Automated Mechanism Design EC-08 Tutorial
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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). -
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. -
Obviousness Around the Clock
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Breitmoser, Yves; Schweighofer-Kodritsch, Sebastian Working Paper Obviousness around the clock WZB Discussion Paper, No. SP II 2019-203 Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Breitmoser, Yves; Schweighofer-Kodritsch, Sebastian (2019) : Obviousness around the clock, WZB Discussion Paper, No. SP II 2019-203, Wissenschaftszentrum Berlin für Sozialforschung (WZB), Berlin This Version is available at: http://hdl.handle.net/10419/195919 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 Yves Breitmoser Sebastian Schweighofer-Kodritsch Obviousness around the clock Put your Research Area and Unit Discussion Paper SP II 2019–203 March 2019 WZB Berlin Social Science Center Research Area Markets and Choice Research Unit Market Behavior Wissenschaftszentrum Berlin für Sozialforschung gGmbH Reichpietschufer 50 10785 Berlin Germany www.wzb.eu Copyright remains with the author(s). -
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 -
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](https://docslib.b-cdn.net/cover/4900/arxiv-1709-04326v4-cs-ai-19-sep-2018-674900.webp)
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. -
Handout 21: “You Can't Handle the Truth!”
CSCE 475/875 Multiagent Systems Handout 21: “You Can’t Handle the Truth!” October 24, 2017 (Based on Shoham and Leyton-Brown 2011) Independent Private Value (IPV) To analyze properties of the various auction protocols, let’s first consider agents’ valuations of goods: their utilities for different allocations of the goods. Auction theory distinguishes between a number of different settings here. One of the best-known and most extensively studied is the independent private value (IPV) setting. In this setting all agents’ valuations are drawn independently from the same (commonly known) distribution, and an agent’s type (or “signal”) consists only of its own valuation, giving itself no information about the valuations of the others. An example where the IPV setting is appropriate is in auctions consisting of bidders with personal tastes who aim to buy a piece of art purely for their own enjoyment. (Note: Also think about the common-value assumption. Resale value!) Second-price, Japanese, and English auctions Theorem 11.1.1 In a second-price auction where bidders have independent private values, truth telling is a dominant strategy. Proof. Assume that all bidders other than i bid in some arbitrary way, and consider i’s best response. First, consider the case where i’s valuation is larger than the highest of the other bidders’ bids. In this case i would win and would pay the next-highest bid amount. Could i be better off by bidding dishonestly in this case? • If i bid higher, i would still win and would still pay the same amount. • If i bid lower, i would either still win and still pay the same amount or lose and pay zero. -
Reinterpreting Mixed Strategy Equilibria: a Unification Of
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics Reinterpreting Mixed Strategy Equilibria: A Unification of the Classical and Bayesian Views∗ Philip J. Reny Arthur J. Robson Department of Economics Department of Economics University of Chicago Simon Fraser University September, 2003 Abstract We provide a new interpretation of mixed strategy equilibria that incor- porates both von Neumann and Morgenstern’s classical concealment role of mixing as well as the more recent Bayesian view originating with Harsanyi. For any two-person game, G, we consider an incomplete information game, , in which each player’s type is the probability he assigns to the event thatIG his mixed strategy in G is “found out” by his opponent. We show that, generically, any regular equilibrium of G can be approximated by an equilibrium of in which almost every type of each player is strictly opti- mizing. This leadsIG us to interpret i’s equilibrium mixed strategy in G as a combination of deliberate randomization by i together with uncertainty on j’s part about which randomization i will employ. We also show that such randomization is not unusual: For example, i’s randomization is nondegen- erate whenever the support of an equilibrium contains cyclic best replies. ∗We wish to thank Drew Fudenberg, Motty Perry and Hugo Sonnenschein for very helpful comments. We are especially grateful to Bob Aumann for a stimulating discussion of the litera- ture on mixed strategies that prompted us to fit our work into that literature and ultimately led us to the interpretation we provide here, and to Hari Govindan whose detailed suggestions per- mitted a substantial simplification of the proof of our main result. -
When Can Limited Randomness Be Used in Repeated Games?
When Can Limited Randomness Be Used in Repeated Games? Pavel Hub´aˇcek∗ Moni Naor† Jonathan Ullman‡ July 19, 2018 Abstract The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely re- peated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudo- random generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist. -
CS395T Agent-Based Electronic Commerce Fall 2003
CS395T Agent-Based Electronic Commerce Fall 2003 Peter Stone Department or Computer Sciences The University of Texas at Austin Week 2a, 9/2/03 Logistics • Mailing list and archives Peter Stone Logistics • Mailing list and archives • Submitting responses to readings Peter Stone Logistics • Mailing list and archives • Submitting responses to readings − no attachments, strange formats − use specific subject line (see web page) Peter Stone Logistics • Mailing list and archives • Submitting responses to readings − no attachments, strange formats − use specific subject line (see web page) − by midnight; earlier raises probability of response Peter Stone Logistics • Mailing list and archives • Submitting responses to readings − no attachments, strange formats − use specific subject line (see web page) − by midnight; earlier raises probability of response • Presentation dates to be assigned soon Peter Stone Logistics • Mailing list and archives • Submitting responses to readings − no attachments, strange formats − use specific subject line (see web page) − by midnight; earlier raises probability of response • Presentation dates to be assigned soon • Any questions? Peter Stone Klemperer • A survey • Purpose: a broad overview of terms, concepts and the types of things that are known Peter Stone Klemperer • A survey • Purpose: a broad overview of terms, concepts and the types of things that are known − Geared more at economists; assumes some terminology − Results stated with not enough information to verify − Apolgies if that was frustrating Peter Stone