Lemma 23 Dynamic equilibria are also Nash equilibria.
Proof. We will show that Separated Dynamic equilibria are also Separated Nash equilibria. The proof for the connected case is very similar and is omitted.
We must have
ψDE(bi)≥ψ(bi) because
ψDE(bi)−ψ(bi) =π(n−bi+ 1, Lmin,Θ, Ii)−πs(n−bi+ 1) +πe(bi)−π(bi, Lmin,Θ, I) or
ψDE(bi)−ψ(bi) =π(bi, Lmin,Θ, Ii)−π(bi, Ls,Θ, Ii) +πe(bi)−π(bi, Lmin,Θ, I) or
ψDE(bi)−ψ(bi) =πe(bi)−π(bi, Ls,Θ, Ii)>0 Similarly, we must have
ψDE(bi)≤ψ(bi)
Lemma 24 Any subgroup of an identity block can deviate to switching identity and linking to the other identity block.
Proof. Consider a network where b1 players have chosen identity characteristicc1 andb2 players have chosen identity characteristicc2. Let us consider a deviation by a subgroupb′1ofb1to identity characteristic c2.
Suppose the block b1 is arranged in such a way that playeri∈B1 is the center of a star withb′1 spikes and another link to the rest of the players. Ifiis the chosen player in any period, then he can retain links withb′1 players on the condition that they switch identity toc2, sever his link with the rest of his block and offer a new link to the other identity block.
If b1 does not already have that form, then we know with positive probability it will attain that form.
Consider any other playerj ∈B1 who wlog forms a single link withj′. In any period ifj is chosen to act, one of his best responses will be to sever link withj′ and offer to link with i. Sincej forms a single link, linking to any player in his identity block is a best response. By A3, this link offer will be acceptable by i. And continuing thus, iwould in some periods end up as the center of a star withb′1 spokes which would then have to choice to deviate to a link with the other identity block.
Lemma 25 The dynamic game will not converge to block sizes b1 ≤b2, if there exists some n3 > b2 such that φ(n3)≥φ(b1)
Proof. If it were so, then n3−b2 players from B1 would migrate to the bigger block to make higher network profitsand higher profits from identity.
Lemma 26 If
π(B1, L1,Θ, I)−π(B1, L2,Θ, I) =π(B2, L1,Θ, I)−π(B2, L2,Θ, I)
then, the dynamic game will converge to separated blocks ofb1≤b2 if for any b′1≥b1 and/or any b′2≥b2, it were true that
π(b′1,0,Θ, Ib′1,n−b′1) +φ(b′1)≤π(b2,0,Θ, I) +φ(b2) π(b′2,0,Θ, In−b′2,b′2) +φ(b′2)≤π(b1,0,Θ, I) +φ(b1)
(Ix,y denotes the identity profile wherexplayers have identityc1 andy players have identityc2)
Proof. From the previous lemmas we know that a move to any partition that involves players migrating from one of the current blocks to the other is not possible. The assumption ensure this to be true for all strategy profiles, in particular for the profile at which no costs are bourne.
Lemma 27 A necessary condition for the dynamic game to converge to connected identity blocks of sizes b1≤b2 if for anyb′1≥b1 and/or anyb′2≥b2, it were true that
φ(b′1)≤φ(b2) φ(b′2)≤φ(b1)
(Ix,y denotes the identity profile wherexplayers have identityc1 andy players have identityc2)
Proof. These conditions must hold because any player/block choosing to deviate to the other identity block will make the same network profits, they will be making the decision based only on the identity benefits.
Lemma 28 If a dynamic equilibrium exists, it is unique.
Proof. Suppose a Separated Dynamic equilibrium exists atb1, b2 as well atn1, n2. We know that either b2< n1< n2< b2 orn1< b1< b2< n2. Suppose the former case is true , then we show thatn1, n2 cannot be a Separated Dynamic equilibrium. This is so because forb1, b2 to be a Separated Dynamic equilibrium, it must be that the profits atn1 are less than profits atb2. In other words, some players from n1will wish to join the other identity block to form a block of size b2. Similarly, in the latter caseb1, b2 cannot be an equilibrium.
Suppose a Connected Dynamic equilibrium exists, then players will keep moving between identities till no improvements can be made on identity benefits.
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