R

1
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 AM

2
Two Vector Games
3
The original liar game
4
Original liar game example
5
Original liar game example
6
Original liar game history
7
A football pool
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1 W W W W W
Bet 2 L W W W W
Bet 3 W L W W W
Bet 4 W W L L L
Bet 5 L L W L L
Bet 6 L L L W L
Bet 7 L L L L W
Payoff a bet with 1 bad
prediction Question. Min bets to guarantee a
payoff?
Ans.7
8
Pathological liar game as a football pool
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1 W
Bet 2 W
Bet 3 W
Bet 4 L
Bet 5 L
Bet 6 L
Carole W
Payoff a bet with 1 bad
prediction Question. Min bets to guarantee a
payoff?
Ans.6
9
Pathological liar game history
Liar Games
Covering Codes
10
Optimal n for Pauls win
11
Sphere bound for both games
12
Converse to sphere bound a counterexample
Y
N
10
6
9
7
7
9
3-weight of possible next states
13
Perfect balancing is winning
16 (4-weight)
8 (3-weight)
4
2
1
14
A balancing theorem for both games
15
Lower bound for the original game
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Upper bound for the pathological game
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Upper bound for the pathological game
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Summary of game bounds
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Unified 1 lie strategy
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Unified 1 lie strategy
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Recall (x,q,1) game as a football pool
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1 W W W
Bet 2 W L W W
Bet 3 W L L L L
Bet 4 L W
Bet 5 L W
Bet 6 L W
Carole W L L L W
Payoff a bet with 1 bad
prediction Question. Min bets to guarantee a
payoff?
Ans.6
22
Bets adaptive Hamming balls
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
Round 2
Round 4
Round 5
Round 3
Round 1
A fixed choice in 0,1 for each yields an
adaptive Hamming ball of radius 1.
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Strategy tree for adaptive betting
W/1
L/0
L/0
L/0
W/1
W/1
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)
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Adaptive code reformulation
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Radius 1 packings within coverings
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Radius 1 packings within coverings
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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 kk(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_at_math.iit.edu http//math.iit.edu/rellis/
vadim_at_trinity.edu http//www.trinity.edu/vadim/
cyan_at_math.tamu.edu http//www.math.tamu.edu/cyan/

(preprints)
28
Lower bound by probabilistic strategy
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Upper bound Stage I, x! y
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Upper bound Stages I (cont) II
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Upper bound Stage III and conclusion
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Exact result for k1
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Exact result for k2
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Linear relaxation and a random walk
If Paul is allowed to choose entries of a to be
real rather than integer, then ax/2 makes the
weight imbalance 0. Example
((n,0,0,0),q,3)-game and random walk on the
integers
35
Covering code formulation
W!1, L!0
C
Equivalent question What is the minimum number of
radius 1 Hamming balls needed to cover the
hypercube Q5?
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Sparse history of covering code density
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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 Pauls 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_at_math.tamu.edu http//www.math.tamu.edu/rel
lis/ vadim_at_trinity.edu http//www.trinity.edu/va
dim/ cyan_at_math.tamu.edu http//www.math.tamu.edu/
cyan/