"Multi-mining" covers two different things that get used interchangeably. One is switching a machine between chains as conditions change. The other is mining two chains with the same work at once. They're worth separating, because only one of them is free.
The revenue side
For proof-of-work, your expected share of what a network pays out is just your share of the work being done:
coins/day = (h / H) × (86400 / T) × B
Where h is your hashrate, H is total network hashrate, T is the target
block time in seconds, and B is the reward per block. Multiply by price P
for revenue:
revenue/day = (h / H) × (86400 / T) × B × P
Your cost, meanwhile, barely moves:
cost/day = kW × 24 × rate
So margin is revenue/day − cost/day. Note which terms you control. You set
h and kW by what hardware you own and how it's tuned. Everything else,
H, B, P, is set by the rest of the world.
Why the target moves
h/H is the whole story. Your hashrate can be perfectly constant while your
revenue falls, because H is the sum of everyone's. When others add machines,
your slice shrinks proportionally. Difficulty adjustment keeps T roughly
fixed by making the work harder as H climbs, so the network's block schedule
stays put and your individual share absorbs the change.
Meanwhile P moves on its own schedule and B steps down on a fixed halving
schedule. Three independent inputs, one output.
Since those three move differently on different chains, the chain with the best
revenue − cost right now is frequently not the one that was best last week.
Switching is just tracking that maximum. It isn't clever; it's arithmetic run
repeatedly.
It also isn't free. Switching costs downtime, transaction fees, and sometimes a minimum payout threshold you have to re-reach on the new chain. A switch only pays if the margin gap is bigger than the cost of moving, which means the right switching frequency is lower than a naive reading of the numbers suggests.
The case where the second chain is free
Merged mining is the more interesting one.
Hashing is a search for a value below a target. If a second chain agrees to accept the first chain's proof of work, then a single hash attempt can be checked against both chains' targets at once. Finding a solution good enough for the parent chain may also satisfy the auxiliary chain: the same work, submitted twice.
The energy accounting is the part worth sitting with:
energy for chain A alone = E
energy for chain A + chain B = E
Not 2E. The marginal energy cost of securing the second chain is
approximately zero, because no additional hashing happened. You did the work
once and it counted twice.
That is a genuinely unusual property. Most ways of getting more output require more input. This one doesn't, which is why the distinction between "switching between chains" and "merged mining" matters more than the shared name suggests.
The short version
Switching optimizes which fixed cost you're paying against, and it's bounded by how much the margin gap exceeds the friction of moving. Merged mining changes the denominator instead: same energy, more chains secured.
Neither makes a machine more efficient. They change what the efficiency is spent
on, which is a different lever than the one in the undervolting
post: that one lowers kW.
These lower the cost of the work by getting more out of it.