博弈论课件 1

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Lecture 1

Introduction to Game Theory,Extensive &Normal Form,Mixed Strategies and Beliefs 2

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Readings

?Watson: Strategy_ An introduction to game theory

–Ch 1‐5:1rd ed p.1-40; 3rd ed p.1‐46.

?Introduction;The Extensive Form;Strategies &the Normal Form;Beliefs,Mixed Strategies and Expecte d Payoffs;General Assumptions and Methodology.?Appendix A:Review of Mathematics p.409‐420.3

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Outline

?Introduction.

?Extensive form representation.

?Strategies.

?Examples.

?Normal form representation.?Mixed strategies.

?Beliefs.

?Expected payoffs.

?General assumptions.

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Game Theory

?Mathematical models of strategic situations :

–Each agent’s behavior affects the well-being of other agents.?Perfect competition and monopoly are not strategic situations.

–In perfect competition no agent considers the specific action of any other individual agent.

–In monopoly the monopolist doesn’t worry about specific consumer’s actions.(Chooses price/quantity based on overall demand.)?Duopoly (2firms producing)is strategic.

–Each firm considers the other’s action when deciding on its own.

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Ch2:The Extensive Form

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Example 1

?A family is on vacation is San Diego.Their young children decide whether to behave or misbehave duri ng breakfast.

?After breakfast the parents decide whether the family goes to Legoland(乐高) or sits quietly in their hotel room for the rest of the day.

?This is a strategic situation.–Each party considers what the other is likely to do (or has already done)when making their own decision.

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Formal Definition of a Game ?A game is (formally):

–A list of players ;

–Specification of all possible actions each player can take and when;

–The players’knowledge ?(what each player knows when he acts);

–How actions lead to outcomes ;

–A specification of preferences over outcomes.?A game can be non-cooperative or cooperative .–Non-cooperative:All decisions are made individually

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Example 1

?Players:

–The children and their parents.

?Actions:

–Children –{Behave,Misbehave}

–Parents –{Legoland after the kids behave, Hotel after the kids behave, Legoland after the kids misbehave, Hotel after the kids misbehave}?Knowledge:

–We typically assume the players know the game and that each player is rational.

–Children –Don’t know anything else.

–Parents –Knows whether the children have behaved or misbehaved.?Outcomes &Preferences:

–We still need to specify these.We’ll do so later.9

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Extensive Form Representation ?One way to represent games is with the extensive form.

–Nodes:Where players choose actions (or the game ends).–Branches:Specific actions.

–Labels: ?Player making a decision (for decision nodes);

?or actions (for branches).

–Payoffs:Represent preferences over outcomes.

–Information sets:Reveal what a player knows when he or she makes a decision.

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Example1:Extensive Form

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Information Sets

?An information set of player i is a collection of player i ’s nodes among which i cannot distinguish.–Perfect information : all information sets in the tree have just one node.

–Imperfect information : not perfect information.12

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Example 1:Information Sets ?The children have one information set.–The initial node.

?Their parents have two information sets.

–When the parents make their decision they knows if the children behaved or misbehaved.?This is why their actions are labeled differently.

?For this example each information set is a single node.

?Each decision is associated with a single information set.13

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Example 1(b)

?The children decide to behave or misbehave.?Their parents decide whether to take the family to Legoland or sit quietly in their hotel room.–But they don’t observe whether the children behaved or misbehaved.?This time the parents only have one information set.–Their decision cannot be contingent on anything.

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Example1(b):Extensive Form

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Example 1(b):Information Sets ?The children have one information set.–The initial node.

?The parents also have one information set.–When the parents make their decision they don’t know if the children behaved or misbehaved.?They don’t know which node they’re at when they makes their decision.

?This is why their actions are labeled the same at both nodes.?Each decision is still associated with a single information set.16

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Strategies

?Strategy:A complete contingent plan for a player in a game.

–Prescribes an action for each of this player’ information sets.

?Example 1–strategy example:

–Children:Behave.

–Parents:Sit quietly in the hotel room if the children behave,and go to Legoland if the children misbehave.?Example 1(b)–strategy example:

–Children: Behave

–Parents: ?

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Notation

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Example 1

?S Children ={Behave,Misbehave},or for short {B ,M }?S Parents ={Legoland if behave and Legoland if misbehave,Legoland if behave and Hotel if misbehave,Hotel if behave and Legoland if misbehave,Hotel if behave and Hotel if misbeha ve}–We can simplify this notation to {LL ’, LH’, HL ’, HH ’}?One specific s is (B , LL ’)

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Example 1(b)

?S Children ={Behave,Misbehave}

?S Parents ={Legoland,Hotel}

?One specific s is (Behave,Legoland)20