GAME THEORY AND
APPLICATIONS
SUBGAME PERFECT
EQUILIBRIUM
Assoc. Prof. Yeşim Kuştepeli
1
•
In dynamic games, the players who move later in a game do so
knowing the moves others have made before them. Those who
move earlier must take this into account in devising their optimal
strategy. Ex. Chess.
•
Ex. Software game
•
Macrosoft’s profits with
no competitor
Macrosoft’s profits
with a competitor
Slick
campaign
simple
campaign
900,000
200,000
Profit in year 2 100,000
800,000
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cost
570,000
200,000
Net profit
430,000
800,000
Profit in year 1
Slick
campaign
simple
campaig
n
Profit in year 1
900,000
200,000
Profit in year 2
50,000
400,000
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cost
570,000
200,000
Net profit
380,000
400,000
Assoc. Prof. Yeşim Kuştepeli
2
Microcorp’s profits if it enters the market
Macrosoft
Slick
campaign
simple
campaign
Profit in year 1
0
0
Profit in year 2
50,000
400,000
Cloning cost
300,000
300,000
Net profit
-250,000
100,000
•Macrosoft’s moves: 1) slick campaign 2) simple campaign
•Microcorp’s moves: 1) enter the market 2) stay out of the market
•Order of play: Macrosoft is the first mover, microcorp is last mover.
Assoc. Prof. Yeşim Kuştepeli
3
• Strategy: complete description of how to play the
game.
• In static games, moves = strategies.
• In dynamic games moves may not equal strategies.
• Game tree : the set of strategies for each player
by specifying both the moves of the players and
also the order in which they choose theie moves
and the information they have when they make
their decisions.
• Node: a decision point for one of the players.
• Branch: a possible move by a player.
Assoc. Prof. Yeşim Kuştepeli
4
Rules of dynamic games:
1)
Every node is immediately preceded by at most
one other node.
2) No path in a tree connects a decision node to
itself.
3) Every node is the successor of a unique initial
node.
4) Every game tree has exactly one initial node.
Assoc. Prof. Yeşim Kuştepeli
5
In our example,
•
Macrosoft has one decision node so its strategy
consists of choosing between “slick” or “simple”
campaign.
•
1)
Microcorp has two decision nodes:
Enter if slick, Stayout if simple, (enter, stayout)
2) Enter if slick, enter if simple, (enter, enter)
3) Stayout if slick, enter if simple, (stayout, enter)
4) Stayout if slick, stayout if simple, (stayout, stayout)
Assoc. Prof. Yeşim Kuştepeli
6
If it was a static game:
Macrosoft
slick
Microcorp
simple
enter
(-250,380) (100,400)
stayout
(0,430)
(0,800)
Assoc. Prof. Yeşim Kuştepeli
7
As a dynamic game:
Macrosoft
Microcorp
slick
simple
(enter, enter)
(-250,380)
(100,400)
(enter, out)
(-250,380)
(0,800)
(out, enter)
(0,430)
(100,400)
(out, out)
(0,430)
(0,800)
Assoc. Prof. Yeşim Kuştepeli
8
• Backward induction: consider the moves
that are last in the game and determine
the best move for the player in each case.
Then, proceed backwards in time again
determinng the best move for the
respective player until the beginning of
the game is reached.
• First mover advantage
Assoc. Prof. Yeşim Kuştepeli
9
Subgames and subgame perfect equilibrium
• In a game with perfect information, a subgame
consists of a subset of nodes and branches of the
original game that, when taken tohether,
consitute a game in themselves. Since a subgame
must be a game, it has a unique initial node, which
is called the subroot of the larger game.
• The first requirement of a subgame is that it
consists of te subroot and all its successors.
• Every game is a trivial subgame of itself.
Assoc. Prof. Yeşim Kuştepeli
10
• A strategy profile is a subgame perfect
equilibrium of a game if this strategy
profile is also a Nash equilibrium for
every proper subgame of the game.
• A strategy profile is a subgame perfect
equilibrium of a dynamic game with
perfect information if and only if it is
the Nash equilibrium selected by
backward induction.
Assoc. Prof. Yeşim Kuştepeli
11