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.