Fig. 4: Dynamics in the Stag Hunt game.

Left panel: in the classical two-player stag hunt game, two individuals decide to hunt for a stag or a hare. If both choose to hunt a stag together, they successfully hunt one. Hunting a stag brings a larger payoff than hunting a hare, but hare hunting does not require any coordination of actions and can be achieved alone. However, the players do not know their partner’s action preferences. In an infinite population, change in the fraction of stag hunters (\(\dot{x}\)) depends on the current value of x. If there are enough stag hunters in a population, hunting a stag is profitable. However, stag-hunting is not favoured if the fraction of stag hunters is low x < x^*. The value of x at which the direction of the change shifts (the unstable internal equilibrium x^*) is represented with an open dot. How do we convert a population of hare hunters to stag hunters? Introducing collective narratives provides a solution. Central panel: In the stag hunt with narratives, the strategy of each individual consists of three elements. The centremost layer (\({a}_{1}^{* }\)) corresponds to the action taken by an individual when they find themselves in a group believing in narrative 1, the options being stag or hare. The middle layer (\({a}_{2}^{* }\)) shows the two possible actions that can be taken in the group believing in narrative 2. The outermost layer (u^*) depicts an individual’s two possible beliefs in narrative 1 or 2. A strategy is then represented as \(({a}_{1}^{* },{a}_{2}^{* },u* )\). Thus, in all, there can be eight strategies. Right panel: like the two-strategy outcome (left panel), the right panel shows the result of the eight-strategy case in an infinitely large, well-mixed population. White points represent the unstable equilibria. If two strategies have the same payoff when played against each other, the change in composition may happen by neutral drift—indicated by the grey dashed lines. Here, we see two paths from a hare hunting population ((H,H,1) or (H,H,2)) to stag-hunting. If an initial population consists of only (H,H,1) individuals, it may be taken over by (H,S,1) individuals by chance. From there, the dynamics would lead to a takeover of (H,S,2) individuals, and the population would end up drifting neutrally between the stag-hunting strategies without the possibility of reverting to hare hunting. A similar transition may also occur if first (H,H,2) and then (S,H,2) individuals takeover the initial population via drift. The plots are generated for N = 5, M = 4, P_H = 1, P_S = 4 with N = 2, M = 2 for the two-player case. The central and right panels have been adapted from Gokhale et al.²⁹.