Rényi-Ulam liar games with a fixed number of lies Robert B. Ellis Illinois Institute of Technology IIT Graduate Seminar, November 9, 2005 coauthors: Vadim Ponomarenko, Trinity University Catherine Yan, Texas A&M Two Vector Games 2 The original liar game 3 Original liar game example 4 Original liar game example 5 Original liar game history 6 A football pool Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 Bet 2 Bet 3 Bet 4 Bet 5 Bet 6 Bet 7 W L W W L L L W W L W L L L W W W L W L L W W W L L W L Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? W W W L L L W Ans.=7 7 Pathological liar game as a football pool Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 Bet 2 Bet 3 Bet 4 Bet 5 Bet 6 Carole W W W L L L W Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6 8 Pathological liar game history Liar Games Covering Codes 9 Optimal n for Paul’s win 10 Sphere bound for both games 11 Converse to sphere bound: a counterexample Y N 10 6 9 7 3-weight of possible next states 7 9 12 Perfect balancing is winning 16 (4-weight) 8 (3-weight) 4 2 1 13 A balancing theorem for both games 14 Lower bound for the original game 15 Upper bound for the pathological game 16 Upper bound for the pathological game 17 Summary of game bounds 18 Unified 1 lie strategy 19 Unified 1 lie strategy 20 Recall: (x,q,1)* game as a football pool Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 Bet 2 Bet 3 Bet 4 Bet 5 Bet 6 Carole W W W L L L W W L L W W W L W W L W L L L L W Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6 21 Bets $ adaptive Hamming balls Round 2 Round 3 Round 4 Round 5 Round 1 A “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways: Root 1 1 0 1 0 All predictions correct Child 1 0 * * * * 1st prediction incorrect Child 2 1 0 * * * 2nd prediction incorrect Child 3 1 1 1 * * 3rd prediction incorrect Child 4 1 1 0 0 * 4th prediction incorrect Child 5 1 1 0 1 1 5th prediction incorrect A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1. 22 Strategy tree for adaptive betting W/1 W/1 L/0 Paths to leaves containing 1: 11111 Root (0 incorrect predictions) 00101 Child 1 (1 incorrect prediction) 10101 Child 2 11001 Child 3 11101 Child 4 11110 Child 5 (1 incorrect prediction) L/0 L/0 W/1 11111 11110 11101 11011 10111 11100 11010 11001 10110 10101 10011 11000 10100 10010 10001 10000 01111 01110 01101 01011 00111 01100 01010 01001 00110 00101 00011 01000 00100 00010 00001 23 00000 Adaptive code reformulation 24 Radius 1 packings within coverings 25 Radius 1 packings within coverings 26 Open directions •Asymmetric Hamming balls and structures for arbitrary communication channels (Spencer, Dumitriu for original game) •Questions occurring in batches (partly solved for original game) •Simultaneous packings and coverings for general k •Passing to k=k(n), such as allowing some fraction of answers to be lies (partly studied by Spencer and Winkler) •Comparisons to random walks and discrete-balancing processes such as chip-firing and the Propp machine Thank you. rellis@math.iit.edu http://math.iit.edu/~rellis/ vadim@trinity.eduhttp://www.trinity.edu/~vadim/ cyan@math.tamu.edu http://www.math.tamu.edu/~cyan/ 27 Lower bound by probabilistic strategy 28 Upper bound: Stage I, x! y’ 29 Upper bound: Stages I (con’t) & II 30 Upper bound: Stage III and conclusion 31 Exact result for k=1 32 Exact result for k=2 33 Linear relaxation and a random walk If Paul is allowed to choose entries of a to be real rather than integer, then a=x/2 makes the weight imbalance 0. Example: ((n,0,0,0),q,3)*-game and random walk on the integers: 34 Covering code formulation 11111 11111 11110 1011111101 11011 10111 01111 C= 11010 11100 1100011001 10110 10101 10011 00100 0001010100 10010 10001 11000 00001 10000 W!1, L!0 01111 01110 01101 01011 00111 01100 01010 01001 00110 00101 00011 01000 00100 00010 00001 00000 Equivalent question What is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q5? 35 Sparse history of covering code density 36 Future directions •Efficient Algorithmic implementations of encoding/decoding using adaptive covering codes •Generalizations of the game to k a function of n •Generalization to an arbitrary communication channel (Carole has t possible responses, and certain responses eliminate Paul’s vector entirely) •Pullback of a directed random walk on the integers with weighted transition probabilities •Generalization of the game to a general weighted, directed graph •Comparison of game to similar processes such as chip-firing and the Propp machine via discrepancy analysis rellis@math.tamu.edu http://www.math.tamu.edu/~rellis/ vadim@trinity.eduhttp://www.trinity.edu/~vadim/ cyan@math.tamu.edu http://www.math.tamu.edu/~cyan/ 37