That is always a good place to start, and could be a good place to end for simplicity. Last time I rang my supplier they were up front they could not match a competitor's particular program but could come close. Since I never had a billing or customer service issue with them and their plans are straight forward and simple with no tricky useage maths, that comfort is worth the relatively small difference per annum (as computed by my spreadsheet based on my actual use) of moving on.
Companies working on "price matching", whether by guarantee or by customer retention policies do not impress me. "We" often do a lot of footwork to find the best deals so going back for a match is a disservice to those companies trying to get our business by sharp pricing, and the lazy company continues to profit.
Since changing suppliers is a phone call or internet form why not reward the company that seeks business? When the high priced company gets enough churn they may be back with a better cost profile next year or the year after, when it suits their business needs.
The Choice concierge service appears to take our emotions and inertia out of the equation and replace them with hard business. It might not be something everyone is comfortable with, but I applaud it and await reports in how it works in practice.