TY - GEN

T1 - Information value of two-prover games

AU - Braverman, Mark

AU - Ko, Young Kun

N1 - Publisher Copyright:
© Mark Braverman and Young Kun Ko.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We introduce a generalization of the standard framework for studying the di culty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) – in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1−)-probability by communicating at most bits of information, then one can win n copies with probability at least 2−O(n). This gives an intuitive explanation why Raz’s counter-example to strong parallel repetition [16] (the odd cycle game) is a counter-example to strong parallel repetition – one can win the odd-cycle game on a cycle of length m by communicating O(m−2)-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1 − )n, then one can win one copy with at least (1 − O())-probability by communicating O() bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the “Chi-Squared Lemma” which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker’s Inequality.

AB - We introduce a generalization of the standard framework for studying the di culty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) – in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1−)-probability by communicating at most bits of information, then one can win n copies with probability at least 2−O(n). This gives an intuitive explanation why Raz’s counter-example to strong parallel repetition [16] (the odd cycle game) is a counter-example to strong parallel repetition – one can win the odd-cycle game on a cycle of length m by communicating O(m−2)-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1 − )n, then one can win one copy with at least (1 − O())-probability by communicating O() bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the “Chi-Squared Lemma” which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker’s Inequality.

UR - http://www.scopus.com/inward/record.url?scp=85041666122&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.ITCS.2018.12

DO - 10.4230/LIPIcs.ITCS.2018.12

M3 - Conference contribution

AN - SCOPUS:85041666122

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 9th Innovations in Theoretical Computer Science, ITCS 2018

A2 - Karlin, Anna R.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 9th Innovations in Theoretical Computer Science, ITCS 2018

Y2 - 11 January 2018 through 14 January 2018

ER -