Test 2 Flashcards
SFGWT
Rules of a TFGWT except axiom 5 May or may not be satisfied
TFGWT rules
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
Solution and value
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
Rules for Supergame
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
Rules of Hypergame
Same as Supergame except they name a SFGWT on first move
Saddle point
A strategy profile in a 0 sum game in which neither player can better their payoff by unilaterally switching profiles
Minimax Theorem
Every 0 sum game has a solution
Mani ax method
Maximin- the maximum of Row’s minimum
Minimax- the minimum of Col’s maximum
Expected value
P(E1)(a2)+p(E2)(a2)
Konig’s Infinityy Lemma
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
Zermelo’s Theorem
- 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
Game Tree Lemma
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
Hypergame theorem 1 positive
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 #
Hypergame Theorem 2 negative
HG is not a SFGWT
HG does not satisfy axiom 4 because it is legal to say HG on 1st move because it is a SFGWT. For move two it can be named again because of the positive version proving it is a SFGWT. This pattern can go on forever failing axiom 4.
