Game Theory Notes

Game Theory

Concept

• Game theory is the study of strategic interactions among economic agents.
• Strategic interactions: a situation in which the pay-off of one economic agent is dependent upon the choices of others.

Essential Elements of a Game

• Players
• Strategies
• Payoffs
• Each player’s goal is to maximize their individual payoff.

Types of Games

Simultaneous vs. Sequential Games

• Simultaneous games are games in which players take strategic actions at the same time without knowing what move the other has chosen, e.g., rock, paper, scissors.
• Sequential games are games where players take turns and move consecutively. One player observes the move of the other player and then makes their play and so on, e.g., chess.

Single-Play vs. Repeated Games

• Single-play games are played once, and then the game is over, e.g., a game of rock, paper, scissors played once to determine a winner.
• Repeated games are simultaneous move games played repeatedly by the same players, e.g., repeated plays of rock, paper, scissors, with the first player to win five times being the winner.

Representation

Normal Form

• A table representing players, strategies, and payoffs.

• Example:

  Player A \ Player B	Sports	Comedy
  Sports	3,2	1,1
  Comedy	0,0	2,3

Extensive Form

• A game tree representing the sequence of moves, players, and payoffs.

• Includes:

  • Nodes (decision points)
  • Players
  • Strategies
  • Payoffs

• Information set: A set of nodes among which a player cannot differentiate.

Dominant Strategies

The Prisoners’ Dilemma

• Example payoff matrix:

  Player A \ Player B	Confess	Deny
  Confess	-3,-3	0,-6
  Deny	-6,0	-1,-1

Definitions

• Dominant strategy: a strategy for which the payoffs are always greater than any other strategy no matter what the opponent does.
• Dominated strategy: a strategy for which the payoffs are always lower than any other strategy no matter what the opponent does.
• Equilibrium in dominant strategies: outcome of a game in which each player is doing the best it can regardless of the actions of its opponent.
  • Each player has a dominant strategy and plays it.

Pareto Efficiency

• Each player has a dominant strategy – Confess, and the game has a dominant strategy equilibrium – (Confess, Confess).
• However, If they both chose Deny, both of them would have been better off. In this case, the equilibrium in dominant strategies is not Pareto efficient.
• Pareto efficiency: a situation where no action or allocation is available that makes one individual better off without making another worse off.

Oligopolies as a Prisoners’ Dilemma

• Example payoff matrix:

  Firm A \ Firm B	High production	Low production
  High production	16,16	20,15
  Low production	15,20	18,18

When Only One Player Has a Dominant Strategy

• Example payoff matrix:

  Firm A \ Firm B	Advertise	Don’t advertise
  Advertise	10,5	15,0
  Don’t advertise	6,8	20,2

Nash Equilibrium

• Nash equilibrium: a set of strategies (or actions): each player is doing the best it can given the actions of its opponent.
• Player A’s choice is optimal, given player B’s choice, and player B’s choice is optimal, given player A’ choice.
• Because each player has no incentive to deviate from its Nash strategy, the strategies are stable.

Pure Coordination Game

• Example payoff matrix:

  Player A \ Player B	Left	Right
  Left	10,10	0,0
  Right	0,0	10,10

Exercise: The Product Choice Problem

• Example payoff matrix:

  Firm A \ Firm B	Crispy	Sweet
  Crispy	-5,-5	10,10
  Sweet	10,10	-5,-5

Battles of the Sexes

• Example payoff matrix:

  Player A \ Player B	Football	Shopping
  Football	10,5	0,0
  Shopping	0,0	5,10

Stag Hunt

• Example payoff matrix:

  Player A \ Player B	Stag	Hare
  Stag	8,8	0,7
  Hare	7,0	5,5

Mixed Strategies

• Pure strategy: strategy in which a player makes a specific choice or takes a specific action.
• Mixed strategy: strategy in which a player makes a random choice among two or more possible actions, based on a set of chosen probabilities.
• Some games do not have any Nash equilibria in pure strategies.
• However, once we allow for mixed strategies, every game has at least one Nash equilibrium.

Battles of the Sexes: More General Case

• Example payoff matrix:

  Player A \ Player B	Football	Shopping
  Football	10,5	0,0
  Shopping	0,0	5,10
  Probability	q	1 – q
  Probability	p	1 – p

• In a mixed strategy equilibrium, player A should have: $10q + 0(1 – q) = 0q + 5(1 – q)$
  $q = \frac{1}{3}$ $(1)$

• Similarly, player B should have: $5p + 0(1 – p) = 0p + 10(1 – p)$
  $p = \frac{2}{3}$ $(2)$

• The mixed-strategy Nash equilibrium is $[(\frac{2}{3},\frac{1}{3}),(\frac{1}{3},\frac{2}{3})]$.

Sequential Games

Backward Induction

• Solving sequential games by considering the optimal decisions of players at each decision point, starting from the end of the game and working backward.
• Example: Extensive form game tree.