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Q: game theory ( Answered 5 out of 5 stars,   0 Comments )
Subject: game theory
Category: Business and Money > Economics
Asked by: vitaminc-ga
List Price: $7.00
Posted: 05 Nov 2002 19:29 PST
Expires: 05 Dec 2002 19:29 PST
Question ID: 100006
Construct an extensive form game in which every equilibrium is subgame
perfect; construct one with exactly one subgame perfect equilibrium.
Subject: Re: game theory
Answered By: rbnn-ga on 09 Nov 2002 13:56 PST
Rated:5 out of 5 stars
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:


          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.
vitaminc-ga rated this answer:5 out of 5 stars
thanks for help^^

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