# Zero Sum Game

An interaction where every benefit to a participant is balanced by an equal detriment to the other participants. As a consequence, a zero sum game is a situation where there has to be at least one loser for every winner and (conversely) at least one winner for every loser.

The physical world can be interpreted as a zero sum game with respect to mass and energy.

A counter example is respect. The respect lost by one wiki user is not won by his opponent. In fair communications both can win respect.

However, relative respect would be a zero-sum game. (For example, two party political systems seem to devolve into this as its usually easier to destroy respect for an opponent than to build respect for yourself, and all you have to do to win is look better than your opponent.)
The following is an example:

```   \ Me|  m1   |  m2
You\  |       |
----+-+-------+-------
y1  | -3\+3 | -6\+6
------+-------+-------
y2  | -9\+9 | -3\+3

```
The game is as follows. You choose a row. I choose a column. We both say (at the same time) what we chose, and look up the number. The first number is what you win, the second number is what I win.

Thus, we can think that you give me the amount it says there in the second entry; in Euros. As an example, if you choose row y2 and I choose column m1 you will have to give me 9 Euros.

Note that the game is not fair. Thus, you will always lose. But zero sum relates to the fact that what you lose is what I win. Thus, a game is zero sum exactly if the two entries in every cell of the matrix sum to zero.

If you need an example of a NonZeroSumGame, you may want to look at the PrisonersDilemma.

There may be cases where the payoffs in each cell don't sum to zero, but the whole game does. For instance, in a game with a random element, the expected value of the game for both players might be zero. Also, the game may have rules as to what moves may follow what other moves, such that any sequence of moves of length 4 (for example) add up to zero. If the game requires 4-move sequences to be completed, then the game would be zero sum. I'm not an expert at GameTheory, so I could be wrong on this definition, but I think these cases apply.

The expected value is irrelevant for whether or not the game is zero sum. But if the game required 4 move sequence, and every 4 move sequence added to zero, that would be a zero sum game. A game is zero sum if the sum across all players is the same before and after the game regardless of what happens in the game.
The simplest examples are competitive head-to-head sports like chess, tennis, basketball and hockey; one team wins and the other loses. If you hurt the other team, your team gains an advantage. A bad example, you may hurt a player into hospital, still don't win the game - zero sum is in results, not in the detail

This seems to be conflating the definition of game in game theory with the colloquial definition. If the aspect of interest is the enjoyment of the participants, these games are not zero sum. On the other hand, if you assign one point for winning, minus one point for losing, and zero points for ties, they are.
Unfortunately, many relationships which could be WinWin are inappropriately identified as zero sum games; advancement within many companies is fixed so that (as an example) exactly one person per business unit can be promoted per year. This leads to backbiting and sabotage within a department as members try to gain advantages over their coworkers.

Likewise, many ReligiousWars are misinterpreted this way, as in EmacsVsVi and SysvVsBsd?.

This isn't zero sum as much as harming your opponents is a viable strategy. That can happen whether or not the game is zero sum.
This is the subject (sort of) of the 1983's Matthew Broderick film WarGames, in which a computer treats everything as a zero sum game, including its simulations of global thermonuclear war. Unfortunately, it's attached to the actual American thermonuclear stockpile. In the end, the intrepid programmer protagonist must race to teach the computer a completely foreign concept: that of futility. A memorable quote:

David: How about Global Thermonuclear War?

WOPR: Wouldn't you prefer a nice game of chess?
and later (after being presented with TicTacToe, and progressing from there to make the connection between the two "games")
WOPR: Strange game. The only winning move is not to play
```   (http://www2.english.uiuc.edu/cybercinema/sounds/wargames_sounds/strange_game.wav)

```