Utility functions are a way of characterizing preference orderings between events. If a preference ordering satisfies certain properties, then there exists a utility function such that its expected value over the events can be used to decide which events are preferred over which other events (see VNMtheorem). Utility values are not defined with respect to anything else, they are not money or happiness or resources. In particular, utilities of different players can’t be compared a priori, without bringing in more structure (for example redistribution of resources in the setup of Kaldor-Hicks improvement establishes a way of comparing utilities of players, see the original comment).
If you add a constant to a utility function, its expected value over some event increases by the same constant. So if one event had greater expected utility than another, it would still be the case after you add the constant. This is the sense in which adding constants or multiplying by positive factors makes no difference.
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.
Utility functions are a way of characterizing preference orderings between events. If a preference ordering satisfies certain properties, then there exists a utility function such that its expected value over the events can be used to decide which events are preferred over which other events (see VNM theorem). Utility values are not defined with respect to anything else, they are not money or happiness or resources. In particular, utilities of different players can’t be compared a priori, without bringing in more structure (for example redistribution of resources in the setup of Kaldor-Hicks improvement establishes a way of comparing utilities of players, see the original comment).
If you add a constant to a utility function, its expected value over some event increases by the same constant. So if one event had greater expected utility than another, it would still be the case after you add the constant. This is the sense in which adding constants or multiplying by positive factors makes no difference.
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.