Victor Palmer (2006)
Deception and Convergence of Opinions Part 2: the Effects of Reproducibility
Journal of Artificial Societies and Social Simulation
vol. 9, no. 1
<http://jasss.soc.surrey.ac.uk/9/1/14.html>
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Received: 31Jul2005 Accepted: 07Oct2005 Published: 31Jan2006
q = f(R_{a } p, a, b) = pa + (1p)(1b)  (1) 
q = f(R_{a } p,e,d,a,b) = ed + (1e)[pa + (1p)(1b)]  (2) 
Figure 1. This figure shows our replication of the Martins results for the infinite article limit. Notice that for small deception (small e and small e variation), the p versus q graph is almost a step function whereas for even for small amounts of deception unity p is never reached. This trend continues monotonically, with less and less steplike behavior observed for increasing e. In all cases a = b = 0.55. 
Figure 2. For two values of a, we plot the number of articles our agent observer must read to be 99% sure that Theory A is correct. For this series of simulations we set q = a since that is always the point of maximum p in the "p versus q" graph. Our simulation did not converge for e > 0.24 for the a = 0.7 case and for e > 0.28 for the a = 0.8 case. 
q = f( {R_{a },R_{a}}  p,e,d,a,b) = (ed)^{2 }+ 2(ed)(1e)[pa + (1p)(1b)] + (1e)^{ 2 }[pa + (1p)(1b)]  (3) 
Where each term is justified below:
(ed)^{2}  The probability that both the article and its replication are deceptions and that both support Theory A 
2(ed)(1e)[a + (1p)(1b)]  The probability that the article is a deception and supports Theory A, but its replication is NOT a deception and also happens to support Theory A. The factor of 2 is to account for an identical term (since ordering is not important) that occurs when the article is honest and its replication is a deception in support of Theory A. 
(1e)^{2}[pa + (1p)(1b)]  Both the article and its replication are honest and both support Theory A. Notice that the factor [pa + (1p)(1b)] is not squared in the expression — as we said earlier, this factor determines the likelihood of a randomly selected experiment coming up in favor of Theory A — once the experiment is selected (by the initial article), its replication will always come out in favor of the same theory. 

Figure 3. The infinite article (2000) limit for 0 and 1 replications and various e. The sharper, steplike curves correspond to the 1replication case. Notice how the 1replication p curve is relatively unchanged as we increase the amount of a priori suspected deception (e). 
We can see that if our agent requires that the articles it reads have been replicated, even once, the "p versus q" curve remains more or less a step function, regardless of the amount of assumed deception. There are no doubt pathological cases (perhaps an a very, very close to 0.5, for example) where requiring a second replication could offer our agent additional advantages, but since in all the cases we tested, stepfunction type behavior was observed for the singlereplication case, we were not able to test this.

Figure 4. The sparse article limit simulation for 0 and 1 replications. For the no replication case (strongly increasing curve), as we increase e, the number of articles that our observer agent must read to be 99% sure that Theory A is correct grows superexponentially. However, for even a single replication, the number of required articles remains more or less constant with increasing e. Notice that once again, our 0replication simulation never converged for several values of e. 
In the cases we tested, the presence of deception had no visible effect on the number of articles required by our agent to reach certainty — while the 0replication case diverged superexponentially with deception as before, the singlereplication case remained relatively constant. However, it seems unreasonable that we should observe a true shift from superexponential scaling to no scaling just by requiring a single replication — instead, we hypothesize that the scaling of our results was still superexponential in nature, only with a drastically smaller scaling constant. In other words, we guess that the number of required articles still grows superexponentially under replication, but just very, very slowly. Again, there are probably some pathological conditions where two or more replications would yield additional scaling benefits, but since we never saw a situation where the singlereplication case scaled at all, this was not testable.
HEGSELMANN, R. and Krause, U. (2002) Opinion Dynamics and Bounded Condence Models, Analysis and Simulation, Journal of Artificial Societies and Social Simulations, vol. 5, no. 3.
JAYNES, E.T. (2003) Probability Theory: The Logic of Science.Cambridge: Cambridge University Press.
LANE, D. (1997) Is What Is Good For Each Best For All? Learning From Others In The Information Contagion Model, in The Economy as an Evolving Complex System I (ed. by Arthur, W.B., Durlauf, S.N. and Lane, D.), pp. 105127. Santa Fe: Perseus Books.
MARTINS, A. (2005) Deception and Convergence of Opinions, Journal of Artificial Societies and Social Simulations, vol. 8, no. 2.
SZNAJDWERON, K. and Sznajd, J. (2000) Opinion Evolution in Closed Communities, Int. J. Mod. Phys. C 11, 1157.
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