What is a word for when you pass the responsibility of something to someone or something else?

Delegate: verb (with object) entrust (a task or responsibility) to another person, typically one who is less senior than oneself.

• She must delegate duties so as to free herself for more important tasks.

• The power delegated to him must never be misused.

• I delegate my responsibility of self-defence to the state.

cede (sēd) TFD

To surrender possession of, especially by treaty. See Synonyms at relinquish. To yield; grant:

As in:

I cede my responsibility of self-defence to the state.

I entrust my responsibility of self-defence to the state.

entrust - verb - "If you entrust something important to someone or entrust them with it, you make them responsible for looking after it or dealing with it."

Example sentences from the web:

• I'll entrust the job to you.
• To your care I entrust the book, the embroidery frame, and the letter upon which I had begun.
• People entrust their money to others, who accept the responsibility to deal with it according to the terms agreed.
• He will entrust more responsibilities in your hands and elevate you to your proper status.

Devolve

transfer (power) to a lower level, especially from central government to local or regional administration.

OR

give an assignment to (a person) to a post, or assign a task to (a person)

Example sentences:

1. Measures to devolve power to a Scottish assembly
2. The representative devolved his duties to his aides while he was in the hospital
3. The U.S. government could devolve a certain responsibility to the states

I relinquish my responsibility of self-defense to the state.

3b : to give over possession or control of : yield - few leaders willingly relinquish power.

"Relinquish." Merriam-Webster.com. Merriam-Webster, n.d. Web. 19 Aug. 2018.

Delegate tends to imply that you have the authority to insist that the delegatee accepts the delegation. It's an offer they can't refuse.

Entrust focuses on the recipient and the thing recieved. This is slightly awkward here because it wouldn't then be self-defense, would it?

Relinquish focuses on the turning over or giving up. I think it works better in this particular case.

Forgo also has the correct sense, but you can't forgo something to some one.

Forsake suggests a totality of abandonment that probably isn't wanted here.

If you take the position that the state legislates the rules and is the one who decides where the responsibility lay, the entire sentence is a bit off. It is the state that causes you to forgo this responsibility. But this can be a matter of degree, and if you are electing to take the fullest advantage of their protection, I think you are choosing to relinquish what discretionary responsibility exists.

Delegate

E.g. She was delegated to do the job.