The direct-sum question is a classical question that asks whether
performing a task on $m$ independent inputs is $m$ times harder
than performing it on a single input. In order to study this question,
Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose
one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output
a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing
the original task on a single instance, and it is natural to ask whether
it is strictly easier or not. In particular, proving that the choice
problem is not easier is necessary for proving a direct-sum theorem,
and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,
if $f$ is a random function then with high probability its corresponding
choice and agreement problem are not asymptotically easier than computing $f$
on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).

Added a proof of Corollary 5.

The direct-sum question is a classical question that asks whether
performing a task on $m$ independent inputs is $m$ times harder
than performing it on a single input. In order to study this question,
Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose
one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output
a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing
the original task on a single instance, and it is natural to ask whether
it is strictly easier or not. In particular, proving that the choice
problem is not easier is necessary for proving a direct-sum theorem,
and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,
if $f$ is a random function then with high probability its corresponding
choice and agreement problem are not much easier than computing $f$
on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).

Corrected an error in the statement of Theorem 4, and changed Corollary 5 to account for the change.

The direct-sum question is a classical question that asks whether
performing a task on $m$ independent inputs is $m$ times harder
than performing it on a single input. In order to study this question,
Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose
one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output
a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing
the original task on a single instance, and it is natural to ask whether
it is strictly easier or not. In particular, proving that the choice
problem is not easier is necessary for proving a direct-sum theorem,
and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,
if $f$ is a random function then with high probability its corresponding
choice and agreement problem are not asymptotically easier than computing $f$
on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).

Added references to [JKS10] and to [CDKLM17].

The direct-sum question is a classical question that asks whether
performing a task on $m$ independent inputs is $m$ times harder
than performing it on a single input. In order to study this question,
Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice problem: Given $m$ distinct instances, choose
one of them and solve it.

* The agreement problem: Given $m$ distinct instances, output
a solution that is correct for at least one of them.

It is easy to see that these problems are no harder than performing
the original task on a single instance, and it is natural to ask whether
it is strictly easier or not. In particular, proving that the choice
problem is not easier is necessary for proving a direct-sum theorem,
and is also related to the KRW composition conjecture (Computational Complexity 5, 3/4, 1995).

In this note, we observe that in a variety of computational models,
if $f$ is a random function then with high probability its corresponding
choice and agreement problem are not asymptotically easier than computing $f$
on a single instance (as long as $m$ is noticeably smaller than $2^{n}$).