Ω
Lecture 5: Extensive Form
Games with Perfect Information
INSE 6441/2 DD - Applied Game
Theory and Mechanism Design
Fall 2014Acknowledgement: Jackson, Leyton-Brown & Shoham
Ω
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Overview
Informal Treatment
Formalization
Best Response and Nash Equilibrium
Subgame Perfection
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Ω Sequential Move Games
Sequential-move games entail strategic
situations in which time is considered and
there is a strict order of play
Players take turns making their moves, and
they know what other players have done
To play well in such games, participants must
use a particular type of interactive thinking
Each player must consider
If I make this move, how will my opponent
respond?
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Ω Sequential Move Games
To make decisions, players need to consider
how their current actions will influence future
actions, both for their rivals and for
themselves
Players thus decide their current moves on
the basis of calculations of future
consequence
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Ω Game Trees
Game trees provide a graphical technique for
displaying and analyzing sequential-move-games
Game trees are referred to as extensive form games
They are very similar to decision trees
Decision trees show all the successive decision
points, or nodes, for a singled decision maker in a
neutral environment
Decision trees also include branches corresponding
to the available choices emerging from each node
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Ω Game Trees
Game trees are joint decision trees for
all of the players in a game
The trees illustrate all of the possible
actions that can be taken by all of the
players and indicate all of the possible
outcomes from the game
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Ω
Initial node or
root
Decision
Nodes
Terminal nodes
Branches
Payoffs
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Example
Ω Example: Uncertainty
If Ann chooses "Go" and then Chris chooses "Risky,"
something happens at random
a fair coin is tossed and the outcome of the game is
determined by whether that coin comes up "heads" or "tails"
This aspect of the game is handled by introducing an
outsider player called "Nature"
It chooses one of two branches each with 50%
probability
The probabilities here are fixed by the type of random
event, a coin toss, but could vary in other
circumstances
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ΩExample: Payoffs
We show the outcome of each sequence of
actions, as the payoffs of the players
For our four players, we list the payoffs in the
alphabetical order (Ann, Bob, Chris, Deb)
The inclusion of a random event (a choice made
by Nature) means that players need to determine
what they get on average:
Expected Payoffs. Example: Ann’s
expected payoff = (0.5X6+0.5X2) = 4
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Ω Strategies
Move: a single action taken by a player at
a node
Players should make plans for the
succession o f moves that they expect to
make in all of the various eventualities that
might arise in the course of a game
Such a plan of action is called a strategy
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Ω Strategies
In the tree example, Bob, Chris, and Deb
each get to move at most once
For them, there is no distinction between a move
and a strategy
But Ann has two opportunities to move, so
her strategy needs a fuller specification
One strategy for her is, "Choose Stop, and
then if Bob chooses 1, choose Down."
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Ω
Solving The Games
Take, for example, a teenager named Carmen who
is deciding whether to smoke
First, she has to decide whether to try smoking at all
If she does try it, she has the further decision of
whether to continue
We illustrate this example as a simple decision tree
The solution is, she should try but not continue
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Ω
We convert the decision tree of the previous example
into a game tree, by distinguishing between the two
players who make the choices at the two nodes
At the initial node, "Today's Carmen" decides whether
to try smoking
If her decision is to try, then the addicted "Future
Carmen" comes into being and chooses whether to
continue or not
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Solving The Games
Ω
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Solving The Games
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Ω
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Solution in this case is:
Today’s Carman should not try smoking
Future Carman should continue smoking
Solving The Games
Ω
Another Example
If Instead of Emily, Nina or Talia moved first, what
would happen?
The advantage is given to the first mover
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Ω
First and Second Mover
Emily gets her best outcome, because she can take
advantage of the opportunity to make the first move
Such first-mover advantage does not exist in all games
Second-mover advantage is possible
Example: Product Price (Catalog ) between competing
companies
First-mover advantage comes from the ability to commit
oneself to an advantageous position and to force the
other players to adapt to it
second-mover advantage comes from the flexibility to
adapt oneself to the others‘ Choice
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Ω
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Overview
Informal Treatment
Formalization
Best Response and Nash Equilibrium
Subgame Perfection
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Ω Formal Definition
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Ω
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Formal Definition
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Ω
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Formal Definition
Ω Example: The Sharing Game
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Ω Pure Strategy
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A pure strategy for a player in a perfect-
information game is a complete specification of
which action to take at each node belonging to
that player.
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Player 1 has 3 pure strategies
Player 2 has 8 pure strategies
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Pure Strategy
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Ω
Pure strategies for player 2:
S2 = CE, CF, DE, DF
Pure strategies for player 1:
S1 = BG, BH, AG, AH
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Example 2
Ω
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Overview
Informal Treatment
Formalization
Best Response and Nash Equilibrium
Subgame Perfection
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Ω
Given our new definition of pure
strategy, we are able to reuse our old
definitions of:
Mixed strategies
Best response
Nash equilibrium
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Nash Equilibrium
ΩInduced Normal Form
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Any extensive form game can be
converted into a normal form game
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Ω Induced Normal Form
This is easy to see, since the players move
sequentially
It is important to notice that we cannot always
perform the transformation from normal form
to extensive form
Matching pennies cannot be written as a perfect-
information extensive form game
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ΩInduced Normal Form
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Pure strategy equilibria:
AG, CF
AH, CF
BH, CE
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Ω
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Overview
Informal Treatment
Formalization
Best Response and Nash Equilibrium
Subgame Perfection
ΩDiscussing Equilibrium
There’s something intuitively wrong with the equilibrium
BH, CE
Why would player 1 ever choose to play H if he got to
the second choice node?
After all, G dominates H for him
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Ω
Discussing Equilibrium
Player 1 does select H to threaten player 2, to
prevent him from choosing F, and so gets 5
However, this seems like a non-credible threat
If player 1 reached his second decision node, would
he really follow through and play H?
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Ω
Formal Definitions
s is a subgame perfect equilibrium (SPE) of G iff for any
subgame G′ of G, the restriction of s to G′ is a Nash
equilibrium of G′
Since G is its own subgame, every SPE is a NE.
This definition rules out “non-credible threats”
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Ω
Example
AG, CF: is subgame perfect
BH, CE: BH is non-credible; not subgame perfect
AH, CF: AH is non-credible, though H is “off-path”
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