Download GAME THEORY AND APPLICATIONS

Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
SUBGAME PERFECT
EQUILIBRIUM
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

Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory

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
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
Macrosoft’s profits
with a competitor
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
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Microcorp’s profits if it enters the market
Macrosoft
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
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
Slick
campaign
•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.
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
Strategy: complete description of how to play the
game.
In static games, moves = strategies.
 In dynamic games moves may not equal strategies.

Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory

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.
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Rules of dynamic games:
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.
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
1)
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
1)
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
In our example,

Macrosoft has one decision node so its strategy
consists of choosing between “slick” or “simple”
campaign.
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)
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IF IT WAS A STATIC GAME:
slick
Microcorp
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
Macrosoft
simple
enter
(-250,380) (100,400)
stayout
(0,430)
(0,800)
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AS A DYNAMIC GAME:
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
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)
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 Backward
 First
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
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.
mover advantage
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SUBGAMES AND SUBGAME PERFECT EQUILIBRIUM
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
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.
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory

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.
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A
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.
Prof. Dr. Yeşim Kuştepeli ECO 4413 Game Theory
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.
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