22.6.3. a Rationale for Mixed Strategy Equilibrium (Difficult)
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Lecture 4 Rationalizability & Nash Equilibrium Road
Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i). -
Learning and Equilibrium
Learning and Equilibrium Drew Fudenberg1 and David K. Levine2 1Department of Economics, Harvard University, Cambridge, Massachusetts; email: [email protected] 2Department of Economics, Washington University of St. Louis, St. Louis, Missouri; email: [email protected] Annu. Rev. Econ. 2009. 1:385–419 Key Words First published online as a Review in Advance on nonequilibrium dynamics, bounded rationality, Nash equilibrium, June 11, 2009 self-confirming equilibrium The Annual Review of Economics is online at by 140.247.212.190 on 09/04/09. For personal use only. econ.annualreviews.org Abstract This article’s doi: The theory of learning in games explores how, which, and what 10.1146/annurev.economics.050708.142930 kind of equilibria might arise as a consequence of a long-run non- Annu. Rev. Econ. 2009.1:385-420. Downloaded from arjournals.annualreviews.org Copyright © 2009 by Annual Reviews. equilibrium process of learning, adaptation, and/or imitation. If All rights reserved agents’ strategies are completely observed at the end of each round 1941-1383/09/0904-0385$20.00 (and agents are randomly matched with a series of anonymous opponents), fairly simple rules perform well in terms of the agent’s worst-case payoffs, and also guarantee that any steady state of the system must correspond to an equilibrium. If players do not ob- serve the strategies chosen by their opponents (as in extensive-form games), then learning is consistent with steady states that are not Nash equilibria because players can maintain incorrect beliefs about off-path play. Beliefs can also be incorrect because of cogni- tive limitations and systematic inferential errors. -
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 .................................. -
More Experiments on Game Theory
More Experiments on Game Theory Syngjoo Choi Spring 2010 Experimental Economics (ECON3020) Game theory 2 Spring2010 1/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Mixed (or Randomized) Strategy In the penalty kick, Left (L) and Right (R) are the pure strategies of the kicker and the goalkeeper. A mixed strategy refers to a probabilistic mixture over pure strategies in which no single pure strategy is played all the time. e.g., kicking left with half of the time and right with the other half of the time. A penalty kick in a soccer game is one example of games of pure con‡icit, essentially called zero-sum games in which one player’s winning is the other’sloss. Experimental Economics (ECON3020) Game theory 2 Spring2010 3/28 The use of pure strategies will result in a loss. What about each player playing heads half of the time? Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins. Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. -
Modeling Strategic Behavior1
Modeling Strategic Behavior1 A Graduate Introduction to Game Theory and Mechanism Design George J. Mailath Department of Economics, University of Pennsylvania Research School of Economics, Australian National University 1September 16, 2020, copyright by George J. Mailath. World Scientific Press published the November 02, 2018 version. Any significant correc- tions or changes are listed at the back. To Loretta Preface These notes are based on my lecture notes for Economics 703, a first-year graduate course that I have been teaching at the Eco- nomics Department, University of Pennsylvania, for many years. It is impossible to understand modern economics without knowl- edge of the basic tools of game theory and mechanism design. My goal in the course (and this book) is to teach those basic tools so that students can understand and appreciate the corpus of modern economic thought, and so contribute to it. A key theme in the course is the interplay between the formal development of the tools and their use in applications. At the same time, extensions of the results that are beyond the course, but im- portant for context are (briefly) discussed. While I provide more background verbally on many of the exam- ples, I assume that students have seen some undergraduate game theory (such as covered in Osborne, 2004, Tadelis, 2013, and Wat- son, 2013). In addition, some exposure to intermediate microeco- nomics and decision making under uncertainty is helpful. Since these are lecture notes for an introductory course, I have not tried to attribute every result or model described. The result is a somewhat random pattern of citations and references. -
Comparing Auction Designs Where Suppliers Have Uncertain Costs and Uncertain Pivotal Status
RAND Journal of Economics Vol. 00, No. 0, Winter 2018 pp. 1–33 Comparing auction designs where suppliers have uncertain costs and uncertain pivotal status ∗ Par¨ Holmberg and ∗∗ Frank A. Wolak We analyze how market design influences bidding in multiunit procurement auctions where suppliers have asymmetric information about production costs. Our analysis is particularly relevant to wholesale electricity markets, because it accounts for the risk that a supplier is pivotal; market demand is larger than the total production capacity of its competitors. With constant marginal costs, expected welfare improves if the auctioneer restricts offers to be flat. We identify circumstances where the competitiveness of market outcomes improves with increased market transparency. We also find that, for buyers, uniform pricing is preferable to discriminatory pricing when producers’ private signals are affiliated. 1. Introduction Multiunit auctions are used to trade commodities, securities, emission permits, and other divisible goods. This article focuses on electricity markets, where producers submit offers before the level of demand and amount of available production capacity are fully known. Due to demand shocks, unexpected outages, transmission-constraints, and intermittent output from renewable energy sources, it often arises that an electricity producer is pivotal, that is, that realized demand is larger than the realized total production capacity of its competitors. A producer that is certain to be pivotal possess a substantial ability to exercise market power because it can withhold output ∗ Research Institute of Industrial Economics (IFN), Associate Researcher of the Energy Policy Research Group (EPRG), University of Cambridge; [email protected]. ∗∗ Program on Energy and Sustainable Development (PESD) and Stanford University; [email protected]. -
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 -
Natural Games
Ntur J A a A A ,,,* , a Department of Biosciences, FI-00014 University of Helsinki, Finland b Institute of Biotechnology, FI-00014 University of Helsinki, Finland c Department of Physics, FI-00014 University of Helsinki, Finland ABSTRACT Behavior in the context of game theory is described as a natural process that follows the 2nd law of thermodynamics. The rate of entropy increase as the payoff function is derived from statistical physics of open systems. The thermodynamic formalism relates everything in terms of energy and describes various ways to consume free energy. This allows us to associate game theoretical models of behavior to physical reality. Ultimately behavior is viewed as a physical process where flows of energy naturally select ways to consume free energy as soon as possible. This natural process is, according to the profound thermodynamic principle, equivalent to entropy increase in the least time. However, the physical portrayal of behavior does not imply determinism. On the contrary, evolutionary equation for open systems reveals that when there are three or more degrees of freedom for behavior, the course of a game is inherently unpredictable in detail because each move affects motives of moves in the future. Eventually, when no moves are found to consume more free energy, the extensive- form game has arrived at a solution concept that satisfies the minimax theorem. The equilibrium is Lyapunov-stable against variation in behavior within strategies but will be perturbed by a new strategy that will draw even more surrounding resources to the game. Entropy as the payoff function also clarifies motives of collaboration and subjective nature of decision making. -
Beyond Truth-Telling: Preference Estimation with Centralized School Choice and College Admissions∗
Beyond Truth-Telling: Preference Estimation with Centralized School Choice and College Admissions∗ Gabrielle Fack Julien Grenet Yinghua He October 2018 (Forthcoming: The American Economic Review) Abstract We propose novel approaches to estimating student preferences with data from matching mechanisms, especially the Gale-Shapley Deferred Acceptance. Even if the mechanism is strategy-proof, assuming that students truthfully rank schools in applications may be restrictive. We show that when students are ranked strictly by some ex-ante known priority index (e.g., test scores), stability is a plausible and weaker assumption, implying that every student is matched with her favorite school/college among those she qualifies for ex post. The methods are illustrated in simulations and applied to school choice in Paris. We discuss when each approach is more appropriate in real-life settings. JEL Codes: C78, D47, D50, D61, I21 Keywords: Gale-Shapley Deferred Acceptance Mechanism, School Choice, College Admissions, Stable Matching, Student Preferences ∗Fack: Universit´eParis-Dauphine, PSL Research University, LEDa and Paris School of Economics, Paris, France (e-mail: [email protected]); Grenet: CNRS and Paris School of Economics, 48 boulevard Jourdan, 75014 Paris, France (e-mail: [email protected]); He: Rice University and Toulouse School of Economics, Baker Hall 246, Department of Economcis MS-22, Houston, TX 77251 (e-mail: [email protected]). For constructive comments, we thank Nikhil Agarwal, Peter Arcidi- acono, Eduardo Azevedo, -
Oligopolistic Competition
Lecture 3: Oligopolistic competition EC 105. Industrial Organization Mattt Shum HSS, California Institute of Technology EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 1 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. Industrial Organization (Mattt Shum HSS,Lecture California 3: Oligopolistic Institute of competition Technology) 2 / 38 Oligopoly Models Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j's actions affect firm i's profits PC: firms are small, so no single firm’s actions affect other firms’ profits Monopoly: only one firm EC 105. -
BAYESIAN EQUILIBRIUM the Games Like Matching Pennies And
BAYESIAN EQUILIBRIUM MICHAEL PETERS The games like matching pennies and prisoner’s dilemma that form the core of most undergrad game theory courses are games in which players know each others’ preferences. Notions like iterated deletion of dominated strategies, and rationalizability actually go further in that they exploit the idea that each player puts him or herself in the shoes of other players and imagines that the others do the same thing. Games and reasoning like this apply to situations in which the preferences of the players are common knowledge. When we want to refer to situations like this, we usually say that we are interested in games of complete information. Most of the situations we study in economics aren’t really like this since we are never really sure of the motivation of the players we are dealing with. A bidder on eBay, for example, doesn’t know whether other bidders are interested in a good they would like to bid on. Once a bidder sees that another has submit a bid, he isn’t sure exactly how much the good is worth to the other bidder. When players in a game don’t know the preferences of other players, we say a game has incomplete information. If you have dealt with these things before, you may have learned the term asymmetric information. This term is less descriptive of the problem and should probably be avoided. The way we deal with incomplete information in economics is to use the ap- proach originally described by Savage in 1954. If you have taken a decision theory course, you probably learned this approach by studying Anscombe and Aumann (last session), while the approach we use in game theory is usually attributed to Harsanyi. -
Arxiv:1709.04326V4 [Cs.AI] 19 Sep 2018
Learning with Opponent-Learning Awareness , Jakob Foerster† ‡ Richard Y. Chen† Maruan Al-Shedivat‡ University of Oxford OpenAI Carnegie Mellon University Shimon Whiteson Pieter Abbeel‡ Igor Mordatch University of Oxford UC Berkeley OpenAI ABSTRACT 1 INTRODUCTION Multi-agent settings are quickly gathering importance in machine Due to the advent of deep reinforcement learning (RL) methods that learning. This includes a plethora of recent work on deep multi- allow the study of many agents in rich environments, multi-agent agent reinforcement learning, but also can be extended to hierar- RL has flourished in recent years. However, most of this recent chical reinforcement learning, generative adversarial networks and work considers fully cooperative settings [13, 14, 34] and emergent decentralised optimization. In all these settings the presence of mul- communication in particular [11, 12, 22, 32, 40]. Considering future tiple learning agents renders the training problem non-stationary applications of multi-agent RL, such as self-driving cars, it is obvious and often leads to unstable training or undesired final results. We that many of these will be only partially cooperative and contain present Learning with Opponent-Learning Awareness (LOLA), a elements of competition and conflict. method in which each agent shapes the anticipated learning of The human ability to maintain cooperation in a variety of com- the other agents in the environment. The LOLA learning rule in- plex social settings has been vital for the success of human societies. cludes an additional term that accounts for the impact of one agent’s Emergent reciprocity has been observed even in strongly adversar- policy on the anticipated parameter update of the other agents.