Here are the four factors that influence rankings for local searches in the local 3-pack, as confirmed by Google:
Search history: In the past how many times has the listing been clicked on by users searching with the keyword.
Relevance: Relevance is how well a local listing matches what someone is searching for. Adding complete and detailed business information can help us better understand your business and match you to relevant search results.
Distance: How far is each potential search result from the location term used in a search? If you don’t specify a location in your search, we’ll calculate distance based on what we know about your location.
Prominence: This describes how well-known or prominent a business can be. This is based on information we have about a business from across the web (like links, articles, and directories). Some places are more prominent in the offline world, and we try to reflect this online as well. For example, famous museums, landmark hotels, or well-known store brands that are familiar to many people are also likely to be more prominent in search results.
Make sure to take these factors into consideration when optimizing for local search results.