Lecture 5A Mixed Strategies and Multiplicity

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Lecture 5AMixed Strategies and Multiplicity

• Not every game has a pure strategy Nash
  equilibrium, and some games have more than one.
  This lecture shows that mixed strategies are
  sometimes a best response, and explains that
  every game has at least one Nash equilibrium in
  pure or mixed strategies.

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Matching pennies

• Not every game has a pure strategy Nash
  equilibrium.
• In this zero sum game, each player chooses and
  then simultaneously reveals the face of a two
  sided coin to the other player.
• The row player wins if the faces on the coins are
  the same, while the column player wins if the
  faces are different.

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The chain of best responses

• If Player 1 plays H, Player 2 should play H, but
  if 2 plays H, 1 should play T.
• Therefore (H,H) is not a Nash equilibrium.
• Using a similar argument we can eliminate the
  other strategy profiles as being a Nash
  equilibrium.
• What then is a solution of the game?

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Avoiding losses in the matching pennies game

• If 2 plays Heads with probability greater than
  1/2, then the expected gain to 1 from playing
  Tails is positive.
• Similarly 1 expects to gain from playing Heads if
  2 plays Heads more than half the time.
• But if 2 randomly picks Heads with probability
  1/2 each round, then the expected profit to 1 is
  zero regardless of his strategy.
• Therefore 2 expects to lose unless he
  independently mixes between heads and tails with
  probability one half.

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The Ware case

• 10 years ago Ware received a patent for Dentosite
  that has since captured 60 percent share in the
  market. National had been the largest supplier of
  material for dental prosthetics before Dentosite
  was introduced.
• A new material FR 8420 was recently developed by
  NASA.
• If Ware develops a new composite with FR 8420 it
  will be a perfect substitute for Dentosite.
• If the technique is feasible then Ware would have
  just as good a chance as National of proving it
  first.
• If Ware develops it first they could extend the
  patent protection to this technique and prevent
  any competitors.

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Strategic considerations

• Wares problem is bound to Nationals.
• Ware does not want to develop a technology that
  would not be used if the competitor does not
  develop it.
• If National develops the technology Ware cannot
  afford to drop out of the race.
• It all depends how people at National see this
  situation. Are Ware and National equally as well
  informed?

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Some facts
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Ware case in the extensive form
Using the facts we can present the case in the
following diagram
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Simplifying the extensive form
Folding back the moves of chance that are related
to developing a new technology we obtain the
following simplification.
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Ware case in the strategic form

• The arrows trace out the best replies.
• As in the Matching Pennies example, there is no
  pure strategy Nash equilibrium in the Ware case.
• Rather than defining the solution as a particular
  cell, we now define the solution as the
  probability of reaching any given cell.

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The probability of Ware choosing in

• Suppose Ware chooses in with probability p.
  Then National is indifferent between the two
  choices if the expected profits are equal.
• The value to National from choosing out is 0,
  and the expected profits to National from
  choosing in are
• -0.401p 1.106(1 p)
• Solving for p we obtain
• -0.401p 1.106(1 p) 0
• gt p .734

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The probability of National choosing in

• Suppose National chooses in with probability q.
  Then Ware is indifferent between the two choices
  if the expected profits are equal.
• The expected value to Ware from choosing in is
• -2.462q - 0.955(1 q)
• The expected value to Ware from choosing out
  is
• -3.015q
• Solving for q we obtain
• 2.462q 0.955(1 q) 3.015q
• gt q .633

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Solution to the Ware case

• If Ware sets p 0.734, then a best response of
  National is to set q 0.633. Likewise if
  National sets q 0.633, a best response of Ware
  is to set p 0.734.
• Therefore the strategy profile p 0.734 and q
  0.633 is a mixed strategy Nash equilibrium. It is
  unique.

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Taxation

• We return to solve the tax auditing game that we
  played in the first lecture of this course.
• For convenience the strategic form of this
  simultaneous move game is presented.

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Best replies in taxation game
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Monitoring by the collection agency

• Equating the expected utility for the collection
  agency
• 10?11 4 ?12 2(1- ?11- ?12) -?11 5?12 3(1-
  ?11- ?12)
• and
• 10?11 4 ?12 2(1- ?11- ?12) 2?12 4(1- ?11-
  ?12)
• Solving these equations in two unknowns we obtain
  the mixed strategy
• ?11 1/120.083
• ?12 1/4 0.250
• ?13 2/3 0.667

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Cheating by the taxpayer

• Equating the expected utility for the taxpayer
  across the different choices
• -12?21 -6?21 - 6?22 - 2(1- ?21- ?22)
• and
• -12?21 -4
• Defining the only strategy that leaves the
  taxpayer indifferent between all three choices is
  threfore
• ?21 1/3 0.333
• ?22 1/6 0.167
• ?23 ½ 0.500

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Mixed strategy Nash equilibrium in the taxation
game
?210.333
?220.167
?230.50
?110.083
?120.25
?130.667
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Existence of Nash equilibrium

• This brings us to the central result of this
  lecture. Consider any finite non-cooperative
  game, that is a game in extensive form with a
  finite number of nodes.
• If there is no pure strategy Nash equilibrium in
  the strategic form of the game, then there is a
  mixed strategy Nash equilibrium.
• In other words, every finite game has at least
  one solution in pure or mixed strategies.

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The threat of bankruptcy

• We consider an industry with weak board of
  directors, an organized workforce and an
  entrenched management.
• Workers and management simultaneously make
  demands on the firms resources.
• If the sum of their demands is less than or equal
  to the total resources of the firm, shareholders
  receive the residual.
• If the sum exceeds the firms total resources,
  then the firm is bankrupted by industrial action.

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Strategic form of bargaining game

• To achieve a bigger share of the gains from
  trade, both sides court disastrous consequences.
• This is sometimes called a game of chicken, or
  attrition.
• We investigate more complicated bargaining games
  in 45-976, Bargaining, Contracts and Strategic
  Investment, the second course in this sequence.

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Best responses illustrated
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Multiple Nash equilibrium

• In this game, there are three pairs of mutual
  best responses.
• The parties coordinate on an allocation of the
  pie without excess demands. Shareholders get
  nothing.
• But any of the three allocations is an
  equilibrium.
• If the labor and management do not coordinate on
  one of the equilibrium, the firm will bankrupt or
  shareholders will receive a dividend.

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Corporate plans

• In every corporation departmental heads jostle
  for influence, promotion and higher compensation.
• This creates rivalries between different
  departments.
• Consequently the goals of departments are seldom
  perfectly aligned with each other.

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Best replies in corporate plans

• There are two Nash equilibrium in this game, one
  pure strategy and the other mixed.
• Is one equilibrium more plausible than the other?
  
• Now suppose the payoff elements in all the corner
  cells were magnified by a factor of a hundred. Is
  there a Jack Welch way?

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Lecture summary

• Not every game of imperfect information has a
  pure strategy equilibrium.
• However every strategic form game has at least
  one pure or mixed strategy solution, and we
  showed how to derive them.
• Strategic uncertainty arises in situations where
  the solution to the game is a mixed strategy.
• When there are multiple Nash equilibrium, other
  criteria might be used to pick amongst them, as
  coordinated by management.