An identity for dual versions of
a chip-moving game
Robert B. Ellis
April 8th, 2011
ISMAA 2011, North Central College
Joint work with Ruoran Wang
2
Motivation I: Binary Search
 Search question: which half of surviving list might x be in?
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a8
a9
a10
Is x>a5? Yes.
a6
a7
Is x>a7? No.
a6
a7
Is x>a6? …
 f(M)=d lg M e rounds to search length M list
3
Motivation I: Binary Search
Change perspectives: Ask “is x a red or a black chip?”
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a8
a9
a10
Is x>a5? Yes.
a6
a7
Is x>a7? No.
a6
a7
Is x>a6? Yes…
eliminated
4
Motivation I: Binary Search with Errors
The twist: Fix e ≥ 0 and allow up to e incorrect responses.
Round
Question
10
Is x black?
Answer
5
Yes
5
1
Position: 0
1
5
5
e =2
Is x black?
2
2
6
3
2
No
6
2
Is x black?
2
4
1
4
Yes
1
eliminated
Motivation II: Random Walk
10
•Chips are divisible
5
•Time-evolution:
10 £ binomial distribution of 0-1 coin flips
5
5
3.75
1.25
What question strategy for binary search
with error best approximates random
walk?
2.5
2.5
3.75
1.25
eliminated
5
6
The Search Game and the Dual Game
M=#chips
n=#rounds
e=max #errors
Each round:
 Paul: numbers chips 1,…,M left-to-right;
odd chips are red, even chips are black.
 Carole: Selects a color/parity and
moves chips of that color/parity
1
2
3
4
5
0
move odds
odd(5,5,0)
2
4
1
3
5
6
8
10
7
9
6
7
8
9
10
1
(5,5,0)
e =2
move evens
1
3
5
2
4
7
9
even(5,5,0)
6
8
10
7
The Search Game and the Dual Game
 Search Game Paul wins iff at most one chip survives after n
rounds.
 Dual Game Paul wins iff at least one chip survives after n
rounds
1
2
3
4
5
0
move odds
odd(5,5,0)
2
4
1
3
5
6
8
10
7
9
6
7
8
9
10
1
(5,5,0)
e =2
move evens
1
3
5
2
4
7
9
even(5,5,0)
6
8
10
8
Game Decision Tree
 M=3, n=3, e=1
gives a depth 3 binary decision tree
move odds
move evens
With these parameters, and with Carole playing adversarially,
Paul always wins the dual game,
but not the search game.
9
Game Definitions and Data
For fixed n>0, e≥0, the (M,n,e)-chip game has initial state
1
2
…
M
chip position:
0
…
1
e
chip state: (M,0,…,0)
P*(n,e) = max{M : Paul can win the (M,n,e)-search game}
K*(n,e) = min{M : Paul can win the (M,n,e)-dual game}
10
Game Definitions and Data
P*(n,e) = max{M : Paul can win the (M,n,e)-search game}
K*(n,e) = min{M : Paul can win the (M,n,e)-dual game}
e\n
0
1
2
3
e\n
0
1
2
3
1
2
1
1
1
1
2
1
1
1
2
4
1
1
1
2
4
2
1
1
3
8
2
1
1
P*(n,e)
4
16
2
1
1
5
32
4
2
1
6
64
8
2
1
7
128
12
2
2
8
256
20
4
2
3
8
2
2
1
K*(n,e)
4
16
4
2
2
5
32
8
2
2
6
64
12
4
2
7
128
20
8
2
8
256
32
12
4
11
Previous Results
(3,3,1)-game tree
Definition.
move odds
move evens
pi(s)=position of the ith chip in state s.
Theorem (Cooper,Ellis `10). In the (M,n,e)-game tree, if leaf state s is
to the left of leaf state t, then for all 1 ≤ j ≤ M,
Corollary. K*(n,e) = minimum M such that
(a chip survives in
the leftmost leaf)
12
Analysis of the Search Game
move odds
(3,3,1)-game tree
move evens
C
A
(2,3,1)-game tree
B
C’
A’
B’
13
Analysis of the Search Game
move odds
(3,3,1)-game tree
move evens
C
A
(2,3,1)-game tree
B
C’
A’
B’
14
Analysis of the Search Game
move odds
(3,3,1)-game tree
move evens
C
A
(2,3,1)-game tree
B
C’
A’
B’
Proof of Main Theorem
15