Common use of Motivating Example Clause in Contracts

Motivating Example. We can write an optimizing contract for each vehicle in the example of Sec. II as follows: C1 = ({U2},{U1}, u2 ∈ {Wait, Pass}, (u1 ∈ {Wait, Pass}) ∧ ((u1, u2) = (▇▇▇▇, ▇▇▇▇)), R1) C2 = ({U1},{U2}, u1 ∈ {Wait, Pass}, (u2 ∈ {Wait, Pass}) ∧ ((u1, u2) = (▇▇▇▇, ▇▇▇▇)), R2) The cooperative and non-cooperative compositions can be computed as follows, where C1⊗ˇ C2 = (Cˇ1,Cˇ2): C1⊗ˆ C2 = (0/ ,{U1,U2}, T, ((u1, u2) ∈ {Wait, Pass}2)∧ ((u1, u2) = (▇▇▇▇, ▇▇▇▇)), R1 + R2) Cˇ1 = ({U2},{U1}, T, ((u1, u2) ∈ {Wait, Pass}2)∧ ((u1, u2) ƒ= (▇▇▇▇, ▇▇▇▇)), R1) path planning objective function Oi(xH ) increases as a vehicle 2 Cˇ = ({U },{U }, T, ((u , u ) ∈ {Wait, Pass}2)∧ approaches the destination and decreases if a vehicle travels off the designated course or collides to another vehicle. ((u1, u2) ƒ= (▇▇▇▇, ▇▇▇▇)), R2) The optimal guarantees with cooperation are G1⊗ˆ 2,max = argmaxu ,u (R1 + R2) s.t. ((u1, u2) ∈ {Wait, Pass}2) ∧

Motivating Example. We can write an optimizing contract for each vehicle in the example of Sec. II as follows: C1 = ({U2},{U1}, u2 ∈ {Wait, Pass}, (u1 ∈ {Wait, Pass}) ∧ ((u1, u2) = =ƒ (▇▇▇▇, ▇▇▇▇)), R1) C2 = ({U1},{U2}, u1 ∈ {Wait, Pass}, (u2 ∈ {Wait, Pass}) ∧ ((u1, u2) = =ƒ (▇▇▇▇, ▇▇▇▇)), R2) The cooperative and non-cooperative compositions can be computed as follows, where C1⊗ˇ C2 = (Cˇ1,Cˇ2): C1⊗ˆ C2 = (0/ ,{U1,U2}, T, ((u1, u2) ∈ {Wait, Pass}2)∧ Fig. 2: A closed system with N vehicles ((u1, u2) = (▇▇▇▇, ▇▇▇▇)), R1 + R2) Cˇ1 = ({U2},{U1}, T, ((u1, u2) ∈ {Wait, Pass}2)∧ path planning objective function Oi(xH ) increases as a vehicle ((u1, u2) ƒ= (▇▇▇▇, ▇▇▇▇)), R1) path planning objective function Oi(xH ) increases as a vehicle i 2 Cˇ = ({U },{U }, T, ((u , u ) ∈ {Wait, Pass}2)∧ approaches the destination and decreases if a vehicle travels off the designated course or collides to another vehicle. ((u1, u2) ƒ= (▇▇▇▇, ▇▇▇▇)), R2) The optimal guarantees with cooperation are G1⊗ˆ 2,max = argmaxu ,u (R1 + R2) s.t. ((u1, u2) ∈ {Wait, Pass}2) ∧