Trust - Peer-to-Peer Computing: Applications, Architecture, Protocols, and Challenges - page 144

porting a malicious peer might be very high. Specifically, the cost of reporting
might be time and/or the risk of possible retaliation. To model the situation,
Tuan assumed that there exists a certain level of incentive for each peer to
report about a malicious peer. However, peers would prefer someone else to
do that (i.e., she is better off if other peers do the reporting). This is a typical
“reporting a crime” situation [Osborne, 2004] where many people witnessed a
crime but any one of them is unwilling to report the crime [Rosenthal, 1964].
In Tuan's model, assume that each peer is satisfied if a malicious peer is
reported and attaches a value s to this. Reporting is costly and the cost is
assumed to be c, where s > c > 0. Thus, there are three possible cases for
each peer: (1) the malicious peer is not reported and the payoff is 0; (2) the
malicious peer is reported by the peer and thus the payoff is s−c; and (3) the
malicious peer is reported by some other peer and thus the payoff is s.
Consider a mixed strategy situation where each peer probabilistically
chooses to report or not. Denote the probability that each peer would re-
port as p. Given a peer, the probability that no one out of (n−1) remaining
peers reports is thus (1−p) n . Consequently, the probability that at least one
peer (out of the remaining (n−1) peers) reports is 1−(1−p) n−1 . By defini-
tion, in equilibrium state [Osborne, 2004] the expected payoff of reporting for
each peer is equal to the expected payoff of not reporting. Thus, we have:

s−c = 0 + s(1−(1−p) n−1 ) (6.11)

Solving this equation gives:

1
n − 1

c
s

p = 1− (6.12)

Now, we can see that the probability that each peer reports about the
malicious peer decreases as the number of reporting peers increases.
Furthermore, Tuan also addressed the issue of voting for exclusion of a
(maliciously believed) peer and provided an analysis of the problem. By mod-
eling the decision process as a Bayesian game [Osborne, 2004], Tuan found
that the possible application of exclusion in P2P system might be dangerous.
In Tuan's Bayesian game model, the voting super peers have some a priori
belief about the type of the peer in question. Now, depending on the availabil-
ity and accuracy of new information about such a suspicious peer, this belief
can be changed. Tuan's analysis showed that, under certain assumptions, the
more the number of voting peers, the more likely that an innocent peer is
excluded from the network.