Focusing Failures in Competitive Environments: Explaining Decision Errors in the Monty Hall Game, the Acquiring a Company Problem, and Multiparty Ultimatums
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Let's Make a Deal – the Monty Hall Problem
Simulation and Statistical Exploration of Data (Let’s Make a Deal – The Monty Hall Problem), using the new ClassPad300-technology Ludwig Paditz, University of Applied Sciences Dresden (FH), Germany Statement of the Problem The Monty Hall problem involves a classical game show situation and is named after Monty Hall*), the long-time host of the TV game show Let's Make a Deal. There are three doors labeled 1, 2, and 3. A car is behind one of the doors, while goats are behind the other two: The rules are as follows: 1. The player selects a door. 2. The host selects a different door and opens it. 3. The host gives the player the option of switching from the original choice to the remaining closed door. 4. The door finally selected by the player is opened and he either wins or loses. By the help of the ClassPad300 we will simulate N (say N=100) such game show situations to see, if the option of switching the door is a good or a bad option to improve the chance to win. The experiments help our students better to understand the randomness and the statistic methods of every day life. *) Monty Hall was born on August 25, 1924 in Canada. Hall attended the University of Manitoba, graduating in 1945, and served in the Canadian Army during World War II. He immigrated to the United States in 1955, and for the next few years worked in radio and television for the NBC and CBS networks. In 1963, Hall began as emcee for the game show Let's Make a Deal, the role that would make him famous. -
Uniqueness and Symmetry in Bargaining Theories of Justice
Philos Stud DOI 10.1007/s11098-013-0121-y Uniqueness and symmetry in bargaining theories of justice John Thrasher Ó Springer Science+Business Media Dordrecht 2013 Abstract For contractarians, justice is the result of a rational bargain. The goal is to show that the rules of justice are consistent with rationality. The two most important bargaining theories of justice are David Gauthier’s and those that use the Nash’s bargaining solution. I argue that both of these approaches are fatally undermined by their reliance on a symmetry condition. Symmetry is a substantive constraint, not an implication of rationality. I argue that using symmetry to generate uniqueness undermines the goal of bargaining theories of justice. Keywords David Gauthier Á John Nash Á John Harsanyi Á Thomas Schelling Á Bargaining Á Symmetry Throughout the last century and into this one, many philosophers modeled justice as a bargaining problem between rational agents. Even those who did not explicitly use a bargaining problem as their model, most notably Rawls, incorporated many of the concepts and techniques from bargaining theories into their understanding of what a theory of justice should look like. This allowed them to use the powerful tools of game theory to justify their various theories of distributive justice. The debates between partisans of different theories of distributive justice has tended to be over the respective benefits of each particular bargaining solution and whether or not the solution to the bargaining problem matches our pre-theoretical intuitions about justice. There is, however, a more serious problem that has effectively been ignored since economists originally J. -
Puzzling Game of Life” Production Notes -Page 1
“The Puzzling Game of Life” Production Notes -Page 1 THE PUZZLING GAME OF LIFE Production Notes A four-act skit, using various game show formats to provide the setting for the main character to progress in her spiritual journey. SUMMARY: Suzy Christian finds herself on a journey in “The Game of Life.” Each game show that she participates in is another step in her spiritual journey, where she faces trials and temptations as well as growth and victory.. THEME: The retreat theme was “God Meant It For Good” and the theme verse was Romans 8:28. The visual theme was puzzle pieces, all fitting together. In this skit, we see puzzling things happening to Suzy Christian, and she doesn’t understand why she is experiencing trials. But the skit attempts to show how God is using even those difficult experiences in our life for our good and the good of others. THEME VERSE: “And we know that all things work together for good to those who love God, to those who are the called according to His purpose.” Romans 8:28 CHARACTERS: (Note: There are only a few main characters in this skit (marked with asterisk (*). All of the remaining characters are minor characters. One person could easily take the role of one of the minor characters in each act. But this is a great opportunity to assign small roles, appearing in only one act, to people and see how they handle being on stage.) *Rod Sterling Narrator *Guy Smiley - obnoxious game show host, who greatly exaggerates some of his lines (which are capitalized in the script) in a sarcastic manner *Suzy Christian - a woman, wife and mother who has recently become a Christian. -
Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
Discrete Probability Distributions Random Variables
Discrete Probability Distributions Random Variables • A variable that associates a number with the outcome of a random experiment is called a random variable. • Notation: random variable is denoted by an uppercase letter, such as X. After the experiment is conducted, the measured value is denoted by a lowercase letter, such a x. Both X and x are shown in italics, e.g., P(X=x). Sec 2‐8 Random Variables 2 Continuous & Discrete Random Variables • A discrete random variable is usually integer number – N ‐ the number of p53 proteins in a cell – D ‐ the number of nucleotides different between two sequences • A continuous random variable is a real number – C=N/V – the concentration of p53 protein in a cell of volume V – Percentage (D/L)*100% of different nucleotides in protein sequences of different lengths L (depending on the set of L’s may be discrete but dense) Sec 2‐8 Random Variables 3 Probability Mass Function (PMF) • I want to compare all 4‐ mers in a pair of human genomes • X – random variable: the number of nucleotide Probability Mass Function for differences in a given 4‐ the # of mismatches in 4‐mers mer P(X =0) = 0.6561 • Probability Mass Function: P(X =1) = 0.2916 f(x) or P(X=x) –the P(X =2) = 0.0486 probability that the # of P(X =3) = 0.0036 P(X =4) = 0.0001 SNPs is exactly equal to x Σx P(X=x)= 1.0000 4 Cumulative Distribution Function (CDF) P(X≤x) P(X>x) 0.6561 0.3439 0.9477 0.0523 0.9963 0.0037 0.9999 0.0001 1.0000 0.0000 CDF: P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.9999 Complementary cumulative distribution function (tail distribution) CCDF: P(X >3) =1‐ P(X ≤ 3) =0.0001 5 Mean or Expected Value of X The mean or expected value of the discrete random variable X, denoted as or EX, is EX xPX x xf x xx • The mean = the weighted average of all possible values of X. -
The Role of Self-Interest in Elite Bargaining
The role of self-interest in elite bargaining Brad L. LeVecka,b, D. Alex Hughesa,c, James H. Fowlera,c,d, Emilie Hafner-Burtona,c, and David G. Victora,1 aLaboratory on International Law and Regulation, School of International Relations and Pacific Studies, cDepartment of Political Science, and dDivision of Medical Genetics, University of California, San Diego, La Jolla, CA 92093; and bDepartment of Political Science, University of California, Merced, CA 95343 Edited by David G. Rand, Yale University, New Haven, CT, and accepted by the Editorial Board November 11, 2014 (received for review May 28, 2014) One of the best-known and most replicated laboratory results in evident in less experienced populations (16, 17). Some research behavioral economics is that bargainers frequently reject low has examined whether leaders are psychologically or physiolog- offers, even when it harms their material self-interest. This finding ically different from nonelite populations—for example, research could have important implications for international negotiations on anxiety levels in leaders (18) and work on how leaders may be on many problems facing humanity today, because models of selected for skills in coordination, leadership, and followership international bargaining assume exactly the opposite: that policy (19). The few existing studies that have empirically examined makers are rational and self-interested. However, it is unknown actual decision making by elites in highly particular roles—such whether elites who engage in diplomatic bargaining will similarly as professional tennis players, soccer players, and traders—sug- reject low offers because past research has been based almost gest that, through a combination of learning and attrition, elites exclusively on convenience samples of undergraduates, members are more likely to exhibit self-interested, profit-maximizing be- of the general public, or small-scale societies rather than highly havior than their less experienced counterparts (20–23). -
Nash Equilibrium
Lecture 3: Nash equilibrium Nash equilibrium: The mathematician John Nash introduced the concept of an equi- librium for a game, and equilibrium is often called a Nash equilibrium. They provide a way to identify reasonable outcomes when an easy argument based on domination (like in the prisoner's dilemma, see lecture 2) is not available. We formulate the concept of an equilibrium for a two player game with respective 0 payoff matrices PR and PC . We write PR(s; s ) for the payoff for player R when R plays 0 s and C plays s, this is simply the (s; s ) entry the matrix PR. Definition 1. A pair of strategies (^sR; s^C ) is an Nash equilbrium for a two player game if no player can improve his payoff by changing his strategy from his equilibrium strategy to another strategy provided his opponent keeps his equilibrium strategy. In terms of the payoffs matrices this means that PR(sR; s^C ) ≤ P (^sR; s^C ) for all sR ; and PC (^sR; sC ) ≤ P (^sR; s^C ) for all sc : The idea at work in the definition of Nash equilibrium deserves a name: Definition 2. A strategy s^R is a best-response to a strategy sc if PR(sR; sC ) ≤ P (^sR; sC ) for all sR ; i.e. s^R is such that max PR(sR; sC ) = P (^sR; sC ) sR We can now reformulate the idea of a Nash equilibrium as The pair (^sR; s^C ) is a Nash equilibrium if and only ifs ^R is a best-response tos ^C and s^C is a best-response tos ^R. -
Bounded Rationality in Bargaining Games: Do Proposers Believe That Responders Reject an Equal Split?
No. 11 UNIVERSITY OF COLOGNE WORKING PAPER SERIES IN ECONOMICS BOUNDED RATIONALITY IN BARGAINING GAMES: DO PROPOSERS BELIEVE THAT RESPONDERS REJECT AN EQUAL SPLIT? BEN GREINER Department of Economics University of Cologne Albertus-Magnus-Platz D-50923 Köln Germany http://www.wiso.uni-koeln.de Bounded Rationality in Bargaining Games: Do Proposers Believe That Responders Reject an Equal Split?∗ Ben Greiner† June 19, 2004 Abstract Puzzled by the experimental results of the ’impunity game’ by Bolton & Zwick (1995) we replicate the game and alter it in a systematic manner. We find that although almost nobody actually rejects an offered equal split in a bargaining game, proposers behave as if there would be a considerably large rejection rate for equal splits. This result is inconsistent with existing models of economic decision making. This includes models of selfish play- ers as well as models of social utility and reciprocity, even when combined with erroneous decision making. Our data suggests that subjects fail to foresee their opponent’s decision even for one step in our simple bargaining games. We consider models of bounded rational decision making such as rules of thumb as explanations for the observed behavioral pattern. Keywords: ultimatum game, dictator game, impunity game, social utility, bounded rationality JEL Classification: C72, C92, D3 ∗I thank Bettina Bartels, Hakan Fink and Ralica Gospodinova for research assistance, the Strategic Interaction Group at MPI Jena for comments and serving as pilot participants, especially Susanne B¨uchner, Sven Fischer, Katinka Pantz and Carsten Schmidt for support in the conduction of the experiment, and Werner G¨uth, Axel Ockenfels, Ro’i Zultan, participants at the European ESA Meeting in Erfurt and at the Brown Bag seminars in Jena and Cologne for discussions and valuable comments. -
The Monty Hall Problem
The Monty Hall Problem Richard D. Gill Mathematical Institute, University of Leiden, Netherlands http://www.math.leidenuniv.nl/∼gill 17 March, 2011 Abstract A range of solutions to the Monty Hall problem is developed, with the aim of connecting popular (informal) solutions and the formal mathematical solutions of in- troductory text-books. Under Riemann's slogan of replacing calculations with ideas, we discover bridges between popular solutions and rigorous (formal) mathematical solutions by using various combinations of symmetry, symmetrization, independence, and Bayes' rule, as well as insights from game theory. The result is a collection of intuitive and informal logical arguments which can be converted step by step into formal mathematics. The Monty Hall problem can be used simultaneously to develop probabilistic intuition and to give a deeper understanding of the paradox, not just to provide a routine exercise in computation of conditional probabilities. Simple insights from game theory can be used to prove probability inequalities. Acknowledgement: this text is the result of an evolution through texts written for three internet encyclopedias (wikipedia.org, citizendium.org, statprob.com) and each time has benefitted from the contributions of many other editors. I thank them all, for all the original ideas in this paper. That leaves only the mistakes, for which I bear responsibility. 1 Introduction Imagine you are a guest in a TV game show. The host, a certain Mr. Monty Hall, shows you three large doors and tells you that behind one of the doors there is a car while behind the other two there are goats. You want to win the car. -
Minimally Acceptable Altruism and the Ultimatum Game
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Rotemberg, Julio J. Working Paper Minimally acceptable altruism and the ultimatum game Working Papers, No. 06-12 Provided in Cooperation with: Federal Reserve Bank of Boston Suggested Citation: Rotemberg, Julio J. (2006) : Minimally acceptable altruism and the ultimatum game, Working Papers, No. 06-12, Federal Reserve Bank of Boston, Boston, MA This Version is available at: http://hdl.handle.net/10419/55606 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu No. 06‐12 Minimally Acceptable Altruism and the Ultimatum Game Julio J. Rotemberg Abstract: I suppose that people react with anger when others show themselves not to be minimally altruistic. With heterogeneous agents, this can account for the experimental results of ultimatum and dictator games. -
Econs 424 % Bargaining Games and an Introduction to Repeated Games
EconS 424 - Bargaining Games and an Introduction to Repeated Games Félix Muñoz-García Washington State University [email protected] March 25, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 1/44 Watson, Ch. 19 # 8 In experimental tests of the ultimatum bargaining game, subjects who propose the split rarely o¤er a tiny share of the surplus to the other party. Furthermore, sometimes subjects reject positive o¤ers. These …ndings seem to contradict our standard analysis of the ultimatum bargaining game. Many scholars conclude that the payo¤s speci…ed in the basic model do not represent the actual preferences of the people who participate in the experiments. In reality, people care about more than their own monetary rewards. Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 2/44 Watson, Ch. 19 # 8 For example, people also act on feelings of spite and the ideal of fairness. Suppose that, in the ultimatum game, the responder’spayo¤ is y + a(y z), where y is the responder’smonetary reward, z is the proposer’smonetary take, and a is a positive constant. That is, the responder cares about how much money he gets and he cares about relative monetary amounts (the di¤erence between the money he gets and the money the other player gets). Assume that the proposer’spayo¤ is as in the basic model. Represent this game in the extensive form, writing the payo¤s in terms of m, the monetary o¤er of the proposer, and the parameter a. Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 3/44 Watson, Ch. -
Bargaining Under a Deadline: Evidence from the Reverse Ultimatum Game ✩
Games and Economic Behavior 45 (2003) 347–368 www.elsevier.com/locate/geb Bargaining under a deadline: evidence from the reverse ultimatum game ✩ Uri Gneezy,a,b Ernan Haruvy,c and Alvin E. Roth d,e,∗ a University of Chicago Graduate School of Business, Chicago, IL, USA b Technion, Israel Institute of Technology, Haifa, Israel c School of Management, University of Texas at Dallas, Richardson, TX, USA d Department of Economics, Harvard University, Cambridge, MA, USA e Harvard Business School, Boston, MA, USA Received 8 January 2003 Abstract We study a “reverse” ultimatum game, in which proposers have multiple chances to offer responders a division of some fixed pie. The game ends if the responder accepts an offer, or if, following a rejection, the proposer decides not to make a better offer. The unique subgame perfect equilibrium gives the proposer the minimum possible payoff. Nevertheless, the experimental results are not too different from those of the standard ultimatum game, although proposers generally receive slightly less than half of the surplus. We use the reverse ultimatum game to study deadlines experimentally. With a deadline, the subgame perfect equilibrium prediction is that the proposer gets the entire surplus. Deadlines are used strategically to influence the outcome, and agreements are reached near the deadline. Strategic considerations are evident in the differences in observed behavior between the deadline and no deadline conditions, even though agreements are substantially less extreme than predicted by perfect equilibrium. 2003 Elsevier Inc. All rights reserved. JEL classification: C7; C9 ✩ Accompanying materials to be found at http://www.utdallas.edu/~eharuvy/deadline/.