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1
The disco is not safe!
2the wimp,
and the tough
man…
The bully,
3
Subgame 1
Subgame 2
Subgame 1
Subgame 2
Full information
4
5
•Separating equilibrium?
– Not fight if the man drinks
beer?
•But then all wimps start
drinking beer and all tough
men drink beer
•Bully payoff = ⅓ < than
always fight (Bully payoff =
⅔)
•Pooling equilibrium?
– Always fight?
•But then Tough man will
always drink beer and Wimp
always eat Quiche
•Is perfect signal!
– Always defer?
•Bully payoff =⅓
•But then Tough man will
always drink beer and Wimp
always eat Quiche
•Is perfect signal!
•Thus no equilibrium in pure
strategies
6
Pr( )B Tough drinks Beer=
Pr( )tough man eats QuicheB− =1
Pr( )wimp drinks BeerQ− =1
Pr( )wimp eats QuicheQ =
Pr( )bully defers at BeerBD =
Pr( )bully fights at BeerBD− =1
Pr( )bully defers at QuicheQD =
Pr( )1 bully at QuicheBD fights− =
| Pr( )W QP Wimp|eats Quiche=
| Pr( )T QP Tough|eats Quiche=
Pr( )man is Tough = 1
3
Pr( )man is Wimp = 2
3
| Pr( )W BP Wimp|drinks Beer=
| Pr( )T BP Tough|drinks Beer=
7
• Go to end of the game
• Bully either defers (D) or attacks (A)
• What is best for bully:
– When he sees a man eating Quiche
– When he sees a man drinking Beer
8
, | | |0 1B Fight T Q W Q W QEPO P P P= × + × =
• Bully sees a man eat Quiche
, | | |1 0B Defer T Q W Q T QEPO P P P= × + × =
• Defer only if:
, ,B Fight B DeferEPO EPO<
| |W Q T QP P↔ <
| |W Q T QP PHow are beliefs and formed?
9
• Bully sees a man eat Quiche, he must update his beliefs
|
( )
( )
1
3
1 2
3 3
1
1
T Q
B
P
B Q
−
=
− +
Total Probability that
a Quiche is eaten
Proportion of Tough
men
Probability that a
Tough man eats
Quiche
Probability we have
here a Tough man
eating Quiche
Probability we have
here a Tough man
eating Quiche
Probability that a
Quiche is eaten by a
Wimp
10
• Bully sees a man eat Quiche, he must update his beliefs
|
( )
2
3
1 2
3 31
W Q
Q
P
B Q
=
− +
Total Probability that
a Quiche is eaten
Proportion of Wimps
Probability that a
Wimp eats Quiche
Probability we have
here a Tough man
eating Quiche
Probability we have
here a Wimp eating
Quiche
Probability that a
Quiche is eaten by a
Wimp
11
, | | |0 1B Fight T Q W Q W QEPO P P P= × + × =
• Bully sees a man eat Quiche
, | | |1 0B Defer T Q W Q T QEPO P P P= × + × =
• Defer only if:
, ,B Fight B DeferEPO EPO<
| |W Q T QP P↔ <
12
| |W Q T QP P<
|
( )
( )
1
3
1 2
3 3
1
1
T Q
B
P
B Q
−
=
− +
|
( )
2
3
1 2
3 31
W Q
Q
P
B Q
=
− +
• Defer only if:
| |W Q T QP P<
( )
( ) ( )
Q B
B Q B Q
−
<
− + − +
2 1
3 3
1 2 1 2
3 3 3 3
1
1 1
( )Q B< −2 1
3 3 1
( )Q B< −2 1
13
1. The bully defers at Quiche if: 2Q < (1 – B).
2. The bully fights at Quiche if: 2Q > (1 – B).
3. The bully randomizes at Quiche if: 2Q = (1 – B).
14
, | | |0 1B Fight T B W B W BEPO P P P= × + × =
• Bully sees a man drink Beer
, | | |1 0B Defer T B W B T BEPO P P P= × + × =
• Defer only if:
, ,B Fight B DeferEPO EPO<
| |W B T BP P↔ <
| |W B T BP PHow are beliefs and formed?
15
• Bully sees a man drink Beer, he must update his beliefs
|
( )
1
3
1 2
3 3 1
T B
B
P
B Q
=
+ −
Total Probability that
a Beer is drunk
Proportion of Tough
men
Probability that a
Tough man drinks
Beer
Probability we have
here a Tough man
drinking Beer
Probability we have
here a Tough man
drinking Beer
Probability that a
Beer is drunk by a
Wimp
16
|
( )
( )
2
3
1 2
3 3
1
1
W B
Q
P
B Q
−
=
+ −
Total Probability that
a Beer is drunk
Proportion of Wimps
Probability that a
Wimp drinks Beer
Probability we have
here a Tough man
drinking Beer
Probability we have
here a Wimp drinking
Beer
Probability that a
Beer is drunk by a
Wimp
• Bully sees a man drink Beer, he must update his beliefs
17
, | | |0 1B Fight T B W B W BEPO P P P= × + × =
• Bully sees a man drink Beer
, | | |1 0B Defer T B W B T BEPO P P P= × + × =
• Defer only if:
, ,B Fight B DeferEPO EPO<
• Bully sees a man drink Beer, he must update his belief
| |W B T BP P↔ <
18
| |W B T BP P<
|
( )
1
3
1 2
3 3 1
T B
B
P
B Q
=
+ −
|
( )
( )
2
3
1 2
3 3
1
1
W B
Q
P
B Q
−
=
+ −
• Defer only if:
| |W B T BP P<
( )
( ) ( )
Q B
B Q B Q
−
<
+ − + −
2 1
3 3
1 2 1 2
3 3 3 3
1
1 1
( )Q B− <2 1
3 31
( )Q B− <2 1
19
a) The bully defers at
Quiche if:
2Q < (1 – B).
a) The bully fights at Quiche
if:
2Q > (1 – B).
a) The bully randomises at
Quiche if:
2Q = (1 – B).
d) The bully defers at beer if
B > 2(1 – Q).
d) The bully fights at beer if
B < 2(1 – Q).
d) The bully randomises at
beer if
B = 2(1 – Q).
20
a) The bully defers at Quiche if: 2Q < (1 – B).
b) The bully fights at Quiche if: 2Q > (1 – B).
c) The bully randomises at Quiche if: 2Q = (1 – B).
d) The bully defers at beer if: B > 2(1 – Q).
e) The bully fights at beer if: B < 2(1 – Q).
f) The bully randomises at beer if: B = 2(1 – Q).
c) The bully randomises at Quiche if: 2Q = (1 – B).
• If true, then B = 1 – 2Q
– but then B < 2(1 – Q) e)
– Thus the bully fights at Beer
• Now the Wimp will thus always eat Quiche: Q=1
– But then in c): 2 = 1 – B: B=-1??? CONTR!
– c) cannot be true
21
a) The bully defers at Quiche if: 2Q < (1 – B).
b) The bully fights at Quiche if: 2Q > (1 – B).
c) The bully randomises at Quiche if: 2Q = (1 – B).
d) The bully defers at beer if: B > 2(1 – Q).
e) The bully fights at beer if: B < 2(1 – Q).
f) The bully randomises at beer if: B = 2(1 – Q).
f) The bully randomises at Beer if: B = 2(1 – Q).
• If true, then 2Q = 2 – B
– but then 2Q > 1 – B b)
– Thus the bully always fights at Quiche
• Now the Tough man will always drink Beer: B=1
– But then in f): 1 = 2(1 – Q) -> Q= ½
• Thus Wimp randomises: drinks beer with probability ½
• What is the payoff for the Wimp?
22
, ( )Wimp Beer B B BEPO D D D= + − =2 0 1 2
,Wimp QuicheEPO = 1
To keep the Wimp randomizing with prob.
½, the exp. Payoffs must be equal. Thus
, ,Wimp Quiche Wimp BeerEPO EPO× =1 1
2 2
BD=1 2
, ,Wimp Quiche Wimp BeerEPO EPO=
BD = 1
2
• The Wimp randomizes with prob ½
between Beer and Quiche
• The Bully randomizes with prob. ½
between fight and defer when he sees
a man drinking beer
• The Bully always fights when he sees
a man eating Quiche
23
• The man’s strategy
– The tough man always drinks beer
– The wimp eats quiche with probability ½ and
drinks beer with probability ½.
• The bully’s strategy
– If the man eats quiche; fight
– If the man drinks beer; fight with probability ½
and defer with probability ½

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The bully, the wimp and the tough man

  • 1. 1 The disco is not safe!
  • 2. 2the wimp, and the tough man… The bully,
  • 3. 3 Subgame 1 Subgame 2 Subgame 1 Subgame 2 Full information
  • 4. 4
  • 5. 5 •Separating equilibrium? – Not fight if the man drinks beer? •But then all wimps start drinking beer and all tough men drink beer •Bully payoff = ⅓ < than always fight (Bully payoff = ⅔) •Pooling equilibrium? – Always fight? •But then Tough man will always drink beer and Wimp always eat Quiche •Is perfect signal! – Always defer? •Bully payoff =⅓ •But then Tough man will always drink beer and Wimp always eat Quiche •Is perfect signal! •Thus no equilibrium in pure strategies
  • 6. 6 Pr( )B Tough drinks Beer= Pr( )tough man eats QuicheB− =1 Pr( )wimp drinks BeerQ− =1 Pr( )wimp eats QuicheQ = Pr( )bully defers at BeerBD = Pr( )bully fights at BeerBD− =1 Pr( )bully defers at QuicheQD = Pr( )1 bully at QuicheBD fights− = | Pr( )W QP Wimp|eats Quiche= | Pr( )T QP Tough|eats Quiche= Pr( )man is Tough = 1 3 Pr( )man is Wimp = 2 3 | Pr( )W BP Wimp|drinks Beer= | Pr( )T BP Tough|drinks Beer=
  • 7. 7 • Go to end of the game • Bully either defers (D) or attacks (A) • What is best for bully: – When he sees a man eating Quiche – When he sees a man drinking Beer
  • 8. 8 , | | |0 1B Fight T Q W Q W QEPO P P P= × + × = • Bully sees a man eat Quiche , | | |1 0B Defer T Q W Q T QEPO P P P= × + × = • Defer only if: , ,B Fight B DeferEPO EPO< | |W Q T QP P↔ < | |W Q T QP PHow are beliefs and formed?
  • 9. 9 • Bully sees a man eat Quiche, he must update his beliefs | ( ) ( ) 1 3 1 2 3 3 1 1 T Q B P B Q − = − + Total Probability that a Quiche is eaten Proportion of Tough men Probability that a Tough man eats Quiche Probability we have here a Tough man eating Quiche Probability we have here a Tough man eating Quiche Probability that a Quiche is eaten by a Wimp
  • 10. 10 • Bully sees a man eat Quiche, he must update his beliefs | ( ) 2 3 1 2 3 31 W Q Q P B Q = − + Total Probability that a Quiche is eaten Proportion of Wimps Probability that a Wimp eats Quiche Probability we have here a Tough man eating Quiche Probability we have here a Wimp eating Quiche Probability that a Quiche is eaten by a Wimp
  • 11. 11 , | | |0 1B Fight T Q W Q W QEPO P P P= × + × = • Bully sees a man eat Quiche , | | |1 0B Defer T Q W Q T QEPO P P P= × + × = • Defer only if: , ,B Fight B DeferEPO EPO< | |W Q T QP P↔ <
  • 12. 12 | |W Q T QP P< | ( ) ( ) 1 3 1 2 3 3 1 1 T Q B P B Q − = − + | ( ) 2 3 1 2 3 31 W Q Q P B Q = − + • Defer only if: | |W Q T QP P< ( ) ( ) ( ) Q B B Q B Q − < − + − + 2 1 3 3 1 2 1 2 3 3 3 3 1 1 1 ( )Q B< −2 1 3 3 1 ( )Q B< −2 1
  • 13. 13 1. The bully defers at Quiche if: 2Q < (1 – B). 2. The bully fights at Quiche if: 2Q > (1 – B). 3. The bully randomizes at Quiche if: 2Q = (1 – B).
  • 14. 14 , | | |0 1B Fight T B W B W BEPO P P P= × + × = • Bully sees a man drink Beer , | | |1 0B Defer T B W B T BEPO P P P= × + × = • Defer only if: , ,B Fight B DeferEPO EPO< | |W B T BP P↔ < | |W B T BP PHow are beliefs and formed?
  • 15. 15 • Bully sees a man drink Beer, he must update his beliefs | ( ) 1 3 1 2 3 3 1 T B B P B Q = + − Total Probability that a Beer is drunk Proportion of Tough men Probability that a Tough man drinks Beer Probability we have here a Tough man drinking Beer Probability we have here a Tough man drinking Beer Probability that a Beer is drunk by a Wimp
  • 16. 16 | ( ) ( ) 2 3 1 2 3 3 1 1 W B Q P B Q − = + − Total Probability that a Beer is drunk Proportion of Wimps Probability that a Wimp drinks Beer Probability we have here a Tough man drinking Beer Probability we have here a Wimp drinking Beer Probability that a Beer is drunk by a Wimp • Bully sees a man drink Beer, he must update his beliefs
  • 17. 17 , | | |0 1B Fight T B W B W BEPO P P P= × + × = • Bully sees a man drink Beer , | | |1 0B Defer T B W B T BEPO P P P= × + × = • Defer only if: , ,B Fight B DeferEPO EPO< • Bully sees a man drink Beer, he must update his belief | |W B T BP P↔ <
  • 18. 18 | |W B T BP P< | ( ) 1 3 1 2 3 3 1 T B B P B Q = + − | ( ) ( ) 2 3 1 2 3 3 1 1 W B Q P B Q − = + − • Defer only if: | |W B T BP P< ( ) ( ) ( ) Q B B Q B Q − < + − + − 2 1 3 3 1 2 1 2 3 3 3 3 1 1 1 ( )Q B− <2 1 3 31 ( )Q B− <2 1
  • 19. 19 a) The bully defers at Quiche if: 2Q < (1 – B). a) The bully fights at Quiche if: 2Q > (1 – B). a) The bully randomises at Quiche if: 2Q = (1 – B). d) The bully defers at beer if B > 2(1 – Q). d) The bully fights at beer if B < 2(1 – Q). d) The bully randomises at beer if B = 2(1 – Q).
  • 20. 20 a) The bully defers at Quiche if: 2Q < (1 – B). b) The bully fights at Quiche if: 2Q > (1 – B). c) The bully randomises at Quiche if: 2Q = (1 – B). d) The bully defers at beer if: B > 2(1 – Q). e) The bully fights at beer if: B < 2(1 – Q). f) The bully randomises at beer if: B = 2(1 – Q). c) The bully randomises at Quiche if: 2Q = (1 – B). • If true, then B = 1 – 2Q – but then B < 2(1 – Q) e) – Thus the bully fights at Beer • Now the Wimp will thus always eat Quiche: Q=1 – But then in c): 2 = 1 – B: B=-1??? CONTR! – c) cannot be true
  • 21. 21 a) The bully defers at Quiche if: 2Q < (1 – B). b) The bully fights at Quiche if: 2Q > (1 – B). c) The bully randomises at Quiche if: 2Q = (1 – B). d) The bully defers at beer if: B > 2(1 – Q). e) The bully fights at beer if: B < 2(1 – Q). f) The bully randomises at beer if: B = 2(1 – Q). f) The bully randomises at Beer if: B = 2(1 – Q). • If true, then 2Q = 2 – B – but then 2Q > 1 – B b) – Thus the bully always fights at Quiche • Now the Tough man will always drink Beer: B=1 – But then in f): 1 = 2(1 – Q) -> Q= ½ • Thus Wimp randomises: drinks beer with probability ½ • What is the payoff for the Wimp?
  • 22. 22 , ( )Wimp Beer B B BEPO D D D= + − =2 0 1 2 ,Wimp QuicheEPO = 1 To keep the Wimp randomizing with prob. ½, the exp. Payoffs must be equal. Thus , ,Wimp Quiche Wimp BeerEPO EPO× =1 1 2 2 BD=1 2 , ,Wimp Quiche Wimp BeerEPO EPO= BD = 1 2 • The Wimp randomizes with prob ½ between Beer and Quiche • The Bully randomizes with prob. ½ between fight and defer when he sees a man drinking beer • The Bully always fights when he sees a man eating Quiche
  • 23. 23 • The man’s strategy – The tough man always drinks beer – The wimp eats quiche with probability ½ and drinks beer with probability ½. • The bully’s strategy – If the man eats quiche; fight – If the man drinks beer; fight with probability ½ and defer with probability ½