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					Overview



                                  Overview




           BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   1
Overview


   Game Theory in Part III completes Game Theory in Part II for
   those games where information is strategically revealed or
   withheld.

   In many games, a player may not know all the information that is
   pertinent for the choice that he has to make at every point in the
   game. His uncertainty may be over variables that are either
   internal or external to the game.




           BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   2
Overview


   A player may be potentially uncertain about what moves the
   other player is making at the same time he makes his own move;
   we call that strategic uncertainty. All the simultaneous move
   games in Part II were simple enough that that uncertainty was
   resolved by eliminating dominated strategies.

   Part III’s Lesson 5 considers games where strategic uncertainty
   remains because uncertainty is not resolved by eliminating
   dominated strategies, and because players’ interests conflict (as in
   sports) so players conceal information about their own moves.
   Lesson 6 considers games where players easily reveal
   information about their own moves because players’ interests
   align (as in setting a standard industry format).


           BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   3
Lesson Overview


   Lesson III.5 Strategic Uncertainty when Interests Conflict
   Example 1: Unpredictable Actions
   Example 2: Mixing with Perfect Conflict
   Example 3: Mixing with Major Conflict
   Example 4: Mixing with Minor Conflict
   Summary
   Review Questions




         BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   4
Example 1: Unpredictable Actions



        Example 1: Unpredictable Actions




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   5
Example 1: Unpredictable Actions


    Comment: Bob Gustavson, professor of health science and men's
    soccer coach at John Brown University in Siloam Springs,
    Arkansas, says “When you consider that a ball can be struck
    anywhere from 60-80 miles per hour, there's not a whole lot of
    time for the goalkeeper to react”. Gustavson says skillful goalies
    use cues from the kicker. They look at where the kicker's plant
    foot is pointing and the posture during the kick. Some even study
    tapes of opponents. But most of all they take a guess — jump left
    or right at the same time the kicker is committing himself to
    kicking left or right.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   6
Example 1: Unpredictable Actions


    Question: Consider a penalty kick in soccer. The goalie either
    jumps left or right after the kicker has committed himself to
    kicking left or right. The kicker’s payoffs are the probability of
    him scoring, and the goalie’s payoffs are the probability of the
    kicker not scoring. Those actions and payoffs define a normal
    form for this Penalty Kick Game. Try to predict strategies or
    recommend strategies.

                                                           Goalie
                                                   Left                Right
                            Left                   .1,.9               .8,.2
       Kicker
                            Right                  .4,.6               .3,.7
            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   7
Example 1: Unpredictable Actions


    Answer: To predict actions or                                         Goalie
    recommend actions, since                                         Left       Right
    the game has simultaneous                             Left       .1,.9      .8,.2
                                      Kicker
                                                          Right      .4,.6      .3,.7
    moves, first look for dominate
    or dominated actions. There are none.

    Then look for a Nash equilibrium. There is none. If the Kicker
    were known to kick Left, the Goalie guards Left. But if the
    Goalie were known to guard Left, the Kicker kicks Right. But if
    the Kicker were known to kick Right, the Goalie guards Right.
    But if the Goalie were known to guard Right, the Kicker kicks
    Left. So there is no Nash equilibrium.



            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict           8
Example 1: Unpredictable Actions


    Finally, look to see if any                                           Goalie
    action can be eliminated because                                 Left       Right
    it is not rationalizable (that is, it                 Left       .1,.9      .8,.2
                                               Kicker
    is not a best response to some                        Right      .4,.6      .3,.7
    action by the other player.

    But all actions are rationalizable.

    On the one hand, it is rational to kick left if the Kicker believes
    the Goalie jumps right. On the other hand, it is rational for the
    Kicker to kick right if he believes the Goalie jumps left.

    Likewise, it is rational for the Goalie to jump left if the Goalie
    believes the Kicker kicks left, and it is rational for the Goalie to
    jump right if the Goalie believes the Kicker kicks right.


            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict           9
Example 1: Unpredictable Actions


    Since there are no dominance                                          Goalie
    solutions and there are no                                       Left       Right
    Nash equilibria for this game               Left                 .1,.9      .8,.2
                                      Kicker
                                               Right                 .4,.6      .3,.7
    of simultaneous moves,
    actions are unpredictable, and game theory has no
    recommendation; either action is acceptable.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict           10
Example 2: Mixing with Perfect Conflict



           Example 2: Mixing with Perfect
                      Conflict




             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   11
Example 2: Mixing with Perfect Conflict


    Comment: Example 1’s choices for the goalie were jump left or
    jump right. Call those actions because, in Example 2, strategies
    are going to be more complicated; they will be probabilities for
    taking specific actions --- say, jump left with probability 0.24 and
    jump right with probability 0.76




             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   12
Example 2: Mixing with Perfect Conflict


    Question: Consider the normal form below for the Penalty Kick
    Game in soccer. Predict strategies or recommend strategies if this
    game is repeated throughout the careers of the kicker and the
    goalie.




                                                            Goalie
                                                    Left                Right
                             Left                   .1,.9               .8,.2
       Kicker
                             Right                  .4,.6               .3,.7
             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   13
Example 2: Mixing with Perfect Conflict


    Answer: If the game were not                              Goalie
    repeated, then since there are no                    Left       Right
    dominance solutions and there               Left     .1,.9      .8,.2
                                        Kicker
                                               Right     .4,.6      .3,.7
    are no Nash equilibria (in pure
    strategies) for this game of simultaneous moves, actions are
    unpredictable, and game theory has no recommendation; either
    action is acceptable. For example, the Kicker could kick Left and
    the Goalie could jump Left.




             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   14
Example 2: Mixing with Perfect Conflict


    But since the game is repeated,                                        Goalie
    actions need to become                                            Left       Right
    unpredictable because                                  Left       .1,.9      .8,.2
                                                Kicker
                                                           Right      .4,.6      .3,.7
    predictable actions can be
    exploited.

    For example, see how predicting actions helps the Goalie. If the
    Kicker chooses Left predictably, the Goalie can choose Left and
    keep the Kicker at payoff .1 and the Goalie at .9; and if the
    Kicker chooses Right predictably, the Goalie can choose Right
    and keep the Kicker at payoff .3 and the Goalie at .7.




             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict           15
Example 2: Mixing with Perfect Conflict


    The Nash equilibrium strategy for the Kicker is the mixed
    strategy for which the Goalie would not benefit if he could
    predict the Kicker’s mixed strategy. Suppose the Goalie predicts
    p and (1-p) are the probabilities the Kicker chooses Left or Right.
    The Goalie expects .9p + .6(1-p) from playing Left, and .2p +
    .7(1-p) from Right. The Goalie does not benefit if those payoffs
    equal, .9p + .6(1-p) = .2p + .7(1-p), or .6 + .3p = .7 - .5p, or
    p = 1/8 = 0.125
                                                            Goalie
                                                    Left                Right
                             Left                   .1,.9               .8,.2
       Kicker
                             Right                  .4,.6               .3,.7
             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   16
Example 2: Mixing with Perfect Conflict


    The Nash equilibrium strategy for the Goalie is the mixed
    strategy for which the Kicker would not benefit if he could
    predict the Goalie’s mixed strategy. Suppose the Kicker predicts
    q and (1-q) are the probabilities the Goalie chooses Left or Right.
    The Kicker expects .1q + .8(1-q) from playing Left, and .4q +
    .3(1-q) from Right. The Kicker does not benefit if those payoffs
    equal, .1q + .8(1-q) = .4q + .3(1-q), or .8 - .7q = .3 + .1q, or
    q = 5/8 = 0.625
                                                            Goalie
                                                    Left                Right
                             Left                   .1,.9               .8,.2
       Kicker
                             Right                  .4,.6               .3,.7
             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   17
Example 2: Mixing with Perfect Conflict


    Comment: Randomizing actions adds strategies (called mixed
    strategies) that solve some games that have no dominance
    solution or Nash Equilibrium (in pure strategies, where all
    probability is on one particular action). For example, in the
    Penalty Kick Game, there was no Nash equilibrium with pure
    strategies, and there were multiple rationalizable pure strategies.
    It turns out that most games have at least one Nash equilibrium in
    mixed strategies.

    In fact, the Penalty Kick Game has a unique Nash equilibrium in
    mixed strategies. While any of the rationalizable strategies would
    be reasonable if the game were played once, if instead the game
    were repeated, then strategies in the unique Nash equilibrium are
    the only way to play that guarantees the other player cannot gain
    even if they used your history to correctly predict your strategy.
             BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   18
Example 3: Mixing with Major Conflict



           Example 3: Mixing with Major
                     Conflict




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   19
Example 3: Mixing with Major Conflict


    Comment: Employers are in conflict with (selfish, amoral)
    workers, who want to steal or shirk (not work, or steal time).
    However, the Work-Shirk Game is not one of total conflict (it is
    not like the Penalty Kick Game) because monitoring workers
    costs the employer but does not help the worker.

    Because of the conflict, the other player exploiting your
    systematic choice of strategy is to your disadvantage, and so there
    is reason to follow mixed strategies in such games.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   20
Example 3: Mixing with Major Conflict


    Question: Consider the Work-Shirk Game for an employee and
    an employer. Suppose if the employee chooses to work, he
    looses $100 of happiness from the effort of working, but he
    yields $400 to his employer. Suppose the employer can monitor
    the employee at a cost of $80. Finally, if the employee chooses to
    not work and the employer chooses to monitor, then the
    employee is not paid, but in every other case (“work” or “not
    monitor”), then the employee is paid $150.

    Predict strategies or recommend strategies if this game is
    repeated daily.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   21
Example 3: Mixing with Major Conflict


    Answer: First, complete the normal form below for the Work-
    Shirk Game. For example, if the employee chooses to work and
    the employer chooses to monitor, then the employee looses $100
    of happiness from the effort of working but is paid $150, and the
    employer gain $400 from his employer but pays $80 for
    monitoring and pays $150 to his employee.



                                                        Employer
                                               Monitor  Trust
                            Work               50,170  50,250
   Employee
                            Shirk               0,-80 150,-150
            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   22
Example 3: Mixing with Major Conflict


    To predict actions or                                               Employer
    recommend actions, since                                       Monitor  Trust
    the game has simultaneous                             Work     50,170  50,250
                                    Employee
                                                          Shirk     0,-80 150,-150
    moves, first look for dominate
    or dominated actions. There are none.

    Then look for a Nash equilibrium in pure strategies. There is
    none. If the Employee were known to Work, the Employer
    Trusts. But if the Employer were known to Trust, the Employee
    Shirks. But if the Employee were known to Shirk, the Employer
    Monitors. But if the Employer were known to Monitor, the
    Employee Works. So there is no Nash equilibrium.



            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict        23
Example 3: Mixing with Major Conflict


    Since the game is repeated,                                         Employer
    actions need to become                                         Monitor  Trust
    unpredictable because                                 Work     50,170  50,250
                                            Employee
                                                          Shirk     0,-80 150,-150
    predictable actions can be
    exploited.

    The Nash equilibrium strategy for the Employee is the mixed
    strategy for which the Employer would not benefit if he could
    predict the Employee’s mixed strategy. Suppose the Employer
    predicts p and (1-p) are the probabilities the Employee chooses
    Work or Shirk. The Employer expects 170p - 80(1-p) from
    playing Monitor, and 250p - 150(1-p) from Trust. The Employer
    does not benefit if those payoffs equal,
    170p - 80(1-p) = 250p - 150(1-p), or -80 + 250p = -150 + 400p,
    or p = 70/150 = 0.467
            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict        24
Example 3: Mixing with Major Conflict


                                                                        Employer
                                                                   Monitor  Trust
                                                          Work     50,170  50,250
                                            Employee
                                                          Shirk     0,-80 150,-150

    The Nash equilibrium strategy for the Employer is the mixed
    strategy for which the Employee would not benefit if he could
    predict the Employer’s mixed strategy. Suppose the Employee
    predicts q and (1-q) are the probabilities the Employer chooses
    Monitor or Trust. The Employee expects 50q + 50(1-q) from
    playing Work, and 0q + 150(1-q) from Shirk. The Employee does
    not benefit if those payoffs equal,
    50q + 50(1-q) = 0q + 150(1-q), or 50 = 150 – 150q,
    or q = 100/150 = 0.667


            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict        25
Example 4: Mixing with Minor Conflict



           Example 4: Mixing with Minor
                     Conflict




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   26
Example 4: Mixing with Minor Conflict


    Comment 1: Blu-ray Disc is designed to supersede the standard
    DVD format. The disc has the same physical dimensions as
    standard DVDs and CDs. The name Blu-ray Disc derives from
    the blue-violet laser used to read the disc. Blu-ray Disc was
    developed by the Blu-ray Disc Association, a group representing
    makers of consumer electronics, computer hardware, and motion
    pictures.

    During the format war over high-definition optical discs, Blu-ray
    competed with the HD DVD format. Toshiba, the main company
    supporting HD DVD, conceded in February 2008, and the format
    war ended. In late 2009, Toshiba released its own Blu-ray Disc
    player.


            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   27
Example 4: Mixing with Minor Conflict


    Comment 2: The format war over high-definition optical discs
    has The Blu-ray Disc Association in some conflict with Toshiba
    since each group has gained expertise and lower costs in
    producing a particular format and, so, each would gain if their
    format were universally adopted. However, the Format War
    game is not one of total conflict (it is not like the Penalty Kick
    Game) or even of major conflict (like the Work-Shirk Game)
    because both players loose most if neither format is universally
    adopted.

    Because conflict is less important than cooperation, the other
    player exploiting your systematic choice of strategy is to your
    advantage because you both want a universal format. So there is
    less reason to follow mixed strategies in such games.



            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   28
Example 4: Mixing with Minor Conflict


    Question: Consider the Format Game for The Blu-ray Disc
    Association and Toshiba. Suppose each player either adopts the
    Blu-ray format or the HD format. Suppose if both adopt the same
    format, then both gain $100 million from customers that value the
    convenience of having a universal format. Suppose if they both
    adopt the Blu-ray format, then The Blu-ray Disc Association
    gains an extra $10 million since their expertise with that format
    gives them lower production costs. Finally, suppose if they both
    adopt the HD format, then Toshiba gains an extra $10 million
    since their expertise with that format gives them lower production
    costs.

    Predict strategies or recommend strategies if this game is
    repeated yearly.



            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   29
Example 4: Mixing with Minor Conflict


    Answer: First, complete the normal form below for the Format
    Game. For example, if The Blu-ray Disc Association and
    Toshiba both adopt HD, then both gain $100 million from
    customers that value the convenience of having a universal
    format, and Toshiba gains an extra $10 million since their
    expertise with the HD format gives them lower production costs.



                                                          Toshiba
                                               Blu-ray                 HD
                          Blu-ray              110,100                 0,0
      Blu-ray
                            HD                   0,0                 100,110
            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   30
Example 4: Mixing with Minor Conflict


    To predict actions or                                                Toshiba
    recommend actions, since                                        Blu-ray       HD
    the game has simultaneous                            Blu-ray    110,100       0,0
                                     Blu-ray
                                                           HD         0,0       100,110
    moves, first look for dominate
    or dominated actions. There are none.

    Then look for a Nash equilibrium in pure strategies. There are
    two. On the one hand, both players choose Blu-ray; on the other
    than, both players choose HD.

    There is also a Nash equilibrium in mixed strategies.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict             31
Example 4: Mixing with Minor Conflict


    The Nash equilibrium mixed                               Toshiba
    strategy for Blu-ray Association                    Blu-ray     HD
    is the mixed strategy for which Blu-ray Blu-ray 110,100         0,0
                                                 HD       0,0     100,110
    Toshiba would not benefit if
    they could predict Blu-ray Association’s mixed strategy. Suppose
    Toshiba predicts p and (1-p) are the probabilities Blu-ray
    Association chooses Blu-ray or HD. Toshiba expects 100p + 0(1-
    p) from playing Blu-ray, and 0p + 110(1-p) from HD. Toshiba
    does not benefit if those payoffs equal,
    100p + 0(1-p) = 0p + 110(1-p), or 100p = 110 - 110p,
    or p = 110/210 = 0.524

    The expected payoff for Toshiba (whatever its strategy) is thus
    100p + 0(1-p) = 0p + 110(1-p) = 52.4

            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   32
Example 4: Mixing with Minor Conflict


    The Nash equilibrium mixed                                Toshiba
    strategy for Toshiba is the mixed                    Blu-ray     HD
    strategy for which Blu-ray                  Blu-ray 110,100      0,0
                                       Blu-ray
                                                  HD       0,0     100,110
    Association would not benefit if
    they could predict Toshiba’s mixed strategy. Suppose Blu-ray
    Association predicts q and (1-q) are the probabilities Toshiba
    chooses Blu-ray or HD. Blu-ray Association expects 110q + 0(1-
    q) from playing Blu-ray, and 0q + 100(1-q) from HD. Blu-ray
    Association does not benefit if those payoffs equal,
    110q + 0(1-q) = 0q + 100(1-q), or 110q = 100 – 100q,
    or q = 100/210 = 0.476

    The expected payoff for Blu-ray Association (whatever its
    strategy) is thus 110q + 0(1-q) = 0q + 100(1-q) = 52.4

            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   33
Example 4: Mixing with Minor Conflict


    Comment: The expected payoff                              Toshiba
    of 52.4 for each player in the                       Blu-ray     HD
    mixed strategy Nash equilibrium Blu-ray Blu-ray 110,100          0,0
                                                  HD       0,0     100,110
    is less than if both players had
    agreed to one format or the other. That is a general lesson in
    games with only minor conflict of interest. The players are better
    off resolving the strategic uncertainty. The remaining lessons
    take up the problem of revealing information.




            BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   34
Summary



                                Summary




          BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   35
Review Questions


   Review Questions
       You should try to answer some of the following questions
      before the next class.
       You will not turn in your answers, but students may request
      to discuss their answers to begin the next class.
       Your upcoming cumulative Final Exam will contain some
      similar questions, so you should eventually consider every
      review question before taking your exams.




         BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict   36
BA 210                                               Introduction to Microeconomics



                    End of Lesson III.5




         BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict       37

				
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