Game Theory Iterated Dominance, Rationalizability

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Game Theory Iterated Dominance, Rationalizability

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

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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

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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

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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)

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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

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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

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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

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(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

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(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)

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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

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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)

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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
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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

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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

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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

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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

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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

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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

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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

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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