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Lecture 5
Dynamic Games of Perfect Competition
Games in which players observe all moves and play iteratively.
A: F A: O
C: F 2,1 0,
C: O 0,0 2,
F O
F O F O
Christina
Alex Alex
2,1 0,0 0,0 1,
Formalizing: Extensive Form
This game has three Nash equilibria: ( F , F ), ( O , 0), and a mixed strategy equilibrium. Supposing Christina moves first, and Alex observes her move before choosing his:
We can write this equilibrium as ( F , ( F , O )) or ( F , F O ).
The extensive form is a formal description of dynamic games, it includes the following: A game tree with nodes connected by edges.
Solving Dynamic Games
A: FF A: FO A: OF A: OO
C: F 2,1 2,1 0,0 0,
C: O 0,0 1,2 0,0 1,
Subgame Perfect Nash Equilibrium (SPNE)
A subgame of a dynamic game of perfect information consists of a non-final node of the game, and all nodes and edges that follow it. (i.e. it corresponds to every node).
A node is a point where a player takes an action. There are final/terminal nodes where the game ends. Each non-final node is assigned only to one player. At each node, the player has a set of actions which correspond to a leading edge. Payoffs are asigned to each node.
We need to restrict the set of NE to get rid of non- credible commitments.
In a finite game, we can find all SPNE by backward induction. Start at the final node, then go backwards. Equilibrium : a strategy profile that satisfies certain conditions. Equilibrium path : list of actions that players take if they play according to equilibrium. Equilibrium payoffs : payoffs that happen if players play according to equilibrium (these are at the end of equilibrium paths).
L R
A B
c d e f
2.1 3,
1.2 1.
5,2 0,0 -1,-1 2,
SPNE?
By backwards induction, we found one SPNE ( Rcf , B ).
