ECOS 3012 Lecture 4
Dynamic games with complete information I
ECOS3012 LECTURE 4
August 31, 2021
Last week
Mixed strategy Nash equilibrium
Strict dominance and weak dominance
Nash’s existence theorem:
Nash equilibrium always exists when players and pure strategies are finite
Plot best response functions to find all NE
Cournot competition (Lecture 2)
Any 2×2 games, looking for mixed strategy NE
Prisoner’s dilemma
Stag hunt
Practice problem 3: Q2
Plot best responses: Prisoner’s dilemma
Prisoner 2
Confess Not Confess
Prisoner 1 Confess -6, -6 0, -9
Not Confess -9, 0 -1, -1
Plot best responses: Stag hunt
Hunter 2
Stag Hare
Hunter 1 Stag 3, 3 0, 2
Hare 2, 0 2, 2
This lecture
Dynamic games with complete information
How to represent a dynamic game
Extensive form and normal form
New solution concept: subgame perfect Nash equilibrium (SPE)
Why?
Good and bad things about it
Examples and applications
Stackelberg competition
Sequential bargaining
Strategic situation: money or grenade
Stage 1: Player 1 has $1000 and chooses between two options:
Keep the money
Give all the money to player 2
Stage 2: Player 2 observes what player 1 does and chooses:
Explode a grenade, both killed
Do nothing
Preferences:
Both prefer to be alive.
Conditional on being alive, having money is better than not.
Important: Player 2 observes Player 1’s choice and then chooses. (Dynamic!)
Money or grenade: extensive form
Money or grenade: normal form
Set of players
Strategy set for each player
Player 1:
Player 2: (A contingent plan)
Payoff functions for each player
Player 2
NE, NE NE, E E, NE E, E
Player 1 Give
Keep
Money or grenade: Nash equilibria
Player 2
NE, NE NE, E E, NE E, E
Player 1 Give 0, 1000 0, 1000 -1000, -1000 -1000, -1000
Keep 1000, 0 -1000, -1000 1000, 0 -1000, -1000
Pure strategy Nash equilibria: (Give, (NE, E)), (Keep, (NE, NE)), (Keep, (E, NE))
Money or grenade: Does NE make sense?
(Give, (NE, E))
(Keep, (NE, NE))
(Keep, (E, NE))
Why would Player 2 choose to explode the grenade if Player 1 gives him the money?
Money or grenade: Does NE make sense?
(Give, (NE, E)): non-credible threat
(Keep, (NE, NE))
(Keep, (E, NE))
Why would Player 2 choose to explode the grenade if Player 1 gives him the money?
In fact, why would player 2 ever explode the grenade? He prefers to be alive.
Money or grenade: Does NE make sense?
(Give, (NE, E)): non-credible threat
(Keep, (NE, NE))
(Keep, (E, NE))
Why would Player 2 choose to explode the grenade if Player 1 gives him the money?
In fact, why would player 2 ever explode the grenade? He prefers to be alive.
Lesson: When it is someone’s turn to act, given what has happened, we want this player to be rational.
Subgame s: extensive form
A proper subgame in an extensive-form game
Begins at a decision node, that is not the game’s first decision node
Include all the decision and terminal nodes following it
Caution: this is a partial (incorrect) definition, since we haven’t learned the information sets. We will revisit this definition.
A game is a subgame of itself.
Subgame perfect Nash equilibrium
A Nash equilibrium is subgame-perfect if the players’ strategies constitute a Nash equilibrium in every subgame.
Subgame 1: Player 2: NE
Subgame 2: Player 2: NE
Nash equilibria:
(Give, (NE, E))
(Keep, (NE, NE)) – Unique SPE
(Keep, (E, NE))
Find SPE of an extensive form game:
Backward induction
Start from the decision nodes that are the closest to the terminal nodes
Find best responses
Move up a level, knowing that if the next level decision node is reached, what outcome it would lead to
Find best responses
Continue until the starting decision node
Report all the branches (like in NE)
SPE: reflections
Rules out strategies that not rational even if the decision node will never be reached.
We cannot have non-credible threat
Reasonable
Enforced at all layers of the game tree
Everybody knows that everybody knows that … at every node the player is rational
May be too strong
Centipede games (See quiz question 1)
Applications: Stackelberg Model of Duopoly
Players: Firm 1 and Firm 2
Stage 1: Firm 1 chooses quantity q1
Stage 2: After observing q1, Firm 2 chooses quantity q2
Payoffs:
P(q1, q2) = 100 – q1 – q2
Ci(qi) = 10qi
Stackelberg vs Cournot
Stackelberg
Cournot
Comparison
First mover advantage:
More information may hurt:
Application: Sequential bargaining
Day 1: $10 to split
Player 1 proposes (player 1’s share, player 2’s share) = (s1, 10 – s1)
Player 2 either accepts (game ends) or rejects (proceed to day 2)
Day 2: $9 to split
Player 2 proposes (player 1’s share, player 2’s share) = (s2, 9 – s2)
Player 1 either accepts (game ends) or rejects (proceed to day 3)
Day 3: $8 in total, each player gets $4
Assume that a player accepts an offer when he is indifferent between acceptance and rejection.
Extensive form Normal form
Extensive form to normal form
A player’s strategy is a contingent plan.
It specifies a feasible action for the player in every contingency in which the player might be called on to act.
Normal form to extensive form
Information set
When the play of the game reaches
a node in the information set, the
player does not know which node in
the information set has been reached
Subgames
Start from a singleton information set
Does not cut information set
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