Games with incomplete information

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Games with incomplete information

• Game Theory, Fall 2006/07
• Steffen Hoernig, FEUNL

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Incomplete Information

• Players have information about themselves which
  other players may not have
• could be strategies and/or payoffs
• modeled by type of payoff
• Examples
• Cost functions of rival firms
• Cooperativeness of other players
• First normal form games
• Later extensive form games

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Types and Beliefs

• Player i is of type ti ?Ti
• Other players have probability distribution over
  types of player i
• Interpretation individual beliefs
• Assumption all have the same distribution over
  types of player i (common prior) gt have
  non-contradicting beliefs
• Will use these beliefs to calculate expected
  utilities!

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Utility and Strategy

• Each player has
• action set Ai
• utility function ui(at),where a (a1,,an), t
  (t1,,tn)
• Strategies
• each type of each player is treated as a separate
  player!
• Strategy si(ti) ?Ai

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Bayes-Nash Equilibrium

• Given rivals strategies s-i, the payoffs of type
  i of player i areU(ai,s-i,ti)
  Et-iu(ai,s-i(t-i)ti,t-i) ?t-i p(t-i)
  u(ai,s-i(t-i)ti,t-i)
• BNE Vector of strategies such that
  U(si(ti),s-i,ti) ? U(ai,s-i,ti) for all ai?Aigt
  each type maximizes his expected utility!
• Comes in pure and mixed strategy versions

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Representation by Extensive Form

• Idea of Harsanyi
• Realization of types represented as unobserved
  choice by a player called naturegt
  representation as imperfect information through
  information sets!
• Do not need to reinvent the (game) wheel
• Can use techniques developed for extensive form
  games

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Example Cournot Duopoly

• Assume firm 1 does not know firm 2 cost
• Inverse demand P a bQ
• Firm 1s beliefs about cost of firm 2
• Some distribution f over cl,ch
• Expected value is Ex(c) ?x(c)f(c)dc
• Given strategy q2(c2) of firm 2, firm 1
  solvesmaxq1 Ec2(a b(q1q2(c2) c1)q1

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Cournot Duopoly (2)

• Since profits linear in q2, simplifies to maxq1
  (a b(q1Ec2q2(c2) c1)q1
• Best response as usualR1(Ec2q2(c2))
  (a-c1)/2 Ec2q2(c2) /2
• Firm 2 of type c2 solves, given strategy q1 of
  firm 1, maxq2 (a b(q1q2) c2)q2gt result is
  q2(c2) (a c2)/2 q1/2

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Cournot Oligopoly (3)

• Bayes-Nash equilibrium
• One quantity for firm 1
• One quantity for each type of firm 2!
• Solution
• Ec2q2(c2) (a Ec2c2)/2 q1/2
• plug into R1

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Cournot Duopoly

• Two-sided uncertainty
• Do for both players as we did for player 1
• BNE one quantity for each type of each player
• (Here this stops to be exciting)
• Lets move on to extensive form games

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Readings and Exercises

• Readings Dutta ch. 21, 22
• Exercises Dutta, ch. 21.2-21.9