An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011

1. 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. 2 Motivation I: Binary Search • Search question: which half of surviving list might x be in? • f(M)=d lg Me rounds to search length M list Is x>a5? Yes. Is x>a7? No. Is x>a6? …

3. Motivation I: Binary Search 3 Is x>a5? Yes. Is x>a7? No. Is x>a6? Yes… eliminated Change perspectives: Ask “is x a red or a black chip?”

4. Motivation I: Binary Search with Errors 4 Round Question Answer 10 Is x black? Yes 5 5 1 Position: 0 1 e =2 Is x black? No 6 5 5 2 2 2 Is x black? Yes 6 4 4 2 2 1 1 3 eliminated The twist: Fix e≥ 0 and allow up to e incorrect responses.

5. 5 Motivation II: Random Walk 10 • Chips are divisible • Time-evolution: 10 £ binomial distribution of 0-1 coin flips • What question strategy for binary search with error best approximates random walk? 5 5 5 2.5 2.5 3.75 3.75 1.25 1.25 eliminated

6. The Search Game and the Dual Game 1 6 2 7 3 8 4 9 5 10 1 3 5 6 2 7 8 4 9 10 6 (5,5,0) 0 1 e =2 move odds move evens odd(5,5,0) even(5,5,0) 2 4 6 1 7 8 3 5 9 10 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

7. The Search Game and the Dual Game 1 6 2 7 3 8 4 9 5 10 1 3 5 6 2 7 8 4 9 10 7 (5,5,0) 0 1 e =2 move odds move evens odd(5,5,0) even(5,5,0) 2 4 6 1 7 8 3 5 9 10 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

8. Game Decision Tree 8 move odds move evens M=3 , n=3, e=1 gives a depth 3 binary decision tree With these parameters, and with Carole playing adversarially, Paul always wins the dual game, but not the search game.

9. Game Definitions and Data 1 2 … … M chip position: 0 1 e chip state: (M,0,…,0) 9 For fixed n>0, e≥0, the (M,n,e)-chip game has initial state 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 10 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}

11. Previous Results move odds move evens 11 (3,3,1)-game tree (a chip survives inthe leftmost leaf) Definition.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

12. Analysis of the Search Game move odds move evens 12 (3,3,1)-game tree (2,3,1)-game tree C A B C’ A’ B’

13. Analysis of the Search Game move odds move evens 13 (3,3,1)-game tree (2,3,1)-game tree C A B C’ A’ B’

14. Analysis of the Search Game move odds move evens 14 (3,3,1)-game tree (2,3,1)-game tree C A B C’ A’ B’

15. Proof of Main Theorem 15