(Game Theory) Lecture notes EC109. Microeconomics (EC109)

EC 109: Microeconomics
Topic 4: Game Theory
Decision Theory: decisions are non-strategic, since actions are independent
Game Theory: mathematical technique to analyze interdependent decision problems
 Interdependent: one person’s actions affect what the other should do
- To define a game, we need:
 Set of players ( N ): parties with decisions to make
 Set of actions for every player i ∈ N ( Ai ): different choices available
 Payoffs for every player i∈ N ( π i): outcomes for each combination of choices
A1 × A2 ×… × A n → R
Normal form:
- Present payoffs in a matrix format



N={1,2}
A1= { w , x } , A2= { y , z }
π i= A1 × A 2 → R
Action y Action z
Action w π 1 ( w , y ) , π 2 (w , y ) π 1 ( w , z ) , π 2 (w , z)
Action z π 1 ( x , y ) , π 2 (x , y ) π 1 ( x , z ) , π 2 (x , z)


 A1 is the row player
 A2 is the column player
- Payoffs must encode everything the players care about:
 Profit
 Long-run consequences of actions
 Appetites for risk
 Enjoyment they get from ‘hurting’ other players
- We must therefore model players as acting to maximize their payoff
- SO, game theory is not a theory of human behavior
 It is a methodology that cannot be disproved by observation
 Empirical content only comes in when game theory is applied
- There are three problems for players in a game:
1. Understanding the game
2. Forming expectations about what the other players will do
3. Finding the best response to what they anticipate other players doing
- Modelling task for game theorists is to:
 Frame real life situations as games
 Develop a solution concept for the situation they want to model
Solution concept: narrows down set of outcomes to reasonable ones
- It has two roles:

,  Positive: make better predictions about how people will play (explain real behavior)
 Normative: give better advice about how you should play the game
Best Response: an action a i that yields the highest payoff, given opponent’s action profile a−i
 Action profile includes actions of all opponents
- Fixing a−i, what is the best a i:
¿
a i =argmax π i (a i , a−i)
ai ∈ Ai

- To form a best response, players must:
 Predict what other players might do
 Find their best response
Dominance
Prisoner’s Dilemma
Confess Stay Quiet
Confess −5 ,−5 0 ,−10
Stay Quiet −10 , 0 −1 ,−1
- For the row player:
 Confess is a best response if they believe the column player will confess
 Confess is also the best response if they believe the column player will stay quiet
- For the column player, the same is true

Dominant strategy: an action that is always a best response, no matter your opponent’s choice
 Weakly dominant strategy: if all a i ' ≠ ai :
π i ( ai , a−i ) ≥ π i ( a'i ,a−i ) for all a−i
AND:
π i ( ai , a−i ) > π i ( ai ' , a−i) for some a−i
 Strongly dominant strategy: if all a i ' ≠ ai :
π i ( ai , a−i ) > π i ( ai ' , a−i) for all a−i
- Not all games have dominant strategies
 But if it exists, it gives strong predictive and normative advice
Dominated strategy: an action a i ' that always gives a lower payoff than a i for any a−i
- Optimal decision for individual is not necessarily the same as the optimal communal decision
Beliefs
- How do we model the way players form beliefs about what other players will do?