普林斯顿大学博弈论讲义10

Eco514—Game Theory

Lecture10:Extensive Games with(Almost)Perfect

Information

Marciano Siniscalchi

October19,1999

Introduction

Beginning with this lecture,we focus our attention on dynamic games.The majority of games of economic interest feature some dynamic component,and most often payoffuncertainty as well.

The analysis of extensive games is challenging in several ways.At the most basic level, describing the possible sequences of events(choices)which define a particular game form is not problematic per se;yet,different formal definitions have been proposed,each with its pros and cons.

Representing the players’information as the play unfolds is nontrivial:to some extent, research on this topic may still be said to be in progress.

The focus of this course will be on solution concepts;in this area,subtle and unexpected difficulties arise,even in simple games.The very representation of players’beliefs as the play unfolds is problematic,at least in games with three or more players.There has been afierce debate on the“right”notion of rationality for extensive games,but no consensus seems to have emerged among theorists.

We shall investigate these issues in due course.Today we begin by analyzing a particu-larly simple class of games,characterized by a natural multistage structure.I should point out that,perhaps partly due to its simplicity,this class encompasses the vast majority of extensive games of economic interest,especially if one allows for payoffuncertainty.We shall return to this point in the next lecture.

Games with Perfect Information

Following OR,we begin with the simplest possible extensive-form game.The basic idea is as follows:play proceeds in stages,and at each stage one(and only one)player chooses an

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action.Sequences of actions are called histories;some histories are terminal,i.e.no further

actions are taken,and players receive their payoffs.Moreover,at each stage every player

gets to observe all previous actions.

Definition1An extensive-form game with perfect information is a tupleΓ=(N,A,H,P,Z,U)

where:

N is a set of players;

A is a set of actions;

H is a collection offinite and countable sequences of elements from A,such that:

(i)∅∈H;

(ii)(a1,...,a k)∈H implies(a1,...,a )∈H for all

(iii)If h=(a1,...,a k,...)and(a1,...,a k)∈H for all k≥1,then h∈H.

Z is the set of terminal histories:that is,(a1,...,a k)∈Z iff(a1,...,a k)∈H and

(a1,...,a k,a)∈H for all a∈A.Also let X=H\Z.All infinite histories are terminal.

P:X→N is the player function,associating with each non-terminal history h∈X the

player P(h)on the move after history h.

U=(U i)i∈N:Z→R is the payofffunction,associating a vector of payoffs to every

terminal history.

I differ from OR in two respects:first,Ifind it useful to specify the set of actions in

the definition of an extensive-form game.Second,at the expense of some(but not much!) generality,I represent preferences among terminal nodes by means of a vN-M utility function.

Interpreting Definition1

A few comments on formal aspects are in order.First,actions are best thought of as move

labels;what really defines the game is the set H of sequences.If one wishes,one can think of

A as a product set(i.e.every player gets her own set of move labels),but this is inessential.

Histories encode all possible partial and complete plays of the gameΓ.Indeed,it is

precisely by spelling out what the possible plays are that we fully describe the game under consideration!

Thus,consider the following game:N={1,2};A={a1,d1,a2,d2,A,D};H={∅,(d1),(a1),(a1,D),(a1, thus,Z={(d1),(a1,D),(a1,A,d2),(a1,A,a2)}and X={∅,(a1),(a1,A),};finally,P(∅)=

P((a1,A))=1,P(a1)=2,and U((d1))=(2,2),U((a1,D))=(1,1),U((a1,A,d1))=(0,0),

U((a1,A,a2))=(3,3).ThenΓ=(N,A,H,Z,P,U)is the game in Figure1.

The empty history is always an element of H,and denotes the initial point of the game.

Part(ii)in the definition of H says that every sub-history of a history h is itself a history in

its own right.Part(iii)is a“limit”definition of infinite histories.Note that infinite histories

are logically required to be terminal.

A key assumption is that,whenever a history h occurs,all players(in particular,Player

P(h))get to observe it.

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