Games (Game Theory)

Relevant Course: Game Theory

Key Ingredients

There are a few key ingredients in a game:

• Players: who are the decision makers?
  • e.g. People, Governments, Companies, Employees
  • Who is controlling the outcome of the game
• Actions: what can the players do?
  • e.g. Enter a bid at an auction, decide whether to strike, what move to make in chess, call a raise, etc.
• Payoffs: what motivates the players? (why is this game even being played in the first place)
  • e.g. Profit, Relationships, Clout, etc.

Representing Games

Normal Form

Also Known As: Matrix Form, Strategic Form

• Lists what payoffs players get as a function of their actions
  • It behaves as if players “moved” simultaneously
  • Strategies can still encode many things
    • not constrained to single-dimensional or binary strategies

Normal Form Representation / Factors

• Finite, $n$-person normal form game: $⟨N,A,u⟩$:
  • Players: $N={\mathbb{N}}_{\le n}=\{1,\dots ,n\}$ is a finite set of $n$, indexed by $i$
  • Action Set for player $i$ is $A_i$:
    • $a=(a_1,\dots ,a_n)\in A=A_1\times \cdots \times A_n$ is an action profile
  • Utility Function or Payoff Function for player $i$: $u_i:A\mapsto \mathbb{R}$
    • $u=(u_1,\dots ,u_n)$ is a profile of utility functions
    • Essentially, assigning a value or payoff to each action $a\in A$ in the action set for each individual

Extensive Form

• Players move sequentially (often represented as a tree)
  • e.g. Chess, Checkers
    • Chess: First white player moves, then Black gets a chance to see how White moved and then react
• Keeps track of what each player knows when he or she makes each decision
  • Poker: bet sequentially—player only knows what they have, what’s on the table, and what was bet

Extensive Form Factors

• Timing: in what order do things happen?
• Information: what do players know when they act?