Among matching rule, do not forget "auction calls", in most markets, you have one at the open and one at the close.
To give you the main reasons to use one matching engine rather than another:
- Auction calls (i.e. fixings) are good to digest a lot of orders in a very short amount of time. It is why after a trading suspension, the trading starts with an auction call. It is also why in London, you have a mid-day auction call on witching days. At my knowledge it is the oldest practice (used on the pits).
- Continuous trading seems good to obtain a progressive relative price valuation ; I mean that it allows market participants to monitor simultaneously different instruments and to buy/sell one relatively to the others.
- Pro-rata matching generates huge sizes at first limits, giving birth to less prices a day (it cannot really prevent the price to move, of course). It gives advantage to market makers and investment banks who can take the risk to buy or sell more than what they need.
- Time priority give advantage to low latency traders (especially when the tick is large with respect to the average bid-ask spread).
Moreover, note the tick size influences the matching process a lot. Some markets implement:
- Internal matching: i.e. when you are a broker and you have an resting buy order on the best bid, but not first in the queue, if you send a sell order, it will match your buy order instead of others that are before you. It implements a "shortcut" of the time priority. It is a cheaper alternative to build a broker crossing network for the intermediaries.
- in some exchanges (like the Spanish exchange), the name of the member to whom orders in the queues belong is available.
If you want more information, read Market Microstructure in Practice. More specifically on pro-rata rules and market descriptions, you have Almgren, R. (2012, August). High-Frequency event analysis in eurex interest rate futures. Technical report and Field, J. and J. Large (2008). Pro-rata matching and one-tick futures markets. CFS working paper 2008,40, Frankfurt.