Disclaimer: none of the views expressed in this post should be taken too seriously
The answer is pretty simple if you live in most of the United States: yes, you should use an umbrella. However, if you are living in Washington, you need to factor in the weird weather patterns and the social ridicule. These are both important elements that can affect your choice in attire/accessories to deal with the weather.
In Washington we have about three types of rain: a little stronger than a drizzle, a decent rain, and “its-below-freezing-oh-dear-god-why-can’t-it-just-snow-this-is-so-much-worse.” It is important to note that in Washington you can rarely expect a decent rain, and instead expect the never ending drizzle that is just strong enough to get you completely soaked. In the standard Washington drizzle your umbrella will work, but it will be far more inconvenient than a raincoat and therefore not worth it. If you try to use an umbrella when it is vertically hailing, you find that it has little to no effect in stopping sleet and therefore not worth it. Finally, on the rare occasion that it does rain decently, an umbrella could be worth it. I write “could,” as you must take into account the ostracization you will face from using an umbrella.
Now you might say: “But wait! I am a professor whose will to live has slowly been drained away due to last minute papers and incomprehensible test answers, and I don’t care about my dignity.” This is a very valid calculation that one must take into account when analyzing the social costs of using an umbrella. If you are a professor whose years of grading papers that students took less time to write than the time you spent writing prompts: yes, an umbrella is worth it (accounting for weather conditions). If you are a first-year college student, who has yet to make any real friends: you want to make real friends eventually and it is therefore not worth it. If you are a second-year college student and you have yet to choose a major: maybe the public shaming will distract you from the crippling anxiety you feel over deciding whether you can really triple major, or just slide by with 5 minors and it is therefore worth it. If you are a third-year college student: you may as well try to keep yourself dry as you contemplate whether not getting above a C+ average and being a philosophy major was possibly a poor life choice and therefore it is worth it. If you are a fourth/fifth-year student: you should know better and it is therefore not worth it. In conclusion, there is a set of circumstances where an umbrella may be the best course of action. However, none of these outcomes are optimal and there is no perfect nash equilibrium except for tourists.