Evolutionary Game Theory: ESS, Convergence Stability, and NIS
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In the Light of Evolution VIII: Darwinian Thinking in the Social Sciences Brian Skyrmsa,1, John C
FROM THE ACADEMY: COLLOQUIUM INTRODUCTION COLLOQUIUM INTRODUCTION In the light of evolution VIII: Darwinian thinking in the social sciences Brian Skyrmsa,1, John C. Aviseb,1, and Francisco J. Ayalab,1 norms are seen as a society’s way of coor- aDepartment of Logic and Philosophy of Science and bDepartment of Ecology and dinating on one of these equilibria. This, Evolutionary Biology, University of California, Irvine, CA 92697 rather than enforcement of out-of-equilibrium behavior, is seen as the function of these Darwinian thinking in the social sciences was Evolutionary game theory was initially norms. Fairness norms, however, may involve inaugurated by Darwin himself in The De- seen by some social scientists as just a way interpersonal comparisons of utility. This is scent of Man and Selection in Relation to to provide a low rationality foundation for true of egalitarian norms, and of the norms Sex high rationality equilibrium concepts. A Nash of proportionality suggested by Aristotle. In (8, 9). Despite various misappropriations ’ of the Darwinian label, true Darwinian think- equilibrium is a rest point of a large class of Binmore s account, the standards for such ing continued in the social sciences. However, adaptive dynamics. If the dynamics converges interpersonal comparisons themselves evolve. toarestpoint,itconvergestoNash.Thismay Komarova(22),inanarticlethatcould with the advent of evolutionary game theory Social Dynamics (10) there has been an explosion of Darwin- nothappeninallgames,butitmayhappen equally well be in ,discusses ian analysis in the social sciences. And it has in large classes of games, depending on the the speed evolution of complex phenotypes in led to reciprocal contributions from the social dynamics in question. -
Econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Banerjee, Abhijit; Weibull, Jörgen W. Working Paper Evolution and Rationality: Some Recent Game- Theoretic Results IUI Working Paper, No. 345 Provided in Cooperation with: Research Institute of Industrial Economics (IFN), Stockholm Suggested Citation: Banerjee, Abhijit; Weibull, Jörgen W. (1992) : Evolution and Rationality: Some Recent Game-Theoretic Results, IUI Working Paper, No. 345, The Research Institute of Industrial Economics (IUI), Stockholm This Version is available at: http://hdl.handle.net/10419/95198 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 Ind ustriens Utred n i ngsi nstitut THE INDUSTRIAL INSTITUTE FOR ECONOMIC AND SOCIAL RESEARCH A list of Working Papers on the last pages No. -
8. Maxmin and Minmax Strategies
CMSC 474, Introduction to Game Theory 8. Maxmin and Minmax Strategies Mohammad T. Hajiaghayi University of Maryland Outline Chapter 2 discussed two solution concepts: Pareto optimality and Nash equilibrium Chapter 3 discusses several more: Maxmin and Minmax Dominant strategies Correlated equilibrium Trembling-hand perfect equilibrium e-Nash equilibrium Evolutionarily stable strategies Worst-Case Expected Utility For agent i, the worst-case expected utility of a strategy si is the minimum over all possible Husband Opera Football combinations of strategies for the other agents: Wife min u s ,s Opera 2, 1 0, 0 s-i i ( i -i ) Football 0, 0 1, 2 Example: Battle of the Sexes Wife’s strategy sw = {(p, Opera), (1 – p, Football)} Husband’s strategy sh = {(q, Opera), (1 – q, Football)} uw(p,q) = 2pq + (1 – p)(1 – q) = 3pq – p – q + 1 We can write uw(p,q) For any fixed p, uw(p,q) is linear in q instead of uw(sw , sh ) • e.g., if p = ½, then uw(½,q) = ½ q + ½ 0 ≤ q ≤ 1, so the min must be at q = 0 or q = 1 • e.g., minq (½ q + ½) is at q = 0 minq uw(p,q) = min (uw(p,0), uw(p,1)) = min (1 – p, 2p) Maxmin Strategies Also called maximin A maxmin strategy for agent i A strategy s1 that makes i’s worst-case expected utility as high as possible: argmaxmin ui (si,s-i ) si s-i This isn’t necessarily unique Often it is mixed Agent i’s maxmin value, or security level, is the maxmin strategy’s worst-case expected utility: maxmin ui (si,s-i ) si s-i For 2 players it simplifies to max min u1s1, s2 s1 s2 Example Wife’s and husband’s strategies -
14.12 Game Theory Lecture Notes∗ Lectures 7-9
14.12 Game Theory Lecture Notes∗ Lectures 7-9 Muhamet Yildiz Intheselecturesweanalyzedynamicgames(withcompleteinformation).Wefirst analyze the perfect information games, where each information set is singleton, and develop the notion of backwards induction. Then, considering more general dynamic games, we will introduce the concept of the subgame perfection. We explain these concepts on economic problems, most of which can be found in Gibbons. 1 Backwards induction The concept of backwards induction corresponds to the assumption that it is common knowledge that each player will act rationally at each node where he moves — even if his rationality would imply that such a node will not be reached.1 Mechanically, it is computed as follows. Consider a finite horizon perfect information game. Consider any node that comes just before terminal nodes, that is, after each move stemming from this node, the game ends. If the player who moves at this node acts rationally, he will choose the best move for himself. Hence, we select one of the moves that give this player the highest payoff. Assigning the payoff vector associated with this move to the node at hand, we delete all the moves stemming from this node so that we have a shorter game, where our node is a terminal node. Repeat this procedure until we reach the origin. ∗These notes do not include all the topics that will be covered in the class. See the slides for a more complete picture. 1 More precisely: at each node i the player is certain that all the players will act rationally at all nodes j that follow node i; and at each node i the player is certain that at each node j that follows node i the player who moves at j will be certain that all the players will act rationally at all nodes k that follow node j,...ad infinitum. -
Proquest Dissertations
Sons of Barabbas or Sons of Descartes? An evolutionary game-theoretic view of psychopathy by Lloyd D. Balbuena A thesis submitted to the Faculty of Graduate Studies and Postdoctoral Affairs in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Cognitive Science Carleton University Ottawa, Ontario ©2010, Lloyd D. Balbuena Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-79631-3 Our file Notre reference ISBN: 978-0-494-79631-3 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. -
On the Behavior of Strategies in Hawk-Dove Game
Journal of Game Theory 2016, 5(1): 9-15 DOI: 10.5923/j.jgt.20160501.02 On the Behavior of Strategies in Hawk-Dove Game Essam EL-Seidy Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt Abstract The emergence of cooperative behavior in human and animal societies is one of the fundamental problems in biology and social sciences. In this article, we study the evolution of cooperative behavior in the hawk dove game. There are some mechanisms like kin selection, group selection, direct and indirect reciprocity can evolve the cooperation when it works alone. Here we combine two mechanisms together in one population. The transformed matrices for each combination are determined. Some properties of cooperation like risk-dominant (RD) and advantageous (AD) are studied. The property of evolutionary stable (ESS) for strategies used in this article is discussed. Keywords Hawk-dove game, Evolutionary game dynamics, of evolutionary stable strategies (ESS), Kin selection, Direct and Indirect reciprocity, Group selection difference between both games has a rather profound effect 1. Introduction on the success of both strategies. In particular, whilst by the Prisoner’s Dilemma spatial structure often facilitates Game theory provides a quantitative framework for cooperation this is often not the case in the hawk–dove game analyzing the behavior of rational players. The theory of ([14]). iterated games in particular provides a systematic framework The Hawk-Dove game ([18]), also known as the snowdrift to explore the players' relationship in a long-term. It has been game or the chicken game, is used to study a variety of topics, an important tool in the behavioral and biological sciences from the evolution of cooperation to nuclear brinkmanship and it has been often invoked by economists, political ([8], [22]). -
Contemporaneous Perfect Epsilon-Equilibria
Games and Economic Behavior 53 (2005) 126–140 www.elsevier.com/locate/geb Contemporaneous perfect epsilon-equilibria George J. Mailath a, Andrew Postlewaite a,∗, Larry Samuelson b a University of Pennsylvania b University of Wisconsin Received 8 January 2003 Available online 18 July 2005 Abstract We examine contemporaneous perfect ε-equilibria, in which a player’s actions after every history, evaluated at the point of deviation from the equilibrium, must be within ε of a best response. This concept implies, but is stronger than, Radner’s ex ante perfect ε-equilibrium. A strategy profile is a contemporaneous perfect ε-equilibrium of a game if it is a subgame perfect equilibrium in a perturbed game with nearly the same payoffs, with the converse holding for pure equilibria. 2005 Elsevier Inc. All rights reserved. JEL classification: C70; C72; C73 Keywords: Epsilon equilibrium; Ex ante payoff; Multistage game; Subgame perfect equilibrium 1. Introduction Analyzing a game begins with the construction of a model specifying the strategies of the players and the resulting payoffs. For many games, one cannot be positive that the specified payoffs are precisely correct. For the model to be useful, one must hope that its equilibria are close to those of the real game whenever the payoff misspecification is small. To ensure that an equilibrium of the model is close to a Nash equilibrium of every possible game with nearly the same payoffs, the appropriate solution concept in the model * Corresponding author. E-mail addresses: [email protected] (G.J. Mailath), [email protected] (A. Postlewaite), [email protected] (L. -
Bargaining with Neighbors: Is Justice Contagious? (1999)
Journal of Philosophy, Inc. Bargaining with Neighbors: Is Justice Contagious? Author(s): Jason Alexander and Brian Skyrms Source: The Journal of Philosophy, Vol. 96, No. 11 (Nov., 1999), pp. 588-598 Published by: Journal of Philosophy, Inc. Stable URL: http://www.jstor.org/stable/2564625 Accessed: 12-05-2016 19:03 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Journal of Philosophy, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Philosophy This content downloaded from 128.91.58.254 on Thu, 12 May 2016 19:03:20 UTC All use subject to http://about.jstor.org/terms 588 THEJOURNAL OF PHILOSOPHY BARGAINING WITH NEIGHBORS: IS JUSTICE CONTAGIOUS ? What is justice? The question is llarder to answer in some cases than in others. We focus on the easiest case of dis- tributive justice. Two individuals are to decide how to dis- tribute a windfall of a certain amount of money. Neither is especially entitled, or especially needy, or especially anything their positions are entirely symmetric. Their utilities derived from the dis- tribution may be taken, for all intents and purposes, simply as the amount of money received. -
Introduction to Game Theory I
Introduction to Game Theory I Presented To: 2014 Summer REU Penn State Department of Mathematics Presented By: Christopher Griffin [email protected] 1/34 Types of Game Theory GAME THEORY Classical Game Combinatorial Dynamic Game Evolutionary Other Topics in Theory Game Theory Theory Game Theory Game Theory No time, doesn't contain differential equations Notion of time, may contain differential equations Games with finite or Games with no Games with time. Modeling population Experimental / infinite strategy space, chance. dynamics under Behavioral Game but no time. Games with motion or competition Theory Generally two player a dynamic component. Games with probability strategic games Modeling the evolution Voting Theory (either induced by the played on boards. Examples: of strategies in a player or the game). Optimal play in a dog population. Examples: Moves change the fight. Chasing your Determining why Games with coalitions structure of a game brother across a room. Examples: altruism is present in or negotiations. board. The evolution of human society. behaviors in a group of Examples: Examples: animals. Determining if there's Poker, Strategic Chess, Checkers, Go, a fair voting system. Military Decision Nim. Equilibrium of human Making, Negotiations. behavior in social networks. 2/34 Games Against the House The games we often see on television fall into this category. TV Game Shows (that do not pit players against each other in knowledge tests) often require a single player (who is, in a sense, playing against The House) to make a decision that will affect only his life. Prize is Behind: 1 2 3 Choose Door: 1 2 3 1 2 3 1 2 3 Switch: Y N Y N Y N Y N Y N Y N Y N Y N Y N Win/Lose: L W W L W L W L L W W L W L W L L W The Monty Hall Problem Other games against the house: • Blackjack • The Price is Right 3/34 Assumptions of Multi-Player Games 1. -
Emergence of Cooperation and Evolutionary Stability in Finite
letters to nature populations may decrease cooperation and lead to more frequent 10. Clutton-Brock, T. H. et al. Selfish sentinels in cooperative mammals. Science 284, 1640–1644 (1999). escalations of conflicts in situations in which cooperation persists in 11. Seyfarth, R. M. & Cheney, D. L. Grooming alliances and reciprocal altruism in vervet monkeys. Nature 308, 541–543 (1984). well-mixed populations. Thus, spatial structure may not be as 12. Milinski, M. Tit for tat in sticklebacks and the evolution of cooperation. Nature 325, 433–435 (1987). universally beneficial for cooperation as previously thought. A 13. Milinski, M., Lu¨thi, J. H., Eggler, R. & Parker, G. A. Cooperation under predation risk: experiments on costs and benefits. Proc. R. Soc. Lond. B 264, 831–837 (1997). 14. Turner, P. E. & Chao, L. Prisoner’s dilemma in an RNA virus. Nature 398, 441–443 (1999). Methods 15. Heinsohn, R. & Parker, C. Complex cooperative strategies in group-territorial African lions. Science Spatial structure 269, 1260–1262 (1995). In our spatially structured populations, individuals are confined to sites on regular 16. Clutton-Brock, T. Breeding together: kin selection and mutualism in cooperative vertebrates. Science 100 £ 100 lattices with periodic boundary conditions, and interact with their neighbours. 296, 69–72 (2002). We used square lattices with N ¼ 4 and N ¼ 8 neighbours, hexagonal lattices (N ¼ 6) and 17. Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, triangular lattices (N ¼ 3). Whenever a site x is updated, a neighbour y is drawn at random Cambridge, UK, 1998). among all N neighbours; the chosen neighbour takes over site x with probability 18. -
Extensive Form Games
Extensive Form Games Jonathan Levin January 2002 1 Introduction The models we have described so far can capture a wide range of strategic environments, but they have an important limitation. In each game we have looked at, each player moves once and strategies are chosen simultaneously. This misses a common feature of many economic settings (and many classic “games” such as chess or poker). Consider just a few examples of economic environments where the timing of strategic decisions is important: • Compensation and Incentives. Firms first sign workers to a contract, then workers decide how to behave, and frequently firms later decide what bonuses to pay and/or which employees to promote. • Research and Development. Firms make choices about what tech- nologies to invest in given the set of available technologies and their forecasts about the future market. • Monetary Policy. A central bank makes decisions about the money supply in response to inflation and growth, which are themselves de- termined by individuals acting in response to monetary policy, and so on. • Entry into Markets. Prior to competing in a product market, firms must make decisions about whether to enter markets and what prod- ucts to introduce. These decisions may strongly influence later com- petition. These notes take up the problem of representing and analyzing these dy- namic strategic environments. 1 2TheExtensiveForm The extensive form of a game is a complete description of: 1. The set of players 2. Who moves when and what their choices are 3. What players know when they move 4. The players’ payoffs as a function of the choices that are made. -
8. Noncooperative Games 8.1 Basic Concepts Theory of Strategic Interaction Definition 8.1. a Strategy Is a Plan of Matching Penn
EC 701, Fall 2005, Microeconomic Theory November 9, 2005 page 351 8. Noncooperative Games 8.1 Basic Concepts How I learned basic game theory in 1986. • Game theory = theory of strategic interaction • Each agent chooses a strategy in order to achieve his goals with • the knowledge that other agents may react to his strategic choice. Definition 8.1. A strategy is a plan of action that specifies what a player will do in every circumstance. EC 701, Fall 2005, Microeconomic Theory November 9, 2005 page 352 Matching Pennies (a zero-sum game) Eva and Esther each places a penny on the table simultaneously. • They each decide in advance whether to put down heads (H )or • tails (T). [This is a choice, NOT a random event as when a coin is flipped.] If both are heads (H ) or both are tails (T), then Eva pays Esther • $1. If one is heads and the other tails, then Esther pays Eva $1. • Esther HT 1 1 H − -1 1 Eva 1 1 T − 1 -1 EC 701, Fall 2005, Microeconomic Theory November 9, 2005 page 353 Esther HT 1 1 H − -1 1 Eva 1 1 T − 1 -1 Eva and Esther are the players. • H and T are strategies,and H , T is each player’s strategy • space. { } 1 and 1 are payoffs. • − Each box corresponds to a strategy profile. • For example, the top-right box represents the strategy profile • H , T h i EC 701, Fall 2005, Microeconomic Theory November 9, 2005 page 354 Deterministic strategies (without random elements) are called • pure strategies.