Econ 805 Advanced Micro Theory 1

1
Econ 805Advanced Micro Theory 1

• Dan Quint
• Fall 2008
• Lecture 1 Sept 2, 2007
• A Quick Review of Game Theory and, in particular,
  Bayesian Games

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Games of complete information

• A static (simultaneous-move) game is defined by
• Players 1, 2, , N
• Action spaces A1, A2, , AN
• Payoff functions ui A1 x x AN ? R
• all of which are assumed to be common knowledge
• In dynamic games, we talk about specifying
  timing, but what we mean is information
• What each player knows at the time he moves
• Typically represented in extensive form (game
  tree)

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Solution concepts for games of complete
information

• Pure-strategy Nash equilibrium s Î A1 x x AN
  s.t.
• ui(si,s-i) ³ ui(si,s-i)
• for all si Î Ai
• for all i Î 1, 2, , N
• In dynamic games, we typically focus on Subgame
  Perfect equilibria
• Profiles where Nash equilibria are also played
  within each branch of the game tree
• Often solvable by backward induction

4
Games of incomplete information

• Example Cournot competition between two firms,
  inverse demand is P 100 Q1 Q2
• Firm 1 has a cost per unit of 25, but doesnt
  know whether firm 2s cost per unit is 20 or 30
• What to do when a players payoff function is not
  common knowledge?

5
John Harsanyis big idea(Games with Incomplete
Information Played By Bayesian Players)

• Transform a game of incomplete information into a
  game of imperfect information parameters of
  game are common knowledge, but not all players
  moves are observed
• Introduce a new player, nature, who determines
  firm 2s marginal cost
• Nature randomizes firm 2 observes natures move
• Firm 1 doesnt observe natures move, so doesnt
  know firm 2s type

Nature
make 2 weak
make 2 strong
Firm 2
Firm 2
Q2
Q2
Firm 1
Q1
Q1
u1 Q1(100 - Q1 - Q2 - 25) u2 Q2(100 - Q1 - Q2
- 30)
u1 Q1(100 - Q1 - Q2 - 25) u2 Q2(100 - Q1 - Q2
- 20)
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Bayesian Nash Equilibrium

• Assign probabilities to natures moves (common
  knowledge)
• Firm 2s pure strategies are maps from his type
  space Weak, Strong to A2 R
• Firm 1 maximizes expected payoff
• in expectation over firm 2s types
• given firm 2s equilibrium strategy

Nature
make 2 weak
make 2 strong
Firm 2
Firm 2
p ½
p ½
Q2
Q2
Q2W
Q2S
Firm 1
Q1
Q1
u1 Q1(100 - Q1 - Q2 - 25) u2 Q2(100 - Q1 - Q2
- 30)
u1 Q1(100 - Q1 - Q2 - 25) u2 Q2(100 - Q1 - Q2
- 20)
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Other players types can enter into a players
payoff function

• In the Cournot example, firm 1 only cares about
  firm 2s type because it affects his action
• In some games, one players type can directly
  enter into another players payoff function
• Poker you dont know what cards your opponent
  has, but they affect both how hell plays the
  hand and whether youll win at showdown
• Either way, in BNE, simply maximize expected
  payoff given opponents strategy and type
  distribution

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Solving the Cournot example, with p ½ that
firm 2 is strong

• Strong firm 2 best-responds by choosing
• Q2S arg maxq q(100-Q1-q-20)
• Maximization gives Q2S (80-Q1)/2
• Weak firm 2 sets
• Q2W arg maxq q(100-Q1-q-30)
• giving Q2W (70-Q1)/2
• Firm 1 maximizes expected profits
• Q1 arg maxq ½q(100-q-Q2S-25)
  ½q(100-q-Q2W-25)
• giving Q1 (75 Q2W/2 Q2S/2)/2
• Solving these simultaneously gives equilibrium
  strategies
• Q1 25, (Q2W, Q2S) (22½ , 27½)

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Formally, for N 2 and finite, independent types

• A static Bayesian game is
• A set of players 1, 2
• A set of possible types T1 t11, t12, , t1K
  and T2 t21, t22, , t2K for each player, and
  a probability for each type p11, , p1K, p21, ,
  p2K
• A set of possible actions Ai for each player
• A payoff function mapping actions and types to
  payoffs for each player
• ui A1 x A2 x T1 x T2 ? R
• A pure-strategy Bayesian Nash Equilibrium is a
  mapping si Ti ? Ai for each player, such that
• for each potential deviation ai Î Ai
• for every type ti Î Ti
• for each player i Î 1,2

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Ex-post versus ex-ante formulations

• With a finite number of types, the following are
  equivalent
• The action si(ti) maximizes ex-post expected
  payoffs for each type ti
• The mapping si Ti ? Ai maximizes ex-ante
  expected payoffs among all such mappings
• I prefer the ex-post formulation for two reasons
• With a continuum of types, the equivalence breaks
  down, since deviating to a worse action at a
  particular type would not change ex-ante expected
  payoffs
• Ex-post optimality is almost always simpler to
  verify

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Auctions are typically modeled as Bayesian games

• Players dont know how badly the other bidders
  want the object
• Assume nature gives each bidder a valuation for
  the object (or information about it) from some
  ex-ante probability distribution that is common
  knowledge
• In BNE, each bidder maximizes his expected
  payoffs, given
• the type distributions of his opponents
• the equilibrium bidding strategies of his
  opponents
• Thursday some common auction formats and the
  baseline model