Subgame Perfect Equilibrium
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Game Tree
• An extensive-form game is defined by a tree
that consists of nodes connected 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 y is a predecessor of z, then it must be
that x is a predecessor of z.
• A tree starts with the initial node and ends at
terminal nodes where 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 set contains 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 (i) the
player has the move at every node in the
information set, and (ii) when the play of the
game reaches a node in the information set,
the player with the move does not know
which node in the information set has been
reached.
 At every decision node in an information set,
each player must (i) have the same set of
feasible actions, and (ii) choose the same
action.
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Subgame
• A subgame in an extensive-form game
(a) begins at some decision node n with
a singleton information set, (b) includes
all the decision and terminal nodes
following n, and (c) 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 NE
• A subgame perfect Nash equilibrium
(SPNE) is a combination of strategies in
a extensive-form which constitutes a
Nash equilibrium in every subgame.
 Since the entire game itself is a
subgame, it is obvious that a SPNE is a
NE, i.e., SPNE is stronger solution
concept than NE.
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Stackelberg Model
The Stackelberg model is a dynamic version of
the Cournot model in which a dominant
firm moves first and a subordinate firm
moves second.
• Firm 1 (a leader) chooses a quantity first
• Firm 2 (a follower) observes the firm 1’s
quantity and then chooses a quantity
 Solve the game backwards!
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Remarks
• A leader never becomes worse off since she
could have achieved Cournot profit level in
the Stackelberg game simply by choosing
the Cournot output.
• A follower does become worse off although
he has more information in the Stackelberg
game than in the Cournot game, i.e., the
rivals output.
• Note that, in single-person decision making,
having more information can never make the
decision maker worse off.
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