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Game Theory Iterated Dominance, Rationalizability

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Title: Game Theory Iterated Dominance, Rationalizability


1
Game Theory Iterated Dominance, Rationalizability
  • Univ. Prof.dr. M.C.W. Janssen
  • University of Vienna
  • Winter semester 2008-9
  • Week 42 (October 13, 15)

2
Dominated Strategies
  • A strategy is strictly dominated if there is
    another (mixed) strategy that is always better,
    no matter what the opponents are choosing
  • Formally, strategy sets for player i are denoted
    by Si and Si (for mixed strategies) pay-offs by
    ui(si ,s-i )
  • Strategy si ? Si is dominated if there exists
    another strategy si ? Si s.t.
  • ui(si ,s-i )lt ui(si ,s-i ) for all s-i ?S-i

3
Example I
  • Why do we explicitly allow for domination by
    mixed strategies?
  • M is not dominated by a pure strategy, but it is
    strictly dominated by a mixed strategy, e.g., by
    (½,0,½)
  • Thus, M is strictly dominated

4
Clarification II
  • Why do we not explicitly allow for other players
    playing mixed strategies?
  • If a strategy is dominated for all pure
    strategies of the other players, then it is also
    dominated for all mixed strategies
  • Vice versa, if a strategy is dominated for all
    mixed strategies of the other players, then it is
    also dominated for all pure strategies (trivial)

5
Example II mixed strategies can also be
dominated
  • Clearly, a mixed strategy given positive weight
    to a dominated pure strategy is itself dominated
  • But, other mixed strategies can be strictly
    dominated as well
  • None of the pure strategies is dominated here,
    but the mixed strategy (½,½,0) is strictly
    dominated by D

6
Example III Second price sealed-bid auction
  • Suppose there are I bidders, each with valuation
    vi for the object.
  • Suppose here valuation is known.
  • Sealed bid implies it is a simultaneous move
    game, si bi second price means highest bidder
    wins and pays second-highest bid
  • Bidding your valuation is dominant strategy,
    i.e., all other strategies are dominated by si
    vi
  • Proof

7
Iterated dominance
  • Let S0i Si and S0i Si and define iteratively
    all strategies that are not dominated yet as Sni
    si ? Sn-1i ? there is no si ? Sn-1i
    s.t.
  • ui(si ,s-i ) gt ui(si ,s-i ) for all s-i ?Sn-1-i
  • S?i ?n Sni
  • S?i is set of all mixed strategies si such that
    there does not exists a si with ui(si ,s-i ) gt
    ui(si ,s-i ) for all s-i ? S?-i
  • The game is dominance solvable if for all players
    S?i and thus contains only one element
  • If game is finite, a finite number of iterations
    will suffice
  • Justification in terms of common knowledge of
    rationality

8
(Iterated) Weak dominance (IEWDS)
  • Same procedure can be applied to weak dominance
  • Formally, strategy si ? Si is weakly dominated
    if there exists another strategy si ? Si s.t.
  • ui(si ,s-i ) ui(si ,s-i ) for all s-i ?S-i
    with strict inequality for some s-i

9
(Iterated) Weak dominance (IEWDS)
  • Order of elimination may matter for final outcome
  • E.g., one may eliminate in this order U (weakly
    dominated by D), then L. Outcome M and D for
    player 1 and R for player 2
  • Or, eliminate in this order M, R and then U.
    Outcome (D,L)

10
Justification for IEDS clear
  • Order of elimination for iterated dominance does
    not matter
  • Main idea if some strategy is strictly dominated
    for a larger set of strategies of the opponent,
    then it is also dominated for a smaller set.
    Hence, if one does not immediately delete a
    strategy, it should be deleted at some point.
  • Also, one does not need to delete immediately
    mixed strategies

11
Application to Cournot oligopoly
  • N firms
  • Linear market demand P(Q) a-bQ
  • Linear cost C(q) cq
  • Firms want to max (P(qiQ-i) qi C(qi)

12
Best response functions for 2 firms
  • What can be eliminated in first round?
  • everything that is a best response cannot be
    eliminated.
  • only possibility is quantities above monopoly
    output
  • fact that they are not best reponses does not
    necessarily imply they are dominated

qM
13
Elimination in rounds I
  • Round 1 All outputs above monopoly level are
    dominated
  • ?p/?q a b(n-1)q-i c 2bqi lt 0 for all q-i
    and qi gt (a c)/2b
  • thus, monopoly output gives always higher profit
    than larger output levels
  • Round 2 for n2, all output levels smaller than
    (a c)/4b ( optimal response to monopoly
    output) are dominated
  • ?p/?q a bq-i c 2bqi gt 0 for all q-i lt (a
    c)/2b and qi lt (a c)/4b

14
Elimination in rounds II
  • Round 2 for ngt2, no output levels are dominated
  • ?p/?q a b(n-1)q-i c 2bqi lt 0 for q-i (a
    c)/2b and qi gt 0
  • Hence, there are strategies for the other players
    that are not dominated such that qi 0 is a
    better response than qi gt 0.
  • Round 3 (and more) for n gt 2, nothing changes
  • Round 3 (for n 2), output levels larger than
    3(a c)/8b are dominated by 3(a c)/8b
  • ?p/?q a bq-i c 2bqi lt 0 for q-i gt (a
    c)/4b and qi gt 3(a c)/8b

15
Elimination in rounds III
  • Round 4, 6, 8, for n2, again low output
    levels are dominated
  • Suppose that q,q are strategies that remain
    after previous round, with q lt qc lt q lt qm and
    q is best response to q
  • Thus, q (a-c)/2b - q/2 and best response to q
    is (a-c)/2b q/2 (a-c)/4b q/4 gt q if, and
    only if, q lt (a-c)/3b qc
  • For any q lt qc the following is true ?p/?q a
    bq-i c 2bqi gt 0 for q-i lt q (a-c)/2b - q/2
    and qi lt (a-c)/4b q/4
  • Hence, strategies qi lt (a-c)/4b q/4 can be
    eliminated at this round
  • Similarly for odd rounds for n 2
  • Thus, for n2 game is dominance solvable

16
Tilman Borgers Maarten C.W. Janssen
  • This result is due to the way of replication of
    the economy, i.e., only the supply side is
    replicated for fixed demand
  • Another (more natural?) way to replace is to say
    that if for a given n, demand is given by P a
    bQ, then we replicate both sides and say that
    for k 2,3,4, there are nk firms and demand is
    P a bkQ
  • Their main result then says that for general cost
    and demand functions and for large k, the Cournot
    game is dominance solvable iff the Cobweb process
    is dynamically stable

17
Relation between IEDS and naïve learning
  • Dominance solvability is highly sophisticated
    eductive reasoning process (CKR)
  • Cobweb process based on very naïve learning as
    also used by Cournot in creating a dynamic story
    to his theory
  • Relation in three parts
  • IEDS and rationalizability
  • Rationalizability and Cournot dynamics
  • Cournot for large N and Cobweb

18
Play Guessing Game
  • Players have to write down a number between 1 and
    100
  • I calculate average and take 2/3 of it
  • Player that is closest to 2/3 of the average wins
  • Which number do you choose?
  • Analyze the game with dominated strategies

19
Rationalizability
  • Let S0i Si and define iteratively for all
    players i Sni si ? Sn-1i ? there is a s-i ?
    product of the convex hull of Sn-1-i s.t. ui(si
    , s-i ) ? ui(si, s-i) for all si ? Sn-1i
  • The set of rationalizable strategies is R?i ?n
    Sni
  • Intuitively, a strategy is rationalizable if it
    is a best response to some belief about the other
    players play, which should be a best response to
    some belief the other players have about what
    their opponents will do, etc.
  • Convex hull is used to smoothen the process and
    is needed as generally, even though two pure
    strategies themselves can be a best response, a
    mixture of them is not
  • The set of rationalizable strategies is non-empty

20
Rationalizability and IEDS
  • If a strategy is dominated, it is never a best
    response and therefore cannot be rationalizable.
    Thus, rationalizability is a stronger notion
  • For n2, the two actually coincide
  • Graphical illustration
  • For N gt 2, rationalizability is stronger
  • Exercise 2.7 FT
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