Zero-sum
Zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). It is so named because when the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero.
Chess is an example of a zero-sum game - it is impossible for both players to win. Zero-sum is a special case of a more general
constant sum where the benefits and losses to all players sum to the same value. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others.
Situations where participants can all gain or suffer together, such as a
country with an
excess of
bananas
trading with another country for their excess of
apples, where both benefit from the transaction, are referred to as
non-zero-sum. Other non-zero-sum games are games in which the sum of gains and losses by the players are always less than what they began with, such as in a game of
poker played in a
casino in which
vigorish is taken by the house.
The concept was first developed in
game theory and consequently zero-sum situations are often called
zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games. Optimal strategies for two-player zero-sum games can often be found using
minimax strategies.
In 1944
John von Neumann and
Oskar Morgenstern proved that any zero-sum game involving
n players is in fact a generalised form of a zero-sum game for two persons, and that any non-zero-sum game for
n players can be reduced to a zero-sum game for
n + 1 players; the (
n + 1) player representing the global profit or loss. This suggests that the zero-sum game for two players forms the essential core of mathematical game theory.
[This paragraph was translated from the French wikipedia article on this subject.]Non-zero-sum situations are an important part of economic activity due to
production,
marginal utility and
value-subjectivity. Most economic situations are non-zero-sum, since valuable goods and services can be created, destroyed, or badly allocated, and any of these will create a net gain or loss. One strategy for non-zero-sum games is
tit for tat.
Zero sum games in economics include many financial markets such as
futures, exchange traded
options and
foreign exchange markets.
If a
farmer succeeds in raising a bumper crop, he will benefit by being able to sell more food and make more money. The consumers he serves benefit as well, because there is more food to go around, so the price per unit of food will be lower. Other farmers who have not had such a good crop might suffer somewhat due to these lower prices, but this cost to other farmers may very well be less than the benefits enjoyed by everyone else, such that overall the bumper crop has created a net benefit. The same argument applies to other types of productive activity.
Trade is a non-zero-sum activity, both because different parties to a trade may place different values on the goods being traded, and because of the law of
comparative advantage.
The non-zero-sum nature of economic transactions contrasts with the zero-sum nature in which they are reflected by the transfer of money. Some see this as a justification for advancing less traditional forms of money.
Altruistic Economics, for example, allows explicit reflection of the non-zero sum nature of economic interactions, by allowing all parties to keep independent records of the value of transactions to them.
The most common or simplistic example is from the subfield of
Social Psychology is the concept of "Social Traps". In some cases we can enhance our collective well being by pursuing our personal interests or parties can pursue mutually destructive behavior as they choose the own ends. Mutually Assured Destruction (MAD) is another example.The quote below is also apropos to this section.
Here is a link to a comprehensive text on the matter of Psychology and Game Theory.
Game Theory & its ApplicationsIt has been theorized by
Robert Wright, among others, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent. As former US President Bill Clinton states:
The more complex societies get and the more complex the networks of interdependence within and beyond community and national borders get, the more people are forced in their own interests to find non-zero-sum solutions. That is, win-win solutions instead of win-lose solutions.... Because we find as our interdependence increases that, on the whole, we do better when other people do better as well - so we have to find ways that we can all win, we have to accommodate each other - Bill Clinton,
Wired interview, December
2000.[
1]
align=bottom|A zero sum game| A | B | C |
|---|
| 1 | 30, -30 | -10, 10 | 20, -20 |
|---|
| 2 | 10, -10 | 20, -20 | -20, 20 |
|---|
A game's
payoff matrix is a convenient way of representation. Consider for example the two-player zero-sum game pictured to the right.
The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
Example: the first player chooses action 2 and the second player chose action B. When the payoff is allocated the first player gains 20 points and the second player loses 20 points.Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?
Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C. If both players take these actions, the first player will win 20 points. But what happens if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?
John von Neumann had the fundamental and surprising insight that
probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum
expected point-loss independent of the opponent's strategy; this leads to a
linear programming problem with a unique solution for each player. This
minimax method can compute provably optimal strategies for all two-player zero-sum games.
For the example given above, it turns out that the first player should choose action 1 with probability 57% and action 2 with 43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C.Player one will then win 2.85 points on average per game.
*
Game*
Group-dynamic game *
Win-win game*
Double-entry book-keeping*
Play zero-sum games online by Elmer G. Wiens.
