Toward the end of that slide deck, in the "Slowing Brute Force" section, there the following slide:

> Argon2 ensures that both defenders and attackers hash passwords on the same platform (x86)

Do they simply mean that currently the most cost effective platform to deliver the necessary computation and memory is x86?

I'm having trouble believing that it is either achievable or desirable to design a password hash that's only efficient on one architecture.

  I'm having trouble believing that it is [...] desirable

To make it less useful to build a NSA-style ASIC hash cracking cluster, like https://en.wikipedia.org/wiki/EFF_DES_cracker

  I'm having trouble believing that it is [...] achievable

The point is not to write an algorithm that is only fast on x86, rather than ARM, POWER, whatever; it's to write an algorithm that uses so much memory that it's not orders-of-magnitude faster to use an ASIC that's just a bunch of compute cores connected to fast buses, rather than a general-purpose processor with megabytes of on-chip cache.

The entire point of "memory-hardness" and "anti-parallelisation" and other such properties is to make it such that the CPU is the most desireable platform to for the password stretching function to run.

Why is this desireable?

This is desireable because your server has a CPU. Your attacker has a CPU, a GPU, perhaps an FPGA, and could possibly manufacture a custom ASIC. You do not want the attacker to be able to cost-effectively bruteforce on any of these platforms any more efficiently than a single CPU core can.

This makes password cracking difficult, and it also means we do not need to have absurd CPU runtimes for our hash, a handful of milliseconds is sufficient. (More is of course better, and for a single user system a delay of half a second isn't noticeable.)

What would occur if we did not design password stretching functions that had this property?

We would end up in the situation we have now, where nonstandard hardware computes hashes faster than we can, and thus instead of taking (a few miliseconds * number of possibilities) it takes (smaller amount of time * number of possibilities). This is undesireable.

This is also why you should never use a typical fast cryptographic hash function like SHA256, SHA3 or RIPEMD160 to hash passwords. This is also why PBKDF2 is considered less secure than say scrypt.