Game Theory Static Bayesian Games

1
Game Theory Static Bayesian Games

• Univ. Prof.dr. M.C.W. Janssen
• University of Vienna
• Winter semester 2009-10
• Week 49 (November 30, December 1)

2
In Many Situations Players do not have pay-off
relevant information

• Buyer and Seller negotiating about price
• Willingness to pay, cost (willingness to sell)
• Oligopoly competition
• Cost of other firms in Bertrand, Cournot
• Auctions
• Value of other bidders
• .
• Here, we consider static games
• Later, sequential games and updating (actions may
  then reveal private information)

3
Framework

• Players can be of different types ?i ? Ti
• A firm can have different cost, a buyer may have
  different willingness to pay
• Strategy player i si ?i ? Ai
• can condition action on private information
• Prior probability of types F(?1,.., ?n)
• Updating if types are correlated p(?-i / ?i )
• If types are uncorrelated, knowing your own type
  does not reveal information about types of other
  players posterior prob. prior prob.

4
Example Auction

• Players valuations are uniformly and
  independently distributed over interval 0,1
  this is the prior distribution, vi ? 0, 1
• Valuations are private information and the action
  each player chooses is her bid bi.
• Strategy is function si vi ? bi bi(vi)
• What is then an equilibrium?

5
Bayes-Nash Equilibrium definition

• For def. 1 we define ui (si (.) ,s-i (.))
  E? ui (si (.) ,s-i (.) ?i )
• Def 1. A strategy combination (si (.) ,s-i (.))
  is a Bayes-Nash equilibrium if ui (si (.) ,s-i
  (.)) ? ui (si (.) ,s-i (.)) for all i and all
  si ?Si
• Def 2 A strategy combination (si (.) ,s-i (.))
  is a Bayes-Nash equilibrium if given s-i each
  type ?i chooses an optimal action (action that
  maximizes expected profits taking p(?-i / ?i ))
• Definitions are equivalent
• Profits of types are independent of each other

6
Examples Battles of the Sexes game I

• Consider battles of the sexes game where there is
  some uncertainty about both players pay-off
• t, t ? 0,x
• What is a strategies? Choose B or F depending on
  value of t, t
• Proposal player 1 chooses B iff t gt t player 2
  chooses F iff t gt t
• Reasonable proposal?

7
Examples Battles of the Sexes game II

• For player 2 it looks as if player 1 chooses B
  with probability (x-t)/x
• Equilibrium? Check for player 2
• Ep(B) (x-t)/x 1 t/x 0
• Ep(F) (x-t)/x 0 t/x (2t)
• B optimal iff t lt -3 x/t
• This has to be equal to t
• Similarly, player 1 B optimal if t gt -3 x/t
  t
• t is solution to t2 3t x 0

8
First-Price Sealed-Bid Auction

• Highest bidder wins and pays his own bid.
• Players valuations are uniformly and
  independently distributed over interval 0,1.
• What is equilibrium with n players?
• Pay-off to player i is (vi-bi)Pr(bi gt max bj)
• Suppose strategies are linear, then Pr(bi gt bj)
  Pr(bi gt aßvj) Pr(vj lt (bi a)/ß) (bi a)/ß
  and Pr(bi gt max bj) (bi a)/ßn-1
• Maximizing pay-off wrt bi yields - (bi
  a)/ßn-1 (n-1) (bi a)/ßn-2 (vi-bi)/ß 0
• Solving yields (bi a) (n-1)(vi-bi) or bi
  ((n-1)via))/n
• Linearity requires a0, ß(n-1)/n

9
Second-price sealed-bid auction

• This we already discussed (week 2)
• Strategies are also bids dependent on valuations
  and dominant strategy is to bid valuation bi
  vi
• This is thus also a Bayes-Nash equilibrium

10
Double Auction set-up

• Sellers (Player 1) cost is c buyers valuation
  (Player 2) is v. Both are uniformly distributed
  over 0,1. Both make a bid. If b1b2 then they
  trade at price p (b1b2)/2 otherwise no trade
• Pay-offs
• Seller (b1b2)/2 c if b1b2 otherwise 0
• Buyer v - (b1b2)/2 if b1b2 otherwise 0
• Without asymmetric information, continuum of
  equilibria where both players bid t ? c,v

11
Double Auction asymmetric info I

• Buyer chooses p2 to maximize
• (v-p2E(p1/p1ltp2)/2) Pr (p1 lt p2)
• Seller chooses p1 to maximize
• (p1E(p2/p2gtp1)/2 - c) Pr (p1 lt p2)
• First, consider again linear strategies p1 a1
  b1c so that E(p1/p1ltp2) (a1p2)/2
• Buyers problem reduces to max
• (v-p2(a1p2)/2/2) (p2-a1)/b1
• Solution p2 (2va1)/3
• Similarly for seller p1 (2c(a2b2))/3

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Double Auction asymmetric info II

• Solution for buyer p2 (2va1)/3 implies that
  b22/3 and a2(a1)/3
• Solution for seller p1 (2c(a2b2))/3 implies
  that b12/3 and a1(a22/3)/3
• Thus, a21/12 and a11/4
• In linear equilibrium trade occurs if v c ¼
• Inefficiency
• Other Bayes-Nash equilibria exist, such as
• Buyer offer price p if v p otherwise offer 0
• seller offer price p if p c otherwise offer
  1
• All other equilibria are also inefficient

13
Purification theorem for Mixed Strategy
equilibrium an illustration

• If t0, mixed strategy equilibrium where player 1
  plays B with prob 2/3
• In Bayes-Nash eq. of private info game t, t ?
  0,x, player 1 plays B is if t gt t, with prob.
• What happens when x becomes very small?
  Probability converges to 2/3

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Purification theorem for Mixed Strategy
equilibrium set-up

• Mixed strategy can be considered as the limit of
  a pure strategy Bayes-Nash equilibrium where
  uncertainty disappears.
• Is this a general phenomenon?
• Harsanyi (1973) yes
• Harsanyis set-up perturb pay-offs as follows
  ?si -1,1 e is positive number (going to 0).
  Then ui (s,?i ) ui (s) e?i (Pi is
  continuous distribution of ?i

15
Purification theorem for Mixed Strategy
equilibrium result

• Best reply is essentially unique and in pure
  strategies
• If ?i is continuously distributed it is rare
  event that best replies coincide for interval of
  ?i values
• Any equilibrium for unperturbed pay-offs ui (s)
  is the limit as e?0 of a sequence of pure
  strategy equilibria of the perturbed game ui (s)
• Holds for pure and mixed strategy equilibria,
• But not for pure strategy equilibria in weakly
  dominated strategies

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Exercises

• Make exercises 6.1
• Pay-offs are either as in the upper matrix or as
  in the lower matrix (see right). The Row player
  knows the pay-offs, Column player does not know.
  Prior probability of each matrix is equally
  likely. What are the Bayes-Nash eq?
• Consider the linear Cournot duopoly where players
  have cost c1 and c2. Player 1 knows all cost, but
  player 2 only knows its own cost and it thinks
  player 1 has high cost cH with prob a and low
  cost cL with prob. 1-a.