# Optimal, distributed decision-making: The case of no communication

## Abstract

We present a combinatorial framework for the study of a natural class of *distributed optimization problems* that involve decision-making by a collection of *n distributed agents* in the presence of incomplete information; such problems were originally considered in a *load balancing* setting by Papadimitriou and Yannakakis (*Proceedings of the 10th Annual ACM Symposium on Principles of Distributed Computing*, pp. 61–64, August 1991). For any given *decision protocol* and assuming *no communication* among the agents, our framework allows to obtain a combinatorial inclusion-exclusion expression for the probability that no “overflow” occurs, called the *winning probability*, in terms of the *volume* of some simple combinatorial polytope.

Within our general framework, we offer a complete resolution to the special cases of *oblivious algorithms*, for which agents do not “look at” their inputs, and *non-oblivious algorithms*, for which they do, of the general optimization problem. In either case, we derive optimality conditions in the form of combinatorial polynomial equations. For oblivious algorithms, we explicitly solve these equations to show that the optimal algorithm is simple and *uniform*, in the sense that agents need not “know” *n*. Most interestingly, we show that optimal non-oblivious algorithms must be *non-uniform*: we demonstrate that the optimality conditions admit different solutions for particular, different “small” values of *n*; however, these solutions improve in terms of the winning probability over the optimal, oblivious algorithm. Our results demonstrate an interesting trade-off between the amount of knowledge used by agents and uniformity for optimal, distributed decision-making with no communication.

## Keywords

Input Vector Decision Algorithm Winning Probability Binary Output General Optimization Problem## Preview

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## References

- 1.K. Arrow,
*The Economics of Information*, Harvard University Press, 1984.Google Scholar - 2.G. Brightwell, T. J. Ott, and P. Winkler, “Target Shooting with Programmed Random Variables,”
*Proceedings of the 24th Annual ACM Symposium on Theory of Computing*, pp. 691–698, May 1992.Google Scholar - 3.P. Fizzano, D. Karger, C. Stein and J. Wein, “Job Scheduling in Rings,”
*Proceedings of the 6th Annual ACM Symposium on Parallel Algorithms and Architectures*, pp. 210–219, June 1994.Google Scholar - 4.X. Deng and C. H. Papadimitriou, “Competitive, Distributed Decision-Making,”
*Proceedings of the 12th IFIP Congress*, pp. 350–356, 1992.Google Scholar - 5.J. M. Hayman, A. A. Lazar, and G. Pacifici, “Joint Scheduling and Admission Control for ATM-Based Switching Nodes,”
*Proceedings of the ACM SIGCOMM*, pp. 223–234, 1992.Google Scholar - 6.P. C. Kanellakis and C. H. Papadimitriou, “The Complexity of Distributed Concur-rency Control,”
*SIAM Journal on Computing*, Vol. 14, No. 1, pp. 52–75, February 1985.zbMATHCrossRefMathSciNetGoogle Scholar - 7.E. Kushilevitz and N. Nisan,
*Communication Complexity*, Cambridge University Press, Cambridge, 1996.Google Scholar - 8.C. H. Papadimitriou, “Computational Aspects of Organization Theory,”
*Proceedings of the 4th Annual European Symposium on Algorithms*, September 1996.Google Scholar - 9.C. H. Papadimitriou and M. Yannakakis, “On the Value of Information in Distributed Decision-Making,”
*Proceedings of the 10th Annual ACM Symposium on Principles of Distributed Computing*, pp. 61–64, August 1991.Google Scholar - 10.C. H. Papadimitriou and M. Yannakakis, “Linear Programming Without the Matrix,”
*Proceedings of the 25th Annual ACM Symposium on Theory of Computing*, pp. 121–129, May 1993.Google Scholar - 11.C. Polychronopoulos and D. Kuck, “Guided Self-Scheduling: A Practical Scheduling Scheme for Parallel Computers,”
*IEEE Transactions on Computers*, Vol. 12, pp. 1425–1439, 1987.CrossRefGoogle Scholar - 12.R. P. Stanley,
*Enumerative Combinatorics: Volume 1*, The Wadsworth & Brooks/Cole Mathematics Series, 1986.Google Scholar