Proof of Lemma 3.2. We show that all the sets listed in lemma 3.2 are absorbing, i.e. once they are there, the dynamics will remain there forever, and all the other states are transient, i.e. there is a positive probability that the dynamics will never move back to them. Suppose, in periodt, the dynamics reaches an arbitrary stateω. Let (k∗, s∗) be a myopic best reply to ω for the individuals who can relocate. Hence, a myopic best reply for the individuals who are currently located in k∗ and cannot relocate is s∗. Denote s′ a myopic best reply to ω for those who are not in location k∗ and cannot relocate. Then, with a positive probability, the individuals in locationℓ∗6=k∗who can relocate will move to locationk∗and plays∗, while those residing in location ℓ∗ who cannot relocate will play s′. The players in locationk∗ will remain there and play s∗. Hence, in t+ 1, individuals in location k∗ will coordinate on s∗, while those in location ℓ∗ will coordinate ons′.
Case I.1: ⌊dkN⌋ < 2N −1 for all k ∈ {1,2}. If s∗ = s′, in t+ 2, all the players will play the same strategy and randomly choose their locations if such an opportunity arises. This corresponds to the set Ω(RR) or Ω(P P).
They are absorbing, because (R, R) and (P, P) are Nash equilibria. Once the dynamics reaches any one of the states in the set Ω(RR)(Ω(P P)), my-opic best reply will always lead players to playR (P), and randomize their location choices given the capacity and mobility constraints.
Ifs∗ 6=s′, players in one locationk∈ {1,2}must coordinate onP. Then, all the players in the other location who have an opportunity to relocate will move to location k and play P until the population in location k reaches
⌊dkN⌋. This corresponds to Ω(P R) or Ω(RP). Once there, myopic best reply will lead all theP-players to stay in the current location and playP. The R-players would have an incentive to move to the other location and play P, but are not allowed because of the constraints. Hence, they will playR in their current location.
Case I.2: ⌊dkN⌋ = 2N −1 and ⌊dℓN⌋ < 2N −1, k, ℓ ∈ {1,2}, k 6= ℓ.
Consider the case where k = 1. If the individuals in location 1 coordinate on R and those in location 2 coordinate on P, the dynamics will move to Ω(RP) and remain there forever as explained in Case 1.
If s∗ =s′ =R, the dynamics will move to the set Ω(RR). Then, with a positive probability, the dynamics will move to a state in Ω(RR) where there is only one individual in location 2, who will playR. In the next period, this player will randomize his strategy in the current location. With a positive probability he will playP, and then the dynamics will move to Ω(RP) and remain there forever.
If s∗ =s′ = P, the dynamics will first move to the set Ω(P P). In this set, once the state where only one individual is in location 2 is reached, the player in location 2 will randomize his strategy. Hence, the dynamics must either remain in Ω(P P) or move to Ω(P R). If the dynamics moves to state Ω(P R), there are 2N−1 players in location 1 and one player in location 2.
Hence, the payoff of the player in location 2 is zero regardless of his choice, and he will randomize his strategy. Hence, the dynamics must either remain at Ω(P R) or move to Ω(P P). Therefore, the set Ω(P R, P P) consisting of Ω(P R) and Ω(P P) is absorbing.
Similarly, if the individuals in location 1 coordinate on P and those in location 2 coordinate on R, the dynamics will move to Ω(P R, P P) and remain there forever.
The argument is symmetric for the case withk= 2.
Case I.3: ⌊dkN⌋ = 2N −1 for k = 1,2. If s∗ = s′ = R, the dynamics will move to Ω(RR). Then, with a positive probability, the dynamics will move to a state where there is only one individual playingRin one location.
After that, thisR-player will randomize his strategy. Hence, with a positive probability, the dynamics will move to Ω(RP) or Ω(P R), and never come back to Ω(RR). In the singleton set Ω(RP) or Ω(P R), the R-player is the single individual in his location. Therefore, he will randomize his strategy.
As a result, the dynamics must either remain at the current state or move to Ω(P P). If the dynamics moves to Ω(P P), with a positive probability, it can reach a state where there is a single player in a location playing P. Then, this player will randomize his strategy, hence leading the dynamics either to remain, or to move to Ω(P R) or Ω(RP). Hence, the unique absorbing set is Ω(P R, P P, RP).
Case I.4: dk= 2and⌊dℓN⌋<2N−1. Ifs∗=s′, with a positive probability, all the players will move to locationkand plays∗. Once there, the dynamics will remain there forever. This corresponds to Ω(P O) or Ω(RO) for d1 = 2, or Ω(OP) or Ω(OR) for d2 = 2. Ifs∗ 6=s′, individuals will coordinate onP in one location, andR in the other location. If the individuals in locationk coordinate on P, while those in location ℓ coordinate on R, all the players in location ℓ will move to k and play P. If the individuals in location k coordinate on R, while those in location ℓ coordinate on P, the dynamics will remain there forever. Therefore, the absorbing sets in this case are Ω(P O), Ω(RO) and Ω(RP).
Case I.5: dk= 2 and⌊dℓN⌋= 2N−1. If s∗ =s′, players in both locations coordinate on the same equilibrium. Then, with a positive probability, all the players will reside in locationkand coordinate on the same equilibrium.
If the individuals in locationk play P, while those in location ℓplay R, all the players in location ℓ will move to location k and play P, and then the dynamics will stay there forever. If the individuals in locationkplayR, while those in location ℓ play P, note that the R-player is the single individual locationk, hence, in the next period, he will randomize his strategy. With a positive probability, he will playP, and then the dynamics will come to the state where all the individuals reside in location k and play P, and remain there forever. Hence, the absorbing sets are Ω(P O) and Ω(RO) for d1 = 2, or Ω(OP) and Ω(OR) for d2 = 2.
Case I.6: d1 =d2 = 2. Similarly to Case I.5, if s∗ =s′, all the players will move to one location and coordinate on s∗. If s∗ 6= s′, all the players will move to the location withP-players and play P. Once there, the dynamics will stay there forever. Hence, the absorbing sets in this case are Ω(RO),
Ω(P O), Ω(OR) and Ω(OP).
Proof of Lemma 3.1. If both locations have identical capacity and mo-bility constraints, we have d1 = d2 = d. Clearly, all the corresponding arguments and results in the proof of Lemma 3.2 hold for this particular case. As a result, we obtain the statements in Lemma 3.1.