Optimal, distributed decision-making: The case of no communication
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.
KeywordsInput Vector Decision Algorithm Winning Probability Binary Output General Optimization Problem
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