Extensive games (with perfect information)
(also referred to as extensive-form games or dynamic games)
DEFINITION An extensive game with perfect information has the following components
• A set N (the set of players).
• A set H of sequences (finite or infinite) satisfying:
– The empty sequence is a member of H
– If (ak )k=1,...,K ∈ H (where K may be infinite) and L < K then (ak )k=1,...,L ∈ H
k
– If an infinite sequence (ak )∞
k=1 ∈ H satisfies (a )k=1,...,L ∈ H for every positive
integer L then (ak )∞
k=1 ∈ H
(Each member of H is called a history; each component of a history is an action
taken by a player.) A history (ak )k=1,...,K ∈ H is called terminal (final) if it is infinite
or if there is no aK+1 such that (ak )k=1,...,K+1 ∈ H. The set of terminal histories is
denoted Z.
• A function P that assigns to each nonterminal history (each member of H\Z) a
member of N . (P is the player function, P (h) being the player who takes an action
after the history h.)
• For each player i ∈ N a preference relation i on Z (the preference relation of
player i).
Denote a history h followed by action a by (h, a).
Interpetation of this definition: After any nonterminal history h Player P (h) chooses
a possible action a, i.e. such an action that (h, a) belongs to H.
Game tree is a convenient method to represent extensive-form games. Think of a
rooted tree in graph-theoretic sense. Actions are represented by branches (or edges) and
histories correspond to paths and induce nodes. Non-terminal histories induce decision
nodes and terminal histories induce end-nodes or end-points or leaves.
Example: mini-ultimatum game
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Strategies
Action 6= Strategy!
Strategy is a plan of action for every contingency.
DEFINITION A strategy of player i ∈ N in an extensive game with perfect information
hN, H, P, i i is a function that assigns an admissible action to each nonterminal history
h ∈ H\Z for which P (h) = i.
A combination of strategies induces an outcome of the game - the terminal history
that will result when they are implemented.
Note that the strategy dictates what to do even at nodes that, under this strategy,
will not be visited.
Any extensive-form game can now be represented in a matrix form.
Example: Entrant game
Thus, also the concept of Nash Equilibrium easily applies to dynamic games – it is the
NE of the appropriate strategic (matrix-form) game.
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Nash is not enough
In extensive-form games players may want to revise their equilibrium strategy as the
game unfolds.
It seems naive to assume that they will not when it is in their best interest to do so it
would be tantamount to believing non-credible threats or promises.
DEFINITION The subgame of the extensive game with perfect information Γ =
hN, H, P, (i )i that follows the history h is the extensive game Γ(h) = hN, H|h , P |h , (i |h )i,
where H|h is the set of sequences h0 of actions for which (h, h0 ) ∈ H, P |h is defined by
P |h (h0 ) = P (h, h0 ) for each h0 ∈ H|h , and i |h is defined by h0 i |h h00 if and only if
(h, h0 ) i (h, h00 ).
(examples)
Every game is its own subgame (following an empty history).
Denote by si |h the strategy induced by si in subgame Γ(h) and by s∗ |h the strategy
profile induced by s∗ .
DEFINITION Strategy profile s∗ constitutes a subgame perfect equilibrium of a game
if s∗ |h is a NE of every subgame Γ(h).
One problem with the SPNE: shouldn’t I give up my belief about rationality of the
other player when he makes a dumb choice? (example)
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Centipede game
Is common knowledge of rationality so rational? (example)
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Timing matters: Cournot vs Stackelberg
n = 2, P = a − Q, Q = q1 + q2 , constant unit cost c1 = c2 .
Profit is given by:
Πi = qi (P − ci ) = qi (a − qi − q−i − ci )
FOC:
∂Πi
= (a − qi − q−i − ci ) − qi = (a − 2qi − q−i − ci ) = 0
∂qi
Thus for any q−i i’s BR is qi =
a−q−i −ci
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If they move simultaneously (Counot), NE is ( a−c
, a−c
).
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If they move sequentially (Stackelberg), there are multiple NE. But the unique SPNE
, a−c
). Total output is different, player 1 is better off (why?), player 2 is worse off.
is ( a−c
2
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Some properties of SPNE
Obviously, every SPNE is a NE but not conversely.
Existence: every finitie extensive game with perfect information has a subgame perfect
equilibrium (can be found by backward induction)
(non)Uniqueness: SPNE is in general not unique. However, it is unique if no player is
indifferent between two end-nodes.
(example: the gardening games)
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Two fairly benign extensions
Random moves: nature choses at some nodes, following a pre-defined distribution
Simultaneous moves: more than one player moves simultaneously at some nodes
Note: with simultaneous moves SPNE may fail to exist (example: matching pennies).
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Forward induction
Example: forward induction in Battle-of-the-Sexes
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