[pallet_broker] Fix adapt_price behaviour at zero (#3636)

This fixes the behaviour of `Linear` which is the default implementation
of the `AdaptPrice` trait in the broker pallet. Previously if cores were
offered but not sold in only one sale, the price would be set to zero
and due to the logic being purely multiplicative, the price would stay
at 0 indefinitely.

This could be further paired with a configurable minimum in the broker
pallet itself, which will be a future PR.

This affects the Rococo and Westend Coretime chains, but Kusama has a
different implementation so this isn't required for the Kusama launch. I
actually thought I opened this a while ago.

---------

Co-authored-by: Bastian Köcher <git@kchr.de>

prdoc/pr_3636.prdoc

@@ -0,0 +1,15 @@
+# Schema: Polkadot SDK PRDoc Schema (prdoc) v1.0.0
+# See doc at https://raw.githubusercontent.com/paritytech/polkadot-sdk/master/prdoc/schema_user.json
+
+title: "[pallet_broker] Fix `Linear::adapt_price` behavior at zero"
+
+doc:
+  - audience: Runtime Dev
+    description: |
+      This fixes the behaviour of `Linear` which is the default implementation of the `AdaptPrice`
+      trait in the broker pallet. Previously if cores were offered but not sold in only one sale,
+      the price would be set to zero and due to the logic being purely multiplicative, the price
+      would stay at 0 indefinitely.
+
+crates:
+  - name: pallet-broker

substrate/frame/broker/src/adapt_price.rs

@@ -19,6 +19,7 @@
 
 use crate::CoreIndex;
 use sp_arithmetic::{traits::One, FixedU64};
+use sp_runtime::Saturating;
 
 /// Type for determining how to set price.
 pub trait AdaptPrice {
@@ -49,14 +50,24 @@ impl AdaptPrice for () {
 pub struct Linear;
 impl AdaptPrice for Linear {
 	fn leadin_factor_at(when: FixedU64) -> FixedU64 {
-		FixedU64::from(2) - when
+		FixedU64::from(2).saturating_sub(when)
 	}
 	fn adapt_price(sold: CoreIndex, target: CoreIndex, limit: CoreIndex) -> FixedU64 {
 		if sold <= target {
-			FixedU64::from_rational(sold.into(), target.into())
+			// Range of [0.5, 1.0].
+			FixedU64::from_rational(1, 2).saturating_add(FixedU64::from_rational(
+				sold.into(),
+				target.saturating_mul(2).into(),
+			))
 		} else {
-			FixedU64::one() +
-				FixedU64::from_rational((sold - target).into(), (limit - target).into())
+			// Range of (1.0, 2].
+
+			// Unchecked math: In this branch we know that sold > target. The limit must be >= sold
+			// by construction, and thus target must be < limit.
+			FixedU64::one().saturating_add(FixedU64::from_rational(
+				(sold - target).into(),
+				(limit - target).into(),
+			))
 		}
 	}
 }
@@ -81,4 +92,23 @@ mod tests {
 			}
 		}
 	}
+
+	#[test]
+	fn linear_bound_check() {
+		// Using constraints from pallet implementation i.e. `limit >= sold`.
+		// Check extremes
+		let limit = 10;
+		let target = 5;
+
+		// Maximally sold: `sold == limit`
+		assert_eq!(Linear::adapt_price(limit, target, limit), FixedU64::from_float(2.0));
+		// Ideally sold: `sold == target`
+		assert_eq!(Linear::adapt_price(target, target, limit), FixedU64::one());
+		// Minimally sold: `sold == 0`
+		assert_eq!(Linear::adapt_price(0, target, limit), FixedU64::from_float(0.5));
+		// Optimistic target: `target == limit`
+		assert_eq!(Linear::adapt_price(limit, limit, limit), FixedU64::one());
+		// Pessimistic target: `target == 0`
+		assert_eq!(Linear::adapt_price(limit, 0, limit), FixedU64::from_float(2.0));
+	}
 }