Okay, so I think I get you now, in the imbalanced game, if the payoff is 100 or 1 as in “Zero Sum” is a misnomer, a rational player will still make the same decision, regardless of the imbalance with the other player, given the resulting preference ordering.
However while this imbalance makes no difference to the players’ decisions, it does make a difference to the total payoff, making it non-zero-sum. I’m having difficulty understanding why values such as happiness or resources cannot be substituted for utility—surely at some point this must happen if game theory is to have any application in the real world. Personally I’m interested in real world applications, but I fully acknowledge my ignorance on this particular aspect.
I find a practical way to look at a zero-sum game is to imagine that each of two players must contribute half of the total payoff in order to play. This takes a game that is constant-sum, and makes it zero-sum, and does so in a way that doesn’t break the constant-sumness. In the case of the imbalanced game, because it is not constant-sum it doesn’t reduce to a zero-sum game in this way, remaining a non-zero-sum game with terrible odds for one player, meaning that a rational player won’t opt in if given the option.
If I’m not mistaken, this is generally what is meant when someone refers to a zero-sum game. In chess for instance you enter a competition with your rating (essentially your bet) and the outcome of the game has either a negative or positive (or no) impact on your rating and an opposite impact on your opponent’s rating.(I’m not exactly sure if chess ratings are calculated as exactly zero-sum, but you get the idea). So, the game is zero-sum. Of course there are outside factors that make it beneficial to both players; enjoyment, brain-exercise, socialising etc which may have positive utility on another level, but the game itself and the resulting rating changes are essentially zero-sum.
This is the sense in which I am using the term “zero-sum”, in the most basic sense for someone to win (relative to their starting point, bet or rating) another must lose by an equal amount.
There is probably a more mathematically succinct way of expressing this, but I don’t have those tools at my disposal at present. Again, thanks for your comments. Please don’t feel the need to continue your labours educating me on this topic, I understand that you clearly have a better understanding of game theory than I do, so I appreciate your time. I should probably continue reading further to level up my understanding. Of course if you feel like continuing the floor is yours.
Okay, so I think I get you now, in the imbalanced game, if the payoff is 100 or 1 as in “Zero Sum” is a misnomer, a rational player will still make the same decision, regardless of the imbalance with the other player, given the resulting preference ordering.
However while this imbalance makes no difference to the players’ decisions, it does make a difference to the total payoff, making it non-zero-sum. I’m having difficulty understanding why values such as happiness or resources cannot be substituted for utility—surely at some point this must happen if game theory is to have any application in the real world. Personally I’m interested in real world applications, but I fully acknowledge my ignorance on this particular aspect.
I find a practical way to look at a zero-sum game is to imagine that each of two players must contribute half of the total payoff in order to play. This takes a game that is constant-sum, and makes it zero-sum, and does so in a way that doesn’t break the constant-sumness. In the case of the imbalanced game, because it is not constant-sum it doesn’t reduce to a zero-sum game in this way, remaining a non-zero-sum game with terrible odds for one player, meaning that a rational player won’t opt in if given the option.
If I’m not mistaken, this is generally what is meant when someone refers to a zero-sum game. In chess for instance you enter a competition with your rating (essentially your bet) and the outcome of the game has either a negative or positive (or no) impact on your rating and an opposite impact on your opponent’s rating.(I’m not exactly sure if chess ratings are calculated as exactly zero-sum, but you get the idea). So, the game is zero-sum. Of course there are outside factors that make it beneficial to both players; enjoyment, brain-exercise, socialising etc which may have positive utility on another level, but the game itself and the resulting rating changes are essentially zero-sum.
This is the sense in which I am using the term “zero-sum”, in the most basic sense for someone to win (relative to their starting point, bet or rating) another must lose by an equal amount.
There is probably a more mathematically succinct way of expressing this, but I don’t have those tools at my disposal at present. Again, thanks for your comments. Please don’t feel the need to continue your labours educating me on this topic, I understand that you clearly have a better understanding of game theory than I do, so I appreciate your time. I should probably continue reading further to level up my understanding. Of course if you feel like continuing the floor is yours.