> This is only an issue if later players have knowledge of the action other players are taking during the turn in progress

You’re assuming full endogeneity. That is never the case. Others’ orders are just one input. Other markets’ books and orders are far more important, particularly for derivatives.

So, "simply" integrate those "other markets" into the turn based system?

The information many hft players operate on has nothing to do with the market in question.

I don't have any issue with HFTs that respond only to external news source, the research being done in news sentiment analysis and how to incorporate that into buys and sells might not be great but the HFT players that I take issue with are those that look at price trends and try to snatch penny advantages out of a perceived momentum, that sort of market play is meaningless.

Perceived momentum? I think your idea of HFT strategies might be pretty far off the mark.

Guessing that the market will continue to go up or down (a.k.a 'perceived momentum') is just a gambling strategy used by day traders. There are no long term successful hft operations based on predict the future strategies like that.