@Stuart, you have misunderstood.
There may be a maximal element among the convex combinations of cyclic preferences, when the VM axioms fail to hold. SSB utility axioms have to hold in this case.
This maximal is a good candidate for choice from the cyclic preferences. So the claim that a violation of VM axioms leads to a money pump is false, even in the presence of cyclic preferences.
Read Fishburn's Nonlinear Preference and Utility Theory (1988) or the very recent Essays in Honor of Fishburn, edited by S. Brams et al.
You should probably start your discussion from Merrill Flood's 1952 article on preference cycles, available from Rand. http://www.rand.org/pubs/authors/f/flood_merrill_m.html
This claim is wrong for two reasons.
The usual money pump argument deals only with the difficulty of making a choice when your preferences are cyclic. The VM axioms have nothing to do with it.
Even in the presence of cyclic choices, there could be a maximal element among the convex combinations of the choices, see Peter Fishburn on SSB Utility, or Skew Symmetric Bilinear Utility.
I read the article, also. The description of the game was a bit short and somewhat ambiguous.
The game is designed to show people who participate why it is hard to maintain collusion or price fixing amongst oligopolies, secret agreements are not enough. It was a good demonstration of the difficulties in maintaining a secret deal. Far better than simply reading about it.
A number of theorists think that price fixing is mystery because the economics of it should make any agreements disappear.
However, there are price fixings in the real world which are regularly prosecuted. So, how are the Ashley's dealt with by those groups?
Page 42-44, Non Linear Preference, "The money-pump concept also reveals a narrow perspective on how choice might be based on preferences, and perhaps a lack of imagination in dealing with cyclic patterns. Although there is no transparent way to make a sensible choice from {p,q,r} when p>q>r>p, nothing prevents a person from considering preferences over the set of convex combinations of p, q, and r. And, if there is a combination in the set, then that persons has an ex ante maximally preferred alternative. As first shown in Kreweras (1961), this indeed can be the case, and we shall consider it later as part of the SSB theory."
Here is a modern paper addressing some of these issues: http://hal.archives-ouvertes.fr/docs/00/08/43/90/PDF/B06008.pdf