The largest and most reputable of which funds publicly support HFT driven trading/execution/market-making. I guess it depends on how much you trust Vanguard. I trust them a lot.
As the number of trading venues increases, discrepancies in prices across those venues will naturally result. The price discrepancies across multiple markets create an opportunity for nimble traders to make a small arbitrage profit by scouring the markets for these discrepancies and eliminating them. As the number of trading venues expands, the number of such arbitrage opportunities increases. So, it is not surprising that we have seen a tremendous increase in trading volume over the past decade, and that the activity is increasingly dominated by "high frequency traders." While Vanguard does not engage in this type of trading, we recognize that such trading has a positive impact on the markets at large, including longer term investors. Such arbitrage trading enables investors to get a fair price across market centers. Vanguard believes that the market structure changes facilitated by the Commission's various regulatory initiatives and the "knitting" together of the marketplace by "high frequency trading," have led to a significant decline in transaction costs for long-term investors over the past ten years through increased liquidity and tighter bid-ask spreads.
and
Various groups have attempted to quantify the reduction in transaction costs over the last ten to fifteen years. The Commission will continue to receive this data throughout the comment period. While the data universally demonstrate a significant reduction in transaction costs over the last ten to fifteen years, the precise percentages vary (estimates have ranged from a reduction of 35% to more than 60%). Vanguard estimates are in this range, and we conservatively estimate that transaction costs have declined 50 bps, or 100 bps round trip. This reduction in transaction costs provides a substantial benefit to investors in the form of higher net returns.
One of the reasons that there are currently so many stock exchanges in the US is that while you aren't allowed to quote at subpenny levels, if you have several exchanges with different levels of make/take fees, you effectively have the ability quote (or remove liquidity) at subpenny price levels net of fees and rebates. In that scenario the proliferation of exchanges is less about competition and more about finding clever ways of negotiating the regulatory environment.
I found it odd given what we saw in the US exchanges was the opposite of that before fragmentation, but the argument is that the network effects of large exchanges reduces the overall costs.
On exactly what front are exchanges competing? Order types? Fees? Volume? Differences in any of these are very often opaque to the customer.
From what I've seen, exchanges are basically functionally equivalent, since they provide a utility-like service. And, as Vanguard points out, more trading venues means higher market frictions and costs.
TBH I'm still trying to forming a strong opinion on the issue, as indicated by "probably" in my initial comment. And I think it's a topic worth discussing.
It doesn't matter if it's opaque to the customer or if every exchange is selling an identical product; there's reason for multiple exchanges to exist to avoid monopoly. Multiple players want to play in the market, that means multiple exchanges. Saying there should only be one exchange is picking a winner and giving them the entire market.
Many of which are made on the behalf of the little guy (through pension or mutual funds).