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Composability
Smartink's decentralized exchange follows the following rules, with 0.28% overall fees, when the initial reserves are
of token XTZ and
of tokens and the user sells
of XTZ to receive
, the following can be observed with
being invariant:
By symmetry, the following formula holds with a sale of
tokens to receive
:
Definition
Composability of such a decentralized exchange means that a trader gets the same price from participating all at once as in a set of small trades. Therefore, traders do not need to strategize how they make trades, which can be complex to achieve in a public blockchain context. They are therefore financially incentivized to quickly perform the largest possible trades, which has a positive impact on the gains of liquidity providers.
Proof
Let
and
such that, with
and
:
That means that with initial reserves of
of token XTZ and
of tokens a user may sell
of XTZ to receive
of tokens and then
of XTZ to receive
of tokens.
Or a user may “directly” sell
of XTZ to receive
of tokens.
Let
such that
Note that
is not necessarily equal to
.
Recall that
is the only real such that:
Hence, with initial reserves
of token XTZ and y of tokens a user may sell
of XTZ to receive
of tokens and then
of XTZ to receive
of tokens.
Or a user may "directly" sell
of XTZ to receive
of tokens.
As a consequence, a trader who sells
of XTZ receives the same amount
he would have received if he had either first sold
and then
or first
and then
. This is composability.
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