Refinement of pairwise stability, an alternative way of modeling network formation, this is known as pairwise nash stability

## Strategic Formation Models¶

• Saw conflict between stability and efficiency
• Can Transfers help?
• Modeling Stability and Dynamics, etc
• Refining pairwise stability
• Dynamic processes
• Forward looking behavior
• Directed Networks
• Fitting such models

## Modeling Stability¶

• Beyond Pairwise Stability - Allowing other deviations

• coordinated deviations
• Existence questions

• Dynamics

• Stochastic Stability

• Forward looking behavior

• Directed Networks

## Nash equilibrium¶

• Myerson's announcement game
• Players simutaneously announce their preferred set of neighbors $S_i$
• g(S) = {ij : j in $S_i$ and i in $S_j$}

• the network forms as a function of the profile, the full vector of all the announcements made by different individuals, are the links such that j was named by i and i was names by j, so this is consensual network formation, you form a relationship if and only if both people named each other.

## Nash Stability¶

• Nash stable, $u_i(g(S)) \ge u_i(g(S'_i, S_{-i}))$ for all i $S_i'$

• $u_i(g(S))$ is a situation where the utility that a given individual get from the network that forms under the announcements that are there is at least as good as anything that could get by changing their announcements($u_i(g(S'_i, S_{-i}))$), so they might want to announce for instance that they can't add some new links, announcements that they didn't make and do better and they can delete some of the announcements they did make and do better.
• So g is Nash stable if and only if no player wants to delete some set of his or her links

• So Nash stability basically looks at a given network and says "Does anybody want to take some subset of links that are there and delete them?". So the set of all Nash equilibria of pure startegy Nash equilibria could be of this game are going to be equivalent to the networks where no player wants to deviate from the links that they have and delete some of them. But it doesn't ask about adding mutually.

Why is this Nash stable? This is sort of a coordination failure. Nobody manages to name anybody else and nobody thinks anybody is going to name it anybody else. So everybody, each Si is equal to the empty set. Nobody names anybody and now if nobody named me I can't form a link anyway so I might as well name the empty set. This is a Nash equilibrium. These two players are getting one. They don't want to deviate and the third player doesn't make any sense, they're happy. This is a Nash equilibrium, everybody's getting one, there's no better payoff they could get. This one is not and why isn't it? Well this person must be announcing. So if you call this player one, player one must be announcing player two. They could deviate and not announce player two and they would be better off because their pay off would go from minus one to one. So this one is the only one that's not finished.