Making Your Life Easier in finding NE in Dynamic
Games of Perfect and Complete Information
David Ronayne
2 December 2011
We spoke about the difference of NE and SPNE (sub-game perfect NE) in class because
the vital thing is to get the intuition across about the difference between the two. Practice
everything on PS3 and you are basically going to guarantee yourself full marks on any
question in the exam on NE and SPNE, given you understand everything. You must clear
up ANY misunderstanding as the differences in these things are subtle and can lead to
errors.
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Finding NE
We spoke about how to find SPNE - use backwards induction. Here I am speaking about
the part of the question which says how to find NE. Take any game tree, for example the
professor and student slapping game we did in class. The tree looks like this:
S
S
N
N
N
S
We can put this back into the tables that we used for NE-finding in simultaneous-move
games with one stage, and find all the NE in the usual way (underlining payoffs). You just
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have to be careful with what you call strategies! Recall in these dynamic games, a strategy
for a player is a list of strategies chosen at each node where it is his turn to decide. Hence
in this game strategies are not simply S and N for the student and the professor (as they
are in the simultaneous-move version of this game).
Now the strategies are S and N for the student (same as the simultaneous-move world as
he only has one decision node in the dynamic setup), but the professor’s are different - his
strategies are now SS, SN, NS and NN. The first letter means what he chooses at the first
decision node of his σ1 and the second letter means what he does at his second decision
node σ2 . So the table in the simultaneous move version had 4 areas as the grid was 2x2,
now the grid is 2x4 because the professor has more strategies in this dynamic version of the
game. The grid you should draw to find the NE is hence:
Professor
Student
S
N
SS
-1,-1
-2,0
SN
-1,-1
1,1
NS
0,-2
-2,0
NN
0,-2
1,1
Now use the ‘red-pen algorithm’ i.e. find the NE by the usual underlining technique. And
lo and behold you find all the NE of this game.
We covered how to find the SPNE extensively in class, using backwards induction on the
tree. Now to practice, do the following questions:
(i) Find all the pure NE of this game
(ii) WIll there be a SPNE for this game? If there will be then will it be the same as one
of the NE? If you think there is a SPNE, find one.
(iii) Argue to yourself (using the tree and the example in question - i.e. picture the game
being played) that one of the NE that are not SPNE have this ‘incredible threat’
flavour to them. Argue to yourself why the other non-SPNE NE is an NE.
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