Game Theory

A theory of interaction between rational agents.

• Institutional Design is the field in public policy & PPE that makes sure game theory & agent incentives are taken into account
• It’s part of economics because it’s about rational agents interacting

Game §

def. Game. Assuming $n$ players:

• ${\mathcal{S}}_i$: Strategy space of player $i$
• $\vec{s}=(s_1,s_2,\dots ,s_n)$: which strategy combination happened.
• $c_1(\vec{s}),c_2(\vec{s}),\dots ,c_n(\vec{s})$: associated cost of $\vec{s}$ happening for each player

Types of Games §

• Timing:
  • Static (=Simultaneous)
  • Dynamic (=Sequential Move)
• Strategy Formulation:
  • Pure Strategy: deterministic mapping from information set to action set
  • Mixed Strategy: probabilistic mapping that depends on a Random Variable
• Information availability:
  • Complete Information
  • Incomplete Information
• Repetition:
  • One-off,
  • Finite-Repetition,
  • Infinite-Repetition
• Payoff structure:
  • Zero-sum: each strategy tuple sums to zero
  • Positive/Negative Sum

Equilibria Types §

→ See Equilibria in Game Theory

• Static, Pure, One-off→ Nash Equilibrium
• Dynamic, Pure → Subgame Perfect Nash Equilibirum
• Static, Incomplete, (Pure or Mixed) → Baysian Nash Equilbilibrium (BNE)
• Dynamic, Incomplete, (Pure or Mixed) → Subgame Perfect Baysian Nash Equilibirum (PBE)

Games Modeled by Game Theory §

(DevonThink) Game Theory List of Games for many types of simultaneous game's payoff matricies.

• Oligopoly Games
  • Cartels and Collusion
  • Bertrand Price Competition
  • Cornout Quantity Competition
• Battle of the Sexes
• Prisoner’s Dillemma
  • Public Good Game
• Signaling Game
• Stag Hunt
• Assurance Game
• Chicken Game
• Rock Paper Scissors
• Traffic Lights
• Traffic Routing

Notation §

• Strategy Tuple of player $A$ is denoted $S^A=(S_1^A,S_2^A)$
• Payoff to player $A$ given strategy $S^A$ by $A$, $S^B$ by player $B$, etc. ⇒ is denoted ${\pi }^A(S^A,S^{\mathbf{B}}\dots )$
  • Alternatively, Cost is given as: $C_A(S^A,\dots )$
• Nash Equilibria are Tuples: $NE=(S_1^A,S_2^B)$
  • In a simultaneous game, all player’s strategy should be specified
  • In a sequential game, the second player (=follower)’s strategy should include the response for all of the first player (=leader)’s moves.
    • Notation: $SPE=(S_1^A;S_1^B,S_2^B)$

Branches of Game Theory §

1. Experimental game theory
2. Evolutionary game theory—using game theory to explain strategies that affect natural selection
3. Applied game theory