11/20/11 Game Theory Lecture 7 Cu5ng cake 1 11/20/11 Alterna;ng offers bargaining by Rubinstein (1982) •  Two players Adam and Boney are trying to divide a cake. •  In ;me 0, Adam makes an offer of to Boney –  If Boney accepts, Adam will get , and Boney –  If Boney rejects, then: •  In ;me 1, Boney makes an offer of to Adam –  If Adam accepts, Boney gets and Adam –  If Adam rejects, he must make another offer in ;me 2 •  The process last un;l somebody accepts •  Since Adam and Boney are impa;ent, they discount future with a discount factor Sta;onary equilibrium with no delay •  No delay – all equilibrium offers are accepted •  Sta+onary – Equilibrium offers do not depend on ;me Let be equilbrium offers: – 
What Boney expects to get, if she rejects Adam’s offer? And hence in equilibrium: – 
Likewise for Adam: – 
2 11/20/11 Impa;ence or natural law? Sta;onary equilibrium with no delay •  There exists at most one sta;onary equilibrium with no delay •  But we must prove, that there exists at least one such equilibrium •  Consider the following strategies: •  Adam: Always offer , accept all offers , if: •  Boney: Always offer , accept all offers , if: 3 11/20/11 Stacjonarna równowaga bez opóźnień •  One‐shot devia+on – from strategy s in a subgame is a strategy which differs from s in only one ac;on in the first decision node of this subgame •  One‐devia+on property – a profile of strategies is SPNE if and only if for any subgame and any player there does not exist a profitable one‐shot devia;on –  This property holds for infinite games if certain condi;ons are sa;sfied. –  Fortunately, these condi;ons are sa;sfied by our game. Sta;onary equilibrium with no delay •  We will prove that the following profile of strategies is SPNE: •  We have to show that there are no subgames such that some player has a profitable one‐shot devia;on •  Subgames star;ng from Adam’s offer: –  If Adam offers •  Boney will accept •  But Adam’s payoff will be lower than in equilibrium –  If Adam offers •  Boney will reject and will offer •  Adam accepts, but his payoff will be smaller 4 11/20/11 Sta;onary equilibrium with no delay •  Subgames star;ng from Adam responding to the offers of Boney: –  If Adam rejects an offer •  he will offer and his payoff will be –  If Adam accepts an offer •  It is not profitable either. •  Likewise, Boney’s strategy is op;mal •  And hence we proved that the following profile of strategies is SPNE: Sta;onary equilibrium with no delay •  Ariel Rubinstein (1982) showed, that this is the only SPNE if players are impa;ent (discount factor <1) –  This equilibrium is efficient •  Bargaining power –  A piece for Adam –  A piece for Boney –  Who is more pa;ent, will get more. •  What if they are impa;ent to the same degree? –  First mover’s advantage, but it vanishes as 5 11/20/11 Prisoners’ dilemma – how to get coopera;on? Pareto op+mum A
B
Equivalent to: Cooperate
Defect
A
(0, 0)
(1, -2)
B
(-2, 1)
(-1, -1)
Nash equilibrium Cooperate
Defect
(R, R)
(S, T)
(T, S)
(U, U)
Where T>R>U>S and R≥(S+T)/2 R – reward, S – sucker, T – tempta;on, U ‐ uncoopera;ve Goal: get coopera+on Get coopera;on in prisoners’ dilemma •  Three ways: –  Iterated game –  Meta‐game –  Experiments 6 11/20/11 1 – Iterated game •  In most real situa;ons, the game is played many ;mes •  Suppose we play the game N ;mes: –  Domino effect: solve by backward induc;on •  Two ways to overcome domino effects: –  Real players rarely conform to strict ra;onality –  We don’t know how many games we are gonna play •  Suppose, p is the probability of next itera;on. We play the first game with probability 1, the next with probability p, the second next with probability p2, etc. 1 – Iterated game •  Grim trigger strategy (GTS): –  Play C in the first game –  If your opponent played C always before, play C –  If your opponnet ever deviated in the past, play D •  Suppose that my opponent plays GTS. –  If I play always C, I will get –  If I play first m ;mes C and then D, I will get 7 11/20/11 1 – Iterated game •  So I should never play D, if for any m: •  Which is equivalent to: •  Example A
B
A
(0, 0)
(1, -2)
B
(-2, 1)
(-1, -1)
2 – Meta‐game •  1 Level: Mrs Column makes her decision dependent on her expecta;on of what strategy will Mr Raw choose 1.  Choose A independent on her expecta;on about Mr Raw’s strategy 2.  Choose the same strategy as she expects Mr Raw to use 3.  Choose the opposite strategy to what she expects about Mr Raw’s strategy 4.  Choose B independent on her expecta;on about Mr Raw’s strategy A
Mr Raw
B
A
B
A (0, 0) (-2, 1)
B (1, -2) (-1, -1)
AA
00
1 -2
Mrs Column
AB
BA
00
-2 1
-1 -1
1 -2
BB
-2 1
-1 -1
8 11/20/11 2 – Meta‐game •  2 Level: Mr Raw decision dependent on his predic;ons about Mrs Column’s strategy: Mr Raw
–  16 strategies e.g. AAAA – always play A; ABAB – play B if your predic;on is that Mrs Column uses strategy AB or BB, otherwise play A AAAA
AAAB
AABA
AABB
ABAA
ABAB
ABBA
ABBB
BAAA
BAAB
BABA
BABB
BBAA
BBAB
BBBA
BBBB
AA
00
00
00
00
00
00
00
00
1 -2
1 -2
1 -2
1 -2
1 -2
1 -2
1 -2
1 -2
Mrs Column
AB
BA
00
-2 1
00
-2 1
00
1 -2
00
1 -2
-1 -1
-2 1
-1 -1
-2 1
-1 -1
1 -2
-1 -1
1 -2
00
-2 1
00
-2 1
00
1 -2
00
1 -2
-1 -1
-2 1
-1 -1
-2 1
-1 -1
1 -2
-1 -1
1 -2
BB
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
-2 1
-1 -1
A
A (0, 0)
B (1, -2)
B
(-2, 1)
(-1, -1)
Mr Raw BABB – Cooperate if and only if you are convinced that your opponent will cooperate if and only if you will cooperate (this strategy weakly dominates all the others) Mrs Column AB – Cooperate if and only if you are convinced that your opponent will cooperate 3 – Experiments ● 
In 1984 Robert Axelrod conducted an experiment: ● 
● 
● 
● 
Specialists were asked to write a computer program implemen;ng some strategy, which were amerwards compe;ng aginst each other – 14 programs/strategies were wrinen The winner – Anatol Rapoport proposed a strategy Tit for Tat Axelrod published the results and proposed programs designed to win with Tit for Tat In the second round there were 62 specialists: ● 
.... and again Rapoport won with unchanged program implemen;ng Tit for Tat 9 11/20/11 3 – Experiments ● 
Tit for tat strategy: ● 
● 
● 
In the first round play Cooperate In every next round play what your opponent played in the previous round. 4 proper;es of a good strategy: ● 
Friendly – starts with coopera;on and does not defect first ● 
Retalia;ng – it should punish devia;on ● 
● 
Forgiving – amer punishing it should be willing to restore coopera;on Transparent – its decisions should be consistent and easy to predict Simplified poker •  Linle John and Linle Margaret play cards in a sugar hut when ugly witch (besom) is not around: –  They both put a candy on a table –  They draw a card from a deck of cards consis;ng solely of aces and kings –  Linle John may raise by 2 candies or pass –  If he passes Linle Margaret takes two candies from the table –  If he raises, then Linle Margaret may check or pass –  If she passes, Linle John takes all candies –  If she checks, then they compare their cards: An ace wins over a king; they share equally in case of a draw. 10 11/20/11 Extensive form game Standard form game 11