Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction (i.e. is less risky).
The payoff matrix in Figure 1 provides a simple two-player, two-strategy example of a game with two pure Nash equilibria.
Like the Prisoner's dilemma, it provides a reason why collective action might fail in the absence of credible commitments.
To calculate the risk factor in our 2x2 game, consider the expected payoff to a player if they play H:
Two separate evolutionary models both support the idea that the risk dominant equilibrium is more likely to occur.
The second model, based on best response strategy revision and mutation, predicts that the risk dominant state is the only stochastically stable equilibrium.
In replicator dynamics, the population game is repeated in sequential generations where subpopulations change based on the success of their chosen strategies.
In best response, players update their strategies to improve expected payoffs in the subsequent generations.
The recognition of Kandori, Mailath & Rob (1993) and Young (1993) was that if the rule to update one's strategy allows for mutation4, and the probability of mutation vanishes, i.e. asymptotically reaches zero over time, the likelihood that the risk dominant equilibrium is reached goes to one, even if it is payoff dominated.3 If players use however rules based on imitation to update their strategies, payoff dominant profiles may emerge as the error rates goes to zero (see e.g. Robson and Vega-Redondo 1996 or Alós-Ferrer and Weidenholzer 2008).