In reality it is rare that Nash equilibrium is achieved instantaneously. Nash equilibrium requires that players are rational and know the payoff functions of all players, that they know their opponents are rational and know the payoff functions, that they know their opponents know, etc. In actual life, these requirements may not be met14.
This poses a problem: will the Nash equilibrium always be closely approximated at least in the long run? If this is the case, does the outcome converge to the Nash equilibrium rapidly and what is the path of the convergence?
To answer this question, we use an evolutionary approach (MAYNARD SMITH ). The idea of evolutionary games began with the idea that animals are genetically programmed to play different pure strategies, and that the genes whose strategies are more successful will have higher reproductive fitness. The population fractions of strategies whose payoff against the current distribution of opponents' play is relative high will tend to grow at a faster rate, and any steady state must be Nash equilibrium. There is no need for the strong requirement of rationality and common knowledge among players.
Evolution can be taken as a metaphor for learning in economics. Individuals respond to different payoffs by modifying their strategies. If we assume inertia in human behavior and costs associated with switching strategies, then the proportion of the population choosing each strategy changes smoothly. In the following, the proportion of drivers taking care is subject to evolutionary pressure over time. The fraction of the population using better performing strategies will increase relative to those using lower payoff strategies. Our main focus will not be on the steady state of evolution, but on the relative speed and the path of convergence to the steady state under different liability rules.
We denote and the proportion of drivers taking high, medium and low level of care at time , . Given the population composition (,), a driver will meet drivers taking high care with probability and will meet drivers taking medium and low care with probability . Under CN (and also NCN, which corresponds to ), at any time , the expected payoff for a driver who takes high care is: