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 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 more or less than what they began with. For example, a game of poker 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.

It has been theorized by Robert Wright, among others, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.

An example

A game's payoff matrix is a convenient way of representation. Consider for example the two-player zero-sum game pictured

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.

Thinking "like a lawyer" also means to define people according to their legal rights, trying to understand, prevent and "resolve" problems by applying legal rules to those rights, usually in a zero-sum manner. It is fundamentally negative, critical, pessimistic, and depersonalizing. This method of thinking is conveyed and understood in law schools as a new and superior way of thinking, not a strictly limited legal tool. These beliefs and thought processes have an atomistic worldview and a zero-sum message about life. Nothing much matters beyond winning or losing, and there is always a loser for each winner.

Not to mention that stupid m u t h a @ # ! * i n g grading curve that law schools have in place.

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