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Universal Computation in the Prisoner's Dilemma Game Brian Nakayama Regis University Regis University ePublications at Regis University All Regis University Theses Spring 2013 Universal Computation in the Prisoner's Dilemma Game Brian Nakayama Regis University Follow this and additional works at: https://epublications.regis.edu/theses Part of the Arts and Humanities Commons Recommended Citation Nakayama, Brian, "Universal Computation in the Prisoner's Dilemma Game" (2013). All Regis University Theses. 596. https://epublications.regis.edu/theses/596 This Thesis - Open Access is brought to you for free and open access by ePublications at Regis University. It has been accepted for inclusion in All Regis University Theses by an authorized administrator of ePublications at Regis University. For more information, please contact [email protected]. Regis University Regis College Honors Theses Disclaimer Use of the materials available in the Regis University Thesis Collection (“Collection”) is limited and restricted to those users who agree to comply with the following terms of use. Regis University reserves the right to deny access to the Collection to any person who violates these terms of use or who seeks to or does alter, avoid or supersede the functional conditions, restrictions and limitations of the Collection. The site may be used only for lawful purposes. The user is solely responsible for knowing and adhering to any and all applicable laws, rules, and regulations relating or pertaining to use of the Collection. All content in this Collection is owned by and subject to the exclusive control of Regis University and the authors of the materials. It is available only for research purposes and may not be used in violation of copyright laws or for unlawful purposes. The materials may not be downloaded in whole or in part without permission of the copyright holder or as otherwise authorized in the “fair use” standards of the U.S. copyright laws and regulations. UNIVERSAL COMPUTATION IN THE PRISONER’S DILEMMA GAME A thesis submitted to Regis College The Honors Program in partial fulfillment of the requirements for Graduation with Honors by Brian Nakayama May 2013 ii Contents 1 The Iterated Prisoner’s Dilemma Game 1 1.1 Introducing the G am e........................................................................................ 1 1.2 The Nash Equilibrium and Iterating the G a m e............................................... 4 2 Cellular Automata Theory 8 2.1 What is Cellular Automata?............................................................................... 8 2.2 Cool Cellular Automata..................................................................................... 10 2.3 Universal Computation and Construction: Self-Replicating Machines . 12 2.4 A Review on Literature for the IPDG CA ..................................................... 15 3 Playing the Prisoner’s Dilemma Game with Memory 18 3.1 Giving Each Player “Instructions” .................................................................. 18 3.2 Constructing the Lattice ..................................................................................... 22 3.3 One Interesting Consequence ............................................................................ 25 4 Patterns, Fractals, and Grouping 27 4.1 Creating Order Out of the Randomness............................................................ 27 4.2 How Groups Form ........................................................................................... 34 4.3 Interesting Results ........................................................................................... 35 5 Proof of Universal Computation 39 5.1 Creating a NAND g a t e ..................................................................................... 39 5.2 Making a Computer ........................................................................................... 43 5.3 What this Proof M e a n s .....................................................................................46 6 Social and Biological Relevance 48 6.1 Possible Implications ........................................................................................ 48 iii 6.2 Conclusion ........................................................................................................ 49 7 Further Ideas for Research 52 8 Bibliography 54 iv List of Figures 1 Axelrod's Inequalities for the PDG .................................................................. 3 2 Equations for the IPDG ..................................................................................... 6 3 Rumor Spreading in a Classroom..................................................................... 8 4 Different Wolfram Rules .................................................................................. 10 5 Von Neumann’s Universal Constructor............................................................ 12 6 A Glider Gun and Gliders in the Game of L ife ............................................... 14 7 Composition of Strategy T ypes........................................................................ 19 8 Converting to Base 10 to Make a Strategy Number ...................................... 22 9 A Few Different types of Neighborhoods......................................................... 23 10 Odds of Having a Single Strategy in a Random Initial S tate......................... 25 11 The Effects of the Second N eig h b o r............................................................... 26 12 A 800x600 Random Initial Configuration ..................................................... 27 13 The Hamming Distance..................................................................................... 29 14 4 Iteration Simulation After 750 R ounds......................................................... 30 15 9 Iteration Simulation After 750 R ounds......................................................... 31 16 Cooperating Instructions over Iteratio n s......................................................... 32 17 Point View ........................................................................................................... 34 18 A Natural G un..................................................................................................... 35 19 An Emergent Sierpinski Triangle ..................................................................... 36 20 The Wave Gun .................................................................................................. 37 21 The Three Iteration Gun .................................................................................. 38 22 Randomness Capable of Universal Computation............................................ 39 23 A Glider .............................................................................................................. 39 24 A “Pin” .............................................................................................................. 40 v 25 A Glider Eater.....................................................................................................40 26 Collision Between Two Gliders........................................................................ 41 27 Glider Hooked on a P in .....................................................................................41 28 A NAND Gate made out of IPDG P lay ers..................................................... 42 29 OR and XOR G ates........................................................................................... 43 30 D Latch with a “0” ........................................................................................... 45 31 D Latch with a “1” ........................................................................................... 46 32 A Glider Gun without p i n s ...............................................................................47 vi Picture Credits • Figure 3: Audience in classroom listening intently to speaker during meeting. Har­ less Todd, U.S. Fish and Wildlife Service. Web 08 Aug, 2012. http://www.public- domain-image.com/people-public-domain-images-pictures/crowd-public-domain-images- pictures/audience-in-classroom-listening-intently-to-speaker-during-meeting.jpg.html • Figure 4(Left): Weisstein, Eric W. “Rule 90.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/Rule90.html • Figure 4(Middle): Weisstein, Eric W. “Rule 30.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/Rule30.html • Figure 4(Right): A Textile Cone Snail. 2005. Richard Ling. Web 08 Aug 2012. http://en.wikipedia.org/wiki/File:Textile_cone.JPG • Figure 5: The Nobili-Pesavento Self-Replicator. 2008. Original automaton: Renato Nobili and Umberto Pesavento. Tape: Tim Hutton; Image: Golly screenshot. Web 22 Sept 2012. http://en.wikipedia.org/wiki/File:Nobili_Pesavento_2reps.png • Figure 6: Created by Brian Nakayama using David Bahr’s Cellular Automaton Ex­ plorer 5.0 • All other images created by Brian Nakayama vii List of Tables 1 The Prisoner’s Dilemma Game in m onths ..................................................... 2 2 The Prisoner’s Dilemma Game........................................................................ 2 3 Cleaning the Dishes P D G .................................................................................. 4 4 3 Memory Transient Strategy for Tit for T a t .................................................. 20 5 3 Memory Asymptotic and Transient Strategy............................................... 21 6 Average Percentage of Initial Cooperators..................................................... 30 7 Average Percentage of Population Points........................................................
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