Extensive Games with Perfect Information II

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Extensive Games with Perfect Information II
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Outline

• One stage deviation principle
• Rubinstein alternative bargaining game
• Excess optimum
• Two paradoxes
• Chain store paradox
• Centipede game
• Review of the mid term exam

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One-deviation property

• Definition One-Deviation property is satisfied
  if no player can increase her payoff by changing
  her action at the start of any subgame in which
  she is the first-mover, given the other player's
  strategies and the rest of her own strategy.
• Proposition (One-Deviation Property). Let G be a
  (finite or infinite horizon) EGPI. The strategy
  profile s is a subgame perfect equilibrium if
  and only if it satisfies the one-deviation
  property.
• proof see Fudenberg and Tirole
• Implication Suppose we are given a strategic
  profile for an EGPI, and we wonder if it is
  indeed an SPE. The proposition says we need only
  check those alternative strategies with one
  deviation.

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Infinite horizon

• For the proposition to hold also for infinite
  horizon games, we need the following very mild
  qualification
• Continuity at infinity A game is continuous at
  infinity if for each player i the utility
  function ui satisfies
• This condition says events in the distant future
  are relatively unimportant. It will be satisfied
  if the overall payoffs are a discounted sum of
  per-period payoffs git(at) and the per-period
  payoffs are uniformly bounded, i.e., for is a B
  such that

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One Stage Deviation Principle (OSDP) an
illustration

• Consider a one-player EGPI.
• You are told that s(A,C,F) is an SPE.
• To check this, according to the definition of
  SPE, you need to check s against ALL other
  strategies.
• However, according to the one deviation
  principle, you just need to fare s against
  three of them (B,C,F), (A,D,F), and (A,C,E).
• So far as each of these alternative strategies
  does not give a strictly higher payoff for the
  subgame in which the deviation occurs, then s is
  in an SPE.

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Rubinstein alternative offer game

• One unit of cake is to be divided
• (x1,x2) be an offer
• Player 1 makes offers in periods t0,2,4,..., and
  Player 2 makes offers in periods t1,3,5,....
  
• An agreement (x1,x2) reached t periods later
  gives discounted payoffs of dtx1 and dtx2 to the
  two players.
• The players are risk neutral the objective of
  each player is to maximize his discounted
  expected payoff.

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An SPE

• Let x(x1,x2)((1/(1d)),(d/(1d))) and
  y(y1,y2)((d/(1d)),(1/(1d))).
• Player 1 always proposes x and accepts any offer
  in which he is paid (d/(1d)) or more.
• Player 2 always proposes y and accepts any offer
  in which he is paid (d/(1d)) or more.
• Proof Check that nobody can benefit from
  unilaterally deviating once. Then by one
  deviation property proposition, nobody can
  benefit from multiple deviations. This indeed is
  an SPE.

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Unique SPE

• Let Gi be a subgame in which i is the proposer.
  Different SPEs give different discounted payoffs
  to player i. Let Mi and mi be the supremum and
  infimum of them.
• We claim that M1m1(1/(1d)) and
  M2m2(1/(1d)).
• If this claim is true, then there is a unique SPE
  (not only unique SPE outcome). (verify this)

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Unique SPE (cont.)

• We now turn to show the claim. It takes a few
  steps.
• Step 1 m21-dM1. Suppose it is 2's turn to make
  an offer. If 1 gets a chance to make an offer in
  the next period, 1 will get at most M1. This most
  optimistic outcome for 1 is worth dM1 now after
  discounting. Hence any offer to 1 giving him
  dM1? now must be accepted by 1, leaving 1-dM1-?
  for player 2. Hence it cannot be the case that
  m2lt1-dM1.

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Unique SPE (cont.)

• Step 2 M11-dm2. Suppose it is 1's turn to make
  an offer. If 2 gets a chance to make an offer in
  the next period, he will get at least m2. Then 2
  will not accept an offer giving him dm2-? right
  now. Hence, if agreement is to be reached
  immediately, 1 cannot make an offer so that
  x2ltdm2 or x1gt1- dm2 1's offer has to satisfy
  x11- dm2. Suppose 1's offer is rejected, the
  supremum of his present value will become
  d(1-m2)(1-m2)1- dm2. Whether 1's offer is
  accepted now or not, we have M11-dm2 .

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Unique SPE

• Step 1 m21-dM1 and Step 2 M11-dm2
• Also, Step 3 m11-dM2 and Step 4 M21-dm1.
• Combining Steps 1 and 2, we have m2 1-dM1
  1-d(1-dm2) (1-d)d²m2. Hence, we have
  m2((1-d)/(1-d²))(1/(1d)). Substituting this
  back to step 2, we have M1(1/(1d)).Taking into
  account of symmetry and the existence of SPE, the
  claim is shown.

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Extension Will Excess Optimum always lead to
delay?

• Consider the case where two risk-neutral players
  are trying to divide a dollar, which is worth 1
  at t0, d?(1/2,1) at t1, and zero afterwards. It
  is also common knowledge that each player
  believes with probability one that he will be
  picked to make an offer at t1 so long as no
  agreement is made at t0. Find the unique SPE.

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Extension Will Excess Optimum always lead to
Delay? (cont)

• Now consider a four-period version of the
  previous game. The dollar is worth 1, d, d², and
  d³ at dates 0, 1, 2, and 3, respectively, where
  d?(1/2,1/v2). The dollar is worth 0 afterwards.
  Assume that each player is always sure that he
  will make all the remaining offers. Find the
  unique SPE.
• See Yildiz (Econometrica, 2004??)

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Chain-store game

• A chain-store (CS) has branches in K cities,
  numbered 1,...,K.
• In each city k there is a single potential
  competitor, player k.
• In period k, competitor k chooses to enter or
  not if so CS either fights or cooperates in that
  period. That period's payoff is
• at every point in the game all players know all
  the actions previously chosen.
• The payoff of competitor k is its payoff in
  period k the payoff of the chain-store in the
  game is the sum of its payoffs in the K cities.

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Chain store paradox

• unique SPE
• k always enters
• CS always cooperate after entry of the competitor
  
• Paradox Suppose upon every time a competitor
  entered in the past, it was fought by the
  chain-store. Shouldn't the competitor who has the
  turn decides not to enter? The SPE says it should
  not!!

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Centipede game
Two players 1 and 2 play a 6 stage centipede game.
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Centipede game

• Using backward induction, we find that the game
  will end immediately.
• unless the number of stages is very small it
  seems unlikely that player 1 would immediately
  choose S at the start of the game.
• After a history in which both a player and his
  opponent have chosen to continue many times in
  the past, how should a player form a belief about
  his opponent's action in the next period? It is
  far from clear.

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Summary

• The notion of Nash equilibrium is not
  particularly useful for the study of EGPI.
• The notion of subgame perfect equilibrium is
  introduced to solve this problem.
• SPE presumes that a history with zero probability
  according to the strategy profile is reached is
  viewed as an outcome of mistakes.
• The prediction of SPE leads to famous paradoxes
  that call for re-examination of assumptions of
  the model.
• One deviation property theorem
• Alternative bargaining model as a building block
  in the modeling of many interesting issues (e.g.,
  outside option, threat point, trade bargaining,
  bargaining between firm and union, the role of
  property rights, debt renegotiation, holdup, etc.)