Zero-sum game

Not to be confused with Empty sum or Zero game.

For other uses, see Zero sum (disambiguation).

In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain (or loss) of utility is exactly balanced by the losses (or gains) of the utility of the other participant(s). If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally (see marginal utility).

In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium.

Definition

	Choice 1	Choice 2
Choice 1	-A, A	B, -B
Choice 2	C, -C	-D, D
Generic zero-sum game

The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game).

Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.

Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.

The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent’s payoff at a favorable cost to himself rather to prefer more than less. The punishing-the-opponent standard can be used in both zero-sum games (i.e. warfare game, Chess) and non-zero-sum games (i.e. pooling selection games).

Solution

For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.

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 or above.

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 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?

Émile Borel and 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 minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably 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 4/7 and action 2 with probability 3/7, while Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.

Solving

The Nash equilibrium for a two-player, zero-sum game can be found by solving a linear programming problem. Suppose a zero-sum game has a payoff matrix $M$ where element $M_{i,j}$ is the payoff obtained when the minimizing player chooses pure strategy $i$ and the maximizing player chooses pure strategy $j$ (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of $M$ is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (see ref. [2], page 740) by solving the following linear program to find a vector $u$:

Minimize:

$\sum_{i} u_i$

Subject to the constraints:

$u$ ≥ 0

$M u$ ≥ 1.

The first constraint says each element of the $u$ vector must be nonnegative, and the second constraint says each element of the $M u$ vector must be at least 1. For the resulting $u$ vector, the inverse of the sum of its elements is the value of the game. Multiplying $u$ by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.

If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.

The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of $M$ (adding a constant so it's positive), then solving the resulting game.

If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.

Universal Solution

If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or incredible after the play is started, such as Poker, there is no NE strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it’s interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.

Non-zero-sum

Economics

Many economic situations are not zero-sum, since valuable goods and services can be created, destroyed, or badly allocated in a number of ways, and any of these will create a net gain or loss of utility to numerous stakeholders. Specifically, all trade is by definition positive sum, because when two parties agree to an exchange each party must consider the goods it is receiving to be more valuable than the goods it is delivering. In fact, all economic exchanges must benefit both parties to the point that each party can overcome its transaction costs, or the transaction would simply not take place.

There is some semantic confusion in addressing exchanges under coercion. If we assume that "Trade X", in which Adam trades Good A to Brian for Good B, does not benefit Adam sufficiently, Adam will ignore Trade X (and trade his Good A for something else in a different positive-sum transaction, or keep it). However, if Brian uses force to ensure that Adam will exchange Good A for Good B, then this says nothing about the original Trade X. Trade X was not, and still is not, positive-sum (in fact, this non-occurring transaction may be zero-sum, if Brian's net gain of utility coincidentally offsets Adam's net loss of utility). What has in fact happened is that a new trade has been proposed, "Trade Y", where Adam exchanges Good A for two things: Good B and escaping the punishment imposed by Brian for refusing the trade. Trade Y is positive-sum, because if Adam wanted to refuse the trade, he theoretically has that option (although it is likely now a much worse option), but he has determined that his position is better served in at least temporarily putting up with the coercion. Under coercion, the coerced party is still doing the best they can under their unfortunate circumstances, and any exchanges they make are positive-sum.

There is additional confusion under asymmetric information. Although many economic theories assume perfect information, economic participants with imperfect or even no information can always avoid making trades that they feel are not in their best interest. Considering transaction costs, then, no zero-sum exchange would ever take place, although asymmetric information can reduce the number of positive-sum exchanges, as occurs in "The Market for Lemons".

Psychology

The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing our personal interests can enhance our collective well-being, but in others personal interest results in mutually destructive behavior.

Complexity

It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.

Extensions

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)th player representing the global profit or loss.

Misunderstandings

Zero-sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games.

Zero-sum mentality

In community psychology "Zero-sum mentality" is a way of thinking that assumes all games are zero-sum: that for every winner there is a loser; for every gain there is a loss.

See also

• Comparative advantage
• Zero-sum fallacy
• Gains from trade
• Free trade
• Bimatrix game

Further reading

• Misstating the Concept of Zero-Sum Games within the Context of Professional Sports Trading Strategies, series Pardon the Interruption (2010-09-23) ESPN, created by Tony Kornheiser and Michael Wilbon, performance by Bill Simmons
• Handbook of Game Theory - volume 2, chapter Zero-sum two-person games, (1994) Elsevier Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759, ISBN 0-444-89427-6
• Power:Its Forms, Bases and Uses (1997) Transaction Publishers, by Dennis Wrong

External links

• Play zero-sum games online by Elmer G. Wiens.
• Game Theory & its Applications - comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)
• A playable zero-sum game and its mixed strategy Nash equilibrium.