An example of Sequential Game with incomplete information
Pedro Silva
Introduction
This note is to help you understand the steps necessary to solve a game of incomplete information like
the one in this week’s homework. This example is different in many ways from the homework exercise,
but the methods can be applied to solve said exercise. I encourage you to read this notes carefully, as
the coverage of such games in our section has been minimal.
In essence, “incomplete information” means that one of the players does not know the preferences of
the other. In the example here, the first player does not know the preferences of second player, so first
player cannot predict with certainty what second player will do. First player has to take into account the
probability that player two will move one way or another.
I first present the game in complete information and solve it by backward induction. I then introduce the
same game under incomplete information. I solve it using a special type of backward induction that
takes into account probabilities.
Game 1: Syria uprising with anti-West rebels. Sequential game with complete information.
Syria is currently in a Civil War between the government and rebel groups. In the game below, the
Syrian government is strong enough to survive the uprising if USA does Nothing. In that case, the Syrian
government will have the power to implement Democracy or to keep the current Autocracy. It will
choose Autocracy because, as we know after many years of Assad’s regime, this is their preferred
outcome. USA has preferences: Democracy > Autocracy > Terrorist safe haven. As long as USA is rational,
it will choose to Do Nothing.
I assume that, if USA decides to support
rebels, rebels will win the war and
become the government. If the rebels
become the government, they would
have the opportunity to transform Syria
into a peaceful democracy or a Terrorist
safe haven (suppose these are mutually
exclusive and exhaustive options). Now
suppose the rebels have been heavily
infiltrated by anti-West terrorist groups.
They will prefer to transform Syria into a
Terrorist safe haven rather than a
peaceful democracy.
1
In a sequential game with complete
information, USA knows that Syria
rebels are infiltrated by anti-West
terrorists. USA infers that if rebels win
the conflict, they would transform Syria
into a Terrorist safe haven with
certainty. By backward induction, USA
knows that supporting rebels will result
in Terrorist Safe Haven (with a payoff
of 0), while doing nothing would result
in the autocracy surviving the conflict
(and a payoff of 1).
The SPE is (Nothing; Terrorist, Autocracy), and the backward induction is illustrated in the figure above.
Game 2: Syria uprising with pro-democracy rebels. Sequential game with complete information.
Now let’s consider a similar game
between USA and Syrian governments,
but now with a group of rebels that
are not infiltrated by anti-West
terrorists, but rather pro-democracy,
anti-terrorist, with leader who avoid
violence eve in times of war, who are
nothing less than freedom fighters.
The game below illustrates this
situation. Notice that this game is
exactly the same as before, except for
that Syria government prefers
Democracy in its top node.
By backward induction, USA knows
that a Syrian government formed by
ex-rebels will choose Democracy with
certainty,
given
the
complete
information it has about the
preferences of the other state. A
rational USA will choose then to
Support the rebels, because it values
Democracy above Autocracy.
The backward induction is illustrated above, with SPE = (Support Rebels; Democracy, Autocracy).
2
Game 3: Syria uprising, unknown rebel preferences. Sequential game with incomplete information.
In the previous two games, USA knew with
certainty the preferences of Syria government.
It is more realistic, however, to suppose that
USA is uncertain about the preferences of the
rebels, and therefore uncertain of the
preferences of a future Syrian government if
rebels win the conflict.
The figure on the top right illustrates what is
going on. USA does not know which game he is
playing. Nature randomly chose the actual type
of rebels, but does not inform to USA. With
probability 𝑝, rebels prefer to transform Syria in
a “Terrorist safe haven”. With probability 1 − 𝑝,
rebels will prefer “Democracy” instead.
USA can use backwards induction to predict the
choices of Syria in each game (figure on the
bottom right). But USA doesn’t know which
game it is playing.
We can think about the decision of USA in
probabilistic terms. If USA chooses to Do
Nothing, then the current government, whose
preferences USA knows pretty well, will keep
the current Autocracy. To “Do Nothing” (DN)
results in a payoff of 1 for USA with certainty.
𝐸𝑃(𝐷𝑁) = 1
If USA chooses to “Support rebels”, Syria will
become a Terrorist safe haven with probability
𝑝 , and it will become a Democracy with
probability 1 − 𝑝. While USA cannot know for
certainty what will happen, it can calculate its
expected payoff if it chooses to “Support rebels”
(SR):
𝐸𝑃(𝑆𝑅) = 𝑝(0) + (1 − 𝑝)(4)
We can now ask the question: for what values
of 𝑝 will USA prefer to “Support rebels” over
“Do Nothing”?
𝐸𝑃(𝑆𝑅) > 𝐸𝑃(𝐷𝑁)
↔
3
𝑝(0) + (1 − 𝑝)(4) > 1
↔
0 + 4 − 4𝑝 > 1
↔
4 − 1 > 4𝑝
↔
4𝑝 < 3
↔
3
𝑝 < = .75
4
So as long as the chance of having anti-West
rebels is smaller than 75%, USA will prefer to
Support Rebels. If the probability of rebels
being anti-West is exactly 75%, USA will be
indifferent. If it is larger, USA will prefer to do
Nothing.
Notice that by knowing 𝑝, we could determine
by probabilistic backward induction what is the
best choice for USA. The figures on the right
illustrate this.
4