bsip38: update rounding

bsip-0038.md

@@ -110,13 +110,16 @@ The result is a rational number as well.
 As described in [BSIP 35](bsip-0035.md), at last we need to convert the
 rational numbers to integers, so rounding is involved.
 
+### The first round
+
 When calculating maximum debt to cover, the goal is to go over the specified
 target but not go too far beyond.
 That said, if a short position got matched with a big limit order, after
 partially filled, its collateral ratio should be **just** higher than specified
-target collateral ratio. Which means if `max_debt_to_cover` has no
+target collateral ratio.
-fractional component (e.g. 5.00 as opposed to 5.23), need to plus it by one
+
-Satoshi; otherwise, need to round it up.
+We may calculate like this: if `max_debt_to_cover` has no fractional component
+(e.g. 5.00 as opposed to 5.23), plus it by one Satoshi; otherwise, round it up.
 An effectively same approach is to round down then add one Satoshi onto the
 result:
 
@@ -124,19 +127,64 @@ result:
 max_debt_to_cover_int = round_down(max_debt_to_cover) + 1
 ```
 
-With `max_debt_to_cover_int` in integer, `max_amount_to_sell_int` in integer should be
+With `max_debt_to_cover_int` in integer, `max_amount_to_sell_int` in integer
-calculated as:
+can be calculated as:
 
 ```
 max_amount_to_sell_int = round_up(max_debt_to_cover_int / match_price)
 ```
 
-Then adjust `max_debt_to_cover_int` to be more accurate with:
+It's worth noting that we need to make sure the 2 integers always pair
+perfectly, they're either the full collateral amount and full debt
+amount, or have:
+
+```
+max_amount_to_sell_int == round_up(max_debt_to_cover_int / match_price)
+max_debt_to_cover_int  == round_down(max_amount_to_sell_int * match_price)
+```
+
+For `max_amount_to_sell_int` above, we can adjust `max_debt_to_cover_int` with:
 
 ```
 max_debt_to_cover_int = round_down(max_amount_to_sell_int * match_price)
 ```
 
+### Review the result
+
+Due to rounding, it's not guaranteed that selling more collateral will
+always result in higher collateral ratio on remaining call order.
+
+On one hand, it's not guaranteed that the pair of integers above will
+meet the requirements: after covered `max_debt_to_cover_int`, it's possible
+that collateral ratio of remaining order is still not higher than `MCR` or
+`target_CR`. In this case, the result is not acceptable. Generally, if we search
+upwards, we can find a new pair that meets the requirements, or hit the order's
+collateral or debt amount.
+
+On the other hand, no matter if the pair above meets the collateral ratio
+requirement, it's possible that there exists a smaller pair which meets the
+requirement. However, it's not easy to find a perfect pair. If we search
+downwards, it's not easy to decide when to stop; if we start from a smaller
+pair then search upwards, it's not easy to decide where to start.
+
+### Favor performance over accuracy
+
+Due to the difficulty mentioned above, in this BSIP we allow imperfect results,
+and don't describe or enforce how exactly to find better results.
+
+The first implementation should be made with efforts. It will become consensus
+after approved by stake holders via voting. It will then be in effect until
+changed or replaced with a new BSIP.
+
+Here are some guidelines about implementation.
+
+When searching upwards, usually the real working pair is not far away, but it's
+not guaranteed due to rounding. For better performance, it's not good to search
+by adding one Satoshi every time. Can use a divergence sequence or other
+sublinear-time algorithm, that means it's possible that some good data will be
+skipped which may result in impefect result.
+
+
 ## Rounding on Order Matching, and Edge Cases
 
 Rounding rules about order matching are defined in [BSIP 35](bsip-0035.md).