Zero-sum game is a
mathematical representation in
game theory and
economic theory
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, therefore the net improvement in benefit of the game is zero.
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 more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if
all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like
poker,
chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
, and
bridge
A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually somethi ...
where one person gains and another person loses, which results in a zero-net benefit for every player. In the markets and financial instruments, futures contracts and options are zero-sum games as well.
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
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality.
The first theorem in this sense is von Neumann's minimax theorem from 1928, which was c ...
which is closely related to
linear programming duality,
or with
Nash equilibrium.
Prisoner's Dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("def ...
is a classical non-zero-sum game.
Definition
The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is
Pareto optimal
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
. 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.
In situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain), they 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 is 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 favourable cost to themselves rather than prefer more over less. The punishing-the-opponent standard can be used in both zero-sum games (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games). The player in the game has a simple enough desire to maximise the profit for them, and the opponent wishes to minimise it.
Solution
For two-player finite zero-sum games, the different
game theoretic solution concept
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most comm ...
s of
Nash equilibrium,
minimax, and
maximin all give the same solution. If the players are allowed to play a
mixed strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
, the game always has an equilibrium.
Example
A game's
payoff matrix
In game theory, normal form is a description of a ''game''. Unlike extensive form, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyin ...
is a convenient representation. Consider these situations as an 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.''
In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and 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. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel was ...
and
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
had the fundamental insight that
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
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 and action 2 with probability , and Blue should assign the probabilities 0, , and to the three actions A, B, and C. Red will then win 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 where element is the payoff obtained when the minimizing player chooses pure strategy and the maximizing player chooses pure strategy (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 is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector :
The first constraint says each element of the vector must be nonnegative, and the second constraint says each element of the vector must be at least 1. For the resulting vector, the inverse of the sum of its elements is the value of the game. Multiplying by that value gives a probability vector, giving the probability that the maximizing player will choose each possible pure strategy.
If the game matrix does not have all positive elements, 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 not affect 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. Alternatively, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of (adding a constant so it is 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 and thus 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 non-credible after the play is started, such as poker, there is no Nash equilibrium 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 concerning starting the game or not.
The most common or simple example from the subfield of
social psychology
Social psychology is the scientific study of how thoughts, feelings, and behaviors are influenced by the real or imagined presence of other people or by social norms. Social psychologists typically explain human behavior as a result of the ...
is the concept of "
social trap
In psychology, a social trap is a conflict of interest or perverse incentive where individuals or a group of people act to obtain short-term individual gains, which in the long run leads to a loss for the group as a whole. Social traps are the caus ...
s". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations, all parties pursuing personal interest results in mutually destructive behaviour.
Copeland's review notes that an n-player non-zero-sum game can be converted into an (n+1)-player zero-sum game, where the n+1st player, denoted the ''fictitious player'', receives the negative of the sum of the gains of the other n-players (the global gain / loss).
Zero-sum three-person games
![The zero-sum three-person game](https://upload.wikimedia.org/wikipedia/commons/c/ce/The_zero-sum_three-person_game.png)
It is clear that there are manifold relationships between players in a zero-sum three-person game, in a zero-sum two-person game, anything one player wins is necessarily lost by the other and vice versa; therefore, there is always an absolute antagonism of interests, and that is similar in the three-person game.
A particular move of a player in a zero-sum three-person game would be assumed to be clearly beneficial to him and may disbenefits to both other players, or benefits to one and disbenefits to the other opponent.
Particularly, parallelism of interests between two players makes a cooperation desirable; it may happen that a player has a choice among various policies: Get into a parallelism interest with another player by adjusting his conduct, or the opposite; that he can choose with which of other two players he prefers to build such parallelism, and to what extent.
The picture on the left shows that a typical example of a zero-sum three-person game. If Player 1 chooses to defence, but Player 2 & 3 chooses to offence, both of them will gain one point. At the same time, Player 2 will lose two-point because points are taken away by other players, and it is evident that Player 2 & 3 has parallelism of interests.
Real life example
Economic benefits of low-cost airlines in saturated markets - net benefits or a zero-sum game
Studies show that the entry of low-cost airlines into the Hong Kong market brought in $671 million in revenue and resulted in an outflow of $294 million.
Therefore, the replacement effect should be considered when introducing a new model, which will lead to economic leakage and injection. Thus introducing new models requires caution. For example, if the number of new airlines departing from and arriving at the airport is the same, the economic contribution to the host city may be a zero-sum game. Because for Hong Kong, the consumption of overseas tourists in Hong Kong is income, while the consumption of Hong Kong residents in opposite cities is outflow. In addition, the introduction of new airlines can also have a negative impact on existing airlines.
Consequently, when a new aviation model is introduced, feasibility tests need to be carried out in all aspects, taking into account the economic inflow and outflow and displacement effects caused by the model.
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
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
and
Oskar Morgenstern
Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decis ...
proved that any non-zero-sum game for ''n'' players is equivalent to a zero-sum game with ''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.
[, chapters 1 & 7]
Politics is sometimes called zero sum
because in common usage the idea of a stalemate is perceived to be "zero sum"; politics and
macroeconomics are not zero sum games, however, because they do not constitute
conserved systems.
Zero-sum thinking
In psychology,
zero-sum thinking
Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...
refers to the perception that a given situation is like a zero-sum game, where one person's gain is equal to another person's loss.
See also
*
Bimatrix game In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrices - matrix A descri ...
*
Comparative advantage
In an economic model, agents have a comparative advantage over others in producing a particular good if they can produce that good at a lower relative opportunity cost or autarky price, i.e. at a lower relative marginal cost prior to trade. Comp ...
*
Dutch disease
In economics, the Dutch disease is the apparent causal relationship between the increase in the economic development of a specific sector (for example natural resources) and a decline in other sectors (like the manufacturing sector or agricultur ...
*
Gains from trade
In economics, gains from trade are the net benefits to economic agents from being allowed an increase in voluntary trading with each other. In technical terms, they are the increase of consumer surplus plus producer surplus from lower tariffs ...
*
Lump of labour fallacy
In economics, the lump of labour fallacy is the misconception that there is a fixed amount of work—a lump of labour—to be done within an economy which can be distributed to create more or fewer jobs. It was considered a fallacy in 1891 by eco ...
*
Positive-sum game
*
No-win situation
A no-win situation, also called a lose-lose situation, is one where a person has choices, but no choice leads to a net gain. For example, if an executioner offers the condemned the choice of death by being hanged, shot, or poisoned, all choices lea ...
References
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
ESPN (originally an initialism for Entertainment and Sports Programming Network) is an American international basic cable sports channel owned by ESPN Inc., owned jointly by The Walt Disney Company (80%) and Hearst Communications (20%). Th ...
, created by
Tony Kornheiser
Anthony Irwin Kornheiser (; born July 13, 1948) is an American television sports talk show host and former sportswriter and columnist. Kornheiser is best known for his endeavors in three forms of media: as a writer for ''The Washington Post'' fro ...
and
Michael Wilbon
Michael Wilbon (; born ) is an American commentator for ESPN and former sportswriter and columnist for ''The Washington Post''. He is an analyst for ESPN and has co-hosted ''Pardon the Interruption'' on ESPN since 2001.
Early life and education ...
, performance by
Bill Simmons
* ''Handbook of Game Theory – volume 2'', chapter ''Zero-sum two-person games'', (1994)
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', ...
Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759,
* ''Power: Its Forms, Bases and Uses'' (1997) Transaction Publishers, by
Dennis Wrong
Dennis Hume Wrong (November 22, 1923 – November 8, 2018) was a Canadian-born American sociologist and emeritus professor in the Department of Sociology at New York University.
Wrong was the author of several books, including two essay collect ...
External links
Play zero-sum games onlineby Elmer G. Wiens.
– comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)
A playable zero-sum gameand its mixed strategy Nash equilibrium.
{{DEFAULTSORT:Zero-Sum
Non-cooperative games
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