On Games of Incomplete Information

We study two-person games of cooperation and multi-prover interactive proof systems. We first consider a two-person game G, which we call a free game, defined as follows. A Boolean function C& is given. Players I and II each pick a random number i and j in private, where 1 d i,j < s, and then each chooses a private numberf(i) and g(j), 1 <f(i), g(j) < s. If &( i,j,f( i), g(j)) = 1, then both players win; otherwise, they lose. The objective of both players is to win collectively. We ask whether, if such a game is played n times in parallel, the probability of winning all the games decays exponentially in n. This question was posed in a more general context by Fortnow [lo], which we will discuss soon. Formally, we define the nth product game G” as the following two-person game. Players I and II each pick a vector of independent random numbers i= ( iI,. . . , i,) and j=(j l,...,jn) in private, 1 <ik, jk<s, and then each chooses a private sequence of numbers fi(g, . . ..f.(fl and gl(j) , . . ., g&3. The goal for both players is to ensure Ai=1 &(&,j&.(i),g.Jj))= 1. We define th e winning probability of the game G to be maxS,, Pr[&( i, j,f( i), g(j)) = 11, where the probability is taken over all randomly and uniformly chosen i,j in the range 1 , . . . . s, and we denote it by w(G). The game G is called nontrivial if its winning probability is neither 0 nor 1. We shall consider only

[1]  L. Fortnow Complexity-Theoretic Aspects of Interactive Proof Systems , 1989 .

[2]  Richard J. Lipton,et al.  On bounded round multiprover interactive proof systems , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[3]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[4]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[5]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[6]  Richard J. Lipton,et al.  Efficient Checking of Computations , 1990, STACS.

[7]  L. Fortnow,et al.  On the power of multi-power interactive protocols , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[8]  Richard J. Lipton,et al.  PSPACE is provable by two provers in one round , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[9]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.

[10]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[11]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[12]  Richard J. Lipton,et al.  Playing Games of Incomplete Information , 1990, Symposium on Theoretical Aspects of Computer Science.

[13]  Uriel Feige On the success probability of the two provers in one-round proof systems , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[14]  Shafi Goldwasser,et al.  Private coins versus public coins in interactive proof systems , 1986, STOC '86.