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슬라이드 1

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					 Poor Bart.. Always
                        Rock.. Nothing beats
plays Rock! I’ll play
                          good old Rock!!
       Paper!




                                  Class 3
   Simultaneous-move games
    • Analytic tool: Game Table
    • Nash equilibrium: mutual best response
 Prisoner’sdilemma games
 Dominant strategies
 Ways to solve the dilemma
 Games  with no pure-strategy equilibrium
 Mixed-strategy equilibrium
 Graphic and algebraic solution
Sensibility of being unpredictable
 Mixed    Strategy
    : A strategy in which a player plays his available (pure)
    strategies with certain probabilities.
 Nash’s   existence theorem
    : If each player in a game has a finite number of pure
    strategies, then there exists at least one equilibrium in
    (possibly) mixed strategies.
∴If there are no pure strategy equilibria, there must be a
 unique mixed strategy equilibrium.
                                Player 2

                               Heads            Tails
     Player 1




                Heads           1,-1            -1,1

                Tails           -1,1            2,-1

• Players either place heads or tails
• Player 1 wins if they match; player 2 wins if they don’t
                                Player 2

                               Heads              Tails
     Player 1




                Heads            1,-1             -1,1

                Tails            -1,1             2,-1

• Player 1’s best strategy is to match Player 2’s strategy;
Player 2’s best strategy is to place the opposite strategy of Player 1.
• There is no equilibrium where both players adopting one (pure)
strategy is optimal. I.e., Given the strategy of the other, the player
always wants to do something else.
 Randomize: play each strategy with some probability
 Play a Mixed Strategy NE
  : A mixed strategy NE is an outcome in which players adopt each
  strategy with some non-negative probability
  Note: A pure strategy NE is a special case of a mixed strategy
  NE
 But how much should you mix?
  Up to the point where the opponent is indifferent between their
  own alternatives.
 Why? If they are not indifferent, then you are too predictable and
  therefore they will use it to their own advantage (so to your own
  disadvantage)
   Suppose that Player 1 plays H w/ probability p and T w/ probability
    1-p
   Payoff of Player 2 from playing H:
         -1  p + 1  (1-p) = 1 - 2p
   Payoff of Player 2 from playing T:
         1  p +[ -1  (1-p)] = 2p - 1
   Player 2 will be indifferent between H and T if and only if player 1
    plays H with the following probability:
         1 – 2p = 2p – 1  4p = 2  p = ½
    So player 1 should play H and T w/ probability ½ each.
   Player 2’s payoff will then be 1 – 2p = 1 – 2 (½) = 0.
   Suppose that Player 2 plays H w/ probability q and T w/ probability
    1-q
   Payoff of Player 1 from playing H:
         1  q + [-1  (1-q)] = 2q - 1
   Payoff of Player 1 from playing T:
         [ -1  q] + 2  (1-q) = 2 – 3q
   Player 1 will be indifferent between H and T if and only if Player 2
    plays H with the following probability:
         2q - 1 = 2 – 3q  5q = 3  q = 3/5 = 0.6.
    So Player 2 should play H w/ prob. 0.6 and T w/ prob. 0.4.
   Player 1’s payoff will be 2q – 1 = 2(0.6) – 1 = 0.2
 Payoffs:
  • Player 1 = 0.2
  • Player 2 = 0
 Whatif Player 1 plays H with any other
 probability?
  • Example: p = 0.45 (play tails more often because matches
    on tails pay 2 instead of 1).
  • Player 2 will always play H, and player 1’s expected
    payoff will be -0.1, less than 0.2 before.
 Sometimes     no pure strategy exists
  • Must randomize actions
 How   to randomize?
  • Make other player indifferent
  • Otherwise, they will take preferred action, and we
   should respond to that
 Examples     of when to randomize
  • Sports
  • Auditing
  • Most of the time when keeping secrets
Venus and Serena Williams play tennis. Each can hit the ball
   in different ways. How should they play in order to win?
 Players: Venus, Serena Williams
 Strategies:
   • Serena: can pass down-the-line (DL) or cross-court
     (CC)
   • Venus: can position herself to receive either DL or
     CC
 Information: Serena’s turn to return the ball
 Payoffs
   • Fraction(%) of times that each player wins the point
 Preferences: Better if wins the point
                              Venus
                         DL             CC
             DL        50, 50       80, 20
Serena
             CC        90, 10       20, 80

   No pure strategy Nash Equilibrium.
 Suppose   Serena can mix her strategies
 p: probability that Serena chooses DL
 1-p: probability that Serena chooses CC

 Suppose   Venus can mix strategies as well
 q: probability that Venus positions herself for DL
 1-q: probability that Venus positions herself for CC
                               Venus
                  DL            CC           q-mix
                                         50q+80(1-q),
         DL     50, 50        80, 20
                                         50q+20(1-q)
                                         90q+20(1-q),
Serena   CC     90, 10        20, 80
                                         10q+80(1-q)
         p-  50p+90(1-p), 80p+20(1-p),
         mix 50p+10(1-p) 20p+80(1-p)
    Note: When p=1, Serena always plays DL
          When p=0, Serena always plays CC
 Use best response analysis to find optimal
  mixing probabilities
 Need to know how one’s expected utility from
  playing a certain strategy changes by different
  probability mixes of the other
  Venus’s                   payoff, success probability, as a function of p.

                100                                        20p+80(1-p) [CC]
Venus’s Success %




                    80                                     50p+10(1-p) [DL]
                                       p=.70




                         0                             1
                              Serena’s p-mix
  Serena’s                payoff, in turn, depends on the choices of q
                                                        50q+80(1-q) [DL]
                                                        90q+20(1-q) [CC]
                     100
Serena’s Success %




                                q = 0.6
                     80




                       0                            1
                            Venus’s q-mix
• Serena’s Best Response    • Venus’s Best Response
Function                    Function
    1                          1

                           0.7
p                          p


    0         0.6      1       0                      1
          q                           q
    1
                                         Mixed Strategy
0.7                                      NE Occurs at
                                         p=.70, q=.60
p



    0          q 0.6         1
        Equilibria, pure or mixed, obtain whenever the best
        response functions intersect
         Mutual Best Responses!
 Players   are indifferent between choosing either
 strategy
Opponent’s indifference property
 : An equilibrium mixed strategy of one player
 in a two-person game has to be such that the
 other player is indifferent among all the pure
 strategies that are actually used in his mixture
 Choose p so as to make Venus indifferent
  between playing either pure strategy
 Why?
  ∵Serena is always worse off if Venus knows how Serena
   will return the ball. By making Venus indifferent
   between the two strategies, Serena can keep Venus
   guessing
 Randomizing just right takes away any ability
 to be taken advantage of
 Serena solves for p that equates Venus’s payoffs from
  DL and CC:
               50p+10(1-p) = 20p+80(1-p)
                        ∴p = .70
 Suppose Serena plays DL with probability .7 and CC
  with probability .3
  Then, Venus’s success rate from
  • DL: 50(.70)+10(.30) = 38%
  • CC: 20(.70)+80(.30) = 38%
 Serena’s   success rate: 62%
 Venus solves for q that equates Serena’s payoffs from
  DL and CC:
              50q+80(1-q) = 90q+20(1-q)
                        ∴q = .60
 Suppose Venus plays DL with probability .6 and CC
  with probability .4
  Then, Serena’s success rate from
  • DL: 50(.60)+80(.40) = 62%
  • CC: 90(.60)+20(.40) = 62%
 Venus’s   success rate: 38%
 NE   in mixed strategies
  : A probability distribution for each player, where the
   distributions are mutual best responses to one another
   in the sense of expectations
 NE   in pure strategies
  : A special case of mixed strategies, where the
   probabilities are chosen from the set {0, 1}
 Why   is this a useful objective?
 • Making your opponent indifferent in expected terms is
   equivalent to minimizing your opponents’ ability to
   recognize and exploit systematic patterns of your own
   behavior
 • In constant-sum games, keeping your opponent
   indifferent is equivalent to keeping yourself indifferent
 Mixed Strategies:
  • If opponent knows what I will do, I will always lose!
  • Randomizing just right takes away any ability to be
    taken advantage of
 Implications (strangely):
   • A player chooses his strategy so as to make his
     opponent indifferent
   • If done right, the other player earns the same payoff
     from either of his strategies
 Employees can either work or shirk. Managers want to
monitor them so that they work, but monitoring is costly.
               What is the optimal monitoring strategy?
 Players:   Employee, Manager
 Strategies:
  • Employee: Work or Shirk
  • Manager: Monitor or Not Monitor
 Information
 : Both players know that working costs effort and
 monitoring costs money
   Payoffs:
    • Employee
      - Salary is $100K unless caught shirking
      - Working hard costs effort worth $50K
    • Manager
      - Value of employee output is $200K
      - $0 if employee doesn’t work
      - Cost of monitoring is $10K
   Preferences: The higher the monetary value, the better
                               Manager
                        Monitor     Not Monitor
             Work      50K, 90K      50K, 100K
Employee
             Shirk      0, -10K     100K, -100K

    ∴ No pure strategy equilibria
 Let p be the probability the employee shirks
 Let q be the probability the manager monitors
       find the expected payoffs using p and q
 First,
 Then, calculate the best response
 Wherever the best responses cross is the
  mixed-strategy equilibrium
 If   employee works:
  • E (payoff/work) = 50q + 50(1-q)
                       = 50
 If   employee shirks:
  • E (payoff/shirk) = 0q + 100(1-q)
                       = 100 – 100q
 E (payoff/work) = E (payoff/shirk)
 50 = 100 – 100q


   Indifferent when q=1/2

   Best strategy for all possible strategies of the manager
    • If q < ½: Shirk
    • If q > ½: Work
    • If q = ½ : Indifferent
 If   manager monitors:
  • E (payoff/monitor) = 90(1-p) -10p
                         = 90 -100p
 If   manager not monitors:
  • E (payoff/not monitor) = 100(1-p)-100p
                           = 100 – 200p
E    (payoff/monitor) = E (payoff/not monitor)
 90   -100p = 100 – 200p

 Indifferent       when p = 1/10

   Best strategy for all possible strategies of the manager
    • If p < 1/10: Not Monitor
    • If p > 1/10: Monitor
    • If p = 1/10 : Indifferent
• Employee’s Best        • Manager’s Best
Response Function        Response Function
    1                       1



p                       p

                        0.1
    0    0.5        1       0                1
         q                        q
            1
Shirk




                                          Mixed Strategy
                                          NE Occurs at
        p
                                          p=.1, q=.5
Work




        0.1
            0                         1
                      q 0.5
                Not Monitor Monitor
 Employee shirks with probability 1/10
 Manager monitors with probability ½
 Employee’s expected payoff:
 (1/10)[½*0+½*100]+(9/10)[½*50+½*50]=50
 Manager’s   expected profit:
 ½[(9/10)*90-(1/10)*10]
 +½[(9/10)*100-(1/10)*100]=80
 Bothplayers are indifferent between any
 mixture over their strategies
  E.g. Employee receives a payoff of $50K
        whether he works (½0+½100=50)
        or shirks (½50+½50=50)
 Regardless  of what employee does, expected
  payoff is the same
 Since a player does not care what mixture she
  uses, she picks the mixture that will make her
  opponent indifferent!
 Motivate   compliance at lower monitoring cost

 Parking   Tickets
 Audits
 Drug   Testing
 Increasein cost to all parties should not change
 the optimal “mix” for that party
 “Not surprisingly, it was not intuitive to them
 that a 10% increase in enforcement costs should
 not be met with an equal decrease in
 enforcement. By reducing the ticketing rate,
 incidence of illegal parking increased by over
 40%.”
 IRS   Commissioner Charles Rossotti:
  • Audits more expensive now than in ’97
  • Number of audits decreased
  • Offshore evasion alone increased to $70billion dollars!
 Recommends:
   As audits get more expensive, need to increase budget
   to keep the number of audits constant!
 Manager’s strategy of monitor ½ of the time
  must mean that there is a 50% chance of
  monitoring in every round
 Cannot just monitor every other day


 Humans   are bad at this! Exploit patterns!
  McDonald and Burger King decide whether to
  put their restaurants in a shopping mall or not.
Their profits depend on the actions of each other.
 Players: McDonald, Burger King
 Strategies: Enter, Not Enter
 Information:
   The market size is such that only one firm entering yields
   positive profits.
 Payoffs:
   If only one firm enters, it earns $300K
   If both firms enter, each earns -$100K
 Preferences:   Better the higher the payoff
                                     BK

                            Enter         Not Enter

             Enter          -1, -1          3, 0
MD
           Not Enter         0, 3           0, 0

     There are two pure strategy equilibria
Are there any mixed strategy equilibria as well?
 MD   chooses probability p of entering, so that
  BK is indifferent between entering and not
  entering
 BK’s payoff from entering:
  p(-1)+(1-p)(3) = 3-4p
 BK’s payoff from not entering:
  p(0)+(1-p)(0) = 0
∴BK indifferent when p=3/4
 BK  chooses probability q of entering, so that
  MD is indifferent between entering and not
  entering
 MD’s payoff from entering:
  q(-1)+(1-q)(3) = 3-4q
 MD’s payoff from not entering:
  q(0)+(1-q)(0) = 0
∴MD indifferent when q=3/4
 Both MD and BK choose Enter with probability
  ¾ and Not Enter with probability ¼
 Note the symmetry in equilibrium since the
  payoff structure are the same
Expected Payoff of BK          Best Response Function of BK

3                                1


                  p=¾           q

0
                        1
-1
                                                p=¾           1
       Payoff from Enter
       Payoff from Not Enter
Expected Payoff of MD          Best Response Function of MD

3                                 1


                  q=¾            p

0
                        1
-1
                                                q=¾           1
       Payoff from Enter
       Payoff from Not Enter
      PSE#1
  1

                     MSE
q=¾




                               PSE#2
                  p=¾      1

      BK’s best response
      MD’s best response
 What  if one of the firms has a competitive
  advantage?
 Suppose if MD is the sole entrant, it earns
  $400K. Otherwise, the game is the same as
  before.
                                  BK
                          Enter      Not Enter
            Enter        -1. -1          4, 0
MD
          Not Enter        0, 3          0, 0

The pure strategy equilibria remains the same.
 What about the mixed strategy equilibrium?
 MD’s  choice of p remains the same since BK’s
  payoffs have not changed
 What about BK’s choice of q?
 BK chooses q so that MD is indifferent
  between entering and not entering:
     q(-1)+(1-q)(4) = 0
     q = 4/5 > 3/4
∴BK’s probability of entering increases
 Suppose   BK does not adjust its probability of
 entering
 What would MD do?
      1                       MD would
q = 4/5                      choose p = 1!!
  q=¾
                            MD’s new b.r.



                   p=¾      1
 If MD chooses p = 1, then BK incurs negative
  profits by keeping q at ¾
  i.e. BK’s profit (q=¾) = ¾(-1) + 0 = -¾
 By increasing q to 4/5, BK induces MD’s
  expected profit when enter to become (4/5)(-1)
  + (1/5)(4) = 0, which is equal to the expected
  profit when not enter
 Again, make the competitor indifferent!
 BK is in a disadvantageous position. But the
 MSE is for BK to increase the probability of
 entering. Why?
 Unreasonable   predictors of one-time human
 interaction

 Reasonablepredictors of long-term proportions
 or multimarket contacts

				
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