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

Player 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

Player 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
   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
 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
 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
 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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