Unperturbed Dynamics

But then, the coradius of the basin of attraction of Ω is CR(Ω) = ^(1−p)c_p(b−c)A. The radius of Ω is larger than its coradius if

pb − c

p(b − c)A > (1 − p)c

p(b − c)A ⇔ p > 2c b + c which is a strictly more demanding condition than p >^c_b.

In Figure 11 we provide a graphical representation of the absorbing sets of the system listed in Theorem 4, while the proof of Theorem 4 is reported in Appendix E and it is conveniently divided into four lemmas.

First, we show that the intuitive defection set (L, 0, 0, 1, 0) is ab-sorbing in the entire parameter space. This abab-sorbing set contains all the states²in which all players adopt the intuitive defection strategy and select a location at random. More precisely, all players are indifferent between staying in their current location, moving to another inhabited location, and moving into an empty location as all these alternatives pro-vide them a null payoff. Despite receiving a null payoff every agent my-opically best replies to the current state of the system by keeping an intu-itive defection strategy as every alternative strategy provides a negative expected payoff. This is the case because every other strategy implies paying with positive probability the cost of cooperation – and eventu-ally a cost of deliberation – without the possibility of receiving a benefit of cooperation as in these states the rate of cooperation is null in both one-shot and anonymous and infinitely repeated prisoners’ dilemmas.

This first result is consistent with the findings in Bear and David G Rand (2016) and Jagau and Veelen (2017) who also find that the intuitive defection strategy can be sustained as an equilibrium in the entire pa-rameter space. In addition, according to our model, intuitive defection is associated with high players’ mobility. More precisely, every distribution of agents ranging from full concentration in one location to complete dis-persion over the various locations is feasible and this is the case because, in an intuitive defection environment, social interactions do not provide any benefit to agents.

Second, we show that dual process defection states (ℓ^∗, 0, 0, 1, k^d) with k^d = (k^d, . . . , k^d)and k^d = p(b − c)G(k^d)are absorbing states if p ∈ (︁0,_c+G(k^cd)(b−c))︁. In a dual process defection absorbing state, all agents play AllD under intuition, deliberate with positive probability, and if they do so then they play optimally. Moreover, all players stay in the same location. In such states, the rate of cooperation is null in

one-2The intuitive defection set is composed by A^Lstates where A^Lcorresponds to the number of ways in which A agents can be distributed over L locations.

Figure 11: Absorbing states according to Theorem 4.Graphical representa-tion of the absorbing sets of the system (case A = 4, L = 2). In the intuitive defection absorbing set, all agents adopt the intuitive defection strategy and are indifferent between staying in one location or another. In a dual process defection [cooperation] absorbing state all agents adopt a dual process fection [cooperation] strategy with the same optimal threshold cost of de-liberation and they all stay in the same location.

shot and anonymous interactions while it is equal to G(k^d)in the case of infinitely repeated prisoners’ dilemmas.

It is worth underlying that dual process defection absorbing states do not necessarily exist and if they do there may be multiple types of dual process defection absorbing states differing not only in the equilibrium location choice but also in the equilibrium threshold cost of deliberation.

This is the case because depending on the actual distribution of delibera-tion costs G(·) there may be none, one, or multiple k^d: k^d= p(b−c)G(k^d).

In general, there are as many dual process defection absorbing states as the number of locations times the number of threshold costs of deliber-ation such that k^d = p(b − c)G(k^d). In case there exist multiple types of dual process defection absorbing states the various types can be or-dered from the least to the most socially desirable by simply comparing the equilibrium threshold cost of deliberation: the higher k^dthe higher the rate of cooperation in infinitely repeated interactions and the higher agents’ expected payoff.

Third, we show that each dual process cooperation state (ℓ^∗, 1, 0, 1, k^c) with k^c = (k^c, . . . , k^c) and k^c = (1 − p)c is an absorbing state if p ∈ (^c_b, 1). In each of the L dual process cooperation absorbing states all agents play TFT under intuition, deliberate with positive probability, and if they do so then they play the dominant strategy of the game they are currently facing. Moreover, all players stay in the same location. In such states, the rate of cooperation is equal to 1−G(k^c)in one-shot and anony-mous interactions while it is equal to 1 in the case of infinitely repeated prisoners’ dilemmas. These are the highest cooperation rates that may characterize an absorbing set of the system. Consequently, dual process cooperation absorbing states are the ones in which agents achieve the highest expected payoff and, thus, are socially optimal.

Also, this result is consistent with the findings in Bear and David G Rand (2016) and Jagau and Veelen (2017): both the parameter space in which such states are absorbing and the condition on the optimal thresh-old cost of deliberation coincide. Moreover, our finding that in dual pro-cess cooperation absorbing states all agents stay in the same location is a natural extension of the results in Bear and David G Rand (2016) and

Jagau and Veelen (2017) given our location choice setting.

Finally, we formally prove that the system under unperturbed dy-namics does not have other absorbing sets than the intuitive defection set, (eventually) the set of dual process defection states, and the set of dual process cooperation states. We do this by showing that starting from a generic state the system will always reach – with a positive probability – one of the aforementioned absorbing sets. But then, we can conclude that there are no absorbing sets other than the ones listed in Theorem 4.

The results in Theorem 4 give rise to a problem of multiplicity of equilibria: for any possible probability that the prisoners’ dilemma is infinitely repeated, p, at least two kinds of equilibria exist and, so, we do not have a clear prediction regarding the behavior of the system despite it being of particular interest given that the different kinds of absorbing sets can always be ranked in terms of social desirability.