Outline

  • Part 1: Background and Fundamentals

    • Definitions and Characteristics of Networks(1,2)
    • Empirical Background(3)

  • Part 2: Network Formation

    • Random Network Models (4,5)
    • Strategic Network Models (6,11)

  • Part 3: Networks and Behavior

    • Diffusion and Learning (7,8)
    • Games on Networks (9)

Games on Networks

  • Decisions to be made

    • not just diffusion
    • not just updating beliefs

  • Complementarities...

    • It's that people care about what other individuals are doing, so there's complementarities

  • "Strategic" Interplay

    • Inter-dependencies

    I want to only buy a certain program if other people are using that same program. So the way in which I write articles depends on what my co-authors are doing, or I want to learn a certain language only if other people are also speaking that language. So there's going to be inter-dependencies between what individuals do. And there could also be situations where I can free ride. So if somebody else buys a new book, I can borrow it from them and maybe then I don't buy it myself. So who I know that's actually bought a book, maybe that affects whether I buy the book, both positively and negatively. So there's strategic inter-dependencies. And you know, the idea of games, people think of games - you know, we're not talking about Monopoly or chess, checkers etc. We're thinking about a situation where there's interactions. And what a given individual is going to do depends on what other individuals are doing, so there is some game aspect to it in that sense. But we're using game theory as a tool to try and understand exactly how behavior relates to network structure.

Games on Networks - Outline

  • Basic Definitions

  • Examples

  • Strategic Complements / Substitutes

  • Equilibrium existence and structure

  • Equilibrium response to network structure

Games on Networks

  • Players on a network - explicitly modeled...

  • Care about actions of neighbors

  • Early literature: How complex is the computation of equilibrium in worst case games?

  • Second branch: what can we say about behavior and how it relates to network structure

Start with a Canonical Special Case

  • Each player chooses action $x_i$ in {0,1}

    • e.g. buy the book or don't buy the book, or learn a language or don't learn a language

  • payoff will depend on

    • how many neighbors choose each action
    • how many neighbors a players has

Definitions

  • Each player chooses action $x_i$ in {0,1}

  • Consider cases where i's payoff is $$u_{d_i}(x_i, m_{N_i})$$

    • depends only on $d_i(g)$ and $m_{N_i(g)}$ - the number of neighbors of i choosing 1
    • i.e. only care about the number of friends taking the action.

Example: Simple Complement

  • agent i is willing to choose 1 if and only if at least t neightbors do

  • payoff action 0: $$u_{d_i}(0, m_{N_i})=0$$

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

    • t: is threshold

Example

  • An agent is willing to take action 1 if and only if at least two neighbors do

Example: Best Shot

  • agent i is willing to choose 1 if and only if no neightbors do
    • let's think of the example where if somebody else one of my friends buys the book, I don't buy the book because now I can borrow it from them, so I'm willing to buy the book if and only if none of my neighbors do.

  • payoff action 0:

    • $u_{d_i}(0, m_{N_i})= 1$ if $m_{N_i} > 0$. If I don't buy the book(action 0), my payoff is 1 if some of my neighbors buys the book($m_{N_i} > 0$), I borror it from them
    • $u_{d_i}(0, m_{N_i})= 0$ if $m_{N_i} = 0$. If none of my neighbors bought the book($m_{N_i} = 0$), I can't borrow it, I get a pay off of zero.

  • Payoff action 1:

    • $u_{d_i}(1, m_{N_i}) = 1 - c$. Stead, I buy it myself(action 1), then the payoff is 1-c
    • c: cost of the book.

My optimal payoff would be I'd love to have one of my friends buy it($1 > 1-c >0$), me not buy it and borrow it from them. My worst payoff is nobody buys it and I don't buy it.

Another Example: Best Shot Public Goods

  • An agent is willing to take action 1 if and only if no neighbors do

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