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0

Somewhat Finite Game Without Ties

Any alternating move game satisfying axioms 1-4. Might or might not satisfy axiom 5.

1

Super game

1 on her first moves, I chooses a TFGWT G.
2 player II goes first in a play of sub game G
3 they continue to play sub game G with II playing I's role and vice versa
4 whoever wins the play of the sub game, wins the play of super game

2

Hyper game

1 on her first movement I chooses a SfGWT G
2 player II goes first in a play of this sub game G.
3 hey continue to play sub game G with II playing I's role and vice versa
4 whoever wins the play of the subgame, wins the play of hyper game.

3

Preliminary definition of Solution for a zero sum game

Let G be a 0 sum game. Suppose:
- Row gas some strategy S that guarantees Row a pay off of at least V
- Col has a strategy T that guarantees Row a pay off of at most V
And these 2 V values are the same
Then we say (S,T,V) is a solution if G. And V is the value of the game.

4

Super game satisfying / hyper game rule 1

1 and 2 in SG/ HG rules tell us that they alternate for first two moves in playing.
Subsequent moves taken are a play of a TFGWT G (subgame). G satisfies all 5 axioms, including 1, so subsequent moves alternate

5

SG satisfying/ HG rule axiom 2

No random mechanisms are used in 1st move of SG/ HG, thanks to A
Subsequent moves take place within a play of G, which is a TFGWT. So G satisfies (2) and there are no randomizing mechanisms in these moves.

6

SG satisfying/ HG rules axiom 3

Suppose we are at the end if A play of SG/HG. Then we are at the end of the play of subgame G, which is is a TFGWT. So g has a unique winnier, which becomes the unique winner of this play of SG/HG, thanks to rule D.

7

SG satisfying/ HG rule axiom 4

A play of SG/HG consists of a play of a TFGWT preceded by a single game remaining move. The play of G ends after finitely many moves because G is a TFGWT. The play of SG/HG is 1 move longer. But 1+ finite number = a finite number, so SG/HG satisfies Axiom 4

8

Why SG is not a TFGWT

Super game does not satisfy axiom 5 because I has infinitely many legal options for her game naming move. hex (1) hex(2)... Hex (n)....
SG violates Axiom 5. So SG is not a TFGWT, so SG is not a legal first move in SG.

9

Hyper game is not a SFGWT

When playing HG, player I and player II, moving alternately, can keep naming HG because it is a SFGWT, which is the requirement for A and B. However this patter can go on infinity long, since it is legal, which violates axiom 4, each play ends after finitely many moves. It also would violate axiom 3 as there cannot be a winner if each plater keeps stating hyper game on their respective move. M