as being 'the most promising young mathematician in the world'. John solved problems in mathematics that many mathematicians deemed not solvable. On the threshold of such a promising and outstanding career, he then went on to suffer through three decades of a devastating form of paranoid schizophrenia. He lost his teaching professions and his job. He refused all medical treatment and spent years in and out of delusional states. Remarkably, in 1994 John won the Nobel Prize in Economic Science for his work on Game Theory. Game Theory is an analytical tool to understand the phenomena behind the way decision-makers interact. Nash's work on Game Theory in the early stages has Nash comparing it to football, pigeon feeding habits and picking up women. It's the bar scene where Nash has his big 'Ah Ha' moment. While Nash and his friends all have their eyes on the same blonde woman, he surprises his colleagues with the question that if we all want the same woman, nobody wins, if we all go after her friends (brunette woman), nobody wins, and thus there must be a solution to ensure that everyone wins. With that, Nash writes a formula...
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...Game Theory Themes 1. Introduction to Game Theory 2. Sequential Games 3. Simultaneous Games 4. Conclusion Introduction to Game Theory Game theory is the branch of decision theory concerned with interdependent decisions. The problems of interest involve multiple participants, each of whom has individual objectives related to a common system or shared resources. Because game theory arose from the analysis of competitive scenarios, the problems are called games and the participants are called players. But these techniques apply to more than just sport, and are not even limited to competitive situations. In short, game theory deals with any problem in which each player’s strategy depends on what the other players do. Situations involving interdependent decisions arise frequently, in all walks of life. A few examples in which game theory could come in handy include: ● Friends choosing where to go have dinner ● Parents trying to get children to behave ● Commuters deciding how to go to work ● Businesses competing in a market ● Diplomats negotiating a treaty ● Gamblers betting in a card game All of these situations call for strategic thinking – making use of available information to devise the best plan to achieve one’s objectives. Perhaps you are already familiar with assessing costs and benefits in order to make informed decisions between several options. Game theory simply extends this concept to interdependent decisions, in which the options being evaluated are functions of...
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