To find an extensive form game where each equilibrium is subgame
perfect, just choose a game in which all payoffs are equal, say (1,1)
.
For example, a game in which player 1 has two choices at the root, and
for which after each choice player 2 only has 2 choices. For each of
the 4 terminal nodes make the payoff (1,1) .
Then obviously any strategy at all is an equilibrium, and any
strategy is a subgame perfect equilibrium.
For the second part of this question, the simplest way to do this
would be this .
The nodes are arranged like this:
0
1 2
3 4 5 6
Player 1 moves from 0 to 1 with move a and 0 to 2 with move b .
Player 2 moves from 1 to 3 with move A, from 1 to 4 with move B, from
2 to 5 with move C, and from 2 to 6 with move D.
Now put payoffs of:
(10,10) at 3
(0,0) at 4
(-10,10) at 5
(-100,-100) at 6 .
There are only two proper subgames, the subgames starting at nodes 1
and 2 . For each of these the only equilibrium strategy is when player
2 chooses A and C respectively.
Similarly, player 1's only equilibrium strategy must be to choose a
with probability 1 and b with probability 0.
Hence the only subgame perfect equilibrium strategy is when player 1
chooses a with probability 1, and player 2 chooses A from node 1 with
probability 1 and C from node 2 with probability 1 .
Note that this is not the only equilibrium strategy, since player 2
could could whatever he wants from node 2 without changing the payoff,
since the whole node is unreachable: it's the only subgame perfect
equilibrium strategy though.
Search Strategy
---------------
Game Theory Evolving, by Herbert Gintis. Princeton University Press,
Princeton, 2000.
A net search on subgame perfect equilibrium turned up little that was
useful. I think for mathematics, paper textbooks still seem to be the
best way to go. |