Set aside the moral1 and general economic2 cases for substantially more immigration. Let's look at it from a purely selfish point of view.
If you cherish or rely on programs like Social Security and Medicare, you NEED increases in immigration right away. Social Security benefits are set by law to be cut by 20-30% in about 7 years if nothing changes. While reform is needed in these programs, you can buy time and options if you have more workers. Immigrants are by far the easiest lever to pull here.
Very related to the first point, declining (collapsing) fertility is a foreboding future for us. We have few options here, but immigration is certainly one at least temporarily. This helps with the workforce directly and with economic growth generally by which each of us (again, thinking selfishly) benefit.
Almost everyone is a buyer of immigrant labor rather than a competing supplier. More immigrants will lower prices and increase quantities of goods and services benefiting us all. Think about it this way: Almost no one objects to native population growth within their respective city from a fear it will reduce economic growth and wellbeing. Often people worry about cultural changes in a there-goes-the-neighborhood refrain, but they admit even with the growing pains it will mean more wealth for the area. The same applies to people who happen to be born in another country.
We can take the last point further. In fact there is good reason to believe they will add more than would native immigrants. To the degree foreigners are more different than native immigrants as compared to the existing population, they bring more to the table. This allows for more specialization and variation. And do not forget that the Alchian-Allen Effect applies here—as with all goods and services, fixed costs of transportation or transaction imply that the higher the fixed cost (in this case the fact that foreign immigration into the U.S. is more expensive than relocation within the U.S.) the higher the quality will be of the good or service being transported or transacted. Bottom line: we get more specialization and higher quality additions to the workforce.
Picking up on the previous point, variety is the spice of life. Nowhere is this more true than in the case of food (see what I did there?). More immigrants not only means more supply of labor for food production. It also means more food variety and quality. Who better to bring [insert your "ethnic" food of choice] to your local market than people who have been cooking and eating that food their whole lives.
Suppose you are a young adult looking for a romantic partner. More immigrants means more options to choose from. Sure, the competition cuts both ways, but it is more benefit than cost similar to the buyer versus seller of labor point above. Imagine you are in a town that has what is for you 10,000 eligible partners (e.g., your a guy looking for a girl aged 18-27 and there are 10,000 such girls in the local area). Finding a match means discovering compatibility3 from this pool. Now if we add 1,000 immigrants who are all women in that age group along with 1,000 men who are potential competition, your available set has increased by 10% as has your competition. But your chances of success have increased on net and probably considerably. Here is roughly how: Imagine it was just five eligible partners (women in my example) and only five other people (men in my example) including yourself competing/seeking a compatible partner. There is a good chance that many of these people will not find a sufficient level of compatibility to match up. If we add a man and a woman to the group, they add potential match making without taking hardly any away.4 This process continues as we scale up the volume of people.
There are plenty of reasons to support much more immigration than is currently allowed. But for those who choose to look at it from a self-centered point of view, I have offered quite a bit here. So, what is there left to fret about?
PS, Bonus points for everyone who can identify how strongly this post leans on and draws from Bryan Caplan including the play on words in the title.
Compatibility is a bit of a shorthand for other limiting factors as well like making a connection in the first place. Even in small towns people have to have the opportunity of time and place matching up to have a chance at a relationship. Growing up at a big high school there were many girls who never got the chance to really meet me and me them.
Where they could be truly costly competition is if the only potential partner among the original five who finds you (a man seeking a woman) sufficiently compatible discovers they would rather be with the new male entrant while the new woman does not find you sufficiently compatible. The solution? Add more people and keep doing so until compatibility is reached.