Unfortunately there is little or no octane increase with LPG as it is currently sold and used in Aust. The minimum octane rating is guaranteed to be 92 MON only. Sometimes it may be more, depending on what is in the mix, but you have to tune to the lowest octane that may end up in the tank. If it were possible to introduce it as a liquid spray, the effective octane may be higher but I don't think that there are too many of such systems around and certainly not cheaply. In the past, and perhaps still in the USA where a lot of the (performance) literature is sourced from, LPG was mostly propane, which does have a higher octane than petrol, hence the misconception.
The change was brought about because it allows the inclusion of other (cheaper) components from the refinery stream - mostly butanes, and because most users of LPG don't modify the engine to take advantage of any extra octane and so it would be wasted.
There is a minor maximum power loss associated with LPG because it is almost always introduced as a vapour, displacing some of the air in the manifold. This is not noticeable in normal driving in my experience - how often do you use anything approaching full throttle during normal driving? If you need or want performance switch to petrol.
In assessing the economic benefit of an LPG conversion be aware that the price at the pump is per litre, not per unit of energy. Petrol has a greater energy content per litre than LPG. Exactly how much depends on what is actually in the LPG (and the petrol). Depending on which LPG system is used, there is likely no fuel cut on decel. or lean cruise either. And you get to carry about an extra 40kg of gas equipment.
I get 13.8l/100km on petrol and 18.2l/100km on LPG, mostly urban driving. I have seen as low as 16.4 and as high as 19l/100km on LPG, same driving conditions. Hence, multiplying the price of gas by 1.3 will give the approximate cost relative to petrol.
The price of LPG will also increase as fuel excise is applied (starting 2008, +2.5cpl/year going to 12.5cpl (all X 1.3 X 1.1 GST

) in 2012) and, more people may switch to using it (it will still be cheaper than petrol), increasing demand.
The actual cost of running dual fuel will depend on what mix of running distance/time is spent on each fuel. Currently, about $12/100km for me, using mostly premium petrol fuels.
I'll make the distinction here that by dual fuel I am referring to two separate fuel systems for each each fuel rather than integrated fuel systems (my term) where both fuels are used concurrently or consequeutively under some circumstances eg. cold start, WOT.
While there is some scope for optimising an engine to run on straight LPG - ignition timing maps, cam profiles and timing more biased to inlet flow, and reducing inlet manifold heating - for an economy application they are not cost effective, (changing the ign. timing maps may be, depending on how they are obtained) so I see little benefit to running straight LPG for economy. Take the extra range available and the virtual impossibility of running out of fuel with dual fuel. (I occasionally run my LPG (but never petrol) tank dry). Running dual fuel systems also reduces the number of possible causes of engine stoppage - only a crank angle sensor or possibly ignition module failure will stop the engine.
In a performance application, it may be cheaper (acknowledging the savings in fuel costs) to use a straight LPG system for fuel and avoid the expense of remapping or replacing the ECM/PCM, upgrading injectors and fuel pump. You are still stuck with regards to ignition timing, although it will be cheaper to change just that than to map/remap fuel as well. If the engine is to be rebuilt anyway, then it may make sense to optimise it for LPG too but the gains won't be dramatic, about on par with a similar petrol engine.
Another way is to run two completely separate fuel system allowing the total volume of available fuel supply to be doubled (or more) at zero extra cost other than mapping. Instead of upgrading to a high volume fuel pump and injectors, the extra fuel can be provided by switching in the petrol at high loads. ZOOM magazine issue 101, pp72-75 has an article on one way this has been done. Other variations have been tried before, including by me.
It would be interesting to try a liquid to air intercooler on a boosted engine, using the intercooler coolant to vapourise the LPG (straight gas). Might be tricky to avoid freezing the convertor at low loads (boost) and so require some integration with the engine cooling system though. I believe that this has also been done before.