11/28/11 Game theory Lecture 8 Segrega4on • 
• 
• 
There is 2 towns: East town (E) and West town (W): each holds 100 th. people Two types of people: Tall (T) and Short (S) (100 th. of each) Rules: simultaneous choice, if there is no room, then randomize to ra4on 1 • 
• 
Slightly more of S start in E Slightly more of T start in W • 
What are the equilibria of this game? U5lity of you 1/2 0 # of your type in your town 50 th. 100 th. 1 11/28/11 Segrega4on •  Two segregated equilibria: –  All Short people choose East town, All Tall people choose West town (stable) –  All Tall people choose East town, All Short people choose West town (stable) •  One integrated equilibrium: –  Of all Tall people 50% choose West town, 50% choose East town (the same for Short people) •  But this equilibrium is not stable •  If we allow dynamic adjustment process, the most probable outcome is segrega4on. –  But it may be beneficial for all people to give up ac4ve choice and have the society randomize for them. –  Or individual randomiza5on – mixed strategies. –  Sociology: seeing segrega4on does not imply preference for segrega4on. Strategic moves ● 
● 
What happens if instead of moving simultaneously without prior communica4on:  
One player moves first  
OR players communicate before making their moves A couple of examples 1) Zero‐sum game – Raw makes a first move A
B
A
(3, -3)
(-1, 1)
B
(0, 0)
(4, -4)
2 11/28/11 Strategic moves 1) Zero‐sum game A
B
Simultaneous: A
(3, -3)
(-1, 1)
B
(0, 0)
(4, -4)
Raw moves first: If Raw chooses A, Col will choose B, if Raw chooses B, Column will choose A. So Raw will choose A, and equilibrium payoffs will be (0,0). – in zerosum games, it doesn’t pay off to be the first !!! Strategic moves 2) Chicken game A
B
A
(3, 3)
(4, 2)
B
(2, 4)
(1, 1)
Simultaneous: Two Equilibria (A,B) [Payoffs (2,4)], and (B,A) [Payoffs (4,2)] One of the players moves first: The one who moves first secures payoff 4. Both players want to be the first. 3 11/28/11 Strategic moves 3) Yet another possibility A
B
A
(2, 3)
(1, 2)
B
(4, 1)
(3, 4)
Simultaneous: Raw’s A dominates B. So equilibrium is (A,A) [payoffs (2,3), not Pareto‐op4mal] Column moves first: Nothing changes Raw moves first: Raw will choose B and then Column will choose B as well – payoffs (3,4) They both want Raw to move first Strategic moves Communica5on – the same what you can achieve by the order of moves can be achieved by prior communica4on But how to commit to something, if they both may commit? (Chicken game) You may for example tell your opponent you will do something and quickly hang up the phone 4 11/28/11 Strategic moves 4) Non‐credible threat: Mr Raw declares that in case of some ac5on by Mrs Column he will take his ac5on that is: ● 
bad for Mrs Column ● 
bad for him as well A
(4, 3)
(2, 1)
A
B
Simultaneous: B
(3, 4)
(1, 2)
The only equilibrium in dominant strategies (A, B) [payoffs (3,4)] Whoever moves first: Nothing changes Column moves first and Raw threatens – if you play B, I will play B: If Column believes in Raw’s threat, she chooses between (A,A) and (B,B) and hence will choose A [payoffs (4,3)] Strategic moves 5) A non‐credible promise –prisoners’ dilemma: Mr Raw declares that in case of some ac5on by Mrs Column he will take his ac5ons that is: ● 
good for Mrs Column ● 
but bad for him Simultaneous: A
B
A
(3, 3)
(5, -1)
B
(-1, 5)
(0, 0)
Equilibrium (B,B) [Payoffs (0,0)] Whoever moves first: Nothing changes Whoever moves first and the second promises – if you play A, I will play A If the second believes in this promise, he/she has a choice between (A,A) and (B,B) , and hence will choose A [payoffs (3,3)] 5 11/28/11 Strategic moves 6) Simultaneous non‐credible threat and non‐credible promise: A
(3, 3)
(4, 0)
A
B
B
(1, 5)
(0, 2)
Simultaneous: Equilibrium (A,B) [payoffs (1,5)] Whoever moves first: Nothing changes. Column moves first and Raw threatens and promises – if you play I will play A, but if you play B, I will play B as well If Column believes in this simultaneous threat and promise, she has the choice between (A,A) and (B,B) and hence will choose A [payoffs (3,3)] Strategic moves ● 
Credibility is the key problem – many ways to make your claim credible are based on decreasing voluntarily your own payoff Making it credible: (A,A). 6) Non‐credible threat and promise: Column moves first and Raw threatens and promises – if you play I will play A, but if you play B, I will play B as well 4) Non‐credible threat: Column moves first and Raw threatens – if you play B, I will play B: Raw has to convince Mrs Column that he will choose (B,B) instead of (A,B) (decrease his payoff of 3 from (A,B) below his payoff of 1 from (B.B) A
B
A (4, 3) (3, 4)
B (2, 1) (1, 2)
5) Non‐credible promise: Whoever moves first and the second promises – if you play A, I will play A Suppose Column is first. Raw has to decrease his payoff of 5 from (B,A) below payoff of 3 from A
B
A (3, 3) (-1, 5)
B (5, -1) (0, 0)
Raw has to decrease his payoff of 1 below 0 (to make his threat credible) and his payoff of 4 below 3 (to make his promise credible) A
B
A (3, 3) (1, 5)
B (4, 0) (0, 2)
2) Chicken game – commiVment to play hawk Whoever is second should decrease payoff of 2 below 1. A
B
A (3, 3) (2, 4)
B (4, 2) (1, 1)
6 11/28/11 Strategic moves – exercises •  In the following games will Mr Raw profit from making one of the following strategic moves?: – 
– 
– 
– 
– 
Moving first or commieng to make certain move; Give the first move to Mrs Column; Make a threat; Make a promise; Make a threat and a promise simultaneously. •  For each game it is possible that one/more than one or none of the above moves will help. •  How can Mr Raw make his strategic moves credible? Strategic moves – exercise 1 A
B
A
(3, 4)
(2, 2)
B
(4, 3)
(1, 1)
7 11/28/11 Strategic moves – exercise 1 A
B
A
(3, 4)
(2, 2)
B
(4, 3)
(1, 1)
What Mr Raw likes more What Mr Raw gets in equilibrium Mr Raw threatens: if you play A, I will play B To make it credible Mr Raw has to decrease his payoff from (A,A) below 2 Strategic moves – exercise 2 A
B
A
(3, 4)
(2, 3)
B
(4, 2)
(1, 1)
8 11/28/11 Strategic moves – exercise 2 A
B
A
(3, 4)
(2, 3)
B
(4, 2)
(1, 1)
What Mr Raw likes more What Mr Raw gets in equilibrium Mr Raw cannot do anything. Strategic moves – exercise 3 A
B
A
(2, 4)
(1, 2)
B
(3, 3)
(4, 1)
9 11/28/11 Strategic moves – exercise 3 A
B
A
(2, 4)
(1, 2)
B
(3, 3)
(4, 1)
What Mr Raw likes more What Mr Raw gets in equilibrium Mr Raw threatens: if you play A, I will play B, and promises: if you play B, I will play A. To make it credible Mr Raw has to decrease his payoff from (A,A) below 1, and has to decrease his payoff from (B,B) below 3. Strategic moves – exercise 4 A
B
A
(2, 2)
(1, 3)
B
(4, 1)
(3, 4)
10 11/28/11 Strategic moves – exercise 4 A
B
A
(2, 2)
(1, 3)
B
(4, 1)
(3, 4)
What Mr Raw likes more (and is feasible) What Mr Raw gets in equilibrium 1) Mr Raw should make the first move or commit to play B To make it credible Mr Raw has to decrease his payoff from (A,A) below 1 and from (A,B) below 3 2) Mr Raw should promise: if you play B, I will play B To make it credible Mr Raw has to decrease his payoff from (A,B) below 3 Strategic moves – exercise 5 A
B
A
(3, 2)
(2, 4)
B
(1, 1)
(4, 3)
11 11/28/11 Strategic moves – exercise 5 A
B
A
(3, 2)
(2, 4)
B
(1, 1)
(4, 3)
What Mr Raw likes more (and is feasible) What Mr Raw gets in equilibrium Mr Raw should give the first move to Mrs Column Kidnapping for ransom A kidnapper holds his vic4m for ransom. It may be represented by extensive‐form game as follows: •  The vic4m may pay the ransom or not. •  Then the kidnapper may kill or release the vic4m. •  If the vic4m is released, she may either inform the police or not. The kidnapper’s „u4lity”: •  from being paid the ransom is +5; •  from the police being informed is ‐2; •  from killing the vic4m is ‐1. The vic4m’s „u4lity”: •  from being killed is ‐10; •  from having paid the ransom is ‐2; •  from informing the police is +1 Both the kidnapper’s and the vic4m’s „u4li4es” are addi4ve. 12 11/28/11 Ques4ons 1.  Draw the game tree 2.  Find the equilibrium 3.  Assume that the vic4m may make credible promises and threats. How will she use it? 4.  Assume that aper the vic4m makes a credible promise or threat, the kidnapper may also formulate a threat or a promise. How will he use it? 5.  If the vic4m cannot formulate credible threats and promises, but the kidnapper can. How will he use it? 6.  In what way in the real world may the par4cipants of this game make their par4cular threats and promises credible? Fishing on Jamaica •  In the fipies, about 200 households lived on Jamaica. They made their living primarily by fishing. •  Davenport, an anthropologist used to study their behavior in the six4es. 13 11/28/11 Jamaica on a map Strategie •  200 mieszkańców. 26 załóg żaglowych dłubanych kanoe, z których każda mogła stosować 3 różne strategie: –  Wewnętrzna – ustawić wszystkie kosze na łowiskach wewnętrznych –  Zewnętrzna – ustawić wszystkie kosze na łowiskach zewnętrznych –  Pośrednia (In‐out) – ustawić część koszy na łowiskach zewnętrznych a pozostałe na wewnętrznych 14 11/28/11 Wady i zalety •  Dopłynięcie do łowisk zewnętrznych czasochłonne, stosujący strategię zewnętrzną lub pośrednią ustawiali mniejszą liczbę koszy •  Aktywność prądów przynosiła rybakom wiele szkód: przesuwanie boj oznaczających kosze, uszkodzenia koszy, zmiany temperatur związanych z ruchem wody zabijało ryby •  Połowy na zewnętrznych łowiskach dawały większe ryby oraz dostępne było więcej gatunków ryb. Ryby z zewnętrznych łowisk dostępne w odpowiednich ilościach mogłyby całkowicie wyprzeć ryby z łowisk wewnętrznych •  Do wypływania na łowiska zewnętrzne potrzebne są lepsze łodzie. Ci co łowią na zewnętrznych łowiskach wygrywają więc zawody żeglarskie zdobywając pres4ż i wartościowe nagrody Collec4ng data •  Davenport collected the data concerning the fishermen average monthly profit depending on the fishing strategies they used to adopt. Rybacy\Prąd Płynie Nie płynie Wewnątrz zatoki 17,3 11,5 Na zewnątrz zatoki ‐4,4 20,6 In‐Out 5,2 17,0 15 11/28/11 Spróbujmy użyć teorii gier •  Problem “złośliwego prądu”; p – “strategia” prądu •  Problem rybaków; q – strategia rybaków Rozwiązanie graficzne problemu prądu 20 15 Rozwiązanie: p=0.31 Wewnątrz Na zewnątrz In‐Out 10 5 Prawdopodob. prądu 0 0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3 0.33 0.36 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.78 0.81 0.84 0.87 0.9 0.93 0.96 0.99 ‐5 16 11/28/11 Maxmin i minimax Strategia prądu
p
1-p
0.31
0.69
funkcja celu
minimalizuj
13.31
Oczekiwana wypłata ze strategii
wewnętrznej
zewnętrznej
in-out
prawdopodobieństwa
13.31
12.79
13.31
1.00
funkcja celu
maksymalizuj
13.31
Oczekiwana wypłata prądu gdy:
płynę
nie płynę
prawdopodobieństwa
13.31
13.31
1.00
<=
<=
<=
=
q1
13.31
13.31
13.31
1.00
Strategia rybaków
q2
0.00
0.67
>=
>=
=
q3
0.33
13.31
13.31
1.00
Raport wrażliwości minimax Microsoft Excel 14.1 Sensitivity Report
Worksheet: [maximinnowe.xlsx]minimax
Report Created: 11/16/2011 12:19:08 PM
Variable Cells
Cell
Name
$B$3 minimalizuj funkcja celu
$C$3 minimalizuj p
$D$3 minimalizuj 1-p
Final
Value
13.3125
0.3125
0.6875
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase Decrease
0
1
1E+30
1
0
0
11.8
5.8
0
0
5.8
11.8
Final
Value
13.3125
12.7875
13.3125
1
Shadow
Constraint Allowable Allowable
Price
R.H. Side Increase Decrease
-0.670454545
0
12.1
0.7
0
0
1E+30
0.525
-0.329545455
0
0.3
12.1
13.3125
1
1E+30
1
Constraints
Cell
$B$6
$B$7
$B$8
$B$9
Name
wewnętrznej funkcja celu
zewnętrznej funkcja celu
in-out funkcja celu
prawdopodobieństwa funkcja celu
17 11/28/11 Raport wrażliwości maximin Microsoft Excel 14.1 Sensitivity Report
Worksheet: [maximinnowe.xlsx]maximin
Report Created: 11/16/2011 12:20:13 PM
Variable Cells
Cell
$B$3
$C$3
$D$3
$E$3
Name
maksymalizuj funkcja celu
maksymalizuj q1
maksymalizuj q2
maksymalizuj q3
Final
Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
13.3125
0
1
1E+30
1
0.670454545
0
0
0.7
12.1
0
-0.525
0
0.525
1E+30
0.329545455
0
0
12.1
0.3
Constraints
Cell
Name
$B$6 płynę funkcja celu
$B$7 nie płynę funkcja celu
$B$8 prawdopodobieństwa funkcja celu
Final
Value
13.3125
13.3125
1
Shadow Constraint Allowable Allowable
Price
R.H. Side Increase Decrease
-0.3125
0
5.8
11.8
-0.6875
0
11.8
5.8
13.3125
1
1E+30
1
Prognoza i obserwacja Zatem teoria gier przewiduje Obserwacje pokazują •  Żaden z rybaków nie ryzykuje łowienia na otwartym morzu •  Strategia 2/3 wewnątrz zatoki, 1/3 in‐out •  Optymalna strategia prądu 31% płynie i 69% nie płynie •  Żaden z rybaków nie ryzykuje łowienia na otwartym morzu •  Strategia 69% wewnątrz i 31% in‐out •  Prąd płynie 25% czasu i nie płynie 75% czasu 18 11/28/11 Teoria decyzji •  Prąd nie jest graczem i gra nie do końca optymalnie, zatem powinniśmy to wykorzystać: •  Wartości oczekiwane zysku: –  Wewnątrz: 0,25 x 17,3 + 0,75 x 11,5 = 12,95 –  Na zewnątrz: 0,25 x (‐4,4) + 0,75 x 20,6 = 14,35 –  In‐out: 0,25 x 5,2 + 0,75 x 17,0 = 14,05 •  Paradox??? Rybacy powinni łowić na zewnątrz Odpowiedź •  Prąd nie rozumuje, ale rybacy są bardzo ostrożni •  Poprzez traktowanie prądu jako przeciwnika, rybacy mogą zagwarantować sobie co najmniej wartość gry (13,3) •  Jeśli prąd będzie płynął z inną częstotliwością niż w najgorszym przypadku, rybacy mogą zarobić tylko więcej •  A gdyby zdecydowali się łowić na zewnątrz a prąd danego roku płynąłby z częstotliwością 35 zamiast 25 % to oczekiwany zysk byłby: 0,35 x (‐4,4) + 0,65 x (20,6)=11,85, czyli mniej niż 13,3 •  Rybacy płacą 1,05 funtów jako rodzaj polisy ubezpieczeniowej 19