Subgame Perfection
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
(Chapters 2, 3, 14 and 15 in Watson (2013))
Simon Fraser University
February 15, 2016
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Example of Imperfect Information
A pedestrian and a mugger.
The mugger (player 1) moves first, has three actions: bring a gun and
show, bring a gun and not show, not bring a gun.
The pedestrian (player 2) moves second, has two actions: surrender,
run.
The pedestrian prefers to surrender if the mugger has a gun, and
prefers to run if the mugger does not have a gun. All else equal, the
mugger prefers no gun to gun.
How do we draw the game tree of such a game?
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Mugger-Pedestrian game
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Imperfect Information
Sometimes, the player taking an action doesn’t know where
he/she is in the tree, because he/she didn’t see how other
players played earlier.
Connect these nodes with a dotted line to represent that fact. Such a
set of nodes is called an information set.
Information set represents what the player knows up to this point.
Thus, the information set is equivalent to a history of plays that is
observable to the player. Each information set also represents a
contingency.
Now, a player’s strategy must specify a course of action for
each information set (rather than each node) where he/she
acts.
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Example (cont’d)
Let’s find pure-strategy profiles that make sense as solutions of this
game.
Common sense that the pedestrian surrenders after the mugger shows
his gun.
Then we solve for the NE of the remaining game.
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Subgames
Idea: some parts of the game tree can stand alone as a game. These
are called subgames.
Example: The game we considered: after the mugger shows his gun.
Definition: a node h’s successors are all the nodes after h, all the
way to the terminal nodes (end of the game tree).
Definition: Suppose you have a game G . A subgame of G consists
of a single non-terminal node and all its successors with the
property that every information set of G is either entirely inside
or entirely outside that set of nodes.
The last part of the definition can be rephrased: no information set of
G contains both nodes inside and nodes outside the set.
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Subgames (II)
Way to remember the definition: think of information sets as spider
webs. Subgames are parts of the tree (except for terminal nodes) that
you can detach by snapping a single branch and without tearing a
web.
How many subgames were there in our example? How many
strategies?
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Subgame-Perfect Equilibrium
Once you understand subgames, the definition of subgame-perfect
equilibrium is simple:
A subgame perfect equilibrium is a strategy profile where a
Nash equilibrium is played in each subgame.
To solve for SPE, do what we have been doing! Start with the small
subgames toward the end of the tree, and solve bigger and bigger
subgames.
Backward induction is a special case of this procedure: in games of
perfect information, every non-terminal node and its successors are a
subgame.
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Battle of the Sexes with an Outside Option
Recall the battle of the sexes (BoS):
Guy
Ballet Hockey
Girl Ballet
3,1
0,0
Hockey
0,0
1,3
The girl has an outside option. She can choose “out”, in which case
she gets 2, while the guy gets 0. If she chooses ”in”, then the guy
(who observes that the girl chose “in”) and the girl play the
simultaneous-move BoS game.
Analyze this game.
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Battle of the Sexes with an Outside Option
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Summaries on Battle of the Sexes with an Outside Option
Solve first the smaller subgame, which is a simultaneous-move BoS
game: there are two pure-strategy NE’s and one mixed-strategy NE in
the subgame. Then solve the girl’s choice of In or Out given each of
these NE’s in the subgame.
There are three SPE’s:
1
2
3
The girl plays (Out, Hockey); the guy plays Hockey.
The girl plays (In, Ballet); the guy plays Ballet.
The girl plays (Out, 34 Ballet + 14 Hockey); the guy plays ( 41 Ballet +
3
4 Hockey).
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Signaling Game: Beer-Quiche
Inspired by the popular 1980s book Real Men Don’t Eat Quiche by
Bruce Feirstein.
Three players: player 1 (strong), player 1 (weak), and player 2.
Player 1, strong or weak, chooses a meal: beer or quiche. Player 2
observes this food choice, and chooses to fight or retreat.
Player 2 cannot tell if player 1 is strong or weak, believes that player
1 is strong with probability 0.9.
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Signaling Game: Beer-Quiche
Inspired by the popular 1980s book Real Men Don’t Eat Quiche by
Bruce Feirstein.
Three players: player 1 (strong), player 1 (weak), and player 2.
Player 1, strong or weak, chooses a meal: beer or quiche. Player 2
observes this food choice, and chooses to fight or retreat.
Player 2 cannot tell if player 1 is strong or weak, believes that player
1 is strong with probability 0.9.
Player 2 gets 1 if he fights the weak player, -1 if he fights the strong
player, 0 if no fight.
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Signaling Game: Beer-Quiche
Inspired by the popular 1980s book Real Men Don’t Eat Quiche by
Bruce Feirstein.
Three players: player 1 (strong), player 1 (weak), and player 2.
Player 1, strong or weak, chooses a meal: beer or quiche. Player 2
observes this food choice, and chooses to fight or retreat.
Player 2 cannot tell if player 1 is strong or weak, believes that player
1 is strong with probability 0.9.
Player 2 gets 1 if he fights the weak player, -1 if he fights the strong
player, 0 if no fight.
The strong player 1 prefers beer; the weak player 1 prefers quiche.
Player 1, strong or weak, gets 2 if Player 2 doesn’t fight him, 0
otherwise; player 1, strong or weak, gets an additional 1 if he eats his
preferred meal.
ECON 302 (SFU)
Lecture 6
February 15, 2016
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Signaling Game: Beer-Quiche
Inspired by the popular 1980s book Real Men Don’t Eat Quiche by
Bruce Feirstein.
Three players: player 1 (strong), player 1 (weak), and player 2.
Player 1, strong or weak, chooses a meal: beer or quiche. Player 2
observes this food choice, and chooses to fight or retreat.
Player 2 cannot tell if player 1 is strong or weak, believes that player
1 is strong with probability 0.9.
Player 2 gets 1 if he fights the weak player, -1 if he fights the strong
player, 0 if no fight.
The strong player 1 prefers beer; the weak player 1 prefers quiche.
Player 1, strong or weak, gets 2 if Player 2 doesn’t fight him, 0
otherwise; player 1, strong or weak, gets an additional 1 if he eats his
preferred meal.
Draw the game tree, identify the subgame(s) and the strategies, and
find the SPE’s.
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Beer-Quiche game
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Summaries on Beer-Quiche game
To find a SPE, start with a conjecture on the strategy of Player 1,
and calculate Player 2’s best responses. For each of these best
responses, verify that both types of Player 1 are best responding in
the conjectured strategy.
There are two pure-strategy SPE’s:
1
(S-Beer, W-Beer, B-Retreat Q-Fight)
2
(S-Quiche, W-Quiche, B-Fight Q-Retreat)
Intuition: the weak player 1 must “camouflage” himself with the
strong player 1 to force player 2 to retreat. This is called a pooling
equilibrium.
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