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Belief system foundations of backward induction

By Larry Bell,2014-11-01 09:58
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Belief system foundations of backward induction

    ANTONIO QUESADA

BELIEF

    SYSTEM

    FOUNDATIONS OF

    BACKWARD INDUCTION

    ABSTRACT. Two

    justifications

    of

    backward induction (BI)

    in

    generic perfect information games

    are

    formulated using

    Bonanno??s (1992;

    Theory

    and

    Decision 33,

    153)

    belief

    systems. The

    first

    justification

    concerns the

    BI

    strategy profile

and

    is

    based

    on

    selecting a

    set

    of

    rational belief systems from

    which

    players have

    to

    choose their

    belief functions. The

    second justification

    concerns the

    BI

    path

    of

    play

    and

    is

    based

    on

    a

    sequential deletion of

    nodes

    that

    are

    inconsistent

    with

    the

    choice

of

    rational belief

    functions.

KEY

    WORDS:

    Backward induction, Bonanno??s belief

    systems, Perfect

    information game,

    Rationality.

1.

    INTRODUCTION Though

    generic

    perfect

    information (PI)

    games

    have

    a

    unique

    backward induction (BI)

    strategy profile

    (and,

    therefore, a

    unique

    BI

    path

    of

    play

    and

a

    unique

    BI

    outcome), Selten

    (1978), Rosenthal (1981), Binmore (1987), Basu

    (1988,

    1990),

    Bicchieri (1989)

    (see

    also

    Dufwenberg and

    Lind??n (1996)), Bonanno (1991)

    (see

    also

    Vilks

    (1994)) and

    Reny

    (1993)

    contributed to

    generate the

    perception that

    BI

    lacked

    firm

    foundations.

    Subsequently,

    Bonanno (1992), Bicchieri

and

    Antonelli (1995),

    Aumann

    (1995,

    1996)

    (see

    also

    Bin-

    more

    (1996)), Samet

    (1996),

    Ben-Porath (1997),

    Balkenborg and

    Winter

    (1997),

    Stalnaker (1998),

    Schuhmacher (1999)

    and

    Bonanno

    (2001)

    have

    provided (epistemic rationality) conditions justifying the

    choice

    of

    the

    BI

    strategy profile

    or

    the

    attainment of

    the

BI

    outcome.

This

    paper

    also

    suggests answers to

    the

    question of

    what

    justifies playing according to

    BI

    in

    generic PI

    games.

    The

    justifications

    are

    formulated using

    Bonanno??s (1992)

    belief

    system

    framework (see

    Section 2

    for

    definitions)

    and

    capture ideas

    very

    similar to

those

    motivating the

    concept

    of

    rationalizable

    strategy profile

    in

    Bernheim

Theory

    and

    Decision 53:

    393?C403, 2002.

    .

    2003

    Kluwer

    Academic Publishers. Printed

    in

    the

    Netherlands.

    ANTONIO QUESADA

(1984)

and

    Pearce (1984). The

    choice of

    the

    belief system framework derives from

    the

    conviction that

    a

    satisfactory

    analysis of

    PI

    games

    (even

    from

    an

    epistemic point

    of

    view)

    need

    not

    resort to

    the

    addition of

    extraneous structure (like

    state

spaces

and

    partitions of

    the

    state

    space) as

    occurs, by

    way

    of

    example, in

    Aumann (1995). In

    fact,

    Battigalli and

    Bonanno (1999) criticize this

    strategy on

    the

    grounds that

    it

    generates an

    artificial epistemic model, in

    the

    sense

    that

    there

    is

    nothing in

    the

model

    indicating that

    one

    is

    referring to

    the

    PI

    game

    to

    which

    the

    new

    elements are

    attached.

In

    addition, introducing more

    structure into

    the

    PI

    game

    can

    contribute to

    make

    the

    analysis unclear, as

    the

    interpretation

    of

    the

    new

    elements have

    to

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