R. Lipton, N. Young. "Simple strategies for large zero-sum games with applications to complexity theory". In Proc. 26th ACM Symposium on the Theory of Computing, 734--740, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Since they do not solve linear programs, they are more efficient and simpler to implement. We apply the technique to design a simple new parallel greedy algorithm for finding sparse near optimal mixed strategies for zero sum games. Such sparse strategies were recently shown to exist by [2] and [9]. Oblivous derandomization furthers the utility of thinking probabilistically when designing approximation algorithms and it strengthens the connection between probabilistic and non probabilistic techniques. To illustrate this we use oblivious derandomization to derive the greedy set cover ....

....from optimal mixed strategies is called the value of the game. We denote it V(P ) arse strate es or zero s a es Generally, the optimal mixed strategies correspond to the primal and dual solutions, respectively, of an (N ) size positive linear program. Recently Althofer [2] and Lipton and Young [9] considered mixed strategies that choose uniformly from a small multi set of pure strategies. They showed that there are always such strategies that are near optimal, even if the size of the multi set is logarithmic in the number of pure strategies available to the opponent. eorem [ Let P in ....

Richard J. Lipton and Neal E. Young. Simple strategies for large zero-sum games with applications to complexity theory. In Proc. of the 26th Ann. ACM Symp. on Theory of Computing, 1994. To appear.

....( vN] It constructs a game where one players moves are sets of inputs of size ffi and the other player s moves are circuits, and the payoff to the second player is the advantage of the circuit on the uniform distribution on the set. This approach is similar to that taken by Lipton and Young [LY]. They were interested in finding a small hard core distribution for the problem; we want to find a large one. 4 First Proof of Existence of HardCore Measures The two proofs give somewhat different and incomparable quantitative bounds, so we state both as distinct lemmas. The quantitative aspect ....

R. Lipton and N. Young, "Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory" , 26 th STOC, 1994, pp.734-740.

....win That is, if game outcome is given by f(h; x) decide whether 9h8xf(h; x) But this is simply the canonical Sigma 2 P complete decision problem QSAT 2 [17] so game learning is Sigma 2 P complete. Similar problems in game theory have been related to the complexity class Sigma 2 P [14, 18]. The problem solved by the strategy learning algorithm is in NP (a formula containing several copies of f , with different opponents inserted, must be satisfied) Satisfying the (p; q) randomization criterion is not much more difficult: the method of almost uniform generation [12] can be used to ....

....would consider mixed strategies that randomize over a set of pure strategies, where such pure strategies come from a class of deterministic strategies like H and X in this paper. It has been shown that near optimal mixed strategies exist that randomize over only a small number of pure strategies [14]. These small mixed strategies are compactly representable, and the probability of a win may be computed exactly in polynomial time for a pair of such strategies. This may make it practical to work with mixed strategies. 8 Conclusion We have shown the existence of a competitive algorithm that is ....

Lipton, L.J. and N.E. Young. (1994) Simple Strategies for Large Zero-sum Games With Applications to Complexity Theory. STOC '94

....DVIP iv PSPACE, is not obvious, even though it might seem so at first glance. The proof of this theorem is the main technical contribution of this paper and uses the idea of representing a prover strategy by a short list strategy (See Section 1.4. 3 for a discussion of the related techniques of [18]) 1.2.2 The power of independent provers We say that the provers are independent if they don t share any initial randomness R P . We note that as with other kinds of proof system, the optimal prover strategies in a double verifier proof are deterministic. Thus the shared randomness of the ....

....is used by any party. We note that our result also holds for protocols that are only zero knowledge with respect to an honest verifier; the lower bound in [7] doesn t hold for this case. 1.4. 3 Techniques Our concept of list strategies bares some similarity to a result of Lipton and Young [18] that in two player games, there exist almost optimal mixed strategies which are a combination of a logarithmic number of pure strategies, and hence that there are almost optimal mixed strategy whose representation is logarithmic in the size of the largest mixed strategies. However, in the [18] ....

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R. Lipton and N. Young. Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory. Proceedings of the Twenty Sixth Annual Symposium on the Theory of Computing, ACM, 1994.

....win That is, if game outcome is given by f(h; x) decide whether 9h8xf(h; x) But this is simply the canonical Sigma 2 P complete decision problem QSAT 2 [70] so game learning is Sigma 2 P complete. Similar problems in game theory have been related to the complexity class Sigma 2 P [58, 71]. Lemma 2 shows that, using our definition of a competitive algorithm, no competitive algorithm exists that solves every game in time polynomial in lg(jHj) lg(jX j) and k with worst case strategy learning algorithms. It is natural to ask 43 whether there exists any type of algorithm that can ....

....1 bits, and a second player strategy x represented with n 2 bits. Assume g returns 1 or 0 indicating whether the first player or second player was the winner, respectively, in time polynomial in n 1 and n 2 . The goal is to find a small mixed strategy (in the same sense used by Lipton and Young [58]) that randomizes uniformly over a polynomially large multiset of pure strategies and obtains a value for the game that is close to optimal. We first consider approximate learning in the case where there is a perfect first player strategy (V = 1) then extend it to games where 0 V 1. 46 ....

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R.J. Lipton and N.E. Young. Simple strategies for large zero-sum games with applications to complexity theory. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing. STOC '94. ACM, 1994.

....returned is a mixed strategy that plays uniformly from Sigma 2 ln m 2ffl 2 Upsilon pure strategies, one for each iteration. The opponent has m pure strategies; is the maximum minus the minimum payoff. The existence of such sparse, near optimal strategies was shown probabilistically [2, 13]; our existence proof of the approximate solution for generalized packing is a generalization of the proof in [13] 2 Related Work Plotkin, Shmoys, and Tardos [16] generalizing a series of works on multicommodity flow [19, 11, 12] gave approximation algorithms for general packing and covering ....

....each iteration. The opponent has m pure strategies; is the maximum minus the minimum payoff. The existence of such sparse, near optimal strategies was shown probabilistically [2, 13] our existence proof of the approximate solution for generalized packing is a generalization of the proof in [13]. 2 Related Work Plotkin, Shmoys, and Tardos [16] generalizing a series of works on multicommodity flow [19, 11, 12] gave approximation algorithms for general packing and covering problems similar to those we consider. For these abstract problems, their results are comparable to those in this ....

Richard J. Lipton and Neal E. Young. Simple strategies for large zero-sum games with applications to complexity theory. In Proc. of the 26th Ann. ACM Symp. on Theory of Computing, 1994. To appear.

....Since they do not solve linear programs, they are more efficient and simpler to implement. We apply the technique to design a simple new parallel greedy algorithm for finding sparse near optimal mixed strategies for zero sum games. Such sparse strategies were recently shown to exist by [2] and [9]. Oblivous derandomization furthers the utility of thinking probabilistically when designing approximation algorithms and it strengthens the connection between probabilistic and non probabilistic techniques. To illustrate this we use oblivious derandomization to derive the greedy set cover ....

....from optimal mixed strategies is called the value of the game. We denote it V(P ) arse strate es or zero s a es Generally, the optimal mixed strategies correspond to the primal and dual solutions, respectively, of an (N ) size positive linear program. Recently Althofer [2] and Lipton and Young [9] considered mixed strategies that choose uniformly from a small multi set of pure strategies. They showed that there are always such strategies that are near optimal, even if the size of the multi set is logarithmic in the number of pure strategies available to the opponent. eorem [ Let P in ....

Richard J. Lipton and Neal E. Young. Simple strategies for large zero-sum games with applications to complexity theory. In Proc. of the 26th Ann. ACM Symp. on Theory of Computing, 1994. To appear.

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R. Lipton, N. Young. "Simple strategies for large zero-sum games with applications to complexity theory". In Proc. 26th ACM Symposium on the Theory of Computing, 734--740, 1994.

No context found.

R. Lipton and N. Young. Simple strategies for large zero-sum games with applications to complexity theory. In Proceedings, ACM Symposium on Theory of Computing, pages 734-- 740, 1994.

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