1. Rényi-Ulam liar games with a fixed number of lies Robert B. Ellis Illinois Institute of Technology University of Illinois at Chicago, October 26, 2005 coauthors: Vadim Ponomarenko, Trinity University Catherine Yan, Texas A&M

2. 2 Two Vector Games

3. 3 The original liar game

4. 4 Original liar game example

5. 5 Original liar game history

6. 6 Round 1Round 2Round 3Round 4Round 5 Bet 1WWWWW Bet 2LWWWW Bet 3WLWWW Bet 4WWLLL Bet 5LLWLL Bet 6LLLWL Bet 7LLLLW Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=7 A football pool

7. 7 Round 1Round 2Round 3Round 4Round 5 Bet 1W Bet 2W Bet 3W Bet 4L Bet 5L Bet 6L CaroleW Pathological liar game as a football pool Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

8. 8 Pathological liar game history Liar GamesCovering Codes

9. 9 Optimal n for Paul’s win

10. 10 Sphere bound for both games

11. 11 Converse to sphere bound: a counterexample 1069779 3-weight of possible next states YN

12. 12 Perfect balancing is winning 16 (4-weight) 8 (3-weight) 4 2 1

13. 13 A balancing theorem for both games

14. 14 Lower bound for the original game

15. 15 Upper bound for the pathological game

16. 16 Upper bound for the pathological game

17. 17 Summary of game bounds

18. 18 Unified 1 lie strategy

19. 19 Unified 1 lie strategy

20. 20 Round 1Round 2Round 3Round 4Round 5 Bet 1WWW Bet 2WLWW Bet 3WLLLL Bet 4LW Bet 5LW Bet 6LW CaroleWLLLW Recall: (x,q,1) * game as a football pool Payoff: a bet with · 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

21. 21 Round 1 Bets $ adaptive Hamming balls A “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways: Root11010All predictions correct Child 10****1 st prediction incorrect Child 210***2 nd prediction incorrect Child 3111**3 rd prediction incorrect Child 41100*4 th prediction incorrect Child 5110115 th prediction incorrect Round 2 Round 3 Round 4 Round 5 A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.

22. 22 Strategy tree for adaptive betting W/1L/0 W/1 L/0 W/1 L/0 Paths to leaves containing 1: 11111Root (0 incorrect predictions) 00101Child 1 (1 incorrect prediction) 10101Child 2  11001Child 3  11101Child 4  11110Child 5 (1 incorrect prediction) 1101110111 11100110101011010011 10100100101000111000 10000 01111 01101010110011101110 01100010100100100110 11111 1110111110 1100110101 00101 00011 00100000100000101000 00000

23. 23 Adaptive code reformulation

24. 24 Radius 1 packings within coverings

25. 25 Radius 1 packings within coverings

26. 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 rellis@math.iit.eduhttp://math.iit.edu/~rellis/ vadim@trinity.eduhttp://www.trinity.edu/~vadim/ cyan@math.tamu.eduhttp://www.math.tamu.edu/~cyan/ Thank you. (preprints)

27. 27 Lower bound by probabilistic strategy

28. 28 Upper bound: Stage I, x ! y’

29. 29 Upper bound: Stages I (con’t) & II

30. 30 Upper bound: Stage III and conclusion

31. 31 Exact result for k=1

32. 32 Exact result for k=2

33. 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. 34 Covering code formulation W ! 1, L ! 0 Equivalent question What is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q 5 ? 11111 10111 11000 01111 00100 00010 00001 C=C=

35. 35 Sparse history of covering code density

36. 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.eduhttp://www.math.tamu.edu/~rellis/ vadim@trinity.eduhttp://www.trinity.edu/~vadim/ cyan@math.tamu.eduhttp://www.math.tamu.edu/~cyan/