When first learning the basic concepts of Game Theory, the notion of a zero-sum game is probably one of the easiest topics to understand. In these types of games, there are generally true winners and losers, but those terms can have different meanings. The real defining characteristic of a zero-sum game is that the sum of all gains by a player or group of players is equal to the sum of all losses for every possible outcome of that game.
Games like checkers and arm-wrestling are simple examples of two-player zero-sum games, since at the end of a standard game, there is a single winner and a single loser with the winner being "up 1 game" and the loser being "down 1 game."
A more general example of a zero-sum game that can have more than two players is poker. At the end of a poker hand, it's possible that two or more people could "tie" for the win. This would still be a zero-sum game since the combined payoffs to all of the winners is equal to the combined amount lost in the hand.
As a side note here, ties are possible in zero-sum games. For instance, in an arm-wrestling match, both players may decide to call it a draw if a sufficient amount of time passes without one being able to defeat the other. This still satisfies the conditions of a zero-sum game since the payoff to both players is the same – that is, neither player gains or loses.
When one player's gain does not necessarily mean another player's loss (and vice versa), the situation becomes more complex. These types of games are referred to as non-zero-sum games, because the gains and the losses in the game do not always add up to zero.
One classic example of a non-zero-sum game is the Prisoners' Dilemma. In this game, even though both players are acting in their own self-interests to determine their best possible outcomes, the gains of one player are not equally offset by the losses of the other.
Another major aspect that sets non-zero-sum games apart from zero-sum games is that non-zero-sum games don't have to be completely competitive – they can contain all sorts of degrees of cooperation. Depending upon how much cooperation is permitted in the game, the strategies of each player can change quite a bit.
Going back to the example of the Prisoners' Dilemma, when the prisoners are not allowed to communicate, the best individual strategies would result in both players confessing and receiving a "medium" prison sentence. However, if the prisoners were allowed to collaborate before making their decision, it's less probable they would make this choice. Instead, they would have much more incentive to make a deal with one another and both choose the "don't confess" option so that each one would receive only a very light prison sentence.
Although zero-sum games are interesting to study and do have a number of applications, many situations in the real world are best modeled by non-zero-sum games. With that being said, the so-called "payoffs" in these games can be pretty hard to quantify.
For example, suppose a small business owner has $1,000 and is trying to decide whether to donate that amount to charity or reinvest it in the business. If the owner decides to put the money back into the company, the charity receives nothing at that time – although it is possible that the charity might realize a long-term gain if that $1,000 investment into the business yields remarkable dividends, allowing the owner to donate a great deal more at a later date.
On the other side of the coin, if the owner decides to donate the $1,000 the charity, it might seem that the business is losing out at first. Yet, that charitable donation could yield a number of benefits for the business, such as improving the reputation of the business in the local community, which could result in a much greater value than either the donation itself or any returns that reinvesting the $1,000 back into the business might yield.
References and Additional Resources:
1. Dixit, Avinash K. and Barry J. Nalebuff. The Art of Strategy: A Game Theorist's Guide to Success in Business and Life. New York: W. W. Norton & Company, 2008.