A.2 Communication Evaluation
A.2.1 Communication Message Evaluation: Snapshot
Figure 1.1: 8 agents, arranged in a circle.
(n − 2 total agents) respond positively 6 times (n − 2). Agent 1 responds positively 5 times (n − 3) and negatively once, sending 6 messages (n − 2). Agent 8 responds
negatively 6 times (n − 2), sending 6 messages each time (n − 2), making 36 ((n − 2)2)
total messages. In addition, each agent broadcasts its quality to every other agent before the procedure begins (n(n−1)). The total messages sent during this worst-case
scenario is n − 2 + (n − 2)(n − 2) + n − 3 + n − 2 + (n − 2)2 + n2 − n = 3n2− 7n − 6.
Adding another agent to the mix causes this to grow to 3n2 − n − 10. This is a
difference of 6n − 4 messages if a new agent is added under the worst case scenario. In the eight agent system, this is the equivalent of adding 44 communication messages. In general, the addition of another agent adds 2n − 1 more messages for the dissenter, 2n − 1 more for the new agent, and one more for each of the other remaining agents
Figure 1.2: 8 agents, issuing proposals. Each proposal is sent to every agent except the agent to the proposer’s right.
(n − 1). The agent which dissents the dissenter’s proposal adds n − 1 more on top of
its one additional message. This snapshot scenario is dominated by n2 for large n.
A.2.2 Communication Message Evaluation: Full-Term. The snapshot worst case scenario presented previously captures a very short time in the grand scheme of the coalition formation mechanism, but provides insight into the full-term behavior. The above scenario is a worst case with a single proposal per agent. However, since
there are ¡n2¢ coalitions with 2 agents, ¡n3¢ coalitions with 3 agents, etc., the scale is
more significant. Of these coalitions, each agent has ¡n−11 ¢ coalitions of size 2 that
contain itself, ¡n−12 ¢ coalitions of size 3 that contain itself, etc. A reasonable estimate
is to assume that each agent must propose about ¡np¢/n coalitions for each coalition
of size p, or 2n−1
Figure 1.3: 8 agents, positive responses. Every agent except agent 8 responds positively, and agent 1 does not respond posi- tively to agent 8.
nonempty coalitions scaled by the number of agents in the collective. Given this and
the fact that each agent can be in one of 2n−1− 1 possible coalitions, then each agent
must respond to each coalition it does not propose, about 2n−1−1−2n−1
n proposals. In
this worst case, each agent must disagree with each coalition it does not propose, and send each disagreement at the exact same time as all other non-proposer members of the coalition. This is the same context as the snapshot evaluation, but based upon the full formation time. Each agent’s negative response is sent to every other member of the proposed coalition. There are the same number of coalitions of size 2 for each agent
as there are of size n-1: n − 1. The average response quantity is (n−1)+(n−1)(n−1)2(n−1) = n
2.
Thus, since each agent need not send a response to itself, the average response quantity
is n
2−1. Therefore, each agent must send (2n−1−1−2 n−1
Figure 1.4: 8 agents, negative responses.
total number of messages in this worst case scenario is the sum of the responses and
the proposals: (2n−1− 1 −2n−1
n )(n2 − 1) + (2 n−1
n )(n2 − 1), which is dominated by n ∗ 2n at large n. With communication failures, the worst case is when each agent re-sends a proposal 9 times to every agent in its proposed coalition, every agent downstream receives it and responds, but the communication fails on response. The 10th attempt is successful. Then the total message quantity is 10 times higher than previously
mentioned, giving (10)((2n−1− 1 −2n−1
n )(n2 − 1) + (2 n−1
n )(n2 − 1)).
This scenario only considers the negotiation phase: the true communication quantity includes n − 1 messages each for announcement of qualities, a quantity of duplicate requests due to communication failure, some messages passed for final approval, and up to n − 1 messages passed to announce a winning coalition. This then drives the communication to 2n − 2 more messages than previously mentioned,
or (10)((2n−1− 1 −2n−1 n )( n 2 − 1) + ( 2n−1 n )( n
2 − 1)) + 2n − 2. This is still dominated by
n ∗ 2n at large n, so the order of the message quantity is unchanged.
A.2.3 Best Case Communication Scenario. If perfect knowledge is available, each agent can determine, without feedback from other agents, whether a coalition will be acceptable. The scope of acceptable coalitions may then be much smaller than the generated coalitions since the agents can filter their results prior to proposing. Consider a 6 agent system where two agents are poor at a task, two are moderate, and two are good. The poor agents would not agree to any proposal, so they submit none. The moderate agents would know that the poor agents would not agree, so they
generate 24−1− 1 = 7 coalitions each instead of 26−1− 1 = 31 coalitions. There are a
total of 11 coalitions: ¡42¢+¡43¢+¡44¢= 6 + 4 + 1 = 11. Each agent is likely to propose
about 3 of their coalitions, all of which are acceptable. with a total of 34 messages to be sent, each agent should have about 8.5 communications. Rounding up gives 9
communication messages per agent, as opposed to a worst case of (2n−1−1−2n−1
n )(n2− 1) + (2n−1 n )( n 2− 1) = (31 − 63 6 )(2) + ( 63
6)(2) = 62. Both of these values include only the
communications from the evaluation stage. The true communication quantity includes 5 messages each for announcement of qualities, a quantity of duplicate requests due to communication failure, some messages passed for final approval, and up to n − 1 messages passed to announce a winning coalition. This analysis must be performed on a case-by-case basis, since the effects of full knowledge cannot be evaluated without specifics about the number of agents that will disagree with a proposed coalition. Under the assumption of full state knowledge, however, the number of communication messages may drop dramatically. If the agents merely have accurate models (and not full state knowledge), the assumption cannot be used and the filtering of potential proposals does not occur. If this is the case, the communication is no better nor worse than any other circumstance without full state knowledge.