Chapter-9

# Chapter-9 - SG with Imperfect Information The Kidnapping...

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SG with Imperfect Information The Kidnapping game with simultaneous ransom and release decisions . 1 4 2 2 5 3 4 1 3 5 Release Release Kill Kill Guy Guy Vivica Do not pay ransom Pay ransom Guy Vivica Do not kidnap Kidnap I II III IV Imperfect information: some information set is not singleton.

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Strategy Guy’s strategy: whether to kidnap; if kidnap whether to release or kill. Guy has to play the same action in the last two decision nodes, since he cannot distinguish these two decision nodes. Can we solve this game by backward induction? Guy’s strategy: whether to kidnap; if kidnap whether to release or kill. Guy has to play the same action in the last two decision nodes, since he cannot distinguish these two decision nodes. Can we solve this game by backward induction?
Subgames Subgame perfection requires optimal play at each information set. However, we cannot start backward induction at decision nodes that are not singleton. Identify subgames, “game within a game.” A subgame - Starts at an information set that is a singleton - Includes all the subsequent branches of the game tree. - Does not contain any broken information set. Subgame perfection requires optimal play at each information set. However, we cannot start backward induction at decision nodes that are not singleton. Identify subgames, “game within a game.” A subgame - Starts at an information set that is a singleton - Includes all the subsequent branches of the game tree. - Does not contain any broken information set.

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1 4 2 2 5 3 4 1 3 5 Release Release Kill Kill Guy Guy Vivica Do not pay ransom Pay ransom Guy Vivica Do not kidnap Kidnap I II III IV Subtree 1 Subtree 2 Subtree 4 Subtree 3
0 2 0 1 1 1 2 3 1 1 2 0 0 3 2 0 4 1 3 0 3 1 4 2 ab xx yy cccc dddd 1 22 33 3

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Subgame Perfect Nash Equilibrium Sub-strategy for a subgame is the part of a strategy that prescribes behavior only for information sets in that subgame. Subgame perfect Nash equilibrium: A strategy profile is a subgame perfect Nash equilibrium if, for every subgame, its sub-strategy profile is a Nash equilibrium. Generalized backward induction : - Find Nash equilibrium for the smallest subgame. - Reduce the original game by the equilibrium payoff in the smallest subgame. - Repeat the procedure until the initial decision node. Sub-strategy for a subgame is the part of a strategy that prescribes behavior only for information sets in that subgame. Subgame perfect Nash equilibrium: A strategy profile is a subgame perfect Nash equilibrium if, for every subgame, its sub-strategy profile is a Nash equilibrium. Generalized backward induction : - Find Nash equilibrium for the smallest subgame. - Reduce the original game by the equilibrium payoff in the smallest subgame.
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Chapter-9 - SG with Imperfect Information The Kidnapping...

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