Lemma 3.2. At any population composition (), the instantaneous growth rate of the proportion of individuals taking high care is the same under CN and NCN. The instantaneous rate of decrease of the proportion of individuals taking low care is greater under CN than under NCN by . The instantaneous rate of decrease of the proportion of individuals taking medium care is greater under NCN than under CN by .
Proof: It is easy to check by using expressions (3.1)-(3.4), and by the definition of replicator dynamics (3.5).
Unfortunately, the above lemma is only a local property and it assumes that we are at the same population composition under CN and NCN. Once the system begins to evolve, the evolution of the system will follow different paths under the two liability rules and the local comparison becomes meaningless. Now we begin to discuss the global convergence property and the convergence path under the two liability rules.
We can look at the path of the dynamics of population composition by looking at the () plane (since ). From the same starting point at time , since the increase rate of is the same under both liability rules and the decrease rate of is slower under a comparative negligence rule, we have: