Wstęp do Teorii Gier. Segregacja Dwa miasta: Est (E) and Ovest (W): każde 100 tys. mieszkańców...

Wstęp do Teorii Gier

Segregacja

• Dwa miasta: Est (E) and Ovest (W): każde 100 tys. mieszkańców• Dwa rodzaje mieszkańców: Dłudzy (T) i Krótcy (S) (po 100 tys.)• Zasady: jednoczesny wybór, jeśli nie ma już miejsca, losowy przydział nadmiaru

osób

• Trochę więcej S startuje w E• Trochę więcej T startuje w W

• Jakie są równowagi w tej grze?

1/2

1

050 th. 100 th.

Twoja użyteczność

# liczba osób takich jak ty w Twoim mieście

Segregacja• Dwie równagi segregujące:

– Wszyscy Krótcy wybierają miasto Est, wszyscy Dłudzy wybierają Ovest (stabilne)– Wszyscy Dłudzy wybierają Est, wszyscy Krótcy wybierają Ovest (stabilne)

• Jedna zintegrowana równowaga:– 50% Długich wybiera Ovest, 50% Długich wybiera Est, to samo dla Krótkich – Równowaga niestabilna

• Jeśli wprowadzimy dynamiczny proces dostosowawczy, najbardziej prawdopodnym wynikiem jest segregacja. – Ale może być opłacalne dla ludzi, aby zrezygnować z aktywnego wyboru.

Społeczeństwo wybierze za nich strategię mieszaną.– Albo indywidualny wybór strategii mieszanej.– Socjologia: to, że widzimy segregację nie musi oznaczać preferencji dla segregacji.

Strategic moves● What happens if instead of moving simultaneously without

prior communication: One player moves first OR players communicate before making their moves

● A couple of examples

1) Zero-sum game – Raw makes a first moveA B

A (3,-3) (0,0)B (-1,1) (4,-4)

Strategic moves1) Zero-sum game

Simultaneous:

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 zero-sum games, it doesn’t pay off to be the first !!!

A BA (3,-3) (0,0)B (-1,1) (4,-4)

Strategic moves

2) Chicken game

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.

A BA (3,3) (2,4)B (4,2) (1,1)

Strategic moves

3) Yet another possibility

Simultaneous:

Raw’s A dominates B. So equilibrium is (A,A) [payoffs (2,3), not Pareto-optimal]

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

A BA (2,3) (4,1)B (1,2) (3,4)

Strategic moves

Communication – the same what you can achieve by the order of moves can be achieved by prior communication

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

Strategic moves4) Non-credible threat:

Mr Raw declares that in case of some action by Mrs Column he will take his action that is:

● bad for Mrs Column

● bad for him as well

Simultaneous:

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)]

A BA (4,3) (3,4)B (2,1) (1,2)

Strategic moves5) A non-credible promise –prisoners’ dilemma:

Mr Raw declares that in case of some action by Mrs Column he will take his actions that is:

● good for Mrs Column

● but bad for him

Simultaneous:

Equilibrium (B,B) [Payoffs (0,0)]


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)]

A BA (3,3) (-1,5)B (5,-1) (0,0)

Strategic moves6) Simultaneous non-credible threat and non-credible promise:

Simultaneous:

Equilibrium (A,B) [payoffs (1,5)]


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)]

A BA (3,3) (1,5)B (4,0) (0,2)

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:

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)

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,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

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)

2) Chicken game – committment to play hawk

Whoever is second should decrease payoff of 2 below 1.

A B

A(4,3)

(3,4)

B(2,1)

(1,2)

A B

A (3,3)(-1,5)

B (5,-1) (0,0)

A B

A(3,3)

(1,5)

B(4,0)

(0,2)

A B

A(3,3)

(2,4)

B(4,2)

(1,1)

Strategic moves – exercises• In the following games will Mr Raw profit from making one of the

following strategic moves?:– Moving first or committing 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 BA (3,4) (4,3)B (2,2) (1,1)


A BA (3,4) (4,3)B (2,2) (1,1)

What Mr Raw likes more

What Mr Raw gets in equilibrium

Mr Raw threatens: if you play A, I will play BTo make it credible Mr Raw has to decrease his payoff from (A,A) below 2


A BA (3,4) (4,2)B (2,3) (1,1)


A BA (3,4) (4,2)B (2,3) (1,1)



Mr Raw cannot do anything.


A BA (2,4) (3,3)B (1,2) (4,1)


A BA (2,4) (3,3)B (1,2) (4,1)



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.


A BA (2,2) (4,1)B (1,3) (3,4)


A BA (2,2) (4,1)B (1,3) (3,4)

What Mr Raw likes more (and is feasible)


1) Mr Raw should make the first move or commit to play BTo make it credible Mr Raw has to decrease his payoff from (A,A) below 1 and from (A,B) below 32) Mr Raw should promise: if you play B, I will play BTo make it credible Mr Raw has to decrease his payoff from (A,B) below 3


A BA (3,2) (1,1)B (2,4) (4,3)


A BA (3,2) (1,1)B (2,4) (4,3)

What Mr Raw likes more (and is feasible)


Mr Raw should give the first move to Mrs Column

Kidnapping for ransomA kidnapper holds his victim for ransom. It may be represented by extensive-form game as follows:

• The victim may pay the ransom or not.

• Then the kidnapper may kill or release the victim.

• If the victim is released, she may either inform the police or not.

The kidnapper’s „utility”:

• from being paid the ransom is +5;

• from the police being informed is -2;

• from killing the victim is -1.

The victim’s „utility”:

• from being killed is -10;

• from having paid the ransom is -2;

• from informing the police is +1

Both the kidnapper’s and the victim’s „utilities” are additive.

Questions

1. Draw the game tree2. Find the equilibrium3. Assume that the victim may make credible promises and

threats. How will she use it?4. Assume that after the victim makes a credible promise or

threat, the kidnapper may also formulate a threat or a promise. How will he use it?

5. If the victim 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 participants of this game make their particular threats and promises credible?

Wstęp do Teorii Gier. Segregacja Dwa miasta: Est (E) and Ovest (W): każde 100 tys. mieszkańców...

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