### Topics

- Games with homogeneous externalities
- Games with local network effects

### Recall

Congestion games had simple sets of payoffs but complicated network structure. There are games that have complicated payoffs with simple network structure.

### Homogeneous Externalities

In classical economics, the demand curve is assumed decreasing and the supply curve increasing. The existence of network effects may create a weird shape of the demand curve.

*N*= [0, 1] continuum of players*S*= {buy, not buy}_{i}*u*(_{i}*S*,_{i}*S*) =_{-i}*u*(_{i}*S*,_{i}*x*) =

*v*—_{i}x*p*if*S*= buy_{i}

0 if*S*= not buy_{i}*v*~ F([0, 1])_{i}*p*> 0- Where
*x*is how many buy,*v*is its value, and_{i}*p*is price.

*Example: Office suites, SNS, etc.*

*x*= 1 — F(*v̅*) where*v̅*is the lowest value of agents who buy.

Claim: If agent with *v _{i}* =

*v̅*is better off buying, then any agent with

*v*>

_{j}*v̅*is also better off buying.

Corollary: We may assume without loss of generality that *x* (those who buy) have the highest values in equilibrium/socially optimal outcome.

- Socially optimal outcome
- Social welfare = *
_{v̅}*∫^{1}[*v*(1 — F(*v̅*)) —*p*] dF(*v*) - Maximizing this with respect to
*v̅*yields the social best.

- Social welfare = *
- Nash equilibrum
- Pooling equilibrum:

If no one has the good (*x*= 0) then no one has incentive to buy (*v** 0 —_{i}*p*< 0). Therefore,*x*= 0 is an equilibrium.^{*} - Separating equilibrium:

If someone buys (*x*> 0), then there exists the lowest type*v̅*who buys. His incentive must balance*v̅x*—*p*= 0.

- Pooling equilibrum:

Note that everyone’s strategy is summarized by *x*. So consider the aggregate best response function BR(*x*) = *x̂*. With *v̅x* — *p* = 0 and *x* = 1 — F(*v̅*), we find:

- BR(
*x*) = 1 — F(*p*/*x*). - Its fixed point
*x*^{*}is an equilibrium.

### Local Network Effects

Some games have both complicated payoff structure and complicated network structure.

*N*= {1, 2, 3}.*S*_{i}= ℝ_{+}= [0, ∞).*u*_{i}(*x*_{i},*x*_{-i}, δ,*G*) =

*i*’s best response satisfies

- BR(
*x*) = max { 𝟘, 𝟙 — δ*G*𝕏}