Zero-sum

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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.

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 generalised 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.

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[edit] 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 good s/he is receiving as being at least fractionally more valuable to him/her than the good s/he is delivering (see also the law of comparative advantage)—to exchange in any other circumstances would not be rational.

[edit] Psychology and non-zero-sum

The most common or simple 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 and can pursue mutually destructive behavior as they choose their own ends. The Mutually Assured Destruction (MAD) doctrine is another example.

[edit] 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]

[edit] 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 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.

[edit] References

  1. ^ Samuel Bowles: Microeconomics: Behavior, Institutions, and Evolution, Princeton University Press, pp. 33–36 (2004) ISBN 0691091633

[edit] External links


 view  Topics in game theory

Definitions

Normal form game · Extensive form game · Cooperative game · Information set · Preference

Equilibrium concepts

Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · ESS · Risk dominance

Strategies

Dominant strategies · Mixed strategy · Tit for tat · Grim trigger

Classes of games

Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design

Games

Prisoner's dilemma · Coordination game · Chicken · 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

Theorems

Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's Theorem

Related topics

Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists