Detailed description of a dynamic game. (i) Players Nature, M, and W. Extensive Form. (ii) Game Tree. (ii) Game Tree

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1 (a) Extensive orm Representation of Dynamic Games (a) Extensive orm Representation of Dynamic Games Example: Man and oman going out for a date Detailed description of a dynamic game ootball () or hopping (), if it does not. Man moves first, and oman makes her choice without seeing Man s move. M (a) Extensive orm Representation of Dynamic Games Example: Man and oman going out for a date Extensive orm Extensive orm consists of (i) Players, M, and M A dummy yplayer who chooses random events M Extensive orm consists of (ii) Game Tree A collection of nodes and branches: It shows how things happen in a dynamic game. (ii) Game Tree A collection of nodes and branches Decision nodes 1

2 (ii) Game Tree - ormal definition of a game tree (lecture) - A collection of nodes and branches Terminal nodes t = 1 t = 2 hy a tree? NOT a tree Even if today s weather is identical (), those points are treated as different states, because their previous histories are different. The most detailed description a tree structure Important lesson #1 Game theory represents a dynamic game by a game tree because Extensive orm consists of (iii) Information ets indicate (a) who moves at each decision node, and (b) what the player observes when s/he moves. it is the most detailed way to describe dynamic events M 2

3 Extensive orm consists of (iii) Information ets The fact that this is an information set of oman means Extensive orm consists of (iii) Information ets that oman moves here, without observing if M chose or. M M Extensive orm consists of (iii) Information ets - ormal definition of information sets (lecture) - If oman observes Man s move, then the information sets look like M Important lesson #2 hat does an information set mean? An information set of player i (at time t) History of past events that i can observe (at time t) hy? one-to-one mapping 3

4 t 2 Information sets of 3 T B 3 History of events that 3 can observe 1 chose t A (but important) technical point Extensive orm assumes that only one player moves at each decision node. Example: hen two players move simultaneously, extensive form assumes that one player moves first and then the other player moves without observing the first player s choice. 1 3 Player 2 b 2 T 1 chose b Player 1 B Extensive orm consists of (iv) Actions available at each decision node - ormal definition of actions available at each information set (lecture) - The outgoing branches indicate the actions available at each decision node M Extensive orm consists of (v) Probabilities of random events Extensive orm consists of (vi) Payoffs (u 1,,u N ) are indicated at the terminal nodes M M 4

5 An Extensive orm describes a dynamic game by specifying items (i) (vi) (i) Players (ii) Game tree (iii) Information sets (iv) Actions available at each moment of time (v) Probabilities of random events (vi) Payoffs More precisely. An Extensive orm consists of ( i=0,,n, T, H 0,,H N, A( ),,, u 1,,u N ) Players Game tree Information sets Actions available at each information set An Extensive orm describes Payoffs Mapping from branches to actions Probabilities of random events who moves when, and what players observe when they move An Extensive orm consists of ( i=0,,n, T, H 0,,H N, A( ),,, u 1,,u N ) Players Game tree Information sets Actions available at each information set Payoffs Mapping from branches to actions Probabilities of random events In practice, we often do not explicitly write down those mathematical objects in our research papers. Instead, we carefully describe a dynamic game by words so that those mathematical objects CAN be ded clearly. (b) hat is a strategy in a dynamic game (in an extensive form)? A strategy of a player specifies which action to take at each information set of the player. A strategy t of Man M A strategy of oman * A common misunderstanding A strategy specifies One action at each information set trategy = actions actually taken (on the path of play) Ex: oman observes Man s action A very simple definition, but this may be very different from your common-sense idea of a strategy. M 5

6 * A common misunderstanding trategy = actions actually taken (on the path of play) Man s strategy is alse! oman s strategy is M * The right definition trategy = a complete contingent action plan to specify what to do, is to specify an action at each information set even after the events that never happen on the path of play. M * The right definition is to specify an action at each information set Important lesson #3 oman s strategy specifies. if Man chose a contingent action plan if Man chose M A strategy in an extensive form (i.e., in a dynamic game) is A complete contingent action plan, which specifies what to do, even after events that never happen on the path of play A complete specification of actions at all information sets If you fail to specify what action is taken here (off the path of play) hy should we de a strategy in that way? M? 6

7 If you fail to specify what action is taken here, you cannot determine what Man should do. M?? There are two ways to represent a dynamic game. 1. Extensive orm 2. Normal orm * Another way to represent a dynamic game s i i i s strategy in the extensive form Contingent action plan i (s 1,,s N ) payoff to player i The set of all extensive form strategies { i, i, i=1,,n} is a Normal orm (or trategic orm) representation of a dynamic game It regards a dynamic game as a simultaneous move game, where players choose their action plans at t = 0. Ex.: Entry Game a b (0, (, 2) c d (-1, -1) (1, 1) Extensive orm a b c 1, -1 0, 2 d 1, 1 0, 2 Normal orm In general, you cannot recover the extensive form from a normal form. The Extensive orm is a more detailed description of a dynamic game than the Normal orm is. - Mixed strategies in an extensive form and Kuhn s Theorem (lecture) - - blank - 7

8 (c) ubgame Perfect Equilibria Now let us see how to solve dynamic games. Q. hich part of a dynamic game can be analyzed as a small game played within the dynamic game? The definition of ubgames A part of an extensive form is a subgame if, and * In our definition, the dynamic game itself is a subgame (because it satisfies (i)-(iii)). A ubgame M Let us understand the meaning of conditions (i) (iii). NOT a subgame M 8

9 You cannot determine what happens in unless you know where it starts M This part cannot be treated as one game M NOT a subgame M This part cannot be treated as a game, because oman does not know if she is in M Conditions for a subgame Now we know why we need (i) and (iii) (otherwise the part of the dynamic game cannot be analyzed as a game) but intuitively what do (i) and (iii) mean? ubgame conditions (i) and (iii) Impliy All players in a subgame observe everything that has happened before the subgame starts. 9

10 NOT a subgame NOT a subgame oman does not observe what Man did before this part starts. M oman does not observe what Man did before this part starts. M ubgame conditions (i) and (ii) Man and oman observe what happened before the subgame starts. A ubgame M Impliy All players in a subgame observe everything that has happened before the subgame starts. Important lesson #4 An intuitive and easy way to find a subgame If all players (who are going to move from now on) observe everything that has happened so far, then a subgame starts! 10