DOCSLIB.ORG

Essays on Networks and Hierarchies

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Essays on Networks and Hierarchies The Pennsylvania State University The Graduate School Department of Economics ESSAYS ON NETWORKS AND HIERARCHIES A Dissertation in Economics by Myeonghwan Cho °c 2008 Myeonghwan Cho Submitted in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy August 2008 The dissertation of Myeonghwan Cho was reviewed and approved¤ by the following: Kalyan Chatterjee Distinguished Professor of Economics and Management Science Dissertation Advisor, Chair of Committee Edward J. Green Professor of Economics James S. Jordan Professor of Economics Anthony M. Kwasnica Associate Professor of Business Economics Robert C. Marshall Professor of Economics Head of the Department of Economics ¤ Signatures are on ¯le in the Graduate School. Abstract This dissertation consists of four chapters. Chapter 1 studies a repeated prisoner's dilemma game in a network in which each agent interacts with his neighbors and cannot observe the actions of other agents who are not directly connected to him. If there is global information processing through a public ran- domization and global communication, it is not di±cult to construct a sequential equilibrium which supports cooperation and satis¯es a re¯nement, called stability, which requires that cooperation resumes after any history. Here, we allow agents to locally communicate with their neighbors and show that it is possible to construct such an equilibrium without global information processing. The role of local communication is to resolve the discrepancy of agents' expectations on their neighbor's future actions. Chapter 2 considers the same environment as does Chapter 1, except that agents are not allowed to communicate with their neighbors but instead can observe a public randomiza- tion. By introducing a public randomization, we also can construct a sequential equilibrium for su±ciently patient agents, which supports cooperation and in which cooperation even- tually resumes after any history. In addition, if there is a small possibility of mistakes, the equilibrium we construct can result in a more e±cient outcome than the trigger strategy equilibrium, even though they produce the same payo®s in the limit. Chapter 3 studies a situation in which agents can make a binding agreement both on the amount of local public goods and on the structure of networks through which they share the bene¯ts of public goods. An agent enjoys the bene¯t of public goods produced by others who are (directly or indirectly) connected to him. There is a cost to maintain a link as well as to produce a public good. Since agents can choose the amount of public goods, the value of a link is endogenously determined. In the analysis, we ¯rst consider two di®erent models of sequential bargaining games through which a contract on allocations is established. In the ¯rst model, we allow agents to propose a pure allocation and show that there is no symmetric stationary perfect equilibrium for su±ciently patient agents. In the iii second model, agents are allowed to propose a distribution on allocations. As a result, we ¯nd a symmetric stationary perfect equilibrium in which probabilistic choices are made on an equivalent class of allocations. Next, we characterize core allocations, which consist of a minimally connected network and an e®ort pro¯le in which at most one agent does not produce the maximum amount of public good. Chapter 4 studies the problem of organizational design in a setting where workers di®er in their abilities and each worker is required to be monitored to prevent shirking. The need to monitor workers generates a hierarchical structure in an organization, which can be represented as a rooted tree. The value of an organization is determined by the wage cost, the total number of productive tasks carried out, and the length of chain of supervision in the organizational hierarchy. In this setting, we construct and characterize an e±cient structure of hierarchy which maximizes the value of the organization. In addition, we show that a hierarchy with the optimal size has a form of balanced rooted tree. iv Table of Contents List of Figures vii List of Tables viii Acknowledgments ix 1 Cooperation in the Repeated Prisoner's Dilemma Game with Local In- teraction and Local Communication 1 1.1 Introduction .................................... 1 1.2 The Model .................................... 5 1.3 Strategy σ¤ .................................... 8 1.4 Belief System ¹ .................................. 13 1.5 Sequential Equilibrium .............................. 15 1.6 Discussion ..................................... 17 1.7 Conclusion .................................... 20 Appendix A: Proof of Lemma 1.3 ........................... 22 Appendix B: Actions in period s ¸ t ......................... 25 Appendix C: Proof of Proposition 1.1 ........................ 28 2 Public Randomization in the Repeated Prisoner's Dilemma Game with Local Interaction 34 2.1 Introduction .................................... 34 2.2 The Model .................................... 36 2.3 Sequential Equilibrium .............................. 40 2.4 Stability and Robustness of σp ......................... 45 2.5 Concluding Remarks ............................... 51 Appendix: Proof of Proposition 2.1 .......................... 52 v 3 Endogenous Formation of Networks for Local Public Goods 57 3.1 Introduction .................................... 57 3.2 Models ....................................... 60 3.3 Sequential Bargaining Games .......................... 62 3.3.1 Non-Existence of Symmetric Stationary Perfect Equilibrium with Pure Proposals ................................. 65 3.3.2 Existence of Symmetric Stationary Perfect Equilibrium with Mixed Proposals ................................. 68 3.4 Allocations in the Core .............................. 74 3.5 Concluding Remarks ............................... 80 Appendix A: Proofs in Section 3.3.1 ......................... 81 Appendix B: Proofs in Section 3.3.2 ......................... 83 Appendix C: Proofs in Section 3.4 .......................... 86 4 E±cient Structure of Organization with Heterogeneous Workers 90 4.1 Introduction .................................... 90 4.2 Model ....................................... 93 ¤ 4.3 E±cient Structure Ga .............................. 97 4.4 Optimal Size of Organization . 104 4.5 Characterization of Structure Rule . 107 4.6 Conclusion .................................... 109 Appendix: Proof of Proposition 4.6 . 110 References 113 vi List of Figures 1.1 σ¤-path conditioning on history h2 ....................... 12 1.2 histories ht and h^t ................................ 14 1.3σ ^-path conditioning on h3 ............................ 18 ¤ 3 ¤ 3 1.4 ®N (σ ; h ) and ®N (σ ; h~ ) ........................... 19 ¤ 2 1.5 ®N (σ ; h ) .................................... 20 1.6 Assumption of small l .............................. 21 1.7 Assumption of small g .............................. 21 3.1 (G; x) and (G0; x0) ................................ 76 0 0 3.2 (x1; x2) satisfying NBi = 0 ........................... 77 3.3 Examples of (G; x) for Proposition 3.6 ..................... 79 4.1 Counting up and down rank systems ...................... 96 ¤ 4.2 Example of Ga .................................. 97 0 4.3 Example of Ga and Ga for Lemma 4.1 ..................... 99 00 0 4.4 Example of Ga, Ga, and Ga for Lemma 4.2 . 100 4.5 Counting down rank system . 103 4.6 Example for the proof of Proposition 4.5 . 107 4.7 Structure rule' ¹ ................................. 109 4.8 Example for the proof of Lemma 4.6 . 112 vii List of Tables 1.1 Payo®s in prisoner's dilemma game with communication ........... 6 2.1 Payo® of prisoner's dilemma game ....................... 37 viii Acknowledgments I am deeply indebted to Professor Kalyan Chatterjee for his invaluable advice and encour- agement during the process of developing this research. Without his guidance and support, I would not have reached this point. I also would like to thank Professor Edward Green, Professor James Jordan, Professor Vijay Krishna, Professor Anthony Kwasnica, Profes- sor Marek Pycia, Professor Neil Wallace, and participants at the 18th Stony Brook Game Theory conference and PET07 conference for their helpful comments. In particular, I am grateful to Professor John Riew for his encouragement. The ¯nancial support of the Robert C. Daniels Graduate Fellowship, the Will E. Mason Memorial Endowment Award, and the David W. and Carolyn P. Grow Graduate Fellowship are gratefully acknowledged. Finally and personally, I would like to thank my parents for their continued love and support throughout my entire life and Sun Young for being with me through the last year at Penn State. ix Chapter 1 Cooperation in the Repeated Prisoner's Dilemma Game with Local Interaction and Local Communication 1.1 Introduction From the results called the folk theorem, it is well known that any outcome that Pareto dominates a Nash equilibrium (or, an individually rational outcome) in a stage game can be supported as an equilibrium in in¯nitely repeated games for su±ciently patient agents. The earliest works on this theorem, such as Friedman (1971) and Fudenberg and Maskin (1986), require that each agent perfectly monitors the actions chosen by all the other agents. After those works, many studies focus on situations in which monitoring is not perfect. For example, Kandori (1992b) and Ellison (1994) study repeated games with a random matching process, where agents are matched randomly in each period and each agent
Load more

Recommended publications
  • Repeated Games
    6.254 : Game Theory with Engineering Applications Lecture 15: Repeated Games Asu Ozdaglar MIT April 1, 2010 1 Game Theory: Lecture 15 Introduction Outline Repeated Games (perfect monitoring) The problem of cooperation Finitely-repeated prisoner's dilemma Infinitely-repeated games and cooperation Folk Theorems Reference: Fudenberg and Tirole, Section 5.1. 2 Game Theory: Lecture 15 Introduction Prisoners' Dilemma How to sustain cooperation in the society? Recall the prisoners' dilemma, which is the canonical game for understanding incentives for defecting instead of cooperating. Cooperate Defect Cooperate 1, 1 −1, 2 Defect 2, −1 0, 0 Recall that the strategy profile (D, D) is the unique NE. In fact, D strictly dominates C and thus (D, D) is the dominant equilibrium. In society, we have many situations of this form, but we often observe some amount of cooperation. Why? 3 Game Theory: Lecture 15 Introduction Repeated Games In many strategic situations, players interact repeatedly over time. Perhaps repetition of the same game might foster cooperation. By repeated games, we refer to a situation in which the same stage game (strategic form game) is played at each date for some duration of T periods. Such games are also sometimes called \supergames". We will assume that overall payoff is the sum of discounted payoffs at each stage. Future payoffs are discounted and are thus less valuable (e.g., money and the future is less valuable than money now because of positive interest rates; consumption in the future is less valuable than consumption now because of time preference). We will see in this lecture how repeated play of the same strategic game introduces new (desirable) equilibria by allowing players to condition their actions on the way their opponents played in the previous periods.
    [Show full text]
  • Scenario Analysis Normal Form Game
    Overview Economics 3030 I. Introduction to Game Theory Chapter 10 II. Simultaneous-Move, One-Shot Games Game Theory: III. Infinitely Repeated Games Inside Oligopoly IV. Finitely Repeated Games V. Multistage Games 1 2 Normal Form Game A Normal Form Game • A Normal Form Game consists of: n Players Player 2 n Strategies or feasible actions n Payoffs Strategy A B C a 12,11 11,12 14,13 b 11,10 10,11 12,12 Player 1 c 10,15 10,13 13,14 3 4 Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “A”. • Then 1 should choose “a”. n Player 1’s best response to “A” is “a”. Player 2 Player 2 Strategy A B C a 12,11 11,12 14,13 Strategy A B C b 11,10 10,11 12,12 a 12,11 11,12 14,13 Player 1 11,10 10,11 12,12 c 10,15 10,13 13,14 b Player 1 c 10,15 10,13 13,14 5 6 1 Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “B”. • Then 1 should choose “a”. n Player 1’s best response to “B” is “a”. Player 2 Player 2 Strategy A B C Strategy A B C a 12,11 11,12 14,13 a 12,11 11,12 14,13 b 11,10 10,11 12,12 11,10 10,11 12,12 Player 1 b c 10,15 10,13 13,14 Player 1 c 10,15 10,13 13,14 7 8 Normal Form Game Dominant Strategy • Regardless of whether Player 2 chooses A, B, or C, Scenario Analysis Player 1 is better off choosing “a”! • “a” is Player 1’s Dominant Strategy (i.e., the • Similarly, if 1 thinks 2 will choose C… strategy that results in the highest payoff regardless n Player 1’s best response to “C” is “a”.
    [Show full text]
  • Repeated Games
    Repeated games Felix Munoz-Garcia Strategy and Game Theory - Washington State University Repeated games are very usual in real life: 1 Treasury bill auctions (some of them are organized monthly, but some are even weekly), 2 Cournot competition is repeated over time by the same group of firms (firms simultaneously and independently decide how much to produce in every period). 3 OPEC cartel is also repeated over time. In addition, players’ interaction in a repeated game can help us rationalize cooperation... in settings where such cooperation could not be sustained should players interact only once. We will therefore show that, when the game is repeated, we can sustain: 1 Players’ cooperation in the Prisoner’s Dilemma game, 2 Firms’ collusion: 1 Setting high prices in the Bertrand game, or 2 Reducing individual production in the Cournot game. 3 But let’s start with a more "unusual" example in which cooperation also emerged: Trench warfare in World War I. Harrington, Ch. 13 −! Trench warfare in World War I Trench warfare in World War I Despite all the killing during that war, peace would occasionally flare up as the soldiers in opposing tenches would achieve a truce. Examples: The hour of 8:00-9:00am was regarded as consecrated to "private business," No shooting during meals, No firing artillery at the enemy’s supply lines. One account in Harrington: After some shooting a German soldier shouted out "We are very sorry about that; we hope no one was hurt. It is not our fault, it is that dammed Prussian artillery" But... how was that cooperation achieved? Trench warfare in World War I We can assume that each soldier values killing the enemy, but places a greater value on not getting killed.
    [Show full text]
  • 2.4 Finitely Repeated Games
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Three Essays on Dynamic Games Permalink https://escholarship.org/uc/item/5hm0m6qm Author Plan, Asaf Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Three Essays on Dynamic Games By Asaf Plan A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Economics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Matthew Rabin, Chair Professor Robert M. Anderson Professor Steven Tadelis Spring 2010 Abstract Three Essays in Dynamic Games by Asaf Plan Doctor of Philosophy in Economics, University of California, Berkeley Professor Matthew Rabin, Chair Chapter 1: This chapter considers a new class of dynamic, two-player games, where a stage game is continuously repeated but each player can only move at random times that she privately observes. A player’s move is an adjustment of her action in the stage game, for example, a duopolist’s change of price. Each move is perfectly observed by both players, but a foregone opportunity to move, like a choice to leave one’s price unchanged, would not be directly observed by the other player. Some adjustments may be constrained in equilibrium by moral hazard, no matter how patient the players are. For example, a duopolist would not jump up to the monopoly price absent costly incentives. These incentives are provided by strategies that condition on the random waiting times between moves; punishing a player for moving slowly, lest she silently choose not to move.
    [Show full text]
  • Spring 2017 Final Exam
    Spring 2017 Final Exam ECONS 424: Strategy and Game Theory Tuesday May 2, 3:10 PM - 5:10 PM Directions : Complete 5 of the 6 questions on the exam. You will have a minimum of 2 hours to complete this final exam. No notes, books, or phones may be used during the exam. Write your name, answers and work clearly on the answer paper provided. Please ask me if you have any questions. Problem 1 (20 Points) [From Lecture Slides]. Consider the following game between two competing auction houses, Christie's and Sotheby's. Each firm charges its customers a commission on the items sold. Customers view each of the auction houses as essentially identical. For this reason, which ever house charges the lowest commission charge will be the one most of the customers want to use. However, if they can cooperate they might be able to make more money without undercutting each other. The stage-game for the competition between the auction houses is shown below. Sotheby's 7% 5% 2% 7% 7, 7 1, 10 -2, 3 Christie's 5% 10, 1 4, 4 1, 2 2% 3, -2 2, 1 0, 0 (A) (2.5 points) List all pure strategy Nash equilibria of the stage-game. (B) (2.5 points) List the preferred cooperative outcome of the stage-game and determine the best payoff a player gains by unilaterally deviating from this outcome. (C) (5 points) Describe the Grim-trigger strategy for the infinitely repeated game between Sotheby's and Christie's. (D) (10 points) If both auction houses have a common discount factor 0 ≤ δ ≤ 1, find the con- dition on this discount factor that will allow the Grim-trigger strategy to be sustained as a Subgame Perfect Nash equilibrium of the infinitely repeated game.
    [Show full text]
  • Collusive Behaviour in Finite Repeated Games with Bonding
    DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA. CALIFORNIA 91125 COLLUSIVE BEHAVIOUR IN FINITE REPEATED GAMES WITH BONDING ..i.c:,1\lUTf OJ: \\' )"� Mukesh Eswaran �� � University of British Columbia � !f �\"'.'. ....., 0 c:i: C" ..... -< Tracy R. Lewis � � California Institute of Technology � � University of British Columbia ,;... <.;: �t--r. ,c.;;:, � SlfALL N\�\'-� SOCIAL SCIENCE WORKING PAPER 46 6 February 1983 COLLUSIVE BEHAVIOUR IN FINITE REPEATED GAMES WITH BONDING Abstract It is well known that it is possible (even with strictly positive discounting) to obtain collusive perfect equilibria in In finite repeated games, it is not possible to enforce infinitely repeated games. However, only noncooperative perfect collusive behaviour using deterrent strategies because of the equilibria exist in finite games. Even though finite games may last "unravel! ing" of cooperative behaviour in the 1 ast period. This paper for a long time, the cooperative behaviour of the players unravels in demonstrates that under certain conditions collusion among the players the final period of play: defection from the cooperative agreement is can be maintained if they can post a bond which they must forfeit if the dominanat strategy in the last period, and backward induction they defect from the cooperative mode. We show that the incentives to renders noncooperative action the dominant strategy in all earlier cooperate increase as the period of interaction grows in that the size periods. This phenomenon of unraveling is unsatisfactory for two of the bond required to deter defection becomes arbitrarily small as reasons. First, it contradicts our intuition that cooperative the number of periods in the game increases.
    [Show full text]
  • MS&E 246: Lecture 10 Repeated Games
    MS&E 246: Lecture 10 Repeated games Ramesh Johari What is a repeated game? A repeated game is: A dynamic game constructed by playing the same game over and over. It is a dynamic game of imperfect information. This lecture • Finitely repeated games • Infinitely repeated games • Trigger strategies • The folk theorem Stage game At each stage, the same game is played: the stage game G. Assume: • G is a simultaneous move game •In G, player i has: •Action set Ai • Payoff Pi(ai, a-i) Finitely repeated games G(K) : G is repeated K times Information sets: All players observe outcome of each stage. What are: strategies? payoffs? equilibria? History and strategies Period t history ht: ht = (a(0), …, a(t-1)) where a(τ) = action profile played at stage τ Strategy si: Choice of stage t action si(ht) ∈ Ai for each history ht i.e. ai(t) = si(ht) Payoffs Assume payoff = sum of stage game payoffs Example: Prisoner’s dilemma Recall the Prisoner’s dilemma: Player 1 defect cooperate defect (1,1) (4,0) Player 2 cooperate (0,4) (2,2) Example: Prisoner’s dilemma Two volunteers Five rounds No communication allowed! Round 1 2 3 4 5 Total Player 1 1 1 1 1 1 5 Player 2 1 1 1 1 1 5 SPNE Suppose aNE is a stage game NE. Any such NE gives a SPNE: NE Player i plays ai at every stage, regardless of history. Question: Are there any other SPNE? SPNE How do we find SPNE of G(K)? Observe: Subgame starting after history ht is identical to G(K - t) SPNE: Unique stage game NE Suppose G has a unique NE aNE Then regardless of period K history hK , last stage has unique NE aNE NE ⇒ At SPNE, si(hK) = ai SPNE: Backward induction At stage K -1, given s-i(·), player i chooses si(hK -1) to maximize: Pi(si(hK -1), s-i(hK -1)) + Pi(s(hK)) payoff at stage K -1 payoff at stage K SPNE: Backward induction At stage K -1, given s-i(·), player i chooses si(hK -1) to maximize: NE Pi(si(hK -1), s-i(hK -1)) + Pi(a ) payoff at stage K -1 payoff at stage K We know: at last stage, aNE is played.
    [Show full text]
  • The Complexity of Nash Equilibria in Infinite Multiplayer Games
    The Complexity of Nash Equilibria in Infinite Multiplayer Games Michael Ummels [email protected] FOSSACS 2008 Michael Ummels – The Complexity of Nash Equilibria in Infinite Multiplayer Games 1 / 13 Infinite Games Let’s play! 2 5 1 4 3 6 Play: π 1, 2, 3, 6, 4, 2, 5, ... Note: No probabilistic vertices! = Michael Ummels – The Complexity of Nash Equilibria in Infinite Multiplayer Games 2 / 13 Winning conditions Question: What is the payoff of a play? Specifiedy b a winning condition for each player: ▶ Büchi condition: Given a set F of vertices, defines the set of all plays π that hit F infinitely often. ▶ Co-Büchi condition: Given a set F of vertices, defines the set of all plays π that hit F only finitely often. ▶ Parity condition: Given a priority function Ω V N, defines the set of all plays π such that the least priority occurring infinitely often is even. ∶ → Player receives payoff 1 if her winning condition is satisfied, otherwise 0. But we are not so much interested in the winner of a certain play, but in the strategic behaviour that can occur. Michael Ummels – The Complexity of Nash Equilibria in Infinite Multiplayer Games 3 / 13 The Classical Case Two-player Zero-sum Games: Games with two players where the winning conditions are complements of each other. (Pure) Determinacy:A two-player zero-sum game is determined (in pure strategies)i f one of the two players has a (pure) winning strategy. Theorem (Martin 1975) Any two-player zero-sum game with a Borel winning condition is determined in pure strategies.
    [Show full text]
  • Repeated Games
    Multi-agent learning Repeated games Multi-agent learning Repeated games Gerard Vreeswijk, Intelligent Systems Group, Computer Science Department, Faculty of Sciences, Utrecht University, The Netherlands. Gerard Vreeswijk. Last modified on February 9th, 2012 at 17:15 Slide 1 Multi-agent learning Repeated games Repeated games: motivation 1. Much interaction in multi-agent systems can be modelled through games. 2. Much learning in multi-agent systems can therefore be modelled through learning in games. 3. Learning in games usually takes place through the (gradual) adaption of strategies (hence, behaviour) in a repeated game. 4. In most repeated games, one game (a.k.a. stage game) is played repeatedly. Possibilities: • A finite number of times. • An indefinite (same: indeterminate) number of times. • An infinite number of times. 5. Therefore, familiarity with the basic concepts and results from the theory of repeated games is essential to understand multi-agent learning. Gerard Vreeswijk. Last modified on February 9th, 2012 at 17:15 Slide 2 Multi-agent learning Repeated games Plan for today • NE in normal form games that are repeated a finite number of times. – Principle of backward induction. • NE in normal form games that are repeated an indefinite number of times. – Discount factor. Models the probability of continuation. – Folk theorem. (Actually many FT’s.) Repeated games generally do have infinitely many Nash equilibria. – Trigger strategy, on-path vs. off-path play, the threat to “minmax” an opponent. This presentation draws heavily on (Peters, 2008). * H. Peters (2008): Game Theory: A Multi-Leveled Approach. Springer, ISBN: 978-3-540-69290-4. Ch. 8: Repeated games.
    [Show full text]
  • Chapter 10 Game Theory: Inside Oligopoly
    Managerial Economics & Business Strategy Chapter 10 Game Theory: Inside Oligopoly McGraw-Hill/Irwin Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved. Overview I. Introduction to Game Theory II. Simultaneous-Move, One-Shot Games III. Infinitely Repeated Games IV. Finitely Repeated Games V. Multistage Games 10-2 Game Environments Players’ planned decisions are called strategies. Payoffs to players are the profits or losses resulting from strategies. Order of play is important: – Simultaneous-move game: each player makes decisions with knowledge of other players’ decisions. – Sequential-move game: one player observes its rival’s move prior to selecting a strategy. Frequency of rival interaction – One-shot game: game is played once. – Repeated game: game is played more than once; either a finite or infinite number of interactions. 10-3 Simultaneous-Move, One-Shot Games: Normal Form Game A Normal Form Game consists of: – Set of players i ∈ {1, 2, … n} where n is a finite number. – Each players strategy set or feasible actions consist of a finite number of strategies. • Player 1’s strategies are S 1 = {a, b, c, …}. • Player 2’s strategies are S2 = {A, B, C, …}. – Payoffs. • Player 1’s payoff: π1(a,B) = 11. • Player 2’s payoff: π2(b,C) = 12. 10-4 A Normal Form Game Player 2 Strategy ABC a 12 ,11 11 ,12 14 ,13 b 11 ,10 10 ,11 12 ,12 Player 1 Player c 10 ,15 10 ,13 13 ,14 10-5 Normal Form Game: Scenario Analysis Suppose 1 thinks 2 will choose “A”. Player 2 Strategy ABC a 12 ,11 11 ,12 14 ,13 b 11 ,10 10 ,11 12 ,12 Player 1 Player c 10 ,15 10 ,13 13 ,14 10-6 Normal Form Game: Scenario Analysis Then 1 should choose “a”.
    [Show full text]
  • Repeated Games EC202 Lectures IX & X
    Repeated Games EC202 Lectures IX & X Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 —Lectures IX & X Jan2011 1/16 Summary Repeated Games: Definitions: Feasible Payoffs Minmax Repeated Game Stage Game Trigger Strategy Main Result: Folk Theorem Examples: Prisoner’sDilemma Nava (LSE) EC202 —Lectures IX & X Jan2011 2/16 Feasible Payoffs Q: What payoffs are feasible in a strategic form game? A: A profile of payoffs is feasible in a strategic form game if can be expressed as a weighed average of payoffs in the game. Definition (Feasible Payoffs) A profile of payoffs wi i N is feasible in a strategic form game f g 2 N, Ai , ui i N if there exists a distribution over profiles of actions p suchf that: g 2 wi = ∑a A p(a)ui (a) for any i N 2 2 Unfeasible payoffs cannot be outcomes of the game Points on the north-east boundary of the feasible set are Pareto effi cient Nava (LSE) EC202 —Lectures IX & X Jan2011 3/16 Minmax Q: What’sthe worst possible payoff that a player can achieve if he chooses according to his best response function? A: The minmax payoff. Definition (Minmax) The (pure strategy) minmax payoff of player i N in a strategic form 2 game N, Ai , ui i N is: f g 2 ui = min max ui (ai , a i ) a i A i ai Ai 2 2 Mixed strategy minmax payoffs satisfy: v i = min max ui (si , s i ) s i si The mixed strategy minmax is not higher than the pure strategy minmax.
    [Show full text]
  • Cooperation in a Repeated Public Goods Game with a Probabilistic Endpoint
    W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2014 Cooperation in a Repeated Public Goods Game with a Probabilistic Endpoint Daniel M. Carlen College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Behavioral Economics Commons, Econometrics Commons, Economic History Commons, Economic Theory Commons, Other Economics Commons, Political Economy Commons, and the Social Statistics Commons Recommended Citation Carlen, Daniel M., "Cooperation in a Repeated Public Goods Game with a Probabilistic Endpoint" (2014). Undergraduate Honors Theses. Paper 34. https://scholarworks.wm.edu/honorstheses/34 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Carlen 1 Cooperation in a Repeated Public Goods Game with a Probabilistic Endpoint A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Arts in the Department of Economics from The College of William and Mary by Daniel Marc Carlen Accepted for __________________________________ (Honors) ________________________________________ Lisa Anderson (Economics), Co-Advisor ________________________________________ Rob Hicks (Economics) Co-Advisor ________________________________________ Christopher Freiman (Philosophy) Williamsburg, VA April 11, 2014 Carlen 2 Acknowledgements Professor Lisa Anderson, the single most important person throughout this project and my academic career. She is the most helpful and insightful thesis advisor that I could have ever expected to have at William and Mary, going far beyond the call of duty and always offering a helping hand.
    [Show full text]