ECON6036 1st semester 05-06

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ECON60361st semester 05-06

• Format of final exam
• Same as the mid term
• Material not covered in final exam
• fixed point theoremboth proof and application
• purification of mixed strategy
• rationalizability, dominance solvability
• repeated game overtaking criteria, limit of
  means criteria, Maskin-Tirole theorem
• Cho-Kreps Intuitive criteria

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• Material covered in final exam
• strategic game
• --Nash equilibrium
• Bayesian game
• --Bayesian equilibrium
• extensive game with perfect information
• --subgame perfect equilibrium
• bargaining game
• repeated gamesubgame perfect equilibrium,
  trigger strategy, minmax value
• extensive game with imperfect information
• --perfect Bayesian equilibrium
• --sequential equilibrium

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Exercise 211.1 (Timing claims on an investment)

• An amount of money accumulates in period t ( 1,
  2, ..., T) its size is 2t.
• In each period t two people simultaneously decide
  whether to claim the money. If only one person
  does so, she gets all the money if both people
  do so, they split the money equally either case,
  the game ends.
• If neither person does so, both people have the
  opportunity to do so in the next period if
  neither person claims the money in period T, each
  person obtains T.
• Each person cares only about the amount of money
  she obtains.
• Formulate this situation as an extensive game
  with perfect information and simultaneous moves,
  and find its SPE.

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Equilibrium immediate claiming

• Claim in the SPE, each player always claims
  money whenever she is asked to move
• Proof When tT, it is the strictly dominant
  action for each to claim (by claiming, she gets T
  rather than 0 if the other also claims she gets
  2T rather than T if the other doesnt) gt each
  always claims money at tT
• Assume each always claims money at tk1,,T
• Then at tk, each claiming is also best response
  (by claiming, she gets k rather than 0 if the
  other also claims she gets 2k rather than (k1)
  if the other doesnt).
• (There exists another SPE in which in the first
  period neither claims money but in any subsequent
  period both claim money.)

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EXERCISE 227.3 (Sequential duel)

• In a sequential duel, two people alternately have
  the opportunity to shoot each other each has an
  infinite supply of bullets.
• On each of her turns, a person may shoot or
  refrain from doing so. Each of person is shots
  hits (and kills) its intended target with
  probability p, (independently of whether any
  other shots hit their targets).
• Each person cares only about her probability of
  survival (not about the other person's survival).
  
• Model this situation as an extensive game with
  perfect information and chance moves.
• Show that the strategy pairs in which neither
  person ever shoots and in which each person
  always shoots are both subgame perfect
  equilibria.

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No shooting SPE

• Claim each never shoots (whether or not somebody
  has ever shot)
• Proof According to the prescripts, eachs
  survival probability is already one, and cannot
  be further increased. Hence, no beneficial
  unilateral deviation.

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Shooting SPE

• Claim each always shoot
• Proof
• We argue that deviating one is not beneficial.
  Suppose now it is player 1s turn to to move in
  period t.
• Let Qlt1 be 1s payoff (survival probability)
  conditional on both players exist in period t1
  and they act according to the prescripts
  thereafter.
• If 1 shoots in period t and both act according to
  the prescripts thereafter, 1s payoff is
  p1(1-p1)Qp1(1-Q)Q. If he does not shoot in
  period t and both act according to the
  prescripts, his payoff is Q. Clearly, the
  deviation is NOT beneficial.

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Example 473.1 (One-sided offers)

• Consider the variant of the bargaining game of
  alternating offers in which only player 1 makes
  proposals.
• In every period, player 1 makes a proposal, which
  player 2 either accepts, ending the game, or
  rejects, leading to the next period, in which
  player 1 makes another proposal.
• Consider the strategy pair in which player 1
  always proposes (x1,1-x1) and player 2 always
  accepts a proposal (y1,y2) if and only if y2
  1-x1.
• Find the value(s) of x1 for which this strategy
  pair is a subgame perfect equilibrium.

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Equilibrium

• SPE 1 always proposes (x1,1-x1) 2 always
  accepts a proposal giving her at least 1-x1 and
  rejects any inferior proposal.
• Claim 1-x10.
• Proof (use one stage deviation)
• Suppose not (so that 1-x1gt0). Consider the
  history in which a proposal (z1,1-z1) is proposed
  so that d(1-x1)lt1-z1lt(1-x1).
• By accepting this offer, 2 gets 1-z1 now. By
  rejecting this offer, 2 will get d(1-x1)lt 1-z1.
• Hence, 2 should accept the proposal which is
  strictly inferior than (x1,1-x1). But according
  to her prescript, she should not accept such an
  inferior proposal. A contradiction.

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EXERCISE 445.1 (Tit-for-tat as a subgame perfect
equilibrium)

• Consider the infinitely repeated Prisoner's
  Dilemma in which the payoffs of the component
  game are those given in the Figure.
• Show that (tit-for-tat, tit-for-tat) is a subgame
  perfect equilibrium of this infinitely repeated
  game with discount factor d if and only if y-x1
  and d 1/x.

C D
C x,x 0,y
D y,0 1,1
Note 1 lt x lt y
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Tit for Tat

• Tit-for-tat do whatever the other did to you in
  the previous period
• Four types of histories to check those ending
  with (C,C), (C,D), (D,C), (D,D).
• Need to show player 1 does not gain by one
  deviation

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(C,C)
C D
C x,x 0,y
D y,0 1,1
13
C,D
C D
C x,x 0,y
D y,0 1,1
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(D,C)
C D
C x,x 0,y
D y,0 1,1
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(D,D)
C D
C x,x 0,y
D y,0 1,1
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To summarize

• Conditions for no deviations
• CC y?(1d)x
• CD y(1d)x
• DC dy1d
• DD dy?1d
• Hence, y(1 d)x and dy1d.

• Finally, d1/x and y-x1.
• Both using tit-for-tat is SPE if and only if
  these two conditions hold.
• Very stringent conditions indeed!

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EXERCISE 331.1 (Selten's horse)

• Find the perfect Bayesian equilibria of the game
  in Figure 331.2 in which each player's strategy
  is pure.
• Hint Find the pure strategy Nash equilibria,
  then determine which is part of a weak sequential
  equilibrium

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c d
C 1,1,1 4,4,0
D 3,3,2 3,3,2
c d
C 1,1,1 0,0,1
D 0,0,0 0,0,0
L
R

• Two pure strat Nash equil (D,c,L) and (C,c,R)
• The 1st one is NOT part of a PBE. Foreseeing 3
  will choose L, 2 should choose d to earn 4 rather
  than c to earn 1. Not sequential rationality.
  Hence not PBE.
• The 2nd one is part of a PBE.