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
<https://www.jasss.org/9/1/14.html>
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Received: 31-Jul-2005 Accepted: 07-Oct-2005 Published: 31-Jan-2006
q = f(Ra | p, a, b) = pa + (1-p)(1-b) | (1) |
q = f(Ra | p,e,d,a,b) = ed + (1-e)[pa + (1-p)(1-b)] | (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 step-like 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( {Ra ,Ra} | p,e,d,a,b) = (ed)2 + 2(ed)(1-e)[pa + (1-p)(1-b)] + (1-e) 2 [pa + (1-p)(1-b)] | (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)(1-e)[a + (1-p)(1-b)] | 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. |
(1-e)2[pa + (1-p)(1-b)] | Both the article and its replication are honest and both support Theory A. Notice that the factor [pa + (1-p)(1-b)] 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. |
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Figure 3. The infinite article (2000) limit for 0 and 1 replications and various e. The sharper, step-like curves correspond to the 1-replication case. Notice how the 1-replication 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, step-function type behavior was observed for the single-replication case, we were not able to test this.
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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 super-exponentially. However, for even a single replication, the number of required articles remains more or less constant with increasing e. Notice that once again, our 0-replication 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 0-replication case diverged super-exponentially with deception as before, the single-replication case remained relatively constant. However, it seems unreasonable that we should observe a true shift from super-exponential scaling to no scaling just by requiring a single replication — instead, we hypothesize that the scaling of our results was still super-exponential in nature, only with a drastically smaller scaling constant. In other words, we guess that the number of required articles still grows super-exponentially 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 single-replication case scaled at all, this was not testable.
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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. 105-127. Santa Fe: Perseus Books.
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