Rényi-Ulam liar games with a fixed number of lies

Robert B. EllisIllinois Institute of Technology

University of Illinois at Chicago, October 26, 2005

coauthors:Vadim Ponomarenko, Trinity University

Catherine Yan, Texas A&M

2

Two Vector Games

3

The original liar game

4

Original liar game example

5

Original liar game history

6

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 predictionQuestion. Min # bets to guarantee a payoff?

Ans.=7

A football pool

7

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

Pathological liar game as a football pool

Payoff: a bet with · 1 bad predictionQuestion. 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

10 6 9 7 7 9

3-weight of possible next states

Y N

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

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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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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

Recall: (x,q,1)* game as a football pool

Payoff: a bet with · 1 bad predictionQuestion. Min # bets to guarantee a payoff?

Ans.=6

21

Rou

nd 1

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

Rou

nd 2

Rou

nd 3

Rou

nd 4

Rou

nd 5

A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.

22

Strategy tree for adaptive betting

W/1 L/0

W/1 L/0 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)

11011 10111

11100 11010 10110 10011

10100 10010 1000111000

10000

01111

01101 01011 0011101110

01100 01010 01001 00110

11111

1110111110

11001 10101

00101 00011

00100 00010 0000101000

00000

23

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 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.edu http://math.iit.edu/~rellis/

vadim@trinity.edu http://www.trinity.edu/~vadim/

cyan@math.tamu.edu http://www.math.tamu.edu/~cyan/

Thank you.

(preprints)

27

Lower bound by probabilistic strategy

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Upper bound: Stage I, x! y’

29

Upper bound: Stages I (con’t) & II

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Upper bound: Stage III and conclusion

31

Exact result for k=1

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Exact result for k=2

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Linear relaxation and a random walkIf 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:

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Covering code formulation11111

11101 11011 1011111110

11100 11010 11001 10110 10101 10011

10100 10010 1000111000

10000

01111

01101 01011 0011101110

01100 01010 01001 00110 00101 00011

00100 00010 0000101000

00000

W!1, L!0

Equivalent questionWhat is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q5?

1111110111

1100001111

001000001000001

C=

35

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 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.edu http://www.trinity.edu/~vadim/

cyan@math.tamu.edu http://www.math.tamu.edu/~cyan/