alt Hacker NewsDisclosure: I have never cross-shopped for any of these services. I use Google Flights, Google Maps for hotel search, and book on the hotel website.
What percentage of the total value is typically saved by those who check different devices, browsers, etc?
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One datapoint me: I never see any difference in flight prices, but I haven't tried VPN-ing into the site from another country. For hotels, prices I find through aggregators can be as much as 50% lower than the hotel's own website price, even when their website claims it's showing me the lowest price.
Anywhere from 45% to 5% IMO. So you can almost always save $5, but sometimes you can save like $50+. That said, another big piece I didn't mention is that often you also are watching the rate day by day because timing is a big piece too.
The most I've seen is about 25%, around £500. The difference was due to deal only offered on a one way ticket when bought through the site of a code-sharing airline.
Then you my friend have been overpaying for flights and hotels. Next time you want a holiday, try the classic techniques - Incognito windows, multiple sites, and then compare it with your Google Flights searching
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For changing browsers, IPs, user agents, etc., it probably rounds to 0%. There are some edge cases booking flights directly from national carriers from within/outside the country. For using third party OTAs for hotels, I'd guess maybe 5% or so on average depending on your usage, but the financial tradeoff isn't directly comparable. You can probably save a bit by booking third party, especially if you haven't bothered signing up for hotel loyalty programs [1]. There are...let's say "discontinuous" long tail issues with booking hotels third party, where probably 95-99% of the time you'll have no problems, but when there is one, you'll have a much better chance of having a very bad time. The price gap on flights is probably considerably smaller (commissions on flights are much lower than commissions on hotels), but if you have any sort of problem (e.g. a flight being delayed when you have a layover), you have a much better chance of getting screwed.
Personally I mostly use Google Flights for flight search (or seats.aero for award travel search) and booking.com for hotel search, and then book directly. I've booked probably 100+ hotels through booking.com and then quit entirely after hitting that long tail bad experience.
[1]: I'm a few years out of date on this and don't know the current legalities, but major OTAs (Booking.com and Expedia, who control probably 80-90% of third party hotel bookings) used to have a "most favored nation" clause that prevented hotels from advertising lower prices through other channels without a previous commercial relationship. Signing up for a loyalty program qualifies, so being logged in for direct hotel bookings can sometimes get you cheaper room rates. (In the US, things like AARP and AAA memberships also work.) I know the EU was pushing back on these clauses but I'm not clear on what the current state of things is.