Dynamic Games
Advanced Microeconomics II
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Dynamic Game
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Each dynamic game can be expressed by a
“game tree.” (it is formally called extensive‐
form representation)
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Dynamic games can also be analyzed in
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Dynamic games can also be analyzed in
strategic form: a strategy in dynamic games is
a complete action plan which prescribes how
the player will act in each possible
contingencies in future.
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Entry and Predation
• There are two firms in the market game: a potential entrant and a monopoly incumbent.
• First, the entrant decides whether or not to enter this monopoly market.
• If the potential entrant stays out, then she gets 0while the monopolist gets a large profit, say 4.
• If the entrant enters the market, then the incumbent must choose whether or not to engage in a price war.
• If he triggers a price war, then both firms suffer (receive ‐1).
• If he accommodates the entrant, then both firms obtain modest profits, say 1each.
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Game Tree Analysis
(1,1) Monopolist ACC.
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(0,4)
(‐1,‐1) Entrant
OUT IN
WAR
Strategic‐Form Analysis
Monopolist Entrant
Price War Accommodate
In -1
1
1 1
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Is (Out, Price War) a reasonable NE?
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-1 1
Out 4
0
4 0
Lessons
• Dynamic games often have multiple Nash equilibria, and some of them do not seem plausible since they rely on non‐crediblethreats.
• By solving games from the backto the forward, we can erase those implausible equilibria.
Backward Induction
• This idea will lead us to the refinement of NE, the subgame perfect Nash equilibrium.
Backward Induction Solution
( , ) Monopolist
ACC. ACC.
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( ,4)
(‐1, ) Entrant
OUT IN IN
WAR
Extensive Form Games
The extensive‐form representation of a game specifies the following 5 elements:
1. The playersin the game. 2. Wheneach player has the move. 3. Whateach player can doat each of her
opportunities to move. 4. Whateach player knowsat ‐‐‐.
5. The payoffreceived by each player for each combination of moves that could be chosen by the players.
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Extensive‐Form Representation
2 U
U
D
(3,1) (1,2)
2 0 0 9 /3 /1 1 Le c t ure 9 9
1 D
D
U’
D’
(2,1)
(0,0) 2
Normal‐Form Representation
• Every dynamic game generates a singlenormal‐ form representation.
1 2 (U,U’) (U,D’) (D,U’) (D,D’)
• A strategy for a player is a complete plan of actionsspecifying a feasible action for the player in every contingency.
2 0 0 9 /3 /1 1 Le c t ure 9 1 0
U 3,1 3,1 1,2 1,2
D 2,1 0,0 2,1 0,0
Extensive‐Form Representation 2
U
U
D
(3,1)
(1,2)
2 0 0 9 /3 /1 1 Le c t ure 9 1 1
1 D
D
U
D
(2,1) (0,0) 2
Normal‐Form Representation 2
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Every dynamic game generates a single
normal‐form representation.
1 2 U D
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Every static game can be expressed in
extensive‐form (with possibly many ways).
U 3,1 1,2
D 2,1 0,0
Extensive‐Form Representation 3
U
U
D
(3,1) (2,1)
2 0 0 9 /3 /1 1 Le c t ure 9 1 3
2
D
D
U
D
(1,2)
(0,0) 1
Game Tree
• An extensive‐form game is defined by a tree that consists of nodesconnected by branches.
• Each branch is an arrow, pointing from one node (a predecessor) to another (a successor).
• For nodes x, y, and z, if x is a predecessor of y and yFor nodes x, y, and z, if x is a predecessor of y and y is a predecessor of z, then it must be that x is a predecessor of z.
• A tree starts with the initial nodeand ends at terminal nodeswhere payoffs are specified.
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Tree Rules
1. Every node is a successor of the initial node. 2. Each node except the initial node has exactly one
immediate predecessor. The initial node has no predecessor.
3. Multiple branches extending from the same node have different action labels.
4. Each information setcontains decision nodes for only one of the players.
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Information Set
• An information set for a player is a collection of decision nodes satisfying that
1. the player has the move at every nodein the information set, and
2. when the play of the game reaches a node in the
i f i h l i h h d
information set, the player with the move does not knowwhich node in the information set has been reached.
At every decision node in an information set, each player must
1. have the sameset of feasible actions, and 2. choose the sameaction.
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Subgame
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A subgame in an extensive‐form game
1. begins at some decision node n with a singletoninformation set,
2. includes all the decision and terminal nodes following n, and
3. does not cut any information sets.
We can analyze a subgame on its own,
separating it from the other part of the game.
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Subgame Perfect Nash Equilibrium
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