I'm studying zero-sum stochastic games for my dissertation and I came across this example https://www.researchgate.net/publication/231786753_A_real-world_stochastic_two-person_game
Consider a television game show where two contestants have to play a dice game. The contestants sit behind a battery of buttons numbered 1,2,...,D, which represent the numbers of the dice that will be thrown by the contestant. In each stage of the game the players press simultaneously one of the buttons; in particular, they cannot observe the other opponent's decision. Afterwards, they throw their dice, which score is dictated as it follows:
- the score is equal to the sum of the outcome of the dice, if none of them showed 1 as outcome;
- the score is null if at least one of the dice showed the outcome 1;
In any case, the result is added to their current total. The first player who reaches a total of G points is the winner. If the players reach the goal at them same turn, then the winner is the one with a higher overall score. In case of a tie, the winner is determined by a toss of a fair coin. At each stage both contestants have full information about their own and the opponent's current total score. The formulation of the game is such that it is zero-sum and stochastic.
I get that it's stochastic but why is it zero-sum? From what I understand, since a stochastic game is a set of normal form games, in order for it to be zero-sum, all games of the set must be zero-sum. However in the example the sum of the payoffs of the two players in general is not zero, since that happens only when both players had 1 in at least one of their dice. What am i doing wrong in my reasoning? Can someone help me clarify this doubt?