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.
[email protected]
http://math.iit.edu/~rellis/
[email protected]://www.trinity.edu/~vadim/
[email protected]
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
[email protected]
http://www.math.tamu.edu/~rellis/
[email protected]://www.trinity.edu/~vadim/
[email protected]
http://www.math.tamu.edu/~cyan/
37