1 Reliable Relaying with Uncertain Knowledge Jiwoong Lee, Jean Walrand Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, California 94720 {porce,wlr}@eecs.berkeley.edu Abstract—The motivation for this paper is to analyze the effect of information uncertainty on the design and performance of protocols. The paper considers two types of situations. The first is when different nodes in the network have bounded knowledge about what other nodes know. The second, called common knowledge about inconsistent beliefs, is when the information is inconsistent but everyone knows it. Situations of bounded or inconsistent information arise naturally in networks because the state of these systems changes and nodes take time to learn of those changes. The specific problem that the paper explores is the relaying of packets in a simple butterfly network. Despite its apparent simplicity, this problem enables to illustrate key features of situations of uncertain knowledge that arise in networks. The paper presents two impossibility facts and one possibility fact, in the latter of which a scheme that enables optimal coordination given persisting imperfection in knowledge is introduced. Index Terms—Relay network, Reliable relaying, Bounded Knowledge, Inconsistent Beliefs, Common Knowledge I. I NTRODUCTION This paper studies the impact of bounded or inconsistent information on the performance of a simple relay network. In a network, nodes typically implement distributed proto- cols for routing, relaying, discovery, leader election, conges- tion control, and other operations. Generally, the nodes have delayed and incomplete information about the state of the network. It is therefore natural to question the impact of this incomplete information on the performance of the protocols. A first line of inquiry considers delays and lack of syn- chrony among the nodes. A representative result is that a distributed Bellman-Ford protocol converges to the shortest paths if messages are eventually delivered between nodes, assuming that the network topology does not change [2]. More general results concern the convergence of parallel and distributed algorithms [3]. A second tread of investigation addresses impossibility theorems for distributed applications. An early result is the impossibility of two generals to agree with certainty when messages they exchange have some probability of not being delivered [5], [7]. Another well-known result is the Byzantine general problem where, using oral messages, loyal generals cannot agree on whether to attack or retreat if at least one third of the generals are traitors [6], [8]. In game theory, a related formulation of the imperfection of information has received considerable attention after the publication of Rubinstein’s electronic mail paper[9]. In that paper, two friends exchange lossy messages to decide whether to go out for coffee. One friend knows that the weather is bad and tries to agree with his friend that they should postpone their going out. Even after a large number of messages, they may end up not making the correct joint decisions. This paper examines similar situations where different nodes should coordinate their actions to prevent a bad outcome. However, because of imperfection of knowledge, the nodes may choose the wrong actions. The paper focuses on a simple example where only one of two nodes in a network should relay a packet to prevent a collision. The difficulty is that the nodes do not know perfectly the two probabilities of success nor what the other node knows. Even after exchanging an arbitrarily large number of ‘link state messages’ the nodes may end up making the same decision of either relaying the packet or not. The goal of the paper is to explore protocols that avoid such pitfalls and are robust with respect to imperfect knowledge. The first part of the paper focuses on the impact of bounded knowledge. The second part studies the situations where the nodes have inconsistent beliefs but they know it as a common knowledge. Many other protocol design problems face similar difficul- ties, such as leader election, routing, and forwarding. The authors hope that the paper will increase awareness of this issue. II. PROBLEM FORMULATION Consider the network shown in Figure 1. There are four S D A B p A p B a b Fig. 1. Butterfly relay network wireless nodes: S, A, B, and D. At time 0, node S broadcasts a packet to increase the chance of delivery to D, and relay nodes A and B receive it correctly. At time 1, the nodes A and B decide to forward the packet with probability a and b,