1 Game Theory Lecture 2 Game Theory Lecture 2

Spieltheorie- Übungen P. Kircher: Dienstag – 09: HS M S. Ludwig: Donnerstag Uhr HS L T. Troeger: Mittwoch – Uhr HS N A. Shaked: Freitag Uhr - HS N

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Theorem (Zermelo, 1913): Either player 1 can force an outcome in T or player 2 can force an outcome in T’ Let G be a finite, two player game of perfect information without chance moves. A reminder

Zermelo’s proof uses Backwards Induction

A game G is strictly Competitive if for any two terminal nodes a,b a  1 b  b  2 a

An application of Zermelo’s theorem to Strictly Competitive Games Let a 1,a 2, ….a n be the terminal nodes of a strictly competitive game (with no chance moves and with perfect information) and let: a n  1 a n-1  1 ….  1 a 2  1 a 1 (i.e. a n  2 a n-1  2 ….  2 a 2  2 a 1 ). ? Then there exists k, n  k  1 s.t. player 1 can force an outcome in  a n, a n-1 … … a k  And player 2 can force an outcome in  a k, a k-1 … … a 1 

? a n  1 a n-1  1..  1 a k  1..  1 a 2  1 a 1 G(s,t)= a k Player 1 has a strategy s which forces an outcome better or equal to a k (  1 ) Player 2 has a strategy t which forces an outcome better or equal to a k (  2 )

Proof : Let w j =  a n, a n-1 … …,a j , j =1,…,n w n+1 =   a n, a n-1 … a j …, a 2, a 1  w1w1 w2w2 wjwj w n+1 wnwn

Proof : Player 1 can force an outcome in W 1 =  a n, a n-1 …,a 1 , and cannot force an outcome in w n+1 = . Let w j =  a n, a n-1 … …,a j , j =1,…,n w n+1 =  w 1, w 2, ….w n,w n+1 can force cannot force can force ?? Let k be the maximal integer s.t. player 1 can force an outcome in W k

Proof : w 1, w 2, … w k, w k+1...,w n+1 Player 1 can force Player 1 cannot force Let k be the maximal integer s.t. player 1 can force an outcome in W k   a n, a n-1 … a k+1, a k …, a 2, a 1  w1w1 w k+1 wkwk Player 2 can force an outcome in T -w k+1 by Zermelo’s theorem

!!!!!!!!!! a n  1 a n-1  1..  1 a k  1..  1 a 2  1 a 1 G(s,t)= a k Player 1 has a strategy s which forces an outcome better or equal to a k (  1 ) Player 2 has a strategy t which forces an outcome better or equal to a k (  2 )

Now consider the implications of this result for the strategic form game s t akak player 1’s strategy s guarantees at least a k player 2’s strategy t guarantees him at least a k i.e. at most a k for player 1

s t akak The point (s,t) is a Saddle point

s t akak Given that player 2 plays t, Player 1 has no better strategy than s strategy s is player 1’s best response to player 2’s strategy t Similarly, strategy t is player 2’s best response to player 1’s strategy s

A pair of strategies (s,t) such that each is a best response to the other is a Nash Equilibrium Awarding the Nobel Prize in Economics John F. Nash Jr. This definition holds for any game, not only for strict competitive ones

LW W WL WW r l R M L Example 3 R L R r L l backwards Induction (Zermelo) r ( l, r ) ( R, ,  )

LW W WL WW r l R M L Example 3 R L R r L l r ( l, r ) ( R, ,  ) All those strategy pairs are Nash equilibria But there are other Nash equilibria ……. ( l, r ) ( L, ,  )

LW W WL WW r l R M L Example 3 R L R r L l r ( l, r ) ( R, ,  ) The strategies obtained by backwards induction Are Sub-Game Perfect equilibria in each sub-game they prescribe a Nash equilibrium

LW W WL WW r l R M L Example 3 R L R r L l r ( l, r ) ( R, ,  ) Whereas, the non Sub-Game Perfect Nash equilibrium prescribes a non equilibrium behavior in some sub-games ( l, r ) ( L, ,  )

A Sub-Game Perfect equilibria prescribes a Nash equilibrium in each sub-game Awarding the Nobel Prize in Economics R. Selten

22 Chance Moves Nature (player 0), chooses randomly, with known probabilities, among some actions. 0  +  +  +  = 1    

Russian Roulette 0 1/ / information set N.S. S.S. S.S. S.S. S.S. S.S. S.S. Payoffs: W (when the other dies, or when the other chose not shoot in his turn) D(when not shooting) L(when dead)

Russian Roulette 0 1/ N.S. S.S. S.S. S.S. S.S. S.S. S.S. Payoffs: W (when the other dies, or when the other did not shoot in his turn) D(when not shooting) L(when dead) W  D  L

Russian Roulette 0 1/ N.S. S.S. S.S. S.S. S.S. S.S. S.S. DDDDDD L 22222

Russian Roulette 0 1/ N.S. S.S. S.S. S.S. S.S. S.S. S.S. DDDDDD L S.S. D L S.S. D S.S. D S.S. D S.S. D