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 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
 ```1) Construct an extensive form game in which every equilibrium is subgame perfect; construct one with exactly one subgame perfect equilibrium.```
 ```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.```
 vitaminc-ga rated this answer: `thanks for help^^`