Wstęp do Teorii Gier

1. Wstęp do TeoriiGier

2. 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? Twoja użyteczność 1 1/2 # liczba osób takich jak ty w Twoim mieście 0 100 th. 50 th.

3. 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.

4. Strategicmoves Whathappensifinstead of movingsimultaneouslywithoutpriorcommunication: One player moves first OR playerscommunicatebeforemakingtheirmoves A couple of examples 1) Zero-sum game – Raw makes a first move

5. Strategicmoves 1) 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 equilibriumpayoffs will be (0,0). – in zero-sum games, itdoesn’tpayoff to be the first !!!

6. Strategicmoves 2) Chickengame Simultaneous: TwoEquilibria (A,B) [Payoffs (2,4)], and (B,A) [Payoffs (4,2)] One of theplayersmoves first: The one whomoves first securespayoff 4. Bothplayers want to be the first.

7. Strategicmoves 3) Yetanotherpossibility Simultaneous: Raw’s A dominates B. So equilibriumis (A,A) [payoffs (2,3), not Pareto-optimal] Columnmoves first: Nothingchanges Raw moves first: Raw will choose B and thenColumn will choose B as well – payoffs (3,4) Theyboth want Raw to move first

8. Strategicmoves Communication – the same whatyoucanachieve by the order of movescan be achieved by priorcommunication But how to commit to something, iftheybothmaycommit? (Chickengame) Youmay for example tell youropponentyou will do something and quickly hang upthephone

9. Strategicmoves 4) Non-crediblethreat: Mr Raw declaresthatincase of some action by MrsColumnhe will take his action thatis: bad for MrsColumn bad for him as well Simultaneous: Theonlyequilibriumin dominant strategies (A, B) [payoffs (3,4)] Whoevermoves first: Nothingchanges Columnmoves first and Raw threatens– ifyou play B, I will play B: IfColumnbelievesinRaw’sthreat, shechoosesbetween (A,A) and (B,B) and hence will choose A [payoffs (4,3)]

10. Strategicmoves 5) A non-crediblepromise –prisoners’ dilemma: Mr Raw declaresthatincase of some action by MrsColumnhe will take his actionsthatis: good for MrsColumn but bad for him Simultaneous: Equilibrium (B,B) [Payoffs (0,0)] Whoevermoves first: Nothingchanges Whoevermoves firstand thesecondpromises– ifyou play A, I will play A Ifthesecondbelievesinthispromise, he/shehas a choicebetween (A,A) and (B,B) , and hence will choose A [payoffs (3,3)]

11. Strategicmoves 6) Simultaneousnon-crediblethreat and non-crediblepromise: Simultaneous: Equilibrium (A,B) [payoffs (1,5)] Whoevermoves first: Nothingchanges. Columnmoves first and Raw threatens and promises– ifyou play I will play A, but ifyou play B, I will play B as well IfColumnbelievesinthissimultaneousthreat and promise, shehasthechoicebetween (A,A) and (B,B) and hence will choose A [payoffs (3,3)]

12. Strategicmoves Credibilityisthekey problem – many ways to make yourclaimcrediblearebased on decreasingvoluntarilyyourownpayoff Makingitcredible: 4) Non-crediblethreat:Columnmoves first and Raw threatens– ifyou play B, I will play B: Raw has to convinceMrsColumnthathe 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-crediblepromise: Whoevermoves first and thesecondpromises– ifyou play A, I will play A SupposeColumnis first. Raw has to decrease his payoff of 5 from (B,A) belowpayoff of 3 from (A,A). 6) Non-crediblethreat and promise: Columnmoves first and Raw threatens and promises – ifyou play I will play A, but ifyou play B, I will play B as well Raw has to decrease his payoff of 1 below 0 (to make his threatcredible) and his payoff of 4 below 3 (to make his promisecredible) 2) Chickengame – committment to play hawk Whoeverissecondshoulddecreasepayoff of 2 below 1.

13. 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?

14. Strategic moves – exercise 1

15. Strategic moves – exercise 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

16. Strategic moves – exercise 2

17. Strategic moves – exercise 2 What Mr Raw likes more What Mr Raw gets in equilibrium Mr Raw cannot do anything.

18. Strategic moves – exercise 3

19. Strategic moves – exercise 3 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.

20. Strategic moves – exercise 4

21. Strategic moves – exercise 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

22. Strategic moves – exercise 5

23. Strategic moves – exercise 5 What Mr Raw likes more (and is feasible) What Mr Raw gets in equilibrium Mr Raw should give the first move to Mrs Column

24. Kidnapping for ransom A kidnapperholdshisvictim for ransom. It may be represented by extensive-form game as follows: • The victimmaypay the ransomor not. • Then the kidnappermaykillorrelease the victim. • If the victimisreleased, shemayeitherinform the police or not. The kidnapper’s „utility”: • from beingpaid the ransomis +5; • from the police beinginformedis -2; • from killing the victimis -1. The victim’s „utility”: • from beingkilledis -10; • from havingpaid the ransomis -2; • from informing the police is +1 Both the kidnapper’s and the victim’s „utilities” areadditive.

25. Questions • Draw the game tree • Find the equilibrium • Assume that the victim may make credible promises and threats. How will she use it? • 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? • If the victim cannot formulate credible threats and promises, but the kidnapper can. How will he use it? • In what way in the real world may the participants of this game make their particular threats and promises credible?