#### Social and Economic Networks: Models and Analysis笔记七：Games on Networks

•  Each player chooses action $$x_i$$ in {0,1}
• payoff will depend on: 1.how many neighbors choose each action. 2. how many neighbors a player has

• Payoff action 0: $$u_{d_i} ( 0 ,m_{N_i} ) = 0$$
• Payoff action 1: $$u_{d_i} ( 1 ,m_{N_i})= -t + m_{N_i}$$

• Payoff action 0: $$u_{d_i}(0, m_{N_i})=1$$ if $$m_{N_i}> 0$$ , $$= 0$$ if $$m_{N_i}$$
• Payoff action 1: $$u_{d_i} (1, m_{N_i} ) = 1-c$$

## 2.Complements/Substitutes

• strategic complements: for all $$d,m\ge m'$$ , we have $$u_d(1,m)-u_d(0,m)\gt u_d(1,m')-u_d(0,m')$$
• strategic substitutes: for all $$d,m\ge m'$$ , we have $$u_d(1,m)-u_d(0,m)\le u_d(1,m')-u_d(0,m')$$

## 3.Equilibrium

Nash equilibrium: Every player’s action is optimal for that player given the actions of others.

## 4.Maximal Independent Set

Independent Set: a set S of nodes such that no two nodes in S are linked;

Maximal: every node in N is either in S or linked to a node in S.

## 5.Complete lattice

Complete Lattice: for every set of equilibria X

•  there exists an equilibrium x’ such that x’≥x for all x in X, and
•  there exists an equilibrium x’‘ such that x’‘≤x for all x in X.

## 6.Beyond 0‐1 choices

$x_i+\sum_{j \text{ in }N_i}x_j \ge x^*, \text{for all i, and if >, then } x_i=0$

•  distributed: $$x^*>x_i > 0$$ for some i’s
•  specialized: for each i either $$x_i=0$$ or $$x_i=x^*$$