Backward Induction
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 2, 2016
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Introduction to Sequential Games
Past two weeks: studied games where players move simultaneously.
However, people/firms often make decisions and interact over time.
Example: Battle of the Sexes
Guy
Ballet Hockey
If simultaneous move:
Girl Ballet
3,1
0,0
Hockey
0,0
1,4
What happens if Girl texts Guy: “I’m going to the ballet, and my
phone is dying. See you there!”
So if Girl has the opportunity to send that text (and, for whatever
reason, Guy doesn’t), will she do it?
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The Extensive Form
A good way to represent a sequential game is the extensive form,
often called a “game tree.”
There is a root node, which represents the beginning of the game.
The root is followed by branches. Branches lead to more nodes, which
are followed by more branches. Each branch out of a node is an
action available at that node.
Each non-terminal node is a place where the specified player has to
make a decision on the branches/actions.
(Sometimes, the Nature makes a probabilistic move at a node; the
difference between Nature and a player is that (1) Nature does not
have any payoff, and (2) the probabilities associated with Nature are
fixed.)
Each terminal node is an outcome: a combination of actions, just
like before.
The numbers below each terminal node are the payoffs from the
outcome corresponding to the node. As usual, the first number is
player 1’s payoff, the second is player 2’s, etc.
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Perfect Information
Today, we study games of perfect information: one player acts at a
time, and each player sees all previous actions.
Simultaneous-move games are NOT games of perfect information
(when at least two players have at least two actions each).
After the quiz, we will look at games that do not have perfect
information.
Example of the latter: playing a prisoner’s dilemma more than once.
Note: don’t confuse perfect information with complete information!
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Strategies in Sequential Games
A player’s strategy specifies a probability distribution over her actions
at each node where she plays, regardless of whether that node is
reached.
In other words, a strategy is a player’s full contingency plan.
In our example, the Guy’s strategy must include what he would do if
the Girl’s chooses “Hockey,” even if he doesn’t expect the Girl to
choose “Hockey.”
Just like before, a strategy profile is a collection of each player’s
strategy. So what strategy profile are we predicting?
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Strategies in Sequential Games
A strategy is a player’s full contingency plan and must specify what
to do in every contingency (= node).
A real-world example: United States presidential line of succession.
1
2
3
4
–
5
6
...
Vice President of the United States
Speaker of the House
President pro tempore of the Senate
Secretary of State
Acting Secretary of Treasury
Secretary of Defense
Attorney General
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Nash Equilibrium in Sequential Games
Is our predicted strategy profile a NE?
Are there other pure-strategy NE?
Guy
B→B B→B B→H B→H
H→B H→H H→B H→H
Girl Ballet
3,1
3,1
0,0
0,0
Hockey
0,0
1,4
0,0
1,4
Are these extra pure-strategy NE realistic?
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Backward Induction
Idea: should require that players play a best-response (given what
they know) at all nodes, even those that are not reached.
Strategy profiles satisfying the above are called subgame-perfect
(Nash) equilibria (SPE or SPNE) in games of complete information.
In perfect information games, solving for SPEs is particularly easy:
just start at the terminal nodes to infer what players will do at the
last step. Given that, figure out what happens at the second-to-last
step, and so on.
This procedure is called backward induction.
When is there a unique SPE in perfect information games?
Is every SPE a NE?
Is every NE a SPE?
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Exercise
Consider Rock-Paper-Scissors, but suppose player 2 sees what player
1 does before acting.
Payoff is 1 for a win, -1 for a loss, and 0 for a tie.
Draw this game in extensive form, and find its SPE(s) using backward
induction.
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Commitment versus Flexibility
In Battle of the Sexes, players gain from committing to a course of
action.
As a result, there is a first-mover advantage: the Guy would like to
threaten to go to the hockey game after the Girl has gone to the
ballet dance, but cannot do so credibly.
As we saw, NE allows for such non-credible threats, while SPE
doesn’t.
Another such example: entry into a market. Incumbent would like to
deter entrant by committing to a price war if competitor enters, but
this is often not credible.
By contrast, in Rock-Paper-Scissors, flexibility creates a
second-mover advantage.
There are also games where neither is the case.
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Application: Stackelberg Model
Back to oligopoly, quantity competition: suppose there are two firms,
and Firm 1 picks quantity before Firm 2. Firm 1 is the industry
leader, and Firm 2 is the follower.
For example, Firm 1 commits to quantity by signing a contract with
distributors, or by buying lots of inputs, etc.
Simplest case: both firms have the same constant marginal cost c,
produce a homogeneous good, and face linear market demand
P = a − bQ.
We use backward induction to solve for a subgame-perfect
equilibrium.
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Application: Stackelberg Model (II)
We know from our analysis of the Cournot model that Firm 2’s best
response to q1 is
a−c
q1
q2 (q1 ) =
−
2b
2
By backward induction, instead of taking as given a constant q2 , Firm
1 will take as given Firm 2’s above best response: Firm 1 knows that
q2 now depends on q1 .
Firm 1’s profit function:
q1 (a − b(q1 + q2 ) − c)
a−c
q1
= q1 (a − b(q1 +
− ) − c)
2b
2
1
2
=
((a − c)q1 − bq1 )
2
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Application: Stackelberg Model (III)
Taking the first-order condition and rearranging gives:
q1 =
a−c
2b
Plugging back into Firm 2’s best response function gives:
q2 =
a−c
4b
Compare to Cournot outcome:
q1 = q2 =
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a−c
3b
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Application: Stackelberg Model (IV)
Stackelberg profits are:
π1 =
1 (a − c)2
1 (a − c)2
, π2 =
8
b
16
b
Compare to Cournot profits:
π1 = π2 =
1 (a − c)2
9
b
Who benefits, and why?
What about the market price?
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Centipede Game
Player 1 and 2 are in a partnership: the opportunities are the
money lying on the table. Assume that in the first round, there
are two piles (one for each player): a pile of $4, and a pile of $1.
Player 1 has two options in the first round, either to stop (and
grab a pile of money), or to continue the partnership. If he stops,
the game ends and he gets $4 while Player 2 gets the remaining
$1. If he continues, the game moves to the second round: the
two piles are doubled (to $8 and $2), and Player 2 face with a
similar decision: stop (the game ends, Player 2 gets $8, and
Player 1 gets $2), or continue (the piles double again, and Player
1 decide at round 3). The game continues for n rounds.
Assume that n = 4. Draw the game tree. Find the SPE by backward
induction.
How many strategies does Player 1 have? Player 2?
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Ultimatum game
There is a dollar to be divided between Alice and Bob. Alice
proposes a division, which is an integer between 0 and 100
denoting the amount that she gets in cents (e.g., 50 cents for
herself), and Bob gets the rest. After learning Alice’s proposal,
Bob responds with an Yes or No. If Bob says yes, Alice’s proposal
is implemented; if Bob says no, both of them get nothing.
How many strategies does Alice have? And Bob?
Draw the game tree.
What are the subgame perfect Nash equilibria?
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Ultimatum game (answers)
Alice has 101 strategies, and Bob has 2101 strategies.
There are two (pure-strategy) subgame perfect equilibria:
1
2
Alice proposes 99; Bob accepts Alice’s proposal if and only if it is
smaller than or equal to 99.
Alice proposes 100; Bob accepts every proposal of Alice.
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