Flashcards in Test 2 Deck (14)

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## TFGWT rules

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1- there are two players, I and II, who alternate

2- no random mechanisms

3- whenever a play ends exactly one winner exists

4- each play ends after finitely many moves

5- in any moment, in any play, there are finitely many options for a next legal play

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## SFGWT

### Rules of a TFGWT except axiom 5 May or may not be satisfied

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## Solution and value

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Row has a pure or mixed strategy S giving Row an expected value of V or more regardless of Col's response.

- same situation except for Col

- then we say (S,T,V) is a solution of the game

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## Rules for Supergame

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1-on first move, I chooses a TFGWT

2- player 2 goes first in a play of the subgame G

3- they continue to play the subgame G with II playing I's role and vice versa

4- whoever wins the play of the subgame wins the play of Supergame

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## Rules of Hypergame

### Same as Supergame except they name a SFGWT on first move

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## Saddle point

### A strategy profile in a 0 sum game in which neither player can better their payoff by unilaterally switching profiles

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## Minimax Theorem

### Every 0 sum game has a solution

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## Mani ax method

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Maximin- the maximum of Row's minimum

Minimax- the minimum of Col's maximum

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## Expected value

### P(E1)(a2)+p(E2)(a2)

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## Konig's Infinityy Lemma

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No tree T can satisfy all 3 properties

1- every fork in T is finite in length

2- every branch in T is finite in length

3- T has infinitely many nodes

Variant 1- every tree satisfying both 1 and 2 violates 3

Variant 2- every tree satisfying 1 and 3 violates 2

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## Zermelo's Theorem

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- if G is any TFGWT then I has a W.S. For G or II does

Proof- by GTL, T has finitely many nodes

- label each terminal node with a * or $ using ax 3

- moves labels up, step-by-step ax 1 and 2

- finitely many steps for backwards induction, GTL

- symbol assigned to top node of T has WS in game G

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## Game Tree Lemma

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For each TFGWT G, G's game tree T has finitely many nodes

Proof- as TFGWT we know axiom 4

- we know plays of G correspond to branches finite (KIL II)

- we also know axiom 5

- option correspond to fork in T, finite (KIL I)

- T satisfies I and II of KIL, III fails- finite nodes altogether

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## Hypergame theorem 1 positive

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HG is a SFGWT

- rules A and B tell us they alternate 1st and 2nd moves, other moves are in SFGWT so alternate

- no random, rules A, subgame is SFGWT no ran

- end of subgame, a SFGWT, unique winner, winner of HG, rule D

- play of HG is a SFGWT preceded by a single game-naming move, play ends after finite moves, play is SG is 1 move longer 1+finite #= finite #

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