1. An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th, 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?  f(M)= d lg M e rounds to search length M list 2 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 5 ? Yes. a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 7 ? No. a6a6 a7a7 Is x>a 6 ? …

3. Motivation I: Binary Search Change perspectives: Ask “is x a red or a black chip?” 3 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 5 ? Yes. a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 7 ? No. a6a6 a7a7 Is x>a 6 ? Yes… eliminated

4. 5 The twist: Fix e ≥ 0 and allow up to e incorrect responses. Motivation I: Binary Search with Errors 4 eliminated Round 1 2 3 Is x black? QuestionAnswer Yes No 10 55 5 2 6 2 2 6 2 11 44 01e =2Position:

5. 5 10 Motivation II: Random Walk 5 2.5 5 5 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? 1.25 3.75 1.25 3.75 eliminated

6. M=#chipsn=#roundse=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 The Search Game and the Dual Game 6 5 4 3 2 1 10 9 8 7 6 4 2 9 8 7 6 5 3 1 move odds 10 9 8 7 6 5 3 1 move evens 4 2 (5,5,0) odd(5,5,0) even(5,5,0) 01e =2

7.  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 The Search Game and the Dual Game 7 5 4 3 2 1 10 9 8 7 6 4 2 9 8 7 6 5 3 1 move odds 10 9 8 7 6 5 3 1 move evens 4 2 (5,5,0) odd(5,5,0) even(5,5,0) 01e =2

8.  M=3, n=3, e=1gives 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. Game Decision Tree 8 move oddsmove evens

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} Game Definitions and Data 9 … M 2 1 … chip state: (M,0,…,0) 01echip position:

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} Game Definitions and Data 10 P*(n,e)P*(n,e) e \ n12345678 0248163264128256 11122481220 211112224 311111122 K*(n,e)K*(n,e) e \ n12345678 0248163264128256 112248122032 2112224812 311122224

11. Definition. p i (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 Previous Results 11 (3,3,1)-game tree (a chip survives in the leftmost leaf)

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

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

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

15. Proof of Main Theorem 15

16. Proof of Main Theorem 16 Let 0<f<1/3 K*(f) := lim n->∞ (1/n)log 2 K * (n,fn) P*(f) := lim n->∞ (1/n)log 2 P * (n,fn) Theorem (Delsarte,Piret). K*(f) = 1-h(f), where h(f) = -f ln f – (1-f) ln(1-f) Corollary (EW`10). P*(f) = 1-h(f/(1-f)) P’(n,fn): allow Paul complete freedom in coloring the chips. (Berlekamp,Zigangirov) P’(f) = K*(f) until f=1/(3+5 1/2 ), then linear until f=1/3. K*K* P*P* P’P’ 0 1/3 0 1 f