Secret Santa List

I have a company with four employees who participate in a Secret Santa
program, where each buys a gift for an employee chosen at random. (For
now, I do not mind if an employee ends up buying a gift for himself.)
How can I make this work with an SQL statement?

Here is my Secret Santa table:

--
create table secretsanta
(giver text,
recipient text,
primary key (giver));

insert into secretsanta (giver) values ('Frank'), ('Joe'), ('Steve'),
('Earl');
--

Here is the SQL statement I am using to populate the "recipient" column:

--
update secretsanta set recipient =
( select giver from secretsanta s2 where not exists (select * from
secretsanta s3 where s3.recipient = s2.giver) order by random() limit 1 );
--

The problem: every time I run this, a single name is chosen at random
and used to populate all the rows. So all four rows will get a
recipient of "Steve" or "Earl" or whatever single name is chosen at random.

I suppose the problem is that the "exists" subquery does not re-evaluate
for each record. How do I prevent this from happening? Can I use a
"lateral" join of some kind, or somehow tell PostgreSQL to not be so
optimized?