Information about Zero Sum

For the X-Files episode, see "Zero Sum".

In game theory, 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 and Go are examples of a zero-sum game: it is impossible for both players to win. Zero-sum can be thought of more generally as 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. In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses is either less than or more than zero.

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 more or less than what they began with. For example, a game of poker, disregarding the house's rake, played in a casino is a zero-sum game unless the pleasure of gambling or the cost of operating a casino is taken into account, making it a non-zero-sum game.

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. In pure strategies, each outcome is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game) ^[1]. Nash equilibria of two-player zero-sum games are exactly pairs of minimax strategies.

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, 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.

Economics and non-zero-sum

Many economic situations are not 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. Assuming the counterparties are acting rationally, any commercial exchange is a non-zero-sum activity, because each party must consider the goods s/he is receiving as being at least fractionally more valuable to him/her than the goods he/she is delivering. Economic exchanges must benefit both parties enough above the zero-sum such that each party can overcome his or her transaction costs.

See also:

• Comparative advantage
• Free trade

Psychology and non-zero-sum

The most common or simple example 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 their own ends.

Complexity and non-zero-sum

It 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]

An example

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 representation. Consider for example the two-player zero-sum game pictured at 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: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue 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?

Red 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, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if Red 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 Red should choose action 1 with probability 57% and action 2 with 43%, while Blue should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Red will then win 2.85 points on average per game.

References

External links

• Play zero-sum games online by Elmer G. Wiens.
• Freeware program to create and solve ZeroSum puzzles with more than 20,000 unique solution puzzles (download all 50,000).
• Game Theory & its Applications - comprehensive text on psychology and game theory.

	Topics in game theory
Definitions	Normal form game Extensive form game Cooperative game Information set Preference
Equilibrium concepts	Nash equilibrium Subgame perfection Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance
Strategies	Dominant strategies Mixed strategy Tit for tat Grim trigger Collusion
Classes of games	Symmetric game Perfect information Dynamic game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Stochastic game Nontransitive game
Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Minority game Rock, Paper, Scissors Pirate game Dictator game Public goods game Nash bargaining game Blotto games War of attrition
Theorems	Minimax theorem Purification theorems Folk theorem Revelation principle Arrow's theorem