DOCSLIB.ORG

More Experiments on Game Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. 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? The use of pure strategies will result in a loss. What about each player playing heads half of the time? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Mixed-Strategy Equilibrium In the matching pennies game, both players playing heads with half probability is the unique Nash equilibrium. On one hand, this strategy is used to not be predictable. On the other hand, given the other’sstrategy, there is no (either pure or mixed) strategy giving higher payo¤s than playing head with half probability. One way of seeing this equilibrium is drawing each player’sbest response: Suppose player 2 chooses (Right) with probability p. Then, player 1’spayo¤ di¤erence betwee (Top) and (Bottom) is (1 2p) ( 1 + 2p) = 2 (1 2p). If p< 1/2, it is optimal for player 1 to choose (Top) with probability 1. Similarly for other cases. Experimental Economics (ECON3020) Game theory 2 Spring2010 5/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 6/28 Classroom Experiment Now let’splay a game in a classroom experiment. Go to http://veconlab.econ.virginia.edu/login.htm. Session name is sjc8. Experimental Economics (ECON3020) Game theory 2 Spring2010 7/28 Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Does there exist a pure-strategy Nash equilibrium? Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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 ..................................
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Dynamics of the Nash Map in the Game of Matching Pennies
    Dynamics of the Nash Map in the Game of Matching Pennies 1R. A. Becker, 2S. K. Chakrabarti, 3W. Geller, 3B. Kitchens and 3M. Misiurewicz Current Version: February 3, 2006 1Department of Economics Indiana University Bloomington, IN 47405 2Department of Economics Indiana University - Purdue University Indianapolis 425 University Blvd. Indianapolis, IN 46202 3Department of Mathematical Sciences Indiana University - Purdue University Indianapolis 402 N. Blackford Street Indianapolis, IN 46202 Abstract: An iterative procedure for the game of Matching Pennies is examined in which players use Nash’s map to respond to mixed strategies of the other players. It is shown that even though the game of Matching Pennies is a game with a unique Nash equilibrium point, the iterative procedure does not lead to convergence to the Nash equilibrium. In fact we find that if the game is played using this iterative procedure then the successive plays of the game will follow an orbit around the equilibrium point. This orbit has period eight and it is the only periodic orbit other than the equilibrium point. 1 Introduction One way Nash [1950a] motivated his formulation of an equilibrium point for noncooperative games was based on a mass action interpretation of the game’s play. He went on to present three alternative proofs of an equilibrium point’s existence. Each mapping giving rise to an equilibrium at a fixed point can also be viewed as defining a discrete time evolutionary dynamical system. The object of our paper is to study the dynamics associated with the better response mapping formulated in the published form of the dissertation (Nash [1951]) for the game Matching Pennies and connect its dynamical properties to learning and innovative evolution in strategic form games.
    [Show full text]
  • Game Theory and Its Practical Applications
    University of Northern Iowa UNI ScholarWorks Presidential Scholars Theses (1990 – 2006) Honors Program 1997 Game theory and its practical applications Angela M. Koos University of Northern Iowa Let us know how access to this document benefits ouy Copyright ©1997 Angela M. Koos Follow this and additional works at: https://scholarworks.uni.edu/pst Part of the Other Economics Commons Recommended Citation Koos, Angela M., "Game theory and its practical applications" (1997). Presidential Scholars Theses (1990 – 2006). 100. https://scholarworks.uni.edu/pst/100 This Open Access Presidential Scholars Thesis is brought to you for free and open access by the Honors Program at UNI ScholarWorks. It has been accepted for inclusion in Presidential Scholars Theses (1990 – 2006) by an authorized administrator of UNI ScholarWorks. For more information, please contact [email protected]. Game Theory and its Practical Applications A Presidential Scholar Senior Thesis University of Northern Iowa by Angela M. Koos Spring 1997 Dr. Ken Brown, 7 Dfrte Thesis and Major Advisor ,~-,, Dr. Ed Rathmell, Date Chair of Presidential Scholars Board Table of Contents Section Page(s) I. Historical Overview 1 I.A. Early Contributions to Game Theory 1 - 3 LB. John von Neumann, the RAND Corporation, and the Arms Race 3 - 7 LC. John Nash 7 - 8 I.D. Other Contributions to Game Theory 9 II. Defining Game Theory 9 - 12 II.A. Formal Representations of Games 12 - 13 II.A. I. Extensive Form 13 - 24 II.A.2. Normal Form 24 - 25 III. The Minimax Theorem 25 - 26 III.A. Preliminary Comments 26 - 27 III.B. The Theorem 27 - 28 IV.
    [Show full text]
  • Uncertainty Aversion in Game Theory: Experimental Evidence
    Uncertainty Aversion in Game Theory: Experimental Evidence Evan M. Calford∗ Purdue University Economics Department Working Paper No. 1291 April 2017 Abstract This paper experimentally investigates the role of uncertainty aversion in normal form games. Theoretically, risk aversion will affect the utility value assigned to realized outcomes while am- biguity aversion affects the evaluation of strategies. In practice, however, utilities over outcomes are unobservable and the effects of risk and ambiguity are confounded. This paper introduces a novel methodology for identifying the effects of risk and ambiguity preferences on behavior in games in a laboratory environment. Furthermore, we also separate the effects of a subject's be- liefs over her opponent's preferences from the effects of her own preferences. The results support the conjecture that both preferences over uncertainty and beliefs over opponent's preferences affect behavior in normal form games. Keywords: Ambiguity Aversion, Game Theory, Experimental Economics, Preferences JEL codes: C92, C72, D81, D83 ∗Department of Economics, Krannert School of Management, Purdue University [email protected]; I am partic- ularly indebted to Yoram Halevy, who introduced me to the concept of ambiguity aversion and guided me throughout this project. Ryan Oprea provided terrific support and advice throughout. Special thanks to Mike Peters and Wei Li are also warranted for their advice and guidance. I also thank Li Hao, Terri Kneeland, Chad Kendall, Tom Wilkening, Guillaume Frechette and Simon Grant for helpful comments and discussion. Funding from the University of British Columbia Faculty of Arts is gratefully acknowledged. 1 1 Introduction In a strategic interaction a rational agent must form subjective beliefs regarding their opponent's behavior.
    [Show full text]
  • Stochastic Games and Their Complexities Public Defense
    Stochastic games and their complexities public defense Marcin Przyby lko University of New Caledonia University of Warsaw 03.04.2019 Supervisors: Teodor Knapik & Damian Niwi´nski Auxiliary supervisor: Micha l Skrzypczak 0 / 22 Stochastic games Games with probabilistic transitions backgammon board 1 / 22 Stochastic games Games with probabilistic transitions ... ... ... ... ... ... graph of configurations 1 / 22 Formal model Games on graphs I move a single token along the lines of a graph e m 0:2 e 0:5 0:8 n 0:5 0:8 a m a 0:2 I decide the outcome based on the history of the token a word over the labels of the vertices 2 / 22 Stochastic branching games I study I two-player zero-sum games e.g. chess I of infinite duration e.g. parity games I with randomness e.g. simple stochastic games I and a form of concurrency. i.e. stochastic branching games by Mio Strong model: subsume many turn-based games extension of Gale-Stewart games and some concurrent games. can encode Blackwell games 3 / 22 Branching game Game G = hB; Φi is played by two players (Eve and Adam) and consists of I a board B the graph where we move the tokens I and an objective Φ: plays(B) ! [0; 1]. that can be seen as a rulebook ... 4 / 22 Branching games, an example Board { (unfolding of a) binary graph, consisting of Adam's, Eve's, Nature's and branching vertices. c n e1 0:8 0:2 e2 a f1 f2 The black token marks the initial vertex.
    [Show full text]